0.07/0.13 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.07/0.14 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s 0.14/0.35 % Computer : n031.cluster.edu 0.14/0.35 % Model : x86_64 x86_64 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.14/0.35 % Memory : 8042.1875MB 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64 0.14/0.35 % CPULimit : 960 0.14/0.35 % WCLimit : 120 0.14/0.35 % DateTime : Tue Sep 1 09:54:41 EDT 2020 0.14/0.35 % CPUTime : 103.19/103.18 % SZS status Theorem 103.19/103.18 % Mode: mode449:USE_SINE=true:SINE_TOLERANCE=5.0:SINE_GENERALITY_THRESHOLD=4:SINE_RANK_LIMIT=1.:SINE_DEPTH=0 103.19/103.18 % Inferences: 554 103.19/103.18 % SZS output start Proof 103.19/103.18 thf(ty_kfst, type, kfst : ($i>$i)). 103.19/103.18 thf(ty_binintersect, type, binintersect : ($i>$i>$i)). 103.19/103.18 thf(ty_singleton, type, singleton : ($i>$o)). 103.19/103.18 thf(ty_subset, type, subset : ($i>$i>$o)). 103.19/103.18 thf(ty_eigen__2, type, eigen__2 : $i). 103.19/103.18 thf(ty_setunion, type, setunion : ($i>$i)). 103.19/103.18 thf(ty_iskpair, type, iskpair : ($i>$o)). 103.19/103.18 thf(ty_eigen__1, type, eigen__1 : $i). 103.19/103.18 thf(ty_eigen__0, type, eigen__0 : $i). 103.19/103.18 thf(ty_cartprod, type, cartprod : ($i>$i>$i)). 103.19/103.18 thf(ty_breln, type, breln : ($i>$i>$i>$o)). 103.19/103.18 thf(ty_emptyset, type, emptyset : $i). 103.19/103.18 thf(ty_setminus, type, setminus : ($i>$i>$i)). 103.19/103.18 thf(ty_kpair, type, kpair : ($i>$i>$i)). 103.19/103.18 thf(ty_eigen__3, type, eigen__3 : $i). 103.19/103.18 thf(ty_descr, type, descr : (($i>$o)>$i)). 103.19/103.18 thf(ty_symdiff, type, symdiff : ($i>$i>$i)). 103.19/103.18 thf(ty_nonempty, type, nonempty : ($i>$o)). 103.19/103.18 thf(ty_dsetconstr, type, dsetconstr : ($i>($i>$o)>$i)). 103.19/103.18 thf(ty_powerset, type, powerset : ($i>$i)). 103.19/103.18 thf(ty_omega, type, omega : $i). 103.19/103.18 thf(ty_ex1, type, ex1 : ($i>($i>$o)>$o)). 103.19/103.18 thf(ty_ksnd, type, ksnd : ($i>$i)). 103.19/103.18 thf(ty_prop2set, type, prop2set : ($o>$i)). 103.19/103.18 thf(ty_exu, type, exu : (($i>$o)>$o)). 103.19/103.18 thf(ty_set2prop, type, set2prop : ($i>$o)). 103.19/103.18 thf(ty_in, type, in : ($i>$i>$o)). 103.19/103.18 thf(ty_binunion, type, binunion : ($i>$i>$i)). 103.19/103.18 thf(ty_dpsetconstr, type, dpsetconstr : ($i>$i>($i>$i>$o)>$i)). 103.19/103.18 thf(ty_setadjoin, type, setadjoin : ($i>$i>$i)). 103.19/103.18 thf(h0, assumption, (![X1:$i>$o]:(![X2:$i]:((X1 @ X2) => (X1 @ (eps__0 @ X1))))),introduced(assumption,[])). 103.19/103.18 thf(eigendef_eigen__3, definition, eigen__3 = (eps__0 @ (^[X1:$i]:(~((((in @ X1) @ eigen__0) => ((in @ (setunion @ ((dsetconstr @ eigen__1) @ (^[X2:$i]:((in @ ((kpair @ X1) @ X2)) @ eigen__2))))) @ eigen__1)))))), introduced(definition,[new_symbols(definition,[eigen__3])])). 103.19/103.18 thf(eigendef_eigen__1, definition, eigen__1 = (eps__0 @ (^[X1:$i]:(~((![X2:$i]:((~(((((breln @ eigen__0) @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ eigen__0) => ((ex1 @ X1) @ (^[X4:$i]:((in @ ((kpair @ X3) @ X4)) @ X2)))))))))) => (![X3:$i]:(((in @ X3) @ eigen__0) => ((in @ (setunion @ ((dsetconstr @ X1) @ (^[X4:$i]:((in @ ((kpair @ X3) @ X4)) @ X2))))) @ X1))))))))), introduced(definition,[new_symbols(definition,[eigen__1])])). 103.19/103.18 thf(eigendef_eigen__0, definition, eigen__0 = (eps__0 @ (^[X1:$i]:(~((![X2:$i]:(![X3:$i]:((~(((((breln @ X1) @ X2) @ X3) => (~((![X4:$i]:(((in @ X4) @ X1) => ((ex1 @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2)))))))))), introduced(definition,[new_symbols(definition,[eigen__0])])). 103.19/103.18 thf(eigendef_eigen__2, definition, eigen__2 = (eps__0 @ (^[X1:$i]:(~(((~(((((breln @ eigen__0) @ eigen__1) @ X1) => (~((![X2:$i]:(((in @ X2) @ eigen__0) => ((ex1 @ eigen__1) @ (^[X3:$i]:((in @ ((kpair @ X2) @ X3)) @ X1)))))))))) => (![X2:$i]:(((in @ X2) @ eigen__0) => ((in @ (setunion @ ((dsetconstr @ eigen__1) @ (^[X3:$i]:((in @ ((kpair @ X2) @ X3)) @ X1))))) @ eigen__1)))))))), introduced(definition,[new_symbols(definition,[eigen__2])])). 103.19/103.18 thf(sP1,plain,sP1 <=> ((![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]:((~(((((breln @ X1) @ X2) @ X3) => (~((![X4:$i]:(((in @ X4) @ X1) => ((ex1 @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => (![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((((breln @ X1) @ X2) @ X3) => (~((![X4:$i]:(((in @ X4) @ X1) => ((ex1 @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2)))))))))))))))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP1])])). 103.19/103.18 thf(sP2,plain,sP2 <=> ((![X1:$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:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ 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=> (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$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))))))) => 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=> (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 @ 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=> ((![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) => 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((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)))))) => sP1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP3])])). 103.19/103.18 thf(sP4,plain,sP4 <=> ((![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:$i]:((~(((((breln @ X1) @ X2) @ X3) => (~((![X4:$i]:(((in @ X4) @ X1) => ((ex1 @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => (![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((((breln @ X1) @ X2) @ X3) => (~((![X4:$i]:(((in @ X4) @ X1) => ((ex1 @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP4])])). 103.19/103.18 thf(sP5,plain,sP5 <=> ((![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:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))) => sP2)))))))),introduced(definition,[new_symbols(definition,[sP5])])). 103.19/103.18 thf(sP6,plain,sP6 <=> ((emptyset = emptyset) => ((![X1:$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)))))) => sP1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP6])])). 103.19/103.18 thf(sP7,plain,sP7 <=> (![X1:$i]:((~(((((breln @ eigen__0) @ eigen__1) @ X1) => (~((![X2:$i]:(((in @ X2) @ eigen__0) => ((ex1 @ eigen__1) @ (^[X3:$i]:((in @ ((kpair @ X2) @ X3)) @ X1)))))))))) => (![X2:$i]:(((in @ X2) @ eigen__0) => ((in @ (setunion @ ((dsetconstr @ eigen__1) @ (^[X3:$i]:((in @ ((kpair @ X2) @ X3)) @ X1))))) @ eigen__1))))),introduced(definition,[new_symbols(definition,[sP7])])). 103.19/103.18 thf(sP8,plain,sP8 <=> ((![X1:$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))))) => sP2))))),introduced(definition,[new_symbols(definition,[sP8])])). 103.19/103.18 thf(sP9,plain,sP9 <=> ((![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 @ 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@ 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)))))) => sP1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP9])])). 103.19/103.18 thf(sP10,plain,sP10 <=> ((![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:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => ((in @ X3) @ X1))))) => sP5))))))))))),introduced(definition,[new_symbols(definition,[sP10])])). 103.19/103.18 thf(sP11,plain,sP11 <=> ((![X1:$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))))) => sP6))))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP11])])). 103.19/103.19 thf(sP12,plain,sP12 <=> ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))) => sP2)),introduced(definition,[new_symbols(definition,[sP12])])). 103.19/103.19 thf(sP13,plain,sP13 <=> ((![X1:$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))))))) => sP3))))))))))),introduced(definition,[new_symbols(definition,[sP13])])). 103.19/103.19 thf(sP14,plain,sP14 <=> ((![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:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ 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(![X4:$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)))))) => sP1))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP14])])). 103.19/103.19 thf(sP15,plain,sP15 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((((breln @ X1) @ X2) @ X3) => (~((![X4:$i]:(((in @ X4) @ X1) => ((ex1 @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => (![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((((breln @ X1) @ X2) @ X3) => (~((![X4:$i]:(((in @ X4) @ X1) => ((ex1 @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2)))))))),introduced(definition,[new_symbols(definition,[sP15])])). 103.19/103.19 thf(sP16,plain,sP16 <=> ((![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]:(~($false)))) = X1)) => sP13)))))))))),introduced(definition,[new_symbols(definition,[sP16])])). 103.19/103.19 thf(sP17,plain,sP17 <=> ((![X1:$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:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => sP9)))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP17])])). 103.19/103.19 thf(sP18,plain,sP18 <=> ((![X1:$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:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ 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(![X4:$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)))))) => sP1)))))))))))))))))))))))))))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP18])])). 103.19/103.19 thf(sP19,plain,sP19 <=> ((![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)))))) => sP1))))))))))))))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP19])])). 103.19/103.19 thf(sP20,plain,sP20 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => sP15),introduced(definition,[new_symbols(definition,[sP20])])). 103.19/103.19 thf(sP21,plain,sP21 <=> ((![X1:$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)))))) => sP1)))))))))))))),introduced(definition,[new_symbols(definition,[sP21])])). 103.19/103.19 thf(sP22,plain,sP22 <=> ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => sP19),introduced(definition,[new_symbols(definition,[sP22])])). 103.19/103.19 thf(sP23,plain,sP23 <=> ((![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))))) => sP6))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP23])])). 103.19/103.19 thf(sP24,plain,sP24 <=> ((![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:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => sP1))),introduced(definition,[new_symbols(definition,[sP24])])). 103.19/103.19 thf(sP25,plain,sP25 <=> ((![X1:$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))))))))) => sP22)))))))))),introduced(definition,[new_symbols(definition,[sP25])])). 103.19/103.19 thf(sP26,plain,sP26 <=> ((![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)))))))))) => sP21)),introduced(definition,[new_symbols(definition,[sP26])])). 103.19/103.19 thf(sP27,plain,sP27 <=> ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))) => sP2),introduced(definition,[new_symbols(definition,[sP27])])). 103.19/103.19 thf(sP28,plain,sP28 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => sP4),introduced(definition,[new_symbols(definition,[sP28])])). 103.19/103.19 thf(sP29,plain,sP29 <=> ((~(((((breln @ eigen__0) @ eigen__1) @ eigen__2) => (~((![X1:$i]:(((in @ X1) @ eigen__0) => ((ex1 @ eigen__1) @ (^[X2:$i]:((in @ ((kpair @ X1) @ X2)) @ eigen__2)))))))))) => (![X1:$i]:(((in @ X1) @ eigen__0) => ((in @ (setunion @ ((dsetconstr @ eigen__1) @ (^[X2:$i]:((in @ ((kpair @ X1) @ X2)) @ eigen__2))))) @ eigen__1)))),introduced(definition,[new_symbols(definition,[sP29])])). 103.19/103.19 thf(sP30,plain,sP30 <=> ((![X1:$o]:(![X2:$i]:(((in @ X2) @ (prop2set @ X1)) => X1))) => sP8),introduced(definition,[new_symbols(definition,[sP30])])). 103.19/103.19 thf(sP31,plain,sP31 <=> ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => sP12)),introduced(definition,[new_symbols(definition,[sP31])])). 103.19/103.19 thf(sP32,plain,sP32 <=> ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => sP13),introduced(definition,[new_symbols(definition,[sP32])])). 103.19/103.19 thf(sP33,plain,sP33 <=> (![X1:$i]:(((in @ X1) @ eigen__0) => (singleton @ ((dsetconstr @ eigen__1) @ (^[X2:$i]:((in @ ((kpair @ X1) @ X2)) @ eigen__2)))))),introduced(definition,[new_symbols(definition,[sP33])])). 103.19/103.19 thf(sP34,plain,sP34 <=> ((![X1:$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))))) => sP6)))))),introduced(definition,[new_symbols(definition,[sP34])])). 103.19/103.19 thf(sP35,plain,sP35 <=> ((![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:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => sP9)))))),introduced(definition,[new_symbols(definition,[sP35])])). 103.19/103.19 thf(sP36,plain,sP36 <=> (![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((((breln @ X1) @ X2) @ X3) => (~((![X4:$i]:(((in @ X4) @ X1) => ((ex1 @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))),introduced(definition,[new_symbols(definition,[sP36])])). 103.19/103.19 thf(sP37,plain,sP37 <=> ((singleton @ ((dsetconstr @ eigen__1) @ (^[X1:$i]:((in @ ((kpair @ eigen__3) @ X1)) @ eigen__2)))) => ((in @ (setunion @ ((dsetconstr @ eigen__1) @ (^[X1:$i]:((in @ ((kpair @ eigen__3) @ X1)) @ eigen__2))))) @ ((dsetconstr @ eigen__1) @ (^[X1:$i]:((in @ ((kpair @ eigen__3) @ X1)) @ eigen__2))))),introduced(definition,[new_symbols(definition,[sP37])])). 103.19/103.19 thf(sP38,plain,sP38 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) = ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => sP10),introduced(definition,[new_symbols(definition,[sP38])])). 103.19/103.19 thf(sP39,plain,sP39 <=> ((![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))))) => sP6))),introduced(definition,[new_symbols(definition,[sP39])])). 103.19/103.19 thf(sP40,plain,sP40 <=> ((![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:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => sP14)))))))))))),introduced(definition,[new_symbols(definition,[sP40])])). 103.19/103.19 thf(sP41,plain,sP41 <=> ((![X1:$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))))))) => sP3))))))),introduced(definition,[new_symbols(definition,[sP41])])). 103.19/103.19 thf(sP42,plain,sP42 <=> ((in @ (setunion @ ((dsetconstr @ eigen__1) @ (^[X1:$i]:((in @ ((kpair @ eigen__3) @ X1)) @ eigen__2))))) @ eigen__1),introduced(definition,[new_symbols(definition,[sP42])])). 103.19/103.19 thf(sP43,plain,sP43 <=> ((![X1:$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:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => sP14)))))))),introduced(definition,[new_symbols(definition,[sP43])])). 103.19/103.19 thf(sP44,plain,sP44 <=> ((![X1:$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)))))) => sP1)),introduced(definition,[new_symbols(definition,[sP44])])). 103.19/103.19 thf(sP45,plain,sP45 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$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)))))) => sP28),introduced(definition,[new_symbols(definition,[sP45])])). 103.19/103.19 thf(sP46,plain,sP46 <=> (((!!) @ (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:$i]:((subset @ X1) @ X1)) => sP35))))))),introduced(definition,[new_symbols(definition,[sP46])])). 103.19/103.19 thf(sP47,plain,sP47 <=> ((in @ eigen__3) @ eigen__0),introduced(definition,[new_symbols(definition,[sP47])])). 103.19/103.19 thf(sP48,plain,sP48 <=> (![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => ((in @ X3) @ X1))))),introduced(definition,[new_symbols(definition,[sP48])])). 103.19/103.19 thf(sP49,plain,sP49 <=> ((![X1:$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)))))))))))))) => sP26)))))))))))),introduced(definition,[new_symbols(definition,[sP49])])). 103.19/103.19 thf(sP50,plain,sP50 <=> ((![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))))))) => sP18))))))),introduced(definition,[new_symbols(definition,[sP50])])). 103.19/103.19 thf(sP51,plain,sP51 <=> ((![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:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => sP24)))))))),introduced(definition,[new_symbols(definition,[sP51])])). 103.19/103.19 thf(sP52,plain,sP52 <=> ((![X1:$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))))))) => sP20))),introduced(definition,[new_symbols(definition,[sP52])])). 103.19/103.19 thf(sP53,plain,sP53 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => sP41),introduced(definition,[new_symbols(definition,[sP53])])). 103.19/103.19 thf(sP54,plain,sP54 <=> ((![X1:$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))))) => sP45)))),introduced(definition,[new_symbols(definition,[sP54])])). 103.19/103.19 thf(sP55,plain,sP55 <=> ((![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)))) => sP46))))),introduced(definition,[new_symbols(definition,[sP55])])). 103.19/103.19 thf(sP56,plain,sP56 <=> ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => sP46),introduced(definition,[new_symbols(definition,[sP56])])). 103.19/103.19 thf(sP57,plain,sP57 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => sP20),introduced(definition,[new_symbols(definition,[sP57])])). 103.19/103.19 thf(sP58,plain,sP58 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => sP26),introduced(definition,[new_symbols(definition,[sP58])])). 103.19/103.19 thf(sP59,plain,sP59 <=> ((![X1:$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:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => sP32)))))))),introduced(definition,[new_symbols(definition,[sP59])])). 103.19/103.19 thf(sP60,plain,sP60 <=> ((![X1:$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:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => sP22)))))),introduced(definition,[new_symbols(definition,[sP60])])). 103.19/103.19 thf(sP61,plain,sP61 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => sP22),introduced(definition,[new_symbols(definition,[sP61])])). 103.19/103.19 thf(sP62,plain,sP62 <=> ((![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))))))) => sP3)))))),introduced(definition,[new_symbols(definition,[sP62])])). 103.19/103.19 thf(sP63,plain,sP63 <=> ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => sP34),introduced(definition,[new_symbols(definition,[sP63])])). 103.19/103.19 thf(sP64,plain,sP64 <=> ((![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)))) => sP40))),introduced(definition,[new_symbols(definition,[sP64])])). 103.19/103.19 thf(sP65,plain,sP65 <=> (![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((((breln @ X1) @ X2) @ X3) => (~((![X4:$i]:(((in @ X4) @ X1) => ((ex1 @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))),introduced(definition,[new_symbols(definition,[sP65])])). 103.19/103.19 thf(sP66,plain,sP66 <=> ((![X1:$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:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => sP32))),introduced(definition,[new_symbols(definition,[sP66])])). 103.19/103.19 thf(sP67,plain,sP67 <=> ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => sP3),introduced(definition,[new_symbols(definition,[sP67])])). 103.19/103.19 thf(sP68,plain,sP68 <=> ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => sP9)),introduced(definition,[new_symbols(definition,[sP68])])). 103.19/103.19 thf(sP69,plain,sP69 <=> ((![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))))) => sP49)),introduced(definition,[new_symbols(definition,[sP69])])). 103.19/103.19 thf(sP70,plain,sP70 <=> ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => sP12),introduced(definition,[new_symbols(definition,[sP70])])). 103.19/103.19 thf(sP71,plain,sP71 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => sP35)),introduced(definition,[new_symbols(definition,[sP71])])). 103.19/103.19 thf(sP72,plain,sP72 <=> ((![X1:$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))))) => sP45))),introduced(definition,[new_symbols(definition,[sP72])])). 103.19/103.19 thf(sP73,plain,sP73 <=> ((~(((((breln @ eigen__0) @ eigen__1) @ eigen__2) => (~((![X1:$i]:(((in @ X1) @ eigen__0) => ((ex1 @ eigen__1) @ (^[X2:$i]:((in @ ((kpair @ X1) @ X2)) @ eigen__2)))))))))) => sP33),introduced(definition,[new_symbols(definition,[sP73])])). 103.19/103.19 thf(sP74,plain,sP74 <=> ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => sP51)),introduced(definition,[new_symbols(definition,[sP74])])). 103.19/103.19 thf(sP75,plain,sP75 <=> (sP48 => sP5),introduced(definition,[new_symbols(definition,[sP75])])). 103.19/103.19 thf(sP76,plain,sP76 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => sP71)),introduced(definition,[new_symbols(definition,[sP76])])). 103.19/103.19 thf(sP77,plain,sP77 <=> ((![X1:$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))))))))) => sP61)))),introduced(definition,[new_symbols(definition,[sP77])])). 103.19/103.19 thf(sP78,plain,sP78 <=> ((![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)))))))) => sP39)),introduced(definition,[new_symbols(definition,[sP78])])). 103.19/103.19 thf(sP79,plain,sP79 <=> ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => sP55),introduced(definition,[new_symbols(definition,[sP79])])). 103.19/103.19 thf(sP80,plain,sP80 <=> ((![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)))))))) => sP58)))))))))),introduced(definition,[new_symbols(definition,[sP80])])). 103.19/103.19 thf(sP81,plain,sP81 <=> (![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))),introduced(definition,[new_symbols(definition,[sP81])])). 103.19/103.19 thf(sP82,plain,sP82 <=> ((![X1:$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))))))) => sP18))),introduced(definition,[new_symbols(definition,[sP82])])). 103.19/103.19 thf(sP83,plain,sP83 <=> (singleton @ ((dsetconstr @ eigen__1) @ (^[X1:$i]:((in @ ((kpair @ eigen__3) @ X1)) @ eigen__2)))),introduced(definition,[new_symbols(definition,[sP83])])). 103.19/103.19 thf(sP84,plain,sP84 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![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))))) => sP45)),introduced(definition,[new_symbols(definition,[sP84])])). 103.19/103.19 thf(sP85,plain,sP85 <=> ((![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:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => sP32)),introduced(definition,[new_symbols(definition,[sP85])])). 103.19/103.19 thf(sP86,plain,sP86 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => sP76)),introduced(definition,[new_symbols(definition,[sP86])])). 103.19/103.19 thf(sP87,plain,sP87 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => sP52),introduced(definition,[new_symbols(definition,[sP87])])). 103.19/103.19 thf(sP88,plain,sP88 <=> ((![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)))))))) => sP63)),introduced(definition,[new_symbols(definition,[sP88])])). 103.19/103.19 thf(sP89,plain,sP89 <=> ((![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))) => sP14))))))),introduced(definition,[new_symbols(definition,[sP89])])). 103.19/103.19 thf(sP90,plain,sP90 <=> ((![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>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => sP87))))))),introduced(definition,[new_symbols(definition,[sP90])])). 103.19/103.19 thf(sP91,plain,sP91 <=> ((![X1:$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))))) => sP87)))),introduced(definition,[new_symbols(definition,[sP91])])). 103.19/103.19 thf(sP92,plain,sP92 <=> ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => sP17)),introduced(definition,[new_symbols(definition,[sP92])])). 103.19/103.19 thf(sP93,plain,sP93 <=> ((in @ (setunion @ ((dsetconstr @ eigen__1) @ (^[X1:$i]:((in @ ((kpair @ eigen__3) @ X1)) @ eigen__2))))) @ ((dsetconstr @ eigen__1) @ (^[X1:$i]:((in @ ((kpair @ eigen__3) @ X1)) @ eigen__2)))),introduced(definition,[new_symbols(definition,[sP93])])). 103.19/103.19 thf(sP94,plain,sP94 <=> ((![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)))))))) => sP58)))))),introduced(definition,[new_symbols(definition,[sP94])])). 103.19/103.19 thf(sP95,plain,sP95 <=> ((![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)))))) => (sP81 => sP24))))))),introduced(definition,[new_symbols(definition,[sP95])])). 103.19/103.19 thf(sP96,plain,sP96 <=> ((![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:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => sP14)))),introduced(definition,[new_symbols(definition,[sP96])])). 103.22/103.20 thf(sP97,plain,sP97 <=> ((![X1:$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:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => sP94))),introduced(definition,[new_symbols(definition,[sP97])])). 103.22/103.20 thf(sP98,plain,sP98 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => sP94),introduced(definition,[new_symbols(definition,[sP98])])). 103.22/103.20 thf(sP99,plain,sP99 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$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)))))) => sP1),introduced(definition,[new_symbols(definition,[sP99])])). 103.22/103.20 thf(sP100,plain,sP100 <=> ((![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))))) => sP87)),introduced(definition,[new_symbols(definition,[sP100])])). 103.22/103.20 thf(sP101,plain,sP101 <=> ((![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:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((in @ X3) @ ((dsetconstr @ X1) @ X2))))))) => sP75))))),introduced(definition,[new_symbols(definition,[sP101])])). 103.22/103.20 thf(sP102,plain,sP102 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => sP23),introduced(definition,[new_symbols(definition,[sP102])])). 103.22/103.20 thf(sP103,plain,sP103 <=> ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => sP9),introduced(definition,[new_symbols(definition,[sP103])])). 103.22/103.20 thf(sP104,plain,sP104 <=> ((![X1:$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:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => sP90)))),introduced(definition,[new_symbols(definition,[sP104])])). 103.22/103.20 thf(sP105,plain,sP105 <=> ((![X1:$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))))))) => sP75)))),introduced(definition,[new_symbols(definition,[sP105])])). 103.22/103.20 thf(sP106,plain,sP106 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => sP18),introduced(definition,[new_symbols(definition,[sP106])])). 103.22/103.20 thf(sP107,plain,sP107 <=> ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => sP56)),introduced(definition,[new_symbols(definition,[sP107])])). 103.22/103.20 thf(sP108,plain,sP108 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => sP61),introduced(definition,[new_symbols(definition,[sP108])])). 103.22/103.20 thf(sP109,plain,sP109 <=> ((![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)))))))) => sP58))))),introduced(definition,[new_symbols(definition,[sP109])])). 103.22/103.20 thf(sP110,plain,sP110 <=> ((![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)))) => sP50)),introduced(definition,[new_symbols(definition,[sP110])])). 103.22/103.20 thf(sP111,plain,sP111 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => sP100),introduced(definition,[new_symbols(definition,[sP111])])). 103.22/103.20 thf(sP112,plain,sP112 <=> ((![X1:$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:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => (sP81 => sP24)))))),introduced(definition,[new_symbols(definition,[sP112])])). 103.22/103.20 thf(sP113,plain,sP113 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => sP64),introduced(definition,[new_symbols(definition,[sP113])])). 103.22/103.20 thf(sP114,plain,sP114 <=> ((![X1:$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:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => sP43))),introduced(definition,[new_symbols(definition,[sP114])])). 103.22/103.20 thf(sP115,plain,sP115 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => sP57),introduced(definition,[new_symbols(definition,[sP115])])). 103.22/103.20 thf(sP116,plain,sP116 <=> ((![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))))) => sP101)),introduced(definition,[new_symbols(definition,[sP116])])). 103.22/103.20 thf(sP117,plain,sP117 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => sP58),introduced(definition,[new_symbols(definition,[sP117])])). 103.22/103.20 thf(sP118,plain,sP118 <=> (sP81 => sP24),introduced(definition,[new_symbols(definition,[sP118])])). 103.22/103.20 thf(sP119,plain,sP119 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) = (~((![X3:$i]:(((in @ X2) @ X3) => (~(((in @ X3) @ X1)))))))))) => (((in @ emptyset) @ omega) => sP116)),introduced(definition,[new_symbols(definition,[sP119])])). 103.22/103.20 thf(sP120,plain,sP120 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => sP106),introduced(definition,[new_symbols(definition,[sP120])])). 103.22/103.20 thf(sP121,plain,sP121 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => sP108),introduced(definition,[new_symbols(definition,[sP121])])). 103.22/103.20 thf(sP122,plain,sP122 <=> (((in @ emptyset) @ omega) => sP116),introduced(definition,[new_symbols(definition,[sP122])])). 103.22/103.20 thf(sP123,plain,sP123 <=> ((![X1:$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)))) => sP88))))))),introduced(definition,[new_symbols(definition,[sP123])])). 103.22/103.20 thf(sP124,plain,sP124 <=> ((![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:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => sP88))),introduced(definition,[new_symbols(definition,[sP124])])). 103.22/103.20 thf(sP125,plain,sP125 <=> ((![X1:$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)))))))) => sP124))),introduced(definition,[new_symbols(definition,[sP125])])). 103.22/103.20 thf(sP126,plain,sP126 <=> ((![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:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => sP118)))),introduced(definition,[new_symbols(definition,[sP126])])). 103.22/103.20 thf(sP127,plain,sP127 <=> (sP47 => sP83),introduced(definition,[new_symbols(definition,[sP127])])). 103.22/103.20 thf(sP128,plain,sP128 <=> ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => sP6)),introduced(definition,[new_symbols(definition,[sP128])])). 103.22/103.20 thf(sP129,plain,sP129 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => sP60)),introduced(definition,[new_symbols(definition,[sP129])])). 103.22/103.20 thf(sP130,plain,sP130 <=> ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => sP53)),introduced(definition,[new_symbols(definition,[sP130])])). 103.22/103.20 thf(sP131,plain,sP131 <=> ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => sP71),introduced(definition,[new_symbols(definition,[sP131])])). 103.22/103.20 thf(sP132,plain,sP132 <=> ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => sP50),introduced(definition,[new_symbols(definition,[sP132])])). 103.22/103.20 thf(sP133,plain,sP133 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => sP45),introduced(definition,[new_symbols(definition,[sP133])])). 103.22/103.20 thf(sP134,plain,sP134 <=> ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => sP96),introduced(definition,[new_symbols(definition,[sP134])])). 103.22/103.20 thf(sP135,plain,sP135 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => sP25),introduced(definition,[new_symbols(definition,[sP135])])). 103.22/103.20 thf(sP136,plain,sP136 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => sP121),introduced(definition,[new_symbols(definition,[sP136])])). 103.22/103.20 thf(sP137,plain,sP137 <=> ((![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:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((in @ X3) @ ((dsetconstr @ X1) @ X2))))))) => sP75))),introduced(definition,[new_symbols(definition,[sP137])])). 103.22/103.20 thf(sP138,plain,sP138 <=> ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => sP123)),introduced(definition,[new_symbols(definition,[sP138])])). 103.22/103.20 thf(sP139,plain,sP139 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => sP87),introduced(definition,[new_symbols(definition,[sP139])])). 103.22/103.20 thf(sP140,plain,sP140 <=> ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => sP38),introduced(definition,[new_symbols(definition,[sP140])])). 103.22/103.20 thf(sP141,plain,sP141 <=> ((omega = omega) => sP17),introduced(definition,[new_symbols(definition,[sP141])])). 103.22/103.20 thf(sP142,plain,sP142 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => sP90),introduced(definition,[new_symbols(definition,[sP142])])). 103.22/103.20 thf(sP143,plain,sP143 <=> ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => sP53),introduced(definition,[new_symbols(definition,[sP143])])). 103.22/103.20 thf(sP144,plain,sP144 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => sP91),introduced(definition,[new_symbols(definition,[sP144])])). 103.22/103.20 thf(sP145,plain,sP145 <=> ((![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)))))))))) => sP117))),introduced(definition,[new_symbols(definition,[sP145])])). 103.22/103.21 thf(sP146,plain,sP146 <=> ((![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))))))) => sP75)),introduced(definition,[new_symbols(definition,[sP146])])). 103.22/103.21 thf(sP147,plain,sP147 <=> ((![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)))) => sP67)))),introduced(definition,[new_symbols(definition,[sP147])])). 103.22/103.21 thf(sP148,plain,sP148 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => sP21),introduced(definition,[new_symbols(definition,[sP148])])). 103.22/103.21 thf(sP149,plain,sP149 <=> ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) = ((in @ X3) @ X2))) => (X1 = X2)))) => sP140),introduced(definition,[new_symbols(definition,[sP149])])). 103.22/103.21 thf(sP150,plain,sP150 <=> ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => sP67)),introduced(definition,[new_symbols(definition,[sP150])])). 103.22/103.21 thf(sP151,plain,sP151 <=> ((![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))))) => sP92)),introduced(definition,[new_symbols(definition,[sP151])])). 103.22/103.21 thf(sP152,plain,sP152 <=> ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => sP39),introduced(definition,[new_symbols(definition,[sP152])])). 103.22/103.21 thf(sP153,plain,sP153 <=> ((![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:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => sP118)),introduced(definition,[new_symbols(definition,[sP153])])). 103.22/103.21 thf(sP154,plain,sP154 <=> ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => sP6),introduced(definition,[new_symbols(definition,[sP154])])). 103.22/103.21 thf(sP155,plain,sP155 <=> (![X1:$i]:(((in @ X1) @ eigen__0) => ((in @ (setunion @ ((dsetconstr @ eigen__1) @ (^[X2:$i]:((in @ ((kpair @ X1) @ X2)) @ eigen__2))))) @ eigen__1))),introduced(definition,[new_symbols(definition,[sP155])])). 103.22/103.21 thf(sP156,plain,sP156 <=> ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => sP16),introduced(definition,[new_symbols(definition,[sP156])])). 103.22/103.21 thf(sP157,plain,sP157 <=> ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => sP66)),introduced(definition,[new_symbols(definition,[sP157])])). 103.22/103.21 thf(sP158,plain,sP158 <=> ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => sP118),introduced(definition,[new_symbols(definition,[sP158])])). 103.22/103.21 thf(sP159,plain,sP159 <=> ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => sP134),introduced(definition,[new_symbols(definition,[sP159])])). 103.22/103.21 thf(sP160,plain,sP160 <=> (![X1:$i]:((~(((((breln @ eigen__0) @ eigen__1) @ X1) => (~((![X2:$i]:(((in @ X2) @ eigen__0) => ((ex1 @ eigen__1) @ (^[X3:$i]:((in @ ((kpair @ X2) @ X3)) @ X1)))))))))) => (![X2:$i]:(((in @ X2) @ eigen__0) => (singleton @ ((dsetconstr @ eigen__1) @ (^[X3:$i]:((in @ ((kpair @ X2) @ X3)) @ X1)))))))),introduced(definition,[new_symbols(definition,[sP160])])). 103.22/103.21 thf(sP161,plain,sP161 <=> ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => sP123),introduced(definition,[new_symbols(definition,[sP161])])). 103.22/103.21 thf(sP162,plain,sP162 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => sP14)),introduced(definition,[new_symbols(definition,[sP162])])). 103.22/103.21 thf(sP163,plain,sP163 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => sP142),introduced(definition,[new_symbols(definition,[sP163])])). 103.22/103.21 thf(sP164,plain,sP164 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => sP129),introduced(definition,[new_symbols(definition,[sP164])])). 103.22/103.21 thf(sP165,plain,sP165 <=> ((![X1:$i]:((subset @ X1) @ X1)) => sP35),introduced(definition,[new_symbols(definition,[sP165])])). 103.22/103.21 thf(sP166,plain,sP166 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => sP162),introduced(definition,[new_symbols(definition,[sP166])])). 103.22/103.21 thf(sP167,plain,sP167 <=> (![X1:$i]:(![X2:$i]:((~(((((breln @ eigen__0) @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ eigen__0) => ((ex1 @ X1) @ (^[X4:$i]:((in @ ((kpair @ X3) @ X4)) @ X2)))))))))) => (![X3:$i]:(((in @ X3) @ eigen__0) => ((in @ (setunion @ ((dsetconstr @ X1) @ (^[X4:$i]:((in @ ((kpair @ X3) @ X4)) @ X2))))) @ X1)))))),introduced(definition,[new_symbols(definition,[sP167])])). 103.22/103.21 thf(sP168,plain,sP168 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => sP76),introduced(definition,[new_symbols(definition,[sP168])])). 103.22/103.21 thf(sP169,plain,sP169 <=> ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => sP14),introduced(definition,[new_symbols(definition,[sP169])])). 103.22/103.21 thf(sP170,plain,sP170 <=> ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => sP68),introduced(definition,[new_symbols(definition,[sP170])])). 103.22/103.21 thf(sP171,plain,sP171 <=> ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => sP110),introduced(definition,[new_symbols(definition,[sP171])])). 103.22/103.21 thf(sP172,plain,sP172 <=> ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => sP107),introduced(definition,[new_symbols(definition,[sP172])])). 103.22/103.21 thf(sP173,plain,sP173 <=> ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))))) => sP30),introduced(definition,[new_symbols(definition,[sP173])])). 103.22/103.21 thf(sP174,plain,sP174 <=> ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => sP170)),introduced(definition,[new_symbols(definition,[sP174])])). 103.22/103.21 thf(sP175,plain,sP175 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => sP144),introduced(definition,[new_symbols(definition,[sP175])])). 103.22/103.21 thf(sP176,plain,sP176 <=> (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => sP66),introduced(definition,[new_symbols(definition,[sP176])])). 103.22/103.21 thf(sP177,plain,sP177 <=> ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => sP150),introduced(definition,[new_symbols(definition,[sP177])])). 103.22/103.21 thf(sP178,plain,sP178 <=> ((![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:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => sP82))),introduced(definition,[new_symbols(definition,[sP178])])). 103.22/103.21 thf(sP179,plain,sP179 <=> ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => sP40)),introduced(definition,[new_symbols(definition,[sP179])])). 103.22/103.21 thf(sP180,plain,sP180 <=> ((![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)))))))))) => sP117)),introduced(definition,[new_symbols(definition,[sP180])])). 103.22/103.21 thf(sP181,plain,sP181 <=> ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => sP51),introduced(definition,[new_symbols(definition,[sP181])])). 103.22/103.21 thf(sP182,plain,sP182 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => sP82),introduced(definition,[new_symbols(definition,[sP182])])). 103.22/103.21 thf(sP183,plain,sP183 <=> ((![X1:$i]:((~((((in @ emptyset) @ X1) => (~((![X2:$i]:((~((((in @ X2) @ omega) => (~(((in @ X2) @ X1)))))) => ((in @ ((setadjoin @ X2) @ X2)) @ X1)))))))) => (![X2:$i]:(((in @ X2) @ omega) => ((in @ X2) @ X1))))) => sP101),introduced(definition,[new_symbols(definition,[sP183])])). 103.22/103.21 thf(sP184,plain,sP184 <=> ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => sP67),introduced(definition,[new_symbols(definition,[sP184])])). 103.22/103.21 thf(sP185,plain,sP185 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => sP117),introduced(definition,[new_symbols(definition,[sP185])])). 103.22/103.21 thf(sP186,plain,sP186 <=> (![X1:$i]:(((in @ X1) @ ((dsetconstr @ eigen__1) @ (^[X2:$i]:((in @ ((kpair @ eigen__3) @ X2)) @ eigen__2)))) => ((in @ X1) @ eigen__1))),introduced(definition,[new_symbols(definition,[sP186])])). 103.22/103.21 thf(sP187,plain,sP187 <=> ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => sP63),introduced(definition,[new_symbols(definition,[sP187])])). 103.22/103.21 thf(sP188,plain,sP188 <=> ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => sP56),introduced(definition,[new_symbols(definition,[sP188])])). 103.22/103.21 thf(sP189,plain,sP189 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => sP98),introduced(definition,[new_symbols(definition,[sP189])])). 103.22/103.21 thf(sP190,plain,sP190 <=> ((![X1:$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:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => sP124)),introduced(definition,[new_symbols(definition,[sP190])])). 103.22/103.21 thf(sP191,plain,sP191 <=> ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => sP88)),introduced(definition,[new_symbols(definition,[sP191])])). 103.22/103.21 thf(sP192,plain,sP192 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => sP32),introduced(definition,[new_symbols(definition,[sP192])])). 103.22/103.21 thf(sP193,plain,sP193 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => sP43),introduced(definition,[new_symbols(definition,[sP193])])). 103.22/103.21 thf(sP194,plain,sP194 <=> ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => sP182),introduced(definition,[new_symbols(definition,[sP194])])). 103.22/103.21 thf(sP195,plain,sP195 <=> ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => sP49),introduced(definition,[new_symbols(definition,[sP195])])). 103.22/103.21 thf(sP196,plain,sP196 <=> (sP93 => sP42),introduced(definition,[new_symbols(definition,[sP196])])). 103.22/103.21 thf(sP197,plain,sP197 <=> ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => sP163),introduced(definition,[new_symbols(definition,[sP197])])). 103.22/103.21 thf(sP198,plain,sP198 <=> ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => sP157),introduced(definition,[new_symbols(definition,[sP198])])). 103.22/103.21 thf(sP199,plain,sP199 <=> ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => sP193),introduced(definition,[new_symbols(definition,[sP199])])). 103.22/103.21 thf(sP200,plain,sP200 <=> ((((breln @ eigen__0) @ eigen__1) @ eigen__2) => (~((![X1:$i]:(((in @ X1) @ eigen__0) => ((ex1 @ eigen__1) @ (^[X2:$i]:((in @ ((kpair @ X1) @ X2)) @ eigen__2)))))))),introduced(definition,[new_symbols(definition,[sP200])])). 103.22/103.21 thf(sP201,plain,sP201 <=> ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((in @ X3) @ ((dsetconstr @ X1) @ X2))))))) => sP75),introduced(definition,[new_symbols(definition,[sP201])])). 103.22/103.21 thf(sP202,plain,sP202 <=> ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => sP124),introduced(definition,[new_symbols(definition,[sP202])])). 103.22/103.21 thf(sP203,plain,sP203 <=> ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => sP92),introduced(definition,[new_symbols(definition,[sP203])])). 103.22/103.21 thf(sP204,plain,sP204 <=> (sP47 => sP42),introduced(definition,[new_symbols(definition,[sP204])])). 103.22/103.21 thf(sP205,plain,sP205 <=> ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => sP198),introduced(definition,[new_symbols(definition,[sP205])])). 103.22/103.21 thf(sP206,plain,sP206 <=> ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => sP88),introduced(definition,[new_symbols(definition,[sP206])])). 103.22/103.21 thf(sP207,plain,sP207 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => sP170),introduced(definition,[new_symbols(definition,[sP207])])). 103.22/103.21 thf(sP208,plain,sP208 <=> ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => sP153),introduced(definition,[new_symbols(definition,[sP208])])). 103.22/103.21 thf(sP209,plain,sP209 <=> (![X1:$i]:(![X2:$i]:((~(((((breln @ eigen__0) @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ eigen__0) => ((ex1 @ X1) @ (^[X4:$i]:((in @ ((kpair @ X3) @ X4)) @ X2)))))))))) => (![X3:$i]:(((in @ X3) @ eigen__0) => (singleton @ ((dsetconstr @ X1) @ (^[X4:$i]:((in @ ((kpair @ X3) @ X4)) @ X2))))))))),introduced(definition,[new_symbols(definition,[sP209])])). 103.22/103.21 thf(sP210,plain,sP210 <=> ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => sP40),introduced(definition,[new_symbols(definition,[sP210])])). 103.22/103.21 thf(sP211,plain,sP211 <=> (![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ ((dsetconstr @ eigen__1) @ X1)) => ((in @ X2) @ eigen__1)))),introduced(definition,[new_symbols(definition,[sP211])])). 103.22/103.21 thf(sP212,plain,sP212 <=> ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => sP60),introduced(definition,[new_symbols(definition,[sP212])])). 103.22/103.21 thf(def_setextAx,definition,(setextAx = (![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) = ((in @ X3) @ X2))) => (X1 = X2)))))). 103.22/103.21 thf(def_emptysetAx,definition,(emptysetAx = (![X1:$i]:(~(((in @ X1) @ emptyset)))))). 103.22/103.21 thf(def_setadjoinAx,definition,(setadjoinAx = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) = ((~((X3 = X1))) => ((in @ X3) @ X2)))))))). 103.22/103.21 thf(def_powersetAx,definition,(powersetAx = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) = (![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1)))))))). 103.22/103.21 thf(def_setunionAx,definition,(setunionAx = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) = (~((![X3:$i]:(((in @ X2) @ X3) => (~(((in @ X3) @ X1)))))))))))). 103.22/103.21 thf(def_omega0Ax,definition,(omega0Ax = ((in @ emptyset) @ omega))). 103.22/103.21 thf(def_omegaSAx,definition,(omegaSAx = (![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))))). 103.22/103.21 thf(def_omegaIndAx,definition,(omegaIndAx = (![X1:$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))))))). 103.22/103.21 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))))))))))))))))))). 103.22/103.21 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))))))))))))))). 103.22/103.21 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:$i]:(((in @ X7) @ X6) => ((in @ X7) @ X4))))) => ((in @ X5) @ X6)))))))))))))))))))))). 103.22/103.21 thf(def_descrp,definition,(descrp = (![X1:$i>$o]:((exu @ X1) => (X1 @ (descr @ X1)))))). 103.22/103.21 thf(def_dsetconstrI,definition,(dsetconstrI = (![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((in @ X3) @ ((dsetconstr @ X1) @ X2))))))))). 103.22/103.21 thf(def_dsetconstrEL,definition,(dsetconstrEL = sP48)). 103.22/103.21 thf(def_dsetconstrER,definition,(dsetconstrER = (![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => (X2 @ X3))))))). 103.22/103.21 thf(def_exuE1,definition,(exuE1 = (![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))))))). 103.22/103.21 thf(def_prop2setE,definition,(prop2setE = (![X1:$o]:(![X2:$i]:(((in @ X2) @ (prop2set @ X1)) => X1))))). 103.22/103.21 thf(def_emptysetE,definition,(emptysetE = (![X1:$i]:(((in @ X1) @ emptyset) => (![X2:$o]:X2))))). 103.22/103.21 thf(def_emptysetimpfalse,definition,(emptysetimpfalse = (![X1:$i]:(~(((in @ X1) @ emptyset)))))). 103.22/103.21 thf(def_notinemptyset,definition,(notinemptyset = (![X1:$i]:(~(((in @ X1) @ emptyset)))))). 103.22/103.21 thf(def_exuE3e,definition,(exuE3e = (![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))))). 103.22/103.21 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))))))). 103.22/103.21 thf(def_emptyI,definition,(emptyI = (![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))))). 103.22/103.21 thf(def_noeltsimpempty,definition,(noeltsimpempty = (![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))))). 103.22/103.21 thf(def_setbeta,definition,(setbeta = (![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))))). 103.22/103.21 thf(def_nonemptyE1,definition,(nonemptyE1 = (![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))))). 103.22/103.21 thf(def_nonemptyI,definition,(nonemptyI = (![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))))). 103.22/103.21 thf(def_nonemptyI1,definition,(nonemptyI1 = (![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))))). 103.22/103.21 thf(def_setadjoinIL,definition,(setadjoinIL = (![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))))). 103.22/103.21 thf(def_emptyinunitempty,definition,(emptyinunitempty = ((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)))). 103.22/103.21 thf(def_setadjoinIR,definition,(setadjoinIR = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))))). 103.22/103.21 thf(def_setadjoinE,definition,(setadjoinE = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))))). 103.22/103.21 thf(def_setadjoinOr,definition,(setadjoinOr = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))))). 103.22/103.21 thf(def_setoftrueEq,definition,(setoftrueEq = (![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)))). 103.22/103.21 thf(def_powersetI,definition,(powersetI = (![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))))). 103.22/103.21 thf(def_emptyinPowerset,definition,(emptyinPowerset = (![X1:$i]:((in @ emptyset) @ (powerset @ X1))))). 103.22/103.21 thf(def_emptyInPowerset,definition,(emptyInPowerset = (![X1:$i]:((in @ emptyset) @ (powerset @ X1))))). 103.22/103.21 thf(def_powersetE,definition,(powersetE = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))))). 103.22/103.21 thf(def_setunionI,definition,(setunionI = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))))). 103.22/103.21 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))))))). 103.22/103.21 thf(def_subPowSU,definition,(subPowSU = (![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))))). 103.22/103.21 thf(def_exuE2,definition,(exuE2 = (![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))))). 103.22/103.21 thf(def_nonemptyImpWitness,definition,(nonemptyImpWitness = (![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))))). 103.22/103.21 thf(def_uniqinunit,definition,(uniqinunit = (![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))))). 103.22/103.21 thf(def_notinsingleton,definition,(notinsingleton = (![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))))). 103.22/103.21 thf(def_eqinunit,definition,(eqinunit = (![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))))). 103.22/103.21 thf(def_singletonsswitch,definition,(singletonsswitch = (![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))). 103.22/103.21 thf(def_upairsetE,definition,(upairsetE = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))))). 103.22/103.21 thf(def_upairsetIL,definition,(upairsetIL = (![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))))). 103.22/103.21 thf(def_upairsetIR,definition,(upairsetIR = (![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))))). 103.22/103.21 thf(def_emptyE1,definition,(emptyE1 = (![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))))). 103.22/103.21 thf(def_vacuousDall,definition,(vacuousDall = (![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))))). 103.22/103.21 thf(def_quantDeMorgan1,definition,(quantDeMorgan1 = (![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))))). 103.22/103.21 thf(def_quantDeMorgan2,definition,(quantDeMorgan2 = (![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))). 103.22/103.21 thf(def_quantDeMorgan3,definition,(quantDeMorgan3 = (![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))). 103.22/103.21 thf(def_quantDeMorgan4,definition,(quantDeMorgan4 = (![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))))). 103.22/103.21 thf(def_prop2setI,definition,(prop2setI = (![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))))). 103.22/103.21 thf(def_prop2set2propI,definition,(prop2set2propI = (![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))))). 103.22/103.21 thf(def_notdexE,definition,(notdexE = (![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))). 103.22/103.21 thf(def_notdallE,definition,(notdallE = (![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))))). 103.22/103.21 thf(def_exuI1,definition,(exuI1 = (![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))))). 103.22/103.21 thf(def_exuI3,definition,(exuI3 = (![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))))). 103.22/103.21 thf(def_exuI2,definition,(exuI2 = (![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))))). 103.22/103.21 thf(def_inCongP,definition,(inCongP = (![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))))). 103.22/103.21 thf(def_in__Cong,definition,(in__Cong = (![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))))). 103.22/103.21 thf(def_exuE3u,definition,(exuE3u = (![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))))). 103.22/103.21 thf(def_exu__Cong,definition,(exu__Cong = (![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))))). 103.22/103.21 thf(def_emptyset__Cong,definition,(emptyset__Cong = (emptyset = emptyset))). 103.22/103.21 thf(def_setadjoin__Cong,definition,(setadjoin__Cong = (![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))))). 103.22/103.21 thf(def_powerset__Cong,definition,(powerset__Cong = (![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))))). 103.22/103.21 thf(def_setunion__Cong,definition,(setunion__Cong = (![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))))). 103.22/103.21 thf(def_omega__Cong,definition,(omega__Cong = (omega = omega))). 103.22/103.21 thf(def_exuEu,definition,(exuEu = (![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))))). 103.22/103.21 thf(def_descr__Cong,definition,(descr__Cong = (![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))))). 103.22/103.21 thf(def_dsetconstr__Cong,definition,(dsetconstr__Cong = (![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))))). 103.22/103.21 thf(def_subsetI1,definition,(subsetI1 = (![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))))). 103.22/103.21 thf(def_eqimpsubset2,definition,(eqimpsubset2 = (![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))))). 103.22/103.21 thf(def_eqimpsubset1,definition,(eqimpsubset1 = (![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))))). 103.22/103.21 thf(def_subsetI2,definition,(subsetI2 = (![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))))). 103.22/103.21 thf(def_emptysetsubset,definition,(emptysetsubset = ((!!) @ (subset @ emptyset)))). 103.22/103.21 thf(def_subsetE,definition,(subsetE = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))))). 103.22/103.21 thf(def_subsetE2,definition,(subsetE2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))))). 103.22/103.21 thf(def_notsubsetI,definition,(notsubsetI = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))))). 103.22/103.21 thf(def_notequalI1,definition,(notequalI1 = (![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))))). 103.22/103.21 thf(def_notequalI2,definition,(notequalI2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))))). 103.22/103.21 thf(def_subsetRefl,definition,(subsetRefl = (![X1:$i]:((subset @ X1) @ X1)))). 103.22/103.21 thf(def_subsetTrans,definition,(subsetTrans = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))))). 103.22/103.21 thf(def_setadjoinSub,definition,(setadjoinSub = (![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))))). 103.22/103.21 thf(def_setadjoinSub2,definition,(setadjoinSub2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))))). 103.22/103.21 thf(def_subset2powerset,definition,(subset2powerset = (![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))))). 103.22/103.21 thf(def_setextsub,definition,(setextsub = (![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))))). 103.22/103.21 thf(def_subsetemptysetimpeq,definition,(subsetemptysetimpeq = (![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))))). 103.22/103.21 thf(def_powersetI1,definition,(powersetI1 = (![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))))). 103.22/103.21 thf(def_powersetE1,definition,(powersetE1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))))). 103.22/103.21 thf(def_inPowerset,definition,(inPowerset = (![X1:$i]:((in @ X1) @ (powerset @ X1))))). 103.22/103.21 thf(def_powersetsubset,definition,(powersetsubset = (![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))))). 103.22/103.21 thf(def_sepInPowerset,definition,(sepInPowerset = (![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))))). 103.22/103.21 thf(def_sepSubset,definition,(sepSubset = (![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))))). 103.22/103.21 thf(def_binunionIL,definition,(binunionIL = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))))). 103.22/103.21 thf(def_upairset2IR,definition,(upairset2IR = (![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))))). 103.22/103.21 thf(def_binunionIR,definition,(binunionIR = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))))). 103.22/103.21 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))))))))). 103.22/103.21 thf(def_binunionE,definition,(binunionE = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))))). 103.22/103.21 thf(def_binunionLsub,definition,(binunionLsub = (![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))))). 103.22/103.21 thf(def_binunionRsub,definition,(binunionRsub = (![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))))). 103.22/103.21 thf(def_binintersectI,definition,(binintersectI = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))))). 103.22/103.21 thf(def_binintersectSubset5,definition,(binintersectSubset5 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))). 103.22/103.21 thf(def_binintersectEL,definition,(binintersectEL = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))))). 103.22/103.21 thf(def_binintersectLsub,definition,(binintersectLsub = (![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))))). 103.22/103.21 thf(def_binintersectSubset2,definition,(binintersectSubset2 = (![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))))). 103.22/103.21 thf(def_binintersectSubset3,definition,(binintersectSubset3 = (![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))))). 103.22/103.21 thf(def_binintersectER,definition,(binintersectER = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))))). 103.22/103.21 thf(def_disjointsetsI1,definition,(disjointsetsI1 = (![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))))). 103.22/103.21 thf(def_binintersectRsub,definition,(binintersectRsub = (![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))))). 103.22/103.21 thf(def_binintersectSubset4,definition,(binintersectSubset4 = (![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))))). 103.22/103.21 thf(def_binintersectSubset1,definition,(binintersectSubset1 = (![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))))). 103.22/103.21 thf(def_bs114d,definition,(bs114d = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))))). 103.22/103.21 thf(def_setminusI,definition,(setminusI = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))))). 103.22/103.21 thf(def_setminusEL,definition,(setminusEL = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))))). 103.22/103.21 thf(def_setminusER,definition,(setminusER = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))))). 103.22/103.21 thf(def_setminusSubset2,definition,(setminusSubset2 = (![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))))). 103.22/103.21 thf(def_setminusERneg,definition,(setminusERneg = (![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))))). 103.22/103.21 thf(def_setminusELneg,definition,(setminusELneg = (![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))))). 103.22/103.21 thf(def_setminusILneg,definition,(setminusILneg = (![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))). 103.22/103.21 thf(def_setminusIRneg,definition,(setminusIRneg = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))). 103.22/103.21 thf(def_setminusLsub,definition,(setminusLsub = (![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))))). 103.22/103.21 thf(def_setminusSubset1,definition,(setminusSubset1 = (![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))))). 103.22/103.21 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))))))))). 103.22/103.21 thf(def_symdiffI1,definition,(symdiffI1 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))))). 103.22/103.21 thf(def_symdiffI2,definition,(symdiffI2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))))). 103.22/103.21 thf(def_symdiffIneg1,definition,(symdiffIneg1 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))))). 103.22/103.21 thf(def_symdiffIneg2,definition,(symdiffIneg2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))))). 103.22/103.21 thf(def_secondinupair,definition,(secondinupair = (![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))))). 103.22/103.21 thf(def_setukpairIL,definition,(setukpairIL = (![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))))). 103.22/103.21 thf(def_setukpairIR,definition,(setukpairIR = (![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))))). 103.22/103.21 thf(def_kpairiskpair,definition,(kpairiskpair = (![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))))). 103.22/103.21 thf(def_kpairp,definition,(kpairp = (![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))))). 103.22/103.21 thf(def_singletonsubset,definition,(singletonsubset = (![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))))). 103.22/103.21 thf(def_singletoninpowerset,definition,(singletoninpowerset = (![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))))). 103.22/103.21 thf(def_singletoninpowunion,definition,(singletoninpowunion = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))))). 103.22/103.21 thf(def_upairset2E,definition,(upairset2E = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))))). 103.22/103.21 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)))))))))). 103.22/103.21 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))))))))))). 103.22/103.21 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))))))))))). 103.22/103.21 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)))))))))))). 103.22/103.21 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)))))))))))). 103.22/103.21 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)))))))))). 103.22/103.21 thf(def_cartprodmempair1,definition,(cartprodmempair1 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))))). 103.22/103.21 thf(def_cartprodmempair,definition,(cartprodmempair = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))))). 103.22/103.21 thf(def_setunionE2,definition,(setunionE2 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))))). 103.22/103.21 thf(def_setunionsingleton1,definition,(setunionsingleton1 = (![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)))). 103.22/103.21 thf(def_setunionsingleton2,definition,(setunionsingleton2 = (![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))))). 103.22/103.21 thf(def_setunionsingleton,definition,(setunionsingleton = (![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)))). 103.22/103.21 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)))))))). 103.22/103.21 thf(def_ex1E1,definition,(ex1E1 = (![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))))). 103.22/103.21 thf(def_ex1I,definition,(ex1I = (![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))))). 103.22/103.21 thf(def_ex1I2,definition,(ex1I2 = (![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))))). 103.22/103.21 thf(def_singletonsuniq,definition,(singletonsuniq = (![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))))). 103.22/103.21 thf(def_setukpairinjL1,definition,(setukpairinjL1 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))))). 103.22/103.21 thf(def_kfstsingleton,definition,(kfstsingleton = (![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))))). 103.22/103.21 thf(def_theprop,definition,(theprop = sP81)). 103.22/103.21 thf(def_kfstpairEq,definition,(kfstpairEq = (![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))))). 103.22/103.21 thf(def_cartprodfstin,definition,(cartprodfstin = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))))). 103.22/103.21 thf(def_setukpairinjL2,definition,(setukpairinjL2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))))). 103.22/103.21 thf(def_setukpairinjL,definition,(setukpairinjL = (![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))))). 103.22/103.21 thf(def_setukpairinjR11,definition,(setukpairinjR11 = (![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))))). 103.22/103.21 thf(def_setukpairinjR12,definition,(setukpairinjR12 = (![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))))). 103.22/103.21 thf(def_setukpairinjR1,definition,(setukpairinjR1 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))))). 103.22/103.21 thf(def_upairequniteq,definition,(upairequniteq = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))))). 103.22/103.21 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)))))))). 103.22/103.21 thf(def_setukpairinjR,definition,(setukpairinjR = (![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))))). 103.22/103.21 thf(def_ksndsingleton,definition,(ksndsingleton = (![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))))). 103.22/103.21 thf(def_ksndpairEq,definition,(ksndpairEq = (![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))))). 103.22/103.21 thf(def_kpairsurjEq,definition,(kpairsurjEq = (![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))))). 103.22/103.21 thf(def_cartprodsndin,definition,(cartprodsndin = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))))). 103.22/103.21 thf(def_cartprodpairmemEL,definition,(cartprodpairmemEL = (![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))))). 103.22/103.21 thf(def_cartprodpairmemER,definition,(cartprodpairmemER = (![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))))). 103.22/103.21 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)))))))))). 103.22/103.21 thf(def_cartprodfstpairEq,definition,(cartprodfstpairEq = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))))). 103.22/103.21 thf(def_cartprodsndpairEq,definition,(cartprodsndpairEq = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))))). 103.22/103.21 thf(def_cartprodpairsurjEq,definition,(cartprodpairsurjEq = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))))). 103.22/103.21 thf(def_dpsetconstrI,definition,(dpsetconstrI = (![X1:$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)))))))))))). 103.22/103.21 thf(def_dpsetconstrSub,definition,(dpsetconstrSub = (![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))))). 103.22/103.21 thf(def_setOfPairsIsBReln,definition,(setOfPairsIsBReln = (![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))))). 103.22/103.21 thf(def_dpsetconstrERa,definition,(dpsetconstrERa = (![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))))). 103.22/103.21 thf(def_dpsetconstrEL1,definition,(dpsetconstrEL1 = (![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))))). 103.22/103.21 thf(def_dpsetconstrEL2,definition,(dpsetconstrEL2 = (![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))))). 103.22/103.21 thf(def_dpsetconstrER,definition,(dpsetconstrER = (![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))). 103.22/103.21 thf(def_func,definition,(func = (^[X1:$i]:(^[X2:$i]:(^[X3:$i]:(~(((((breln @ X1) @ X2) @ X3) => (~((![X4:$i]:(((in @ X4) @ X1) => ((ex1 @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))))))))))))). 103.22/103.21 thf(def_funcImageSingleton,definition,(funcImageSingleton = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))). 103.22/103.21 thf(apProp,conjecture,sP149). 103.22/103.21 thf(h1,negated_conjecture,(~(sP149)),inference(assume_negation,[status(cth)],[apProp])). 103.22/103.21 thf(1,plain,(~(sP211) | sP186),inference(all_rule,[status(thm)],[])). 103.22/103.21 thf(2,plain,(~(sP186) | sP196),inference(all_rule,[status(thm)],[])). 103.22/103.21 thf(3,plain,((~(sP196) | ~(sP93)) | sP42),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(4,plain,(~(sP48) | sP211),inference(all_rule,[status(thm)],[])). 103.22/103.21 thf(5,plain,(~(sP81) | sP37),inference(all_rule,[status(thm)],[])). 103.22/103.21 thf(6,plain,((~(sP37) | ~(sP83)) | sP93),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(7,plain,(~(sP33) | sP127),inference(all_rule,[status(thm)],[])). 103.22/103.21 thf(8,plain,((~(sP127) | ~(sP47)) | sP83),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(9,plain,(~(sP65) | sP209),inference(all_rule,[status(thm)],[])). 103.22/103.21 thf(10,plain,(~(sP209) | sP160),inference(all_rule,[status(thm)],[])). 103.22/103.21 thf(11,plain,(~(sP160) | sP73),inference(all_rule,[status(thm)],[])). 103.22/103.21 thf(12,plain,((~(sP73) | sP200) | sP33),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(13,plain,(sP204 | ~(sP42)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(14,plain,(sP204 | sP47),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(15,plain,(sP155 | ~(sP204)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__3])). 103.22/103.21 thf(16,plain,(sP29 | ~(sP155)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(17,plain,(sP29 | ~(sP200)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(18,plain,(sP7 | ~(sP29)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__2])). 103.22/103.21 thf(19,plain,(sP167 | ~(sP7)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__1])). 103.22/103.21 thf(20,plain,(sP36 | ~(sP167)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__0])). 103.22/103.21 thf(21,plain,(sP15 | ~(sP36)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(22,plain,(sP15 | sP65),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(23,plain,(sP20 | ~(sP15)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(24,plain,(sP57 | ~(sP20)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(25,plain,(sP115 | ~(sP57)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(26,plain,(sP52 | ~(sP115)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(27,plain,(sP87 | ~(sP52)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(28,plain,(sP139 | ~(sP87)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(29,plain,(sP100 | ~(sP139)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(30,plain,(sP111 | ~(sP100)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(31,plain,(sP91 | ~(sP111)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(32,plain,(sP144 | ~(sP91)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(33,plain,(sP175 | ~(sP144)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(34,plain,(sP90 | ~(sP175)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(35,plain,(sP142 | ~(sP90)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(36,plain,(sP163 | ~(sP142)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(37,plain,(sP197 | ~(sP163)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(38,plain,(sP104 | ~(sP197)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(39,plain,(sP4 | ~(sP104)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(40,plain,(sP28 | ~(sP4)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(41,plain,(sP45 | ~(sP28)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(42,plain,(sP133 | ~(sP45)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(43,plain,(sP84 | ~(sP133)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(44,plain,(sP72 | ~(sP84)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(45,plain,(sP54 | ~(sP72)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(46,plain,(sP1 | ~(sP54)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(47,plain,(sP99 | ~(sP1)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(48,plain,(sP44 | ~(sP99)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(49,plain,(sP24 | ~(sP44)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(50,plain,(sP118 | ~(sP24)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(51,plain,(sP118 | sP81),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(52,plain,(sP158 | ~(sP118)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(53,plain,(sP153 | ~(sP158)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(54,plain,(sP208 | ~(sP153)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(55,plain,(sP126 | ~(sP208)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(56,plain,(sP112 | ~(sP126)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(57,plain,(sP95 | ~(sP112)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(58,plain,(sP51 | ~(sP95)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(59,plain,(sP181 | ~(sP51)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(60,plain,(sP74 | ~(sP181)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(61,plain,(sP21 | ~(sP74)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(62,plain,(sP148 | ~(sP21)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(63,plain,(sP26 | ~(sP148)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(64,plain,(sP58 | ~(sP26)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(65,plain,(sP117 | ~(sP58)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(66,plain,(sP185 | ~(sP117)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(67,plain,(sP180 | ~(sP185)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(68,plain,(sP145 | ~(sP180)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(69,plain,(sP109 | ~(sP145)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(70,plain,(sP94 | ~(sP109)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(71,plain,(sP98 | ~(sP94)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(72,plain,(sP189 | ~(sP98)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(73,plain,(sP97 | ~(sP189)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(74,plain,(sP80 | ~(sP97)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(75,plain,(sP49 | ~(sP80)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(76,plain,(sP195 | ~(sP49)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(77,plain,(sP69 | ~(sP195)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(78,plain,(sP19 | ~(sP69)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(79,plain,(sP22 | ~(sP19)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(80,plain,(sP61 | ~(sP22)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(81,plain,(sP108 | ~(sP61)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(82,plain,(sP121 | ~(sP108)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(83,plain,(sP136 | ~(sP121)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(84,plain,(sP77 | ~(sP136)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(85,plain,(sP60 | ~(sP77)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(86,plain,(sP212 | ~(sP60)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(87,plain,(sP129 | ~(sP212)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(88,plain,(sP164 | ~(sP129)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(89,plain,(sP25 | ~(sP164)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(90,plain,(sP135 | ~(sP25)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(91,plain,(sP18 | ~(sP135)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(92,plain,(sP106 | ~(sP18)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(93,plain,(sP120 | ~(sP106)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(94,plain,(sP82 | ~(sP120)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(95,plain,(sP182 | ~(sP82)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(96,plain,(sP194 | ~(sP182)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(97,plain,(sP178 | ~(sP194)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(98,plain,(sP50 | ~(sP178)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(99,plain,(sP132 | ~(sP50)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(100,plain,(sP110 | ~(sP132)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(101,plain,(sP171 | ~(sP110)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(102,plain,(sP14 | ~(sP171)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(103,plain,(sP169 | ~(sP14)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(104,plain,(sP162 | ~(sP169)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(105,plain,(sP166 | ~(sP162)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(106,plain,(sP96 | ~(sP166)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(107,plain,(sP134 | ~(sP96)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(108,plain,(sP159 | ~(sP134)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(109,plain,(sP89 | ~(sP159)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(110,plain,(sP43 | ~(sP89)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(111,plain,(sP193 | ~(sP43)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(112,plain,(sP199 | ~(sP193)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(113,plain,(sP114 | ~(sP199)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(114,plain,(sP40 | ~(sP114)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(115,plain,(sP210 | ~(sP40)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(116,plain,(sP179 | ~(sP210)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(117,plain,(sP64 | ~(sP179)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(118,plain,(sP113 | ~(sP64)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(119,plain,(sP9 | ~(sP113)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(120,plain,(sP103 | ~(sP9)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(121,plain,(sP68 | ~(sP103)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(122,plain,(sP170 | ~(sP68)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(123,plain,(sP207 | ~(sP170)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(124,plain,(sP174 | ~(sP207)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(125,plain,(sP35 | ~(sP174)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(126,plain,(sP165 | ~(sP35)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(127,plain,(sP71 | ~(sP165)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(128,plain,(sP131 | ~(sP71)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(129,plain,(sP76 | ~(sP131)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(130,plain,(sP168 | ~(sP76)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(131,plain,(sP86 | ~(sP168)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(132,plain,(sP46 | ~(sP86)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(133,plain,(sP56 | ~(sP46)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(134,plain,(sP188 | ~(sP56)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(135,plain,(sP107 | ~(sP188)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(136,plain,(sP172 | ~(sP107)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(137,plain,(sP55 | ~(sP172)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(138,plain,(sP79 | ~(sP55)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(139,plain,(sP17 | ~(sP79)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(140,plain,(sP141 | ~(sP17)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(141,plain,(sP92 | ~(sP141)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(142,plain,(sP203 | ~(sP92)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(143,plain,(sP151 | ~(sP203)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(144,plain,(sP6 | ~(sP151)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(145,plain,(sP154 | ~(sP6)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(146,plain,(sP128 | ~(sP154)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(147,plain,(sP39 | ~(sP128)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(148,plain,(sP152 | ~(sP39)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(149,plain,(sP78 | ~(sP152)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(150,plain,(sP34 | ~(sP78)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(151,plain,(sP63 | ~(sP34)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(152,plain,(sP187 | ~(sP63)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(153,plain,(sP88 | ~(sP187)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(154,plain,(sP206 | ~(sP88)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(155,plain,(sP191 | ~(sP206)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(156,plain,(sP124 | ~(sP191)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(157,plain,(sP202 | ~(sP124)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(158,plain,(sP190 | ~(sP202)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(159,plain,(sP125 | ~(sP190)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(160,plain,(sP123 | ~(sP125)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(161,plain,(sP161 | ~(sP123)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(162,plain,(sP138 | ~(sP161)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(163,plain,(sP23 | ~(sP138)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(164,plain,(sP102 | ~(sP23)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(165,plain,(sP11 | ~(sP102)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(166,plain,(sP3 | ~(sP11)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(167,plain,(sP67 | ~(sP3)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(168,plain,(sP184 | ~(sP67)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(169,plain,(sP150 | ~(sP184)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(170,plain,(sP177 | ~(sP150)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(171,plain,(sP147 | ~(sP177)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(172,plain,(sP62 | ~(sP147)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(173,plain,(sP41 | ~(sP62)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(174,plain,(sP53 | ~(sP41)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(175,plain,(sP143 | ~(sP53)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(176,plain,(sP130 | ~(sP143)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(177,plain,(sP13 | ~(sP130)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(178,plain,(sP32 | ~(sP13)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(179,plain,(sP192 | ~(sP32)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(180,plain,(sP85 | ~(sP192)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(181,plain,(sP66 | ~(sP85)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(182,plain,(sP176 | ~(sP66)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(183,plain,(sP157 | ~(sP176)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(184,plain,(sP198 | ~(sP157)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(185,plain,(sP205 | ~(sP198)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(186,plain,(sP59 | ~(sP205)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(187,plain,(sP16 | ~(sP59)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(188,plain,(sP156 | ~(sP16)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(189,plain,(sP2 | ~(sP156)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(190,plain,(sP27 | ~(sP2)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(191,plain,(sP12 | ~(sP27)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(192,plain,(sP70 | ~(sP12)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(193,plain,(sP31 | ~(sP70)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(194,plain,(sP8 | ~(sP31)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(195,plain,(sP30 | ~(sP8)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(196,plain,(sP173 | ~(sP30)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(197,plain,(sP5 | ~(sP173)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(198,plain,(sP75 | ~(sP5)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(199,plain,(sP75 | sP48),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(200,plain,(sP201 | ~(sP75)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(201,plain,(sP146 | ~(sP201)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(202,plain,(sP137 | ~(sP146)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(203,plain,(sP105 | ~(sP137)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(204,plain,(sP101 | ~(sP105)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(205,plain,(sP183 | ~(sP101)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(206,plain,(sP116 | ~(sP183)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(207,plain,(sP122 | ~(sP116)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(208,plain,(sP119 | ~(sP122)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(209,plain,(sP10 | ~(sP119)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(210,plain,(sP38 | ~(sP10)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(211,plain,(sP140 | ~(sP38)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(212,plain,(sP149 | ~(sP140)),inference(prop_rule,[status(thm)],[])). 103.22/103.21 thf(213,plain,$false,inference(prop_unsat,[status(thm),assumptions([h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,h1])). 103.22/103.21 thf(214,plain,$false,inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[213,h0])). 103.22/103.21 thf(0,theorem,sP149,inference(contra,[status(thm),contra(discharge,[h1])],[213,h1])). 103.22/103.21 % SZS output end Proof 103.22/103.21 EOF