0.08/0.13 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.08/0.14 % Command : run_Leo-III---1.5 %s %d 0.14/0.35 % Computer : n004.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 : Thu Jul 2 10:53:14 EDT 2020 0.14/0.35 % CPUTime : 0.14/0.36 % Leo-III: Strategy 1 (for '60') 135.41/58.48 % [INFO] Running in sequential loop mode. 135.41/58.48 % [CONFIG] Using configuration: timeout(60) with strategy 135.41/58.48 % [INFO] eprover registered as external prover. 135.41/58.48 % [INFO] cvc4 registered as external prover. 135.41/58.48 % [INFO] Parsing finished. Scanning for conjecture ... 135.41/58.48 % [INFO] Found a conjecture and 0 axioms. Running axiom selection ... 135.41/58.48 % [INFO] Axiom selection finished. Selected 0 axioms (removed 0 axioms). 135.41/58.48 % [INFO] Type checking passed. Searching for refutation ... 135.41/58.48 % [INFO] Killing All external provers ... 135.41/58.48 % Time passed: 57689ms 135.41/58.48 % Effective reasoning time: 15020ms 135.41/58.48 % Solved by strategy 135.41/58.48 % Axioms used in derivation (0): 135.41/58.48 % No. of inferences in proof: 4 135.41/58.48 % No. of processed clauses: 20 135.41/58.48 % No. of generated clauses: 88 135.41/58.48 % No. of forward subsumed clauses: 8 135.41/58.48 % No. of backward subsumed clauses: 0 135.41/58.48 % No. of ground rewrite rules in store: 4 135.41/58.48 % No. of non-ground rewrite rules in store: 5 135.41/58.48 % No. of positive (non-rewrite) units in store: 0 135.41/58.48 % No. of negative (non-rewrite) units in store: 7 135.41/58.48 % No. of choice functions detected: 0 135.41/58.48 % No. of choice instantiations: 0 135.41/58.48 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 57689 ms resp. 15020 ms w/o parsing 135.41/58.48 % SZS output start CNFRefutation for /export/starexec/sandbox/benchmark/theBenchmark.p 135.41/58.48 thf(in_type, type, in: ($i > ($i > $o))). 135.41/58.48 thf(setextAx_type, type, setextAx: $o). 135.41/58.48 thf(setextAx_def, definition, (setextAx = (! [A:$i,B:$i]: ((! [C:$i]: ((in @ C @ A) = (in @ C @ B))) => (A = B))))). 135.41/58.48 thf(emptyset_type, type, emptyset: $i). 135.41/58.48 thf(emptysetAx_type, type, emptysetAx: $o). 135.41/58.48 thf(emptysetAx_def, definition, (emptysetAx = (! [A:$i]: ~ (in @ A @ emptyset)))). 135.41/58.48 thf(setadjoin_type, type, setadjoin: ($i > ($i > $i))). 135.41/58.48 thf(setadjoinAx_type, type, setadjoinAx: $o). 135.41/58.48 thf(setadjoinAx_def, definition, (setadjoinAx = (! [A:$i,B:$i,C:$i]: ((in @ C @ (setadjoin @ A @ B)) = ((C = A) | (in @ C @ B)))))). 135.41/58.48 thf(powerset_type, type, powerset: ($i > $i)). 135.41/58.48 thf(powersetAx_type, type, powersetAx: $o). 135.41/58.48 thf(powersetAx_def, definition, (powersetAx = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) = (! [C:$i]: ((in @ C @ B) => (in @ C @ A))))))). 135.41/58.48 thf(setunion_type, type, setunion: ($i > $i)). 135.41/58.48 thf(setunionAx_type, type, setunionAx: $o). 135.41/58.48 thf(setunionAx_def, definition, (setunionAx = (! [A:$i,B:$i]: ((in @ B @ (setunion @ A)) = (? [C:$i]: ((in @ B @ C) & (in @ C @ A))))))). 135.41/58.48 thf(omega_type, type, omega: $i). 135.41/58.48 thf(omega0Ax_type, type, omega0Ax: $o). 135.41/58.48 thf(omega0Ax_def, definition, (omega0Ax = (in @ emptyset @ omega))). 135.41/58.48 thf(omegaSAx_type, type, omegaSAx: $o). 135.41/58.48 thf(omegaSAx_def, definition, (omegaSAx = (! [A:$i]: ((in @ A @ omega) => (in @ (setadjoin @ A @ A) @ omega))))). 135.41/58.48 thf(omegaIndAx_type, type, omegaIndAx: $o). 135.41/58.48 thf(omegaIndAx_def, definition, (omegaIndAx = (! [A:$i]: (((in @ emptyset @ A) & ! [B:$i]: (((in @ B @ omega) & (in @ B @ A)) => (in @ (setadjoin @ B @ B) @ A))) => (! [B:$i]: ((in @ B @ omega) => (in @ B @ A))))))). 135.41/58.48 thf(replAx_type, type, replAx: $o). 135.41/58.48 thf(replAx_def, definition, (replAx = (! [A:($i > ($i > $o)),B:$i]: ((! [C:$i]: ((in @ C @ B) => (exu @ (A @ C)))) => (? [C:$i]: ! [D:$i]: ((in @ D @ C) = (? [E:$i]: ((in @ E @ B) & (A @ E @ D))))))))). 135.41/58.48 thf(foundationAx_type, type, foundationAx: $o). 135.41/58.48 thf(foundationAx_def, definition, (foundationAx = (! [A:$i]: ((? [B:$i]: (in @ B @ A)) => (? [B:$i]: ((in @ B @ A) & ~ (? [C:$i]: ((in @ C @ B) & (in @ C @ A))))))))). 135.41/58.48 thf(wellorderingAx_type, type, wellorderingAx: $o). 135.41/58.48 thf(wellorderingAx_def, definition, (wellorderingAx = (! [A:$i]: ? [B:$i]: (! [C:$i]: ((in @ C @ B) => (! [D:$i]: ((in @ D @ C) => (in @ D @ A)))) & ! [C:$i,D:$i]: (((in @ C @ A) & (in @ D @ A)) => ((! [E:$i]: ((in @ E @ B) => ((in @ C @ E) = (in @ D @ E)))) => (C = D))) & ! [C:$i,D:$i]: (((in @ C @ B) & (in @ D @ B)) => (! [E:$i]: ((in @ E @ C) => (in @ E @ D)) | ! [E:$i]: ((in @ E @ D) => (in @ E @ C)))) & ! [C:$i]: ((! [D:$i]: ((in @ D @ C) => (in @ D @ A)) & ? [D:$i]: (in @ D @ C)) => (? [D:$i,E:$i]: ((in @ D @ B) & (in @ E @ C) & ~ (? [F:$i]: ((in @ F @ D) & (in @ F @ C))) & ! [F:$i]: ((in @ F @ B) => (! [G:$i]: ((in @ G @ F) => (in @ G @ D)) | (in @ E @ F)))))))))). 135.41/58.48 thf(descr_type, type, descr: (($i > $o) > $i)). 135.41/58.48 thf(descrp_type, type, descrp: $o). 135.41/58.48 thf(descrp_def, definition, (descrp = (! [A:($i > $o)]: ((exu @ (A)) => (A @ (descr @ (A))))))). 135.41/58.48 thf(dsetconstr_type, type, dsetconstr: ($i > (($i > $o) > $i))). 135.41/58.48 thf(dsetconstrI_type, type, dsetconstrI: $o). 135.41/58.48 thf(dsetconstrI_def, definition, (dsetconstrI = (! [A:$i,B:($i > $o),C:$i]: ((in @ C @ A) => ((B @ C) => (in @ C @ (dsetconstr @ A @ (B)))))))). 135.41/58.48 thf(dsetconstrEL_type, type, dsetconstrEL: $o). 135.41/58.48 thf(dsetconstrEL_def, definition, (dsetconstrEL = (! [A:$i,B:($i > $o),C:$i]: ((in @ C @ (dsetconstr @ A @ (B))) => (in @ C @ A))))). 135.41/58.48 thf(dsetconstrER_type, type, dsetconstrER: $o). 135.41/58.48 thf(dsetconstrER_def, definition, (dsetconstrER = (! [A:$i,B:($i > $o),C:$i]: ((in @ C @ (dsetconstr @ A @ (B))) => (B @ C))))). 135.41/58.48 thf(exuE1_type, type, exuE1: $o). 135.41/58.48 thf(exuE1_def, definition, (exuE1 = (! [A:($i > $o)]: ((exu @ (A)) => (? [B:$i]: ((A @ B) & ! [C:$i]: ((A @ C) => (B = C)))))))). 135.41/58.48 thf(prop2set_type, type, prop2set: ($o > $i)). 135.41/58.48 thf(prop2setE_type, type, prop2setE: $o). 135.41/58.48 thf(prop2setE_def, definition, (prop2setE = (! [A:$o,B:$i]: ((in @ B @ (prop2set @ A)) => (A))))). 135.41/58.48 thf(emptysetE_type, type, emptysetE: $o). 135.41/58.48 thf(emptysetE_def, definition, (emptysetE = (! [A:$i]: ((in @ A @ emptyset) => (! [B:$o]: (B)))))). 135.41/58.48 thf(emptysetimpfalse_type, type, emptysetimpfalse: $o). 135.41/58.48 thf(emptysetimpfalse_def, definition, (emptysetimpfalse = (! [A:$i]: ((in @ A @ emptyset) => ($false))))). 135.41/58.48 thf(notinemptyset_type, type, notinemptyset: $o). 135.41/58.48 thf(notinemptyset_def, definition, (notinemptyset = (! [A:$i]: ~ (in @ A @ emptyset)))). 135.41/58.48 thf(exuE3e_type, type, exuE3e: $o). 135.41/58.48 thf(exuE3e_def, definition, (exuE3e = (! [A:($i > $o)]: ((exu @ (A)) => (? [B:$i]: (A @ B)))))). 135.41/58.48 thf(setext_type, type, setext: $o). 135.41/58.48 thf(setext_def, definition, (setext = (! [A:$i,B:$i]: ((! [C:$i]: ((in @ C @ A) => (in @ C @ B))) => ((! [C:$i]: ((in @ C @ B) => (in @ C @ A))) => (A = B)))))). 135.41/58.48 thf(emptyI_type, type, emptyI: $o). 135.41/58.48 thf(emptyI_def, definition, (emptyI = (! [A:$i]: ((! [B:$i]: ~ (in @ B @ A)) => (A = emptyset))))). 135.41/58.48 thf(noeltsimpempty_type, type, noeltsimpempty: $o). 135.41/58.48 thf(noeltsimpempty_def, definition, (noeltsimpempty = (! [A:$i]: ((! [B:$i]: ~ (in @ B @ A)) => (A = emptyset))))). 135.41/58.48 thf(setbeta_type, type, setbeta: $o). 135.41/58.48 thf(setbeta_def, definition, (setbeta = (! [A:$i,B:($i > $o),C:$i]: ((in @ C @ A) => ((in @ C @ (dsetconstr @ A @ (B))) = (B @ C)))))). 135.41/58.48 thf(nonemptyE1_type, type, nonemptyE1: $o). 135.41/58.48 thf(nonemptyE1_def, definition, (nonemptyE1 = (! [A:$i]: ((nonempty @ A) => (? [B:$i]: (in @ B @ A)))))). 135.41/58.48 thf(nonemptyI_type, type, nonemptyI: $o). 135.41/58.48 thf(nonemptyI_def, definition, (nonemptyI = (! [A:$i,B:($i > $o),C:$i]: ((in @ C @ A) => ((B @ C) => (nonempty @ (dsetconstr @ A @ (B)))))))). 135.41/58.48 thf(nonemptyI1_type, type, nonemptyI1: $o). 135.41/58.48 thf(nonemptyI1_def, definition, (nonemptyI1 = (! [A:$i]: ((? [B:$i]: (in @ B @ A)) => (nonempty @ A))))). 135.41/58.48 thf(setadjoinIL_type, type, setadjoinIL: $o). 135.41/58.48 thf(setadjoinIL_def, definition, (setadjoinIL = (! [A:$i,B:$i]: (in @ A @ (setadjoin @ A @ B))))). 135.41/58.48 thf(emptyinunitempty_type, type, emptyinunitempty: $o). 135.41/58.48 thf(emptyinunitempty_def, definition, (emptyinunitempty = (in @ emptyset @ (setadjoin @ emptyset @ emptyset)))). 135.41/58.48 thf(setadjoinIR_type, type, setadjoinIR: $o). 135.41/58.48 thf(setadjoinIR_def, definition, (setadjoinIR = (! [A:$i,B:$i,C:$i]: ((in @ C @ B) => (in @ C @ (setadjoin @ A @ B)))))). 135.41/58.48 thf(setadjoinE_type, type, setadjoinE: $o). 135.41/58.48 thf(setadjoinE_def, definition, (setadjoinE = (! [A:$i,B:$i,C:$i]: ((in @ C @ (setadjoin @ A @ B)) => (! [D:$o]: (((C = A) => (D)) => (((in @ C @ B) => (D)) => (D)))))))). 135.41/58.48 thf(setadjoinOr_type, type, setadjoinOr: $o). 135.41/58.48 thf(setadjoinOr_def, definition, (setadjoinOr = (! [A:$i,B:$i,C:$i]: ((in @ C @ (setadjoin @ A @ B)) => ((C = A) | (in @ C @ B)))))). 135.41/58.48 thf(setoftrueEq_type, type, setoftrueEq: $o). 135.41/58.48 thf(setoftrueEq_def, definition, (setoftrueEq = (! [A:$i]: ((dsetconstr @ A @ (^ [B:$i]: ($true))) = A)))). 135.41/58.48 thf(powersetI_type, type, powersetI: $o). 135.41/58.48 thf(powersetI_def, definition, (powersetI = (! [A:$i,B:$i]: ((! [C:$i]: ((in @ C @ B) => (in @ C @ A))) => (in @ B @ (powerset @ A)))))). 135.41/58.48 thf(emptyinPowerset_type, type, emptyinPowerset: $o). 135.41/58.48 thf(emptyinPowerset_def, definition, (emptyinPowerset = (! [A:$i]: (in @ emptyset @ (powerset @ A))))). 135.41/58.48 thf(emptyInPowerset_type, type, emptyInPowerset: $o). 135.41/58.48 thf(emptyInPowerset_def, definition, (emptyInPowerset = (! [A:$i]: (in @ emptyset @ (powerset @ A))))). 135.41/58.48 thf(powersetE_type, type, powersetE: $o). 135.41/58.48 thf(powersetE_def, definition, (powersetE = (! [A:$i,B:$i,C:$i]: ((in @ B @ (powerset @ A)) => ((in @ C @ B) => (in @ C @ A)))))). 135.41/58.48 thf(setunionI_type, type, setunionI: $o). 135.41/58.48 thf(setunionI_def, definition, (setunionI = (! [A:$i,B:$i,C:$i]: ((in @ B @ C) => ((in @ C @ A) => (in @ B @ (setunion @ A))))))). 135.41/58.48 thf(setunionE_type, type, setunionE: $o). 135.41/58.48 thf(setunionE_def, definition, (setunionE = (! [A:$i,B:$i]: ((in @ B @ (setunion @ A)) => (! [C:$o]: ((! [D:$i]: ((in @ B @ D) => ((in @ D @ A) => (C)))) => (C))))))). 135.41/58.48 thf(subPowSU_type, type, subPowSU: $o). 135.41/58.48 thf(subPowSU_def, definition, (subPowSU = (! [A:$i,B:$i]: ((in @ B @ A) => (in @ B @ (powerset @ (setunion @ A))))))). 135.41/58.48 thf(exuE2_type, type, exuE2: $o). 135.41/58.48 thf(exuE2_def, definition, (exuE2 = (! [A:($i > $o)]: ((exu @ (A)) => (? [B:$i]: ! [C:$i]: ((A @ C) = (C = B))))))). 135.41/58.48 thf(nonemptyImpWitness_type, type, nonemptyImpWitness: $o). 135.41/58.48 thf(nonemptyImpWitness_def, definition, (nonemptyImpWitness = (! [A:$i]: ((nonempty @ A) => (? [B:$i]: ((in @ B @ A) & ($true))))))). 135.41/58.48 thf(uniqinunit_type, type, uniqinunit: $o). 135.41/58.48 thf(uniqinunit_def, definition, (uniqinunit = (! [A:$i,B:$i]: ((in @ A @ (setadjoin @ B @ emptyset)) => (A = B))))). 135.41/58.48 thf(notinsingleton_type, type, notinsingleton: $o). 135.41/58.48 thf(notinsingleton_def, definition, (notinsingleton = (! [A:$i,B:$i]: ((A != B) => (~ (in @ B @ (setadjoin @ A @ emptyset))))))). 135.41/58.48 thf(eqinunit_type, type, eqinunit: $o). 135.41/58.48 thf(eqinunit_def, definition, (eqinunit = (! [A:$i,B:$i]: ((A = B) => (in @ A @ (setadjoin @ B @ emptyset)))))). 135.41/58.48 thf(singletonsswitch_type, type, singletonsswitch: $o). 135.41/58.48 thf(singletonsswitch_def, definition, (singletonsswitch = (! [A:$i,B:$i]: ((in @ A @ (setadjoin @ B @ emptyset)) => (in @ B @ (setadjoin @ A @ emptyset)))))). 135.41/58.48 thf(upairsetE_type, type, upairsetE: $o). 135.41/58.48 thf(upairsetE_def, definition, (upairsetE = (! [A:$i,B:$i,C:$i]: ((in @ C @ (setadjoin @ A @ (setadjoin @ B @ emptyset))) => ((C = A) | (C = B)))))). 135.41/58.48 thf(upairsetIL_type, type, upairsetIL: $o). 135.41/58.48 thf(upairsetIL_def, definition, (upairsetIL = (! [A:$i,B:$i]: (in @ A @ (setadjoin @ A @ (setadjoin @ B @ emptyset)))))). 135.41/58.48 thf(upairsetIR_type, type, upairsetIR: $o). 135.41/58.48 thf(upairsetIR_def, definition, (upairsetIR = (! [A:$i,B:$i]: (in @ B @ (setadjoin @ A @ (setadjoin @ B @ emptyset)))))). 135.41/58.48 thf(emptyE1_type, type, emptyE1: $o). 135.41/58.48 thf(emptyE1_def, definition, (emptyE1 = (! [A:$i,B:($i > $o)]: ((? [C:$i]: ((in @ C @ A) & (B @ C))) => (((dsetconstr @ A @ (B)) = emptyset) => ($false)))))). 135.41/58.48 thf(vacuousDall_type, type, vacuousDall: $o). 135.41/58.48 thf(vacuousDall_def, definition, (vacuousDall = (! [A:($i > $o),B:$i]: ((in @ B @ emptyset) => (A @ B))))). 135.41/58.48 thf(quantDeMorgan1_type, type, quantDeMorgan1: $o). 135.41/58.48 thf(quantDeMorgan1_def, definition, (quantDeMorgan1 = (! [A:$i,B:($i > $o)]: ((~ (! [C:$i]: ((in @ C @ A) => (B @ C)))) => (? [C:$i]: ((in @ C @ A) & ~ (B @ C))))))). 135.41/58.48 thf(quantDeMorgan2_type, type, quantDeMorgan2: $o). 135.41/58.48 thf(quantDeMorgan2_def, definition, (quantDeMorgan2 = (! [A:$i,B:($i > $o)]: ((! [C:$i]: ((in @ C @ A) => (~ (B @ C)))) => (~ (? [C:$i]: ((in @ C @ A) & (B @ C)))))))). 135.41/58.48 thf(quantDeMorgan3_type, type, quantDeMorgan3: $o). 135.41/58.48 thf(quantDeMorgan3_def, definition, (quantDeMorgan3 = (! [A:$i,B:($i > $o)]: ((~ (? [C:$i]: ((in @ C @ A) & (B @ C)))) => (! [C:$i]: ((in @ C @ A) => (~ (B @ C)))))))). 135.41/58.48 thf(quantDeMorgan4_type, type, quantDeMorgan4: $o). 135.41/58.48 thf(quantDeMorgan4_def, definition, (quantDeMorgan4 = (! [A:$i,B:($i > $o)]: ((? [C:$i]: ((in @ C @ A) & ~ (B @ C))) => (~ (! [C:$i]: ((in @ C @ A) => (B @ C)))))))). 135.41/58.48 thf(prop2setI_type, type, prop2setI: $o). 135.41/58.48 thf(prop2setI_def, definition, (prop2setI = (! [A:$o]: ((A) => (in @ emptyset @ (prop2set @ A)))))). 135.41/58.48 thf(set2prop_type, type, set2prop: ($i > $o)). 135.41/58.48 thf(prop2set2propI_type, type, prop2set2propI: $o). 135.41/58.48 thf(prop2set2propI_def, definition, (prop2set2propI = (! [A:$o]: ((A) => (set2prop @ (prop2set @ A)))))). 135.41/58.48 thf(notdexE_type, type, notdexE: $o). 135.41/58.48 thf(notdexE_def, definition, (notdexE = (! [A:$i,B:($i > $o)]: ((~ (? [C:$i]: ((in @ C @ A) & (B @ C)))) => (! [C:$i]: ((in @ C @ A) => (~ (B @ C)))))))). 135.41/58.48 thf(notdallE_type, type, notdallE: $o). 135.41/58.48 thf(notdallE_def, definition, (notdallE = (! [A:$i,B:($i > $o)]: ((~ (! [C:$i]: ((in @ C @ A) => (B @ C)))) => (? [C:$i]: ((in @ C @ A) & ~ (B @ C))))))). 135.41/58.48 thf(exuI1_type, type, exuI1: $o). 135.41/58.48 thf(exuI1_def, definition, (exuI1 = (! [A:($i > $o)]: ((? [B:$i]: ((A @ B) & ! [C:$i]: ((A @ C) => (B = C)))) => (exu @ (A)))))). 135.41/58.48 thf(exuI3_type, type, exuI3: $o). 135.41/58.48 thf(exuI3_def, definition, (exuI3 = (! [A:($i > $o)]: ((? [B:$i]: (A @ B)) => ((! [B:$i,C:$i]: ((A @ B) => ((A @ C) => (B = C)))) => (exu @ (A))))))). 135.41/58.48 thf(exuI2_type, type, exuI2: $o). 135.41/58.48 thf(exuI2_def, definition, (exuI2 = (! [A:($i > $o)]: ((? [B:$i]: ! [C:$i]: ((A @ C) = (C = B))) => (exu @ (A)))))). 135.41/58.48 thf(inCongP_type, type, inCongP: $o). 135.41/58.48 thf(inCongP_def, definition, (inCongP = (! [A:$i,B:$i]: ((A = B) => (! [C:$i,D:$i]: ((C = D) => ((in @ C @ A) => (in @ D @ B)))))))). 135.41/58.48 thf(in__Cong_type, type, in__Cong: $o). 135.41/58.48 thf(in__Cong_def, definition, (in__Cong = (! [A:$i,B:$i]: ((A = B) => (! [C:$i,D:$i]: ((C = D) => ((in @ C @ A) = (in @ D @ B)))))))). 135.41/58.48 thf(exuE3u_type, type, exuE3u: $o). 135.41/58.48 thf(exuE3u_def, definition, (exuE3u = (! [A:($i > $o)]: ((exu @ (A)) => (! [B:$i,C:$i]: ((A @ B) => ((A @ C) => (B = C)))))))). 135.41/58.48 thf(exu__Cong_type, type, exu__Cong: $o). 135.41/58.48 thf(exu__Cong_def, definition, (exu__Cong = (! [A:($i > $o),B:($i > $o)]: ((! [C:$i,D:$i]: ((C = D) => ((A @ C) = (B @ D)))) => ((exu @ (A)) = (exu @ (B))))))). 135.41/58.48 thf(emptyset__Cong_type, type, emptyset__Cong: $o). 135.41/58.48 thf(emptyset__Cong_def, definition, (emptyset__Cong = (emptyset = emptyset))). 135.41/58.48 thf(setadjoin__Cong_type, type, setadjoin__Cong: $o). 135.41/58.48 thf(setadjoin__Cong_def, definition, (setadjoin__Cong = (! [A:$i,B:$i]: ((A = B) => (! [C:$i,D:$i]: ((C = D) => ((setadjoin @ A @ C) = (setadjoin @ B @ D)))))))). 135.41/58.48 thf(powerset__Cong_type, type, powerset__Cong: $o). 135.41/58.48 thf(powerset__Cong_def, definition, (powerset__Cong = (! [A:$i,B:$i]: ((A = B) => ((powerset @ A) = (powerset @ B)))))). 135.41/58.48 thf(setunion__Cong_type, type, setunion__Cong: $o). 135.41/58.48 thf(setunion__Cong_def, definition, (setunion__Cong = (! [A:$i,B:$i]: ((A = B) => ((setunion @ A) = (setunion @ B)))))). 135.41/58.48 thf(omega__Cong_type, type, omega__Cong: $o). 135.41/58.48 thf(omega__Cong_def, definition, (omega__Cong = (omega = omega))). 135.41/58.48 thf(exuEu_type, type, exuEu: $o). 135.41/58.48 thf(exuEu_def, definition, (exuEu = (! [A:($i > $o)]: ((exu @ (A)) => (! [B:$i,C:$i]: ((A @ B) => ((A @ C) => (B = C)))))))). 135.41/58.48 thf(descr__Cong_type, type, descr__Cong: $o). 135.41/58.48 thf(descr__Cong_def, definition, (descr__Cong = (! [A:($i > $o),B:($i > $o)]: ((! [C:$i,D:$i]: ((C = D) => ((A @ C) = (B @ D)))) => ((exu @ (A)) => ((exu @ (B)) => ((descr @ (A)) = (descr @ (B))))))))). 135.41/58.48 thf(dsetconstr__Cong_type, type, dsetconstr__Cong: $o). 135.41/58.48 thf(dsetconstr__Cong_def, definition, (dsetconstr__Cong = (! [A:$i,B:$i]: ((A = B) => (! [C:($i > $o),D:($i > $o)]: ((! [E:$i]: ((in @ E @ A) => (! [F:$i]: ((in @ F @ B) => ((E = F) => ((C @ E) = (D @ F))))))) => ((dsetconstr @ A @ (C)) = (dsetconstr @ B @ (D))))))))). 135.41/58.48 thf(subset_type, type, subset: ($i > ($i > $o))). 135.41/58.48 thf(subsetI1_type, type, subsetI1: $o). 135.41/58.48 thf(subsetI1_def, definition, (subsetI1 = (! [A:$i,B:$i]: ((! [C:$i]: ((in @ C @ A) => (in @ C @ B))) => (subset @ A @ B))))). 135.41/58.48 thf(eqimpsubset2_type, type, eqimpsubset2: $o). 135.41/58.48 thf(eqimpsubset2_def, definition, (eqimpsubset2 = (! [A:$i,B:$i]: ((A = B) => (subset @ B @ A))))). 135.41/58.48 thf(eqimpsubset1_type, type, eqimpsubset1: $o). 135.41/58.48 thf(eqimpsubset1_def, definition, (eqimpsubset1 = (! [A:$i,B:$i]: ((A = B) => (subset @ A @ B))))). 135.41/58.48 thf(subsetI2_type, type, subsetI2: $o). 135.41/58.48 thf(subsetI2_def, definition, (subsetI2 = (! [A:$i,B:$i]: ((! [C:$i]: ((in @ C @ A) => (in @ C @ B))) => (subset @ A @ B))))). 135.41/58.48 thf(emptysetsubset_type, type, emptysetsubset: $o). 135.41/58.48 thf(emptysetsubset_def, definition, (emptysetsubset = (! [A:$i]: (subset @ emptyset @ A)))). 135.41/58.48 thf(subsetE_type, type, subsetE: $o). 135.41/58.48 thf(subsetE_def, definition, (subsetE = (! [A:$i,B:$i,C:$i]: ((subset @ A @ B) => ((in @ C @ A) => (in @ C @ B)))))). 135.41/58.48 thf(subsetE2_type, type, subsetE2: $o). 135.41/58.48 thf(subsetE2_def, definition, (subsetE2 = (! [A:$i,B:$i,C:$i]: ((subset @ A @ B) => ((~ (in @ C @ B)) => (~ (in @ C @ A))))))). 135.41/58.48 thf(notsubsetI_type, type, notsubsetI: $o). 135.41/58.48 thf(notsubsetI_def, definition, (notsubsetI = (! [A:$i,B:$i,C:$i]: ((in @ C @ A) => ((~ (in @ C @ B)) => (~ (subset @ A @ B))))))). 135.41/58.48 thf(notequalI1_type, type, notequalI1: $o). 135.41/58.48 thf(notequalI1_def, definition, (notequalI1 = (! [A:$i,B:$i]: ((~ (subset @ A @ B)) => (A != B))))). 135.41/58.48 thf(notequalI2_type, type, notequalI2: $o). 135.41/58.48 thf(notequalI2_def, definition, (notequalI2 = (! [A:$i,B:$i,C:$i]: ((in @ C @ A) => ((~ (in @ C @ B)) => (A != B)))))). 135.41/58.48 thf(subsetRefl_type, type, subsetRefl: $o). 135.41/58.48 thf(subsetRefl_def, definition, (subsetRefl = (! [A:$i]: (subset @ A @ A)))). 135.41/58.48 thf(subsetTrans_type, type, subsetTrans: $o). 135.41/58.48 thf(subsetTrans_def, definition, (subsetTrans = (! [A:$i,B:$i,C:$i]: ((subset @ A @ B) => ((subset @ B @ C) => (subset @ A @ C)))))). 135.41/58.48 thf(setadjoinSub_type, type, setadjoinSub: $o). 135.41/58.48 thf(setadjoinSub_def, definition, (setadjoinSub = (! [A:$i,B:$i]: (subset @ B @ (setadjoin @ A @ B))))). 135.41/58.48 thf(setadjoinSub2_type, type, setadjoinSub2: $o). 135.41/58.48 thf(setadjoinSub2_def, definition, (setadjoinSub2 = (! [A:$i,B:$i,C:$i]: ((subset @ A @ C) => (subset @ A @ (setadjoin @ B @ C)))))). 135.41/58.48 thf(subset2powerset_type, type, subset2powerset: $o). 135.41/58.48 thf(subset2powerset_def, definition, (subset2powerset = (! [A:$i,B:$i]: ((subset @ A @ B) => (in @ A @ (powerset @ B)))))). 135.41/58.48 thf(setextsub_type, type, setextsub: $o). 135.41/58.48 thf(setextsub_def, definition, (setextsub = (! [A:$i,B:$i]: ((subset @ A @ B) => ((subset @ B @ A) => (A = B)))))). 135.41/58.48 thf(subsetemptysetimpeq_type, type, subsetemptysetimpeq: $o). 135.41/58.48 thf(subsetemptysetimpeq_def, definition, (subsetemptysetimpeq = (! [A:$i]: ((subset @ A @ emptyset) => (A = emptyset))))). 135.41/58.48 thf(powersetI1_type, type, powersetI1: $o). 135.41/58.48 thf(powersetI1_def, definition, (powersetI1 = (! [A:$i,B:$i]: ((subset @ B @ A) => (in @ B @ (powerset @ A)))))). 135.41/58.48 thf(powersetE1_type, type, powersetE1: $o). 135.41/58.48 thf(powersetE1_def, definition, (powersetE1 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (subset @ B @ A))))). 135.41/58.48 thf(inPowerset_type, type, inPowerset: $o). 135.41/58.48 thf(inPowerset_def, definition, (inPowerset = (! [A:$i]: (in @ A @ (powerset @ A))))). 135.41/58.48 thf(powersetsubset_type, type, powersetsubset: $o). 135.41/58.48 thf(powersetsubset_def, definition, (powersetsubset = (! [A:$i,B:$i]: ((subset @ A @ B) => (subset @ (powerset @ A) @ (powerset @ B)))))). 135.41/58.48 thf(sepInPowerset_type, type, sepInPowerset: $o). 135.41/58.48 thf(sepInPowerset_def, definition, (sepInPowerset = (! [A:$i,B:($i > $o)]: (in @ (dsetconstr @ A @ (B)) @ (powerset @ A))))). 135.41/58.48 thf(sepSubset_type, type, sepSubset: $o). 135.41/58.48 thf(sepSubset_def, definition, (sepSubset = (! [A:$i,B:($i > $o)]: (subset @ (dsetconstr @ A @ (B)) @ A)))). 135.41/58.48 thf(binunion_type, type, binunion: ($i > ($i > $i))). 135.41/58.48 thf(binunionIL_type, type, binunionIL: $o). 135.41/58.48 thf(binunionIL_def, definition, (binunionIL = (! [A:$i,B:$i,C:$i]: ((in @ C @ A) => (in @ C @ (binunion @ A @ B)))))). 135.41/58.48 thf(upairset2IR_type, type, upairset2IR: $o). 135.41/58.48 thf(upairset2IR_def, definition, (upairset2IR = (! [A:$i,B:$i]: (in @ B @ (setadjoin @ A @ (setadjoin @ B @ emptyset)))))). 135.41/58.48 thf(binunionIR_type, type, binunionIR: $o). 135.41/58.48 thf(binunionIR_def, definition, (binunionIR = (! [A:$i,B:$i,C:$i]: ((in @ C @ B) => (in @ C @ (binunion @ A @ B)))))). 135.41/58.48 thf(binunionEcases_type, type, binunionEcases: $o). 135.41/58.48 thf(binunionEcases_def, definition, (binunionEcases = (! [A:$i,B:$i,C:$i,D:$o]: ((in @ C @ (binunion @ A @ B)) => (((in @ C @ A) => (D)) => (((in @ C @ B) => (D)) => (D))))))). 135.41/58.48 thf(binunionE_type, type, binunionE: $o). 135.41/58.48 thf(binunionE_def, definition, (binunionE = (! [A:$i,B:$i,C:$i]: ((in @ C @ (binunion @ A @ B)) => ((in @ C @ A) | (in @ C @ B)))))). 135.41/58.48 thf(binunionLsub_type, type, binunionLsub: $o). 135.41/58.48 thf(binunionLsub_def, definition, (binunionLsub = (! [A:$i,B:$i]: (subset @ A @ (binunion @ A @ B))))). 135.41/58.48 thf(binunionRsub_type, type, binunionRsub: $o). 135.41/58.48 thf(binunionRsub_def, definition, (binunionRsub = (! [A:$i,B:$i]: (subset @ B @ (binunion @ A @ B))))). 135.41/58.48 thf(binintersect_type, type, binintersect: ($i > ($i > $i))). 135.41/58.48 thf(binintersectI_type, type, binintersectI: $o). 135.41/58.48 thf(binintersectI_def, definition, (binintersectI = (! [A:$i,B:$i,C:$i]: ((in @ C @ A) => ((in @ C @ B) => (in @ C @ (binintersect @ A @ B))))))). 135.41/58.48 thf(binintersectSubset5_type, type, binintersectSubset5: $o). 135.41/58.48 thf(binintersectSubset5_def, definition, (binintersectSubset5 = (! [A:$i,B:$i,C:$i]: ((subset @ C @ A) => ((subset @ C @ B) => (subset @ C @ (binintersect @ A @ B))))))). 135.41/58.48 thf(binintersectEL_type, type, binintersectEL: $o). 135.41/58.48 thf(binintersectEL_def, definition, (binintersectEL = (! [A:$i,B:$i,C:$i]: ((in @ C @ (binintersect @ A @ B)) => (in @ C @ A))))). 135.41/58.48 thf(binintersectLsub_type, type, binintersectLsub: $o). 135.41/58.48 thf(binintersectLsub_def, definition, (binintersectLsub = (! [A:$i,B:$i]: (subset @ (binintersect @ A @ B) @ A)))). 135.41/58.48 thf(binintersectSubset2_type, type, binintersectSubset2: $o). 135.41/58.48 thf(binintersectSubset2_def, definition, (binintersectSubset2 = (! [A:$i,B:$i]: ((subset @ A @ B) => ((binintersect @ A @ B) = A))))). 135.41/58.48 thf(binintersectSubset3_type, type, binintersectSubset3: $o). 135.41/58.48 thf(binintersectSubset3_def, definition, (binintersectSubset3 = (! [A:$i,B:$i]: (((binintersect @ A @ B) = B) => (subset @ B @ A))))). 135.41/58.48 thf(binintersectER_type, type, binintersectER: $o). 135.41/58.48 thf(binintersectER_def, definition, (binintersectER = (! [A:$i,B:$i,C:$i]: ((in @ C @ (binintersect @ A @ B)) => (in @ C @ B))))). 135.41/58.48 thf(disjointsetsI1_type, type, disjointsetsI1: $o). 135.41/58.48 thf(disjointsetsI1_def, definition, (disjointsetsI1 = (! [A:$i,B:$i]: ((~ (? [C:$i]: ((in @ C @ A) & (in @ C @ B)))) => ((binintersect @ A @ B) = emptyset))))). 135.41/58.48 thf(binintersectRsub_type, type, binintersectRsub: $o). 135.41/58.48 thf(binintersectRsub_def, definition, (binintersectRsub = (! [A:$i,B:$i]: (subset @ (binintersect @ A @ B) @ B)))). 135.41/58.48 thf(binintersectSubset4_type, type, binintersectSubset4: $o). 135.41/58.48 thf(binintersectSubset4_def, definition, (binintersectSubset4 = (! [A:$i,B:$i]: ((subset @ B @ A) => ((binintersect @ A @ B) = B))))). 135.41/58.48 thf(binintersectSubset1_type, type, binintersectSubset1: $o). 135.41/58.48 thf(binintersectSubset1_def, definition, (binintersectSubset1 = (! [A:$i,B:$i]: (((binintersect @ A @ B) = A) => (subset @ A @ B))))). 135.41/58.48 thf(bs114d_type, type, bs114d: $o). 135.41/58.48 thf(bs114d_def, definition, (bs114d = (! [A:$i,B:$i,C:$i]: ((binintersect @ A @ (binunion @ B @ C)) = (binunion @ (binintersect @ A @ B) @ (binintersect @ A @ C)))))). 135.41/58.48 thf(setminus_type, type, setminus: ($i > ($i > $i))). 135.41/58.48 thf(setminusI_type, type, setminusI: $o). 135.41/58.48 thf(setminusI_def, definition, (setminusI = (! [A:$i,B:$i,C:$i]: ((in @ C @ A) => ((~ (in @ C @ B)) => (in @ C @ (setminus @ A @ B))))))). 135.41/58.48 thf(setminusEL_type, type, setminusEL: $o). 135.41/58.48 thf(setminusEL_def, definition, (setminusEL = (! [A:$i,B:$i,C:$i]: ((in @ C @ (setminus @ A @ B)) => (in @ C @ A))))). 135.41/58.48 thf(setminusER_type, type, setminusER: $o). 135.41/58.48 thf(setminusER_def, definition, (setminusER = (! [A:$i,B:$i,C:$i]: ((in @ C @ (setminus @ A @ B)) => (~ (in @ C @ B)))))). 135.41/58.48 thf(setminusSubset2_type, type, setminusSubset2: $o). 135.41/58.48 thf(setminusSubset2_def, definition, (setminusSubset2 = (! [A:$i,B:$i]: ((subset @ A @ B) => ((setminus @ A @ B) = emptyset))))). 135.41/58.48 thf(setminusERneg_type, type, setminusERneg: $o). 135.41/58.48 thf(setminusERneg_def, definition, (setminusERneg = (! [A:$i,B:$i,C:$i]: ((~ (in @ C @ (setminus @ A @ B))) => ((in @ C @ A) => (in @ C @ B)))))). 135.41/58.48 thf(setminusELneg_type, type, setminusELneg: $o). 135.41/58.48 thf(setminusELneg_def, definition, (setminusELneg = (! [A:$i,B:$i,C:$i]: ((~ (in @ C @ (setminus @ A @ B))) => ((~ (in @ C @ B)) => (~ (in @ C @ A))))))). 135.41/58.48 thf(setminusILneg_type, type, setminusILneg: $o). 135.41/58.48 thf(setminusILneg_def, definition, (setminusILneg = (! [A:$i,B:$i,C:$i]: ((~ (in @ C @ A)) => (~ (in @ C @ (setminus @ A @ B))))))). 135.41/58.48 thf(setminusIRneg_type, type, setminusIRneg: $o). 135.41/58.48 thf(setminusIRneg_def, definition, (setminusIRneg = (! [A:$i,B:$i,C:$i]: ((in @ C @ B) => (~ (in @ C @ (setminus @ A @ B))))))). 135.41/58.48 thf(setminusLsub_type, type, setminusLsub: $o). 135.41/58.48 thf(setminusLsub_def, definition, (setminusLsub = (! [A:$i,B:$i]: (subset @ (setminus @ A @ B) @ A)))). 135.41/58.48 thf(setminusSubset1_type, type, setminusSubset1: $o). 135.41/58.48 thf(setminusSubset1_def, definition, (setminusSubset1 = (! [A:$i,B:$i]: (((setminus @ A @ B) = emptyset) => (subset @ A @ B))))). 135.41/58.48 thf(symdiff_type, type, symdiff: ($i > ($i > $i))). 135.41/58.48 thf(symdiffE_type, type, symdiffE: $o). 135.41/58.48 thf(symdiffE_def, definition, (symdiffE = (! [A:$i,B:$i,C:$i]: ((in @ C @ (symdiff @ A @ B)) => (! [D:$o]: (((in @ C @ A) => ((~ (in @ C @ B)) => (D))) => (((~ (in @ C @ A)) => ((in @ C @ B) => (D))) => (D)))))))). 135.41/58.48 thf(symdiffI1_type, type, symdiffI1: $o). 135.41/58.48 thf(symdiffI1_def, definition, (symdiffI1 = (! [A:$i,B:$i,C:$i]: ((in @ C @ A) => ((~ (in @ C @ B)) => (in @ C @ (symdiff @ A @ B))))))). 135.41/58.48 thf(symdiffI2_type, type, symdiffI2: $o). 135.41/58.48 thf(symdiffI2_def, definition, (symdiffI2 = (! [A:$i,B:$i,C:$i]: ((~ (in @ C @ A)) => ((in @ C @ B) => (in @ C @ (symdiff @ A @ B))))))). 135.41/58.48 thf(symdiffIneg1_type, type, symdiffIneg1: $o). 135.41/58.48 thf(symdiffIneg1_def, definition, (symdiffIneg1 = (! [A:$i,B:$i,C:$i]: ((in @ C @ A) => ((in @ C @ B) => (~ (in @ C @ (symdiff @ A @ B)))))))). 135.41/58.48 thf(symdiffIneg2_type, type, symdiffIneg2: $o). 135.41/58.48 thf(symdiffIneg2_def, definition, (symdiffIneg2 = (! [A:$i,B:$i,C:$i]: ((~ (in @ C @ A)) => ((~ (in @ C @ B)) => (~ (in @ C @ (symdiff @ A @ B)))))))). 135.41/58.48 thf(iskpair_type, type, iskpair: ($i > $o)). 135.41/58.48 thf(secondinupair_type, type, secondinupair: $o). 135.41/58.48 thf(secondinupair_def, definition, (secondinupair = (! [A:$i,B:$i]: (in @ B @ (setadjoin @ A @ (setadjoin @ B @ emptyset)))))). 135.41/58.48 thf(setukpairIL_type, type, setukpairIL: $o). 135.41/58.48 thf(setukpairIL_def, definition, (setukpairIL = (! [A:$i,B:$i]: (in @ A @ (setunion @ (setadjoin @ (setadjoin @ A @ emptyset) @ (setadjoin @ (setadjoin @ A @ (setadjoin @ B @ emptyset)) @ emptyset))))))). 135.41/58.48 thf(setukpairIR_type, type, setukpairIR: $o). 135.41/58.48 thf(setukpairIR_def, definition, (setukpairIR = (! [A:$i,B:$i]: (in @ B @ (setunion @ (setadjoin @ (setadjoin @ A @ emptyset) @ (setadjoin @ (setadjoin @ A @ (setadjoin @ B @ emptyset)) @ emptyset))))))). 135.41/58.48 thf(kpairiskpair_type, type, kpairiskpair: $o). 135.41/58.48 thf(kpairiskpair_def, definition, (kpairiskpair = (! [A:$i,B:$i]: (iskpair @ (setadjoin @ (setadjoin @ A @ emptyset) @ (setadjoin @ (setadjoin @ A @ (setadjoin @ B @ emptyset)) @ emptyset)))))). 135.41/58.48 thf(kpair_type, type, kpair: ($i > ($i > $i))). 135.41/58.48 thf(kpairp_type, type, kpairp: $o). 135.41/58.48 thf(kpairp_def, definition, (kpairp = (! [A:$i,B:$i]: (iskpair @ (kpair @ A @ B))))). 135.41/58.48 thf(cartprod_type, type, cartprod: ($i > ($i > $i))). 135.41/58.48 thf(singletonsubset_type, type, singletonsubset: $o). 135.41/58.48 thf(singletonsubset_def, definition, (singletonsubset = (! [A:$i,B:$i]: ((in @ B @ A) => (subset @ (setadjoin @ B @ emptyset) @ A))))). 135.41/58.48 thf(singletoninpowerset_type, type, singletoninpowerset: $o). 135.41/58.48 thf(singletoninpowerset_def, definition, (singletoninpowerset = (! [A:$i,B:$i]: ((in @ B @ A) => (in @ (setadjoin @ B @ emptyset) @ (powerset @ A)))))). 135.41/58.48 thf(singletoninpowunion_type, type, singletoninpowunion: $o). 135.41/58.48 thf(singletoninpowunion_def, definition, (singletoninpowunion = (! [A:$i,B:$i,C:$i]: ((in @ C @ A) => (in @ (setadjoin @ C @ emptyset) @ (powerset @ (binunion @ A @ B))))))). 135.41/58.48 thf(upairset2E_type, type, upairset2E: $o). 135.41/58.48 thf(upairset2E_def, definition, (upairset2E = (! [A:$i,B:$i,C:$i]: ((in @ C @ (setadjoin @ A @ (setadjoin @ B @ emptyset))) => ((C = A) | (C = B)))))). 135.41/58.48 thf(upairsubunion_type, type, upairsubunion: $o). 135.41/58.48 thf(upairsubunion_def, definition, (upairsubunion = (! [A:$i,B:$i,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ B) => (subset @ (setadjoin @ C @ (setadjoin @ D @ emptyset)) @ (binunion @ A @ B)))))))). 135.41/58.48 thf(upairinpowunion_type, type, upairinpowunion: $o). 135.41/58.48 thf(upairinpowunion_def, definition, (upairinpowunion = (! [A:$i,B:$i,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ B) => (in @ (setadjoin @ C @ (setadjoin @ D @ emptyset)) @ (powerset @ (binunion @ A @ B))))))))). 135.41/58.48 thf(ubforcartprodlem1_type, type, ubforcartprodlem1: $o). 135.41/58.48 thf(ubforcartprodlem1_def, definition, (ubforcartprodlem1 = (! [A:$i,B:$i,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ B) => (subset @ (setadjoin @ (setadjoin @ C @ emptyset) @ (setadjoin @ (setadjoin @ C @ (setadjoin @ D @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ A @ B))))))))). 135.41/58.48 thf(ubforcartprodlem2_type, type, ubforcartprodlem2: $o). 135.41/58.48 thf(ubforcartprodlem2_def, definition, (ubforcartprodlem2 = (! [A:$i,B:$i,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ B) => (in @ (setadjoin @ (setadjoin @ C @ emptyset) @ (setadjoin @ (setadjoin @ C @ (setadjoin @ D @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ A @ B)))))))))). 135.41/58.48 thf(ubforcartprodlem3_type, type, ubforcartprodlem3: $o). 135.41/58.48 thf(ubforcartprodlem3_def, definition, (ubforcartprodlem3 = (! [A:$i,B:$i,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ B) => (in @ (kpair @ C @ D) @ (powerset @ (powerset @ (binunion @ A @ B)))))))))). 135.41/58.48 thf(cartprodpairin_type, type, cartprodpairin: $o). 135.41/58.48 thf(cartprodpairin_def, definition, (cartprodpairin = (! [A:$i,B:$i,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ B) => (in @ (kpair @ C @ D) @ (cartprod @ A @ B)))))))). 135.41/58.48 thf(cartprodmempair1_type, type, cartprodmempair1: $o). 135.41/58.48 thf(cartprodmempair1_def, definition, (cartprodmempair1 = (! [A:$i,B:$i,C:$i]: ((in @ C @ (cartprod @ A @ B)) => (? [D:$i]: ((in @ D @ A) & ? [E:$i]: ((in @ E @ B) & (C = (kpair @ D @ E))))))))). 135.41/58.48 thf(cartprodmempair_type, type, cartprodmempair: $o). 135.41/58.48 thf(cartprodmempair_def, definition, (cartprodmempair = (! [A:$i,B:$i,C:$i]: ((in @ C @ (cartprod @ A @ B)) => (iskpair @ C))))). 135.41/58.48 thf(setunionE2_type, type, setunionE2: $o). 135.41/58.48 thf(setunionE2_def, definition, (setunionE2 = (! [A:$i,B:$i]: ((in @ B @ (setunion @ A)) => (? [C:$i]: ((in @ C @ A) & (in @ B @ C))))))). 135.41/58.48 thf(setunionsingleton1_type, type, setunionsingleton1: $o). 135.41/58.48 thf(setunionsingleton1_def, definition, (setunionsingleton1 = (! [A:$i]: (subset @ (setunion @ (setadjoin @ A @ emptyset)) @ A)))). 135.41/58.48 thf(setunionsingleton2_type, type, setunionsingleton2: $o). 135.41/58.48 thf(setunionsingleton2_def, definition, (setunionsingleton2 = (! [A:$i]: (subset @ A @ (setunion @ (setadjoin @ A @ emptyset)))))). 135.41/58.48 thf(setunionsingleton_type, type, setunionsingleton: $o). 135.41/58.48 thf(setunionsingleton_def, definition, (setunionsingleton = (! [A:$i]: ((setunion @ (setadjoin @ A @ emptyset)) = A)))). 135.41/58.48 thf(singleton_type, type, singleton: ($i > $o)). 135.41/58.48 thf(singletonprop_type, type, singletonprop: $o). 135.41/58.48 thf(singletonprop_def, definition, (singletonprop = (! [A:$i,B:($i > $o)]: ((! [C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ A) => ((B @ C) => ((B @ D) => (C = D))))))) => ((? [C:$i]: ((in @ C @ A) & (B @ C))) => (singleton @ (dsetconstr @ A @ (B)))))))). 135.41/58.48 thf(ex1_type, type, ex1: ($i > (($i > $o) > $o))). 135.41/58.48 thf(ex1E1_type, type, ex1E1: $o). 135.41/58.48 thf(ex1E1_def, definition, (ex1E1 = (! [A:$i,B:($i > $o)]: ((ex1 @ A @ (B)) => (? [C:$i]: ((in @ C @ A) & (B @ C))))))). 135.41/58.48 thf(ex1I_type, type, ex1I: $o). 135.41/58.48 thf(ex1I_def, definition, (ex1I = (! [A:$i,B:($i > $o),C:$i]: ((in @ C @ A) => ((B @ C) => ((! [D:$i]: ((in @ D @ A) => ((B @ D) => (D = C)))) => (ex1 @ A @ (B)))))))). 135.41/58.48 thf(ex1I2_type, type, ex1I2: $o). 135.41/58.48 thf(ex1I2_def, definition, (ex1I2 = (! [A:$i,B:($i > $o)]: ((! [C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ A) => ((B @ C) => ((B @ D) => (C = D))))))) => ((? [C:$i]: ((in @ C @ A) & (B @ C))) => (ex1 @ A @ (B))))))). 135.41/58.48 thf(singletonsuniq_type, type, singletonsuniq: $o). 135.41/58.48 thf(singletonsuniq_def, definition, (singletonsuniq = (! [A:$i,B:$i]: (((setadjoin @ A @ emptyset) = (setadjoin @ B @ emptyset)) => (A = B))))). 135.41/58.48 thf(setukpairinjL1_type, type, setukpairinjL1: $o). 135.41/58.48 thf(setukpairinjL1_def, definition, (setukpairinjL1 = (! [A:$i,B:$i,C:$i]: ((in @ (setadjoin @ C @ emptyset) @ (setadjoin @ (setadjoin @ A @ emptyset) @ (setadjoin @ (setadjoin @ A @ (setadjoin @ B @ emptyset)) @ emptyset))) => (A = C))))). 135.41/58.48 thf(kfstsingleton_type, type, kfstsingleton: $o). 135.41/58.48 thf(kfstsingleton_def, definition, (kfstsingleton = (! [A:$i]: ((iskpair @ A) => (singleton @ (dsetconstr @ (setunion @ A) @ (^ [B:$i]: (in @ (setadjoin @ B @ emptyset) @ A)))))))). 135.41/58.48 thf(theprop_type, type, theprop: $o). 135.41/58.48 thf(theprop_def, definition, (theprop = (! [A:$i]: ((singleton @ A) => (in @ (setunion @ A) @ A))))). 135.41/58.48 thf(kfst_type, type, kfst: ($i > $i)). 135.41/58.48 thf(kfstpairEq_type, type, kfstpairEq: $o). 135.41/58.48 thf(kfstpairEq_def, definition, (kfstpairEq = (! [A:$i,B:$i]: ((kfst @ (kpair @ A @ B)) = A)))). 135.41/58.48 thf(cartprodfstin_type, type, cartprodfstin: $o). 135.41/58.48 thf(cartprodfstin_def, definition, (cartprodfstin = (! [A:$i,B:$i,C:$i]: ((in @ C @ (cartprod @ A @ B)) => (in @ (kfst @ C) @ A))))). 135.41/58.48 thf(setukpairinjL2_type, type, setukpairinjL2: $o). 135.41/58.48 thf(setukpairinjL2_def, definition, (setukpairinjL2 = (! [A:$i,B:$i,C:$i,D:$i]: (((setadjoin @ (setadjoin @ A @ emptyset) @ (setadjoin @ (setadjoin @ A @ (setadjoin @ B @ emptyset)) @ emptyset)) = (setadjoin @ (setadjoin @ C @ emptyset) @ (setadjoin @ (setadjoin @ C @ (setadjoin @ D @ emptyset)) @ emptyset))) => (A = C))))). 135.41/58.48 thf(setukpairinjL_type, type, setukpairinjL: $o). 135.41/58.48 thf(setukpairinjL_def, definition, (setukpairinjL = (! [A:$i,B:$i,C:$i,D:$i]: (((kpair @ A @ B) = (kpair @ C @ D)) => (A = C))))). 135.41/58.48 thf(setukpairinjR11_type, type, setukpairinjR11: $o). 135.41/58.48 thf(setukpairinjR11_def, definition, (setukpairinjR11 = (! [A:$i,B:$i]: ((A = B) => ((setadjoin @ A @ (setadjoin @ B @ emptyset)) = (setadjoin @ A @ emptyset)))))). 135.41/58.48 thf(setukpairinjR12_type, type, setukpairinjR12: $o). 135.41/58.48 thf(setukpairinjR12_def, definition, (setukpairinjR12 = (! [A:$i,B:$i]: ((A = B) => ((setadjoin @ (setadjoin @ A @ emptyset) @ (setadjoin @ (setadjoin @ A @ (setadjoin @ B @ emptyset)) @ emptyset)) = (setadjoin @ (setadjoin @ A @ emptyset) @ emptyset)))))). 135.41/58.48 thf(setukpairinjR1_type, type, setukpairinjR1: $o). 135.41/58.48 thf(setukpairinjR1_def, definition, (setukpairinjR1 = (! [A:$i,B:$i,C:$i,D:$i]: (((setadjoin @ (setadjoin @ A @ emptyset) @ (setadjoin @ (setadjoin @ A @ (setadjoin @ B @ emptyset)) @ emptyset)) = (setadjoin @ (setadjoin @ C @ emptyset) @ (setadjoin @ (setadjoin @ C @ (setadjoin @ D @ emptyset)) @ emptyset))) => ((C = D) => (B = D)))))). 135.41/58.48 thf(upairequniteq_type, type, upairequniteq: $o). 135.41/58.48 thf(upairequniteq_def, definition, (upairequniteq = (! [A:$i,B:$i,C:$i]: (((setadjoin @ A @ (setadjoin @ B @ emptyset)) = (setadjoin @ C @ emptyset)) => (A = B))))). 135.41/58.48 thf(setukpairinjR2_type, type, setukpairinjR2: $o). 135.41/58.48 thf(setukpairinjR2_def, definition, (setukpairinjR2 = (! [A:$i,B:$i,C:$i,D:$i]: (((setadjoin @ (setadjoin @ A @ emptyset) @ (setadjoin @ (setadjoin @ A @ (setadjoin @ B @ emptyset)) @ emptyset)) = (setadjoin @ (setadjoin @ C @ emptyset) @ (setadjoin @ (setadjoin @ C @ (setadjoin @ D @ emptyset)) @ emptyset))) => (B = D))))). 135.41/58.48 thf(setukpairinjR_type, type, setukpairinjR: $o). 135.41/58.48 thf(setukpairinjR_def, definition, (setukpairinjR = (! [A:$i,B:$i,C:$i,D:$i]: (((kpair @ A @ B) = (kpair @ C @ D)) => (B = D))))). 135.41/58.48 thf(ksndsingleton_type, type, ksndsingleton: $o). 135.41/58.48 thf(ksndsingleton_def, definition, (ksndsingleton = (! [A:$i]: ((iskpair @ A) => (singleton @ (dsetconstr @ (setunion @ A) @ (^ [B:$i]: (A = (kpair @ (kfst @ A) @ B))))))))). 135.41/58.48 thf(ksnd_type, type, ksnd: ($i > $i)). 135.41/58.48 thf(ksndpairEq_type, type, ksndpairEq: $o). 135.41/58.48 thf(ksndpairEq_def, definition, (ksndpairEq = (! [A:$i,B:$i]: ((ksnd @ (kpair @ A @ B)) = B)))). 135.41/58.48 thf(kpairsurjEq_type, type, kpairsurjEq: $o). 135.41/58.48 thf(kpairsurjEq_def, definition, (kpairsurjEq = (! [A:$i]: ((iskpair @ A) => ((kpair @ (kfst @ A) @ (ksnd @ A)) = A))))). 135.41/58.48 thf(cartprodsndin_type, type, cartprodsndin: $o). 135.41/58.48 thf(cartprodsndin_def, definition, (cartprodsndin = (! [A:$i,B:$i,C:$i]: ((in @ C @ (cartprod @ A @ B)) => (in @ (ksnd @ C) @ B))))). 135.41/58.48 thf(cartprodpairmemEL_type, type, cartprodpairmemEL: $o). 135.41/58.48 thf(cartprodpairmemEL_def, definition, (cartprodpairmemEL = (! [A:$i,B:$i,C:$i,D:$i]: ((in @ (kpair @ C @ D) @ (cartprod @ A @ B)) => (in @ C @ A))))). 135.41/58.48 thf(cartprodpairmemER_type, type, cartprodpairmemER: $o). 135.41/58.48 thf(cartprodpairmemER_def, definition, (cartprodpairmemER = (! [A:$i,B:$i,C:$i,D:$i]: ((in @ (kpair @ C @ D) @ (cartprod @ A @ B)) => (in @ D @ B))))). 135.41/58.48 thf(cartprodmempaircEq_type, type, cartprodmempaircEq: $o). 135.41/58.48 thf(cartprodmempaircEq_def, definition, (cartprodmempaircEq = (! [A:$i,B:$i,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ B) => ((kpair @ C @ D) = (kpair @ C @ D)))))))). 135.41/58.48 thf(cartprodfstpairEq_type, type, cartprodfstpairEq: $o). 135.41/58.48 thf(cartprodfstpairEq_def, definition, (cartprodfstpairEq = (! [A:$i,B:$i,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ B) => ((kfst @ (kpair @ C @ D)) = C))))))). 135.41/58.48 thf(cartprodsndpairEq_type, type, cartprodsndpairEq: $o). 135.41/58.48 thf(cartprodsndpairEq_def, definition, (cartprodsndpairEq = (! [A:$i,B:$i,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ B) => ((ksnd @ (kpair @ C @ D)) = D))))))). 135.41/58.48 thf(cartprodpairsurjEq_type, type, cartprodpairsurjEq: $o). 135.41/58.48 thf(cartprodpairsurjEq_def, definition, (cartprodpairsurjEq = (! [A:$i,B:$i,C:$i]: ((in @ C @ (cartprod @ A @ B)) => ((kpair @ (kfst @ C) @ (ksnd @ C)) = C))))). 135.41/58.48 thf(breln_type, type, breln: ($i > ($i > ($i > $o)))). 135.41/58.48 thf(dpsetconstr_type, type, dpsetconstr: ($i > ($i > (($i > ($i > $o)) > $i)))). 135.41/58.48 thf(dpsetconstrI_type, type, dpsetconstrI: $o). 135.41/58.48 thf(dpsetconstrI_def, definition, (dpsetconstrI = (! [A:$i,B:$i,C:($i > ($i > $o)),D:$i]: ((in @ D @ A) => (! [E:$i]: ((in @ E @ B) => ((C @ D @ E) => (in @ (kpair @ D @ E) @ (dpsetconstr @ A @ B @ (C)))))))))). 135.41/58.48 thf(dpsetconstrSub_type, type, dpsetconstrSub: $o). 135.41/58.48 thf(dpsetconstrSub_def, definition, (dpsetconstrSub = (! [A:$i,B:$i,C:($i > ($i > $o))]: (subset @ (dpsetconstr @ A @ B @ (C)) @ (cartprod @ A @ B))))). 135.41/58.48 thf(setOfPairsIsBReln_type, type, setOfPairsIsBReln: $o). 135.41/58.48 thf(setOfPairsIsBReln_def, definition, (setOfPairsIsBReln = (! [A:$i,B:$i,C:($i > ($i > $o))]: (breln @ A @ B @ (dpsetconstr @ A @ B @ (C)))))). 135.41/58.48 thf(dpsetconstrERa_type, type, dpsetconstrERa: $o). 135.41/58.48 thf(dpsetconstrERa_def, definition, (dpsetconstrERa = (! [A:$i,B:$i,C:($i > ($i > $o)),D:$i]: ((in @ D @ A) => (! [E:$i]: ((in @ E @ B) => ((in @ (kpair @ D @ E) @ (dpsetconstr @ A @ B @ (C))) => (C @ D @ E)))))))). 135.41/58.48 thf(dpsetconstrEL1_type, type, dpsetconstrEL1: $o). 135.41/58.48 thf(dpsetconstrEL1_def, definition, (dpsetconstrEL1 = (! [A:$i,B:$i,C:($i > ($i > $o)),D:$i,E:$i]: ((in @ (kpair @ D @ E) @ (dpsetconstr @ A @ B @ (C))) => (in @ D @ A))))). 135.41/58.48 thf(dpsetconstrEL2_type, type, dpsetconstrEL2: $o). 135.41/58.48 thf(dpsetconstrEL2_def, definition, (dpsetconstrEL2 = (! [A:$i,B:$i,C:($i > ($i > $o)),D:$i,E:$i]: ((in @ (kpair @ D @ E) @ (dpsetconstr @ A @ B @ (C))) => (in @ E @ B))))). 135.41/58.48 thf(dpsetconstrER_type, type, dpsetconstrER: $o). 135.41/58.48 thf(dpsetconstrER_def, definition, (dpsetconstrER = (! [A:$i,B:$i,C:($i > ($i > $o)),D:$i,E:$i]: ((in @ (kpair @ D @ E) @ (dpsetconstr @ A @ B @ (C))) => (C @ D @ E))))). 135.41/58.48 thf(func_type, type, func: ($i > ($i > ($i > $o)))). 135.41/58.48 thf(funcSet_type, type, funcSet: ($i > ($i > $i))). 135.41/58.48 thf(funcImageSingleton_type, type, funcImageSingleton: $o). 135.41/58.48 thf(funcImageSingleton_def, definition, (funcImageSingleton = (! [A:$i,B:$i,C:$i]: ((func @ A @ B @ C) => (! [D:$i]: ((in @ D @ A) => (singleton @ (dsetconstr @ B @ (^ [E:$i]: (in @ (kpair @ D @ E) @ C)))))))))). 135.41/58.48 thf(apProp_type, type, apProp: $o). 135.41/58.48 thf(apProp_def, definition, (apProp = (! [A:$i,B:$i,C:$i]: ((func @ A @ B @ C) => (! [D:$i]: ((in @ D @ A) => (in @ (setunion @ (dsetconstr @ B @ (^ [E:$i]: (in @ (kpair @ D @ E) @ C)))) @ B))))))). 135.41/58.48 thf(ap_type, type, ap: ($i > ($i > ($i > ($i > $i))))). 135.41/58.48 thf(app_type, type, app: $o). 135.41/58.48 thf(app_def, definition, (app = (! [A:$i,B:$i,C:$i]: ((func @ A @ B @ C) => (! [D:$i]: ((in @ D @ A) => (in @ (ap @ A @ B @ C @ D) @ B))))))). 135.41/58.48 thf(infuncsetfunc_type, type, infuncsetfunc: $o). 135.41/58.48 thf(infuncsetfunc_def, definition, (infuncsetfunc = (! [A:$i,B:$i,C:$i]: ((in @ C @ (funcSet @ A @ B)) => (func @ A @ B @ C))))). 135.41/58.48 thf(ap2p_type, type, ap2p: $o). 135.41/58.48 thf(ap2p_def, definition, (ap2p = (! [A:$i,B:$i,C:$i]: ((in @ C @ (funcSet @ A @ B)) => (! [D:$i]: ((in @ D @ A) => (in @ (ap @ A @ B @ C @ D) @ B))))))). 135.41/58.48 thf(funcinfuncset_type, type, funcinfuncset: $o). 135.41/58.48 thf(funcinfuncset_def, definition, (funcinfuncset = (! [A:$i,B:$i,C:$i]: ((func @ A @ B @ C) => (in @ C @ (funcSet @ A @ B)))))). 135.41/58.48 thf(lamProp_type, type, lamProp: $o). 135.41/58.48 thf(lamProp_def, definition, (lamProp = (! [A:$i,B:$i,C:($i > $i)]: ((! [D:$i]: ((in @ D @ A) => (in @ (C @ D) @ B))) => (func @ A @ B @ (dpsetconstr @ A @ B @ (^ [D:$i]: (= @ $i @ (C @ D))))))))). 135.41/58.48 thf(lam_type, type, lam: ($i > ($i > (($i > $i) > $i)))). 135.41/58.48 thf(lamp_type, type, lamp: $o). 135.41/58.48 thf(lamp_def, definition, (lamp = (! [A:$i,B:$i,C:($i > $i)]: ((! [D:$i]: ((in @ D @ A) => (in @ (C @ D) @ B))) => (func @ A @ B @ (lam @ A @ B @ (C))))))). 135.41/58.48 thf(lam2p_type, type, lam2p: $o). 135.41/58.48 thf(lam2p_def, definition, (lam2p = (! [A:$i,B:$i,C:($i > $i)]: ((! [D:$i]: ((in @ D @ A) => (in @ (C @ D) @ B))) => (in @ (lam @ A @ B @ (C)) @ (funcSet @ A @ B)))))). 135.41/58.48 thf(brelnall1_type, type, brelnall1: $o). 135.41/58.48 thf(brelnall1_def, definition, (brelnall1 = (! [A:$i,B:$i,C:$i]: ((breln @ A @ B @ C) => (! [D:($i > $o)]: ((! [E:$i]: ((in @ E @ A) => (! [F:$i]: ((in @ F @ B) => ((in @ (kpair @ E @ F) @ C) => (D @ (kpair @ E @ F))))))) => (! [E:$i]: ((in @ E @ C) => (D @ E))))))))). 135.41/58.48 thf(brelnall2_type, type, brelnall2: $o). 135.41/58.48 thf(brelnall2_def, definition, (brelnall2 = (! [A:$i,B:$i,C:$i]: ((breln @ A @ B @ C) => (! [D:($i > $o)]: ((! [E:$i]: ((in @ E @ A) => (! [F:$i]: ((in @ F @ B) => ((in @ (kpair @ E @ F) @ C) => (D @ (kpair @ E @ F))))))) => (! [E:$i]: ((in @ E @ C) => (D @ E))))))))). 135.41/58.48 thf(ex1E2_type, type, ex1E2: $o). 135.41/58.48 thf(ex1E2_def, definition, (ex1E2 = (! [A:$i,B:($i > $o)]: ((ex1 @ A @ (B)) => (! [C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ A) => ((B @ C) => ((B @ D) => (C = D))))))))))). 135.41/58.48 thf(funcGraphProp1_type, type, funcGraphProp1: $o). 135.41/58.48 thf(funcGraphProp1_def, definition, (funcGraphProp1 = (! [A:$i,B:$i,C:$i]: ((func @ A @ B @ C) => (! [D:$i]: ((in @ D @ A) => (in @ (kpair @ D @ (ap @ A @ B @ C @ D)) @ C))))))). 135.41/58.48 thf(funcGraphProp3_type, type, funcGraphProp3: $o). 135.41/58.48 thf(funcGraphProp3_def, definition, (funcGraphProp3 = (! [A:$i,B:$i,C:$i]: ((in @ C @ (funcSet @ A @ B)) => (! [D:$i]: ((in @ D @ A) => (in @ (kpair @ D @ (ap @ A @ B @ C @ D)) @ C))))))). 135.41/58.48 thf(funcGraphProp2_type, type, funcGraphProp2: $o). 135.41/58.48 thf(funcGraphProp2_def, definition, (funcGraphProp2 = (! [A:$i,B:$i,C:$i]: ((func @ A @ B @ C) => (! [D:$i]: ((in @ D @ A) => (! [E:$i]: ((in @ E @ B) => ((in @ (kpair @ D @ E) @ C) => ((ap @ A @ B @ C @ D) = E)))))))))). 135.41/58.48 thf(funcextLem_type, type, funcextLem: $o). 135.41/58.48 thf(funcextLem_def, definition, (funcextLem = (! [A:$i,B:$i,C:$i]: ((func @ A @ B @ C) => (! [D:$i]: ((func @ A @ B @ D) => ((! [E:$i]: ((in @ E @ A) => ((ap @ A @ B @ C @ E) = (ap @ A @ B @ D @ E)))) => (! [E:$i]: ((in @ E @ A) => (! [F:$i]: ((in @ F @ B) => ((in @ (kpair @ E @ F) @ D) => (in @ (kpair @ E @ F) @ C))))))))))))). 135.41/58.48 thf(funcGraphProp4_type, type, funcGraphProp4: $o). 135.41/58.48 thf(funcGraphProp4_def, definition, (funcGraphProp4 = (! [A:$i,B:$i,C:$i]: ((in @ C @ (funcSet @ A @ B)) => (! [D:$i]: ((in @ D @ A) => (! [E:$i]: ((in @ E @ B) => ((in @ (kpair @ D @ E) @ C) => ((ap @ A @ B @ C @ D) = E)))))))))). 135.41/58.48 thf(subbreln_type, type, subbreln: $o). 135.41/58.48 thf(subbreln_def, definition, (subbreln = (! [A:$i,B:$i,C:$i]: ((breln @ A @ B @ C) => (! [D:$i]: ((breln @ A @ B @ D) => ((! [E:$i]: ((in @ E @ A) => (! [F:$i]: ((in @ F @ B) => ((in @ (kpair @ E @ F) @ C) => (in @ (kpair @ E @ F) @ D)))))) => (subset @ C @ D)))))))). 135.41/58.48 thf(eqbreln_type, type, eqbreln: $o). 135.41/58.48 thf(eqbreln_def, definition, (eqbreln = (! [A:$i,B:$i,C:$i]: ((breln @ A @ B @ C) => (! [D:$i]: ((breln @ A @ B @ D) => ((! [E:$i]: ((in @ E @ A) => (! [F:$i]: ((in @ F @ B) => ((in @ (kpair @ E @ F) @ C) => (in @ (kpair @ E @ F) @ D)))))) => ((! [E:$i]: ((in @ E @ A) => (! [F:$i]: ((in @ F @ B) => ((in @ (kpair @ E @ F) @ D) => (in @ (kpair @ E @ F) @ C)))))) => (C = D))))))))). 135.41/58.48 thf(funcext_type, type, funcext: $o). 135.41/58.48 thf(funcext_def, definition, (funcext = (! [A:$i,B:$i,C:$i]: ((func @ A @ B @ C) => (! [D:$i]: ((func @ A @ B @ D) => ((! [E:$i]: ((in @ E @ A) => ((ap @ A @ B @ C @ E) = (ap @ A @ B @ D @ E)))) => (C = D)))))))). 135.41/58.48 thf(funcext2_type, type, funcext2: $o). 135.41/58.48 thf(funcext2_def, definition, (funcext2 = (! [A:$i,B:$i,C:$i]: ((in @ C @ (funcSet @ A @ B)) => (! [D:$i]: ((in @ D @ (funcSet @ A @ B)) => ((! [E:$i]: ((in @ E @ A) => ((ap @ A @ B @ C @ E) = (ap @ A @ B @ D @ E)))) => (C = D)))))))). 135.41/58.48 thf(ap2apEq1_type, type, ap2apEq1: $o). 135.41/58.48 thf(ap2apEq1_def, definition, (ap2apEq1 = (! [A:$i,B:$i,C:$i]: ((in @ C @ (funcSet @ A @ B)) => (! [D:$i]: ((in @ D @ A) => ((ap @ A @ B @ C @ D) = (ap @ A @ B @ C @ D)))))))). 135.41/58.48 thf(ap2apEq2_type, type, ap2apEq2: $o). 135.41/58.48 thf(ap2apEq2_def, definition, (ap2apEq2 = (! [A:$i,B:$i,C:$i]: ((func @ A @ B @ C) => (! [D:$i]: ((in @ D @ A) => ((ap @ A @ B @ C @ D) = (ap @ A @ B @ C @ D)))))))). 135.41/58.48 thf(beta1_type, type, beta1: $o). 135.41/58.48 thf(beta1_def, definition, (beta1 = (! [A:$i,B:$i,C:($i > $i)]: ((! [D:$i]: ((in @ D @ A) => (in @ (C @ D) @ B))) => (! [D:$i]: ((in @ D @ A) => ((ap @ A @ B @ (lam @ A @ B @ (C)) @ D) = (C @ D)))))))). 135.41/58.48 thf(eta1_type, type, eta1: $o). 135.41/58.48 thf(eta1_def, definition, (eta1 = (! [A:$i,B:$i,C:$i]: ((func @ A @ B @ C) => ((lam @ A @ B @ (ap @ A @ B @ C)) = C))))). 135.41/58.48 thf(lam2lamEq_type, type, lam2lamEq: $o). 135.41/58.48 thf(lam2lamEq_def, definition, (lam2lamEq = (! [A:$i,B:$i,C:($i > $i)]: ((! [D:$i]: ((in @ D @ A) => (in @ (C @ D) @ B))) => ((lam @ A @ B @ (C)) = (lam @ A @ B @ (C))))))). 135.41/58.48 thf(beta2_type, type, beta2: $o). 135.41/58.48 thf(beta2_def, definition, (beta2 = (! [A:$i,B:$i,C:($i > $i)]: ((! [D:$i]: ((in @ D @ A) => (in @ (C @ D) @ B))) => (! [D:$i]: ((in @ D @ A) => ((ap @ A @ B @ (lam @ A @ B @ (C)) @ D) = (C @ D)))))))). 135.41/58.48 thf(eta2_type, type, eta2: $o). 135.41/58.48 thf(eta2_def, definition, (eta2 = (! [A:$i,B:$i,C:$i]: ((in @ C @ (funcSet @ A @ B)) => ((lam @ A @ B @ (ap @ A @ B @ C)) = C))))). 135.41/58.48 thf(iffalseProp1_type, type, iffalseProp1: $o). 135.41/58.48 thf(iffalseProp1_def, definition, (iffalseProp1 = (! [A:$i,B:$o,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ A) => ((~ (B)) => (in @ D @ (dsetconstr @ A @ (^ [E:$i]: (((B) & (E = C)) | (~ (B) & (E = D))))))))))))). 135.41/58.48 thf(iffalseProp2_type, type, iffalseProp2: $o). 135.41/58.48 thf(iffalseProp2_def, definition, (iffalseProp2 = (! [A:$i,B:$o,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ A) => ((~ (B)) => ((dsetconstr @ A @ (^ [E:$i]: (((B) & (E = C)) | (~ (B) & (E = D))))) = (setadjoin @ D @ emptyset))))))))). 135.41/58.48 thf(iftrueProp1_type, type, iftrueProp1: $o). 135.41/58.48 thf(iftrueProp1_def, definition, (iftrueProp1 = (! [A:$i,B:$o,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ A) => ((B) => (in @ C @ (dsetconstr @ A @ (^ [E:$i]: (((B) & (E = C)) | (~ (B) & (E = D))))))))))))). 135.41/58.48 thf(iftrueProp2_type, type, iftrueProp2: $o). 135.41/58.48 thf(iftrueProp2_def, definition, (iftrueProp2 = (! [A:$i,B:$o,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ A) => ((B) => ((dsetconstr @ A @ (^ [E:$i]: (((B) & (E = C)) | (~ (B) & (E = D))))) = (setadjoin @ C @ emptyset))))))))). 135.41/58.48 thf(ifSingleton_type, type, ifSingleton: $o). 135.41/58.48 thf(ifSingleton_def, definition, (ifSingleton = (! [A:$i,B:$o,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ A) => (singleton @ (dsetconstr @ A @ (^ [E:$i]: (((B) & (E = C)) | (~ (B) & (E = D)))))))))))). 135.41/58.48 thf(if_type, type, if: ($i > ($o > ($i > ($i > $i))))). 135.41/58.48 thf(ifp_type, type, ifp: $o). 135.41/58.48 thf(ifp_def, definition, (ifp = (! [A:$i,B:$o,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ A) => (in @ (if @ A @ B @ C @ D) @ A))))))). 135.41/58.48 thf(theeq_type, type, theeq: $o). 135.41/58.48 thf(theeq_def, definition, (theeq = (! [A:$i]: ((singleton @ A) => (! [B:$i]: ((in @ B @ A) => ((setunion @ A) = B))))))). 135.41/58.48 thf(iftrue_type, type, iftrue: $o). 135.41/58.48 thf(iftrue_def, definition, (iftrue = (! [A:$i,B:$o,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ A) => ((B) => ((if @ A @ B @ C @ D) = C)))))))). 135.41/58.48 thf(iffalse_type, type, iffalse: $o). 135.41/58.48 thf(iffalse_def, definition, (iffalse = (! [A:$i,B:$o,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ A) => ((~ (B)) => ((if @ A @ B @ C @ D) = D)))))))). 135.41/58.48 thf(iftrueorfalse_type, type, iftrueorfalse: $o). 135.41/58.48 thf(iftrueorfalse_def, definition, (iftrueorfalse = (! [A:$i,B:$o,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ A) => (in @ (if @ A @ B @ C @ D) @ (setadjoin @ C @ (setadjoin @ D @ emptyset))))))))). 135.41/58.48 thf(binintersectT_lem_type, type, binintersectT_lem: $o). 135.41/58.48 thf(binintersectT_lem_def, definition, (binintersectT_lem = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (in @ (binintersect @ B @ C) @ (powerset @ A)))))))). 135.41/58.48 thf(binunionT_lem_type, type, binunionT_lem: $o). 135.41/58.48 thf(binunionT_lem_def, definition, (binunionT_lem = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (in @ (binunion @ B @ C) @ (powerset @ A)))))))). 135.41/58.48 thf(powersetT_lem_type, type, powersetT_lem: $o). 135.41/58.48 thf(powersetT_lem_def, definition, (powersetT_lem = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (in @ (powerset @ B) @ (powerset @ (powerset @ A))))))). 135.41/58.48 thf(setminusT_lem_type, type, setminusT_lem: $o). 135.41/58.48 thf(setminusT_lem_def, definition, (setminusT_lem = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (in @ (setminus @ B @ C) @ (powerset @ A)))))))). 135.41/58.48 thf(complementT_lem_type, type, complementT_lem: $o). 135.41/58.48 thf(complementT_lem_def, definition, (complementT_lem = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (in @ (setminus @ A @ B) @ (powerset @ A)))))). 135.41/58.48 thf(setextT_type, type, setextT: $o). 135.41/58.48 thf(setextT_def, definition, (setextT = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => ((! [D:$i]: ((in @ D @ A) => ((in @ D @ B) => (in @ D @ C)))) => ((! [D:$i]: ((in @ D @ A) => ((in @ D @ C) => (in @ D @ B)))) => (B = C))))))))). 135.41/58.48 thf(subsetTI_type, type, subsetTI: $o). 135.41/58.48 thf(subsetTI_def, definition, (subsetTI = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => ((! [D:$i]: ((in @ D @ A) => ((in @ D @ B) => (in @ D @ C)))) => (subset @ B @ C)))))))). 135.41/58.48 thf(powersetTI1_type, type, powersetTI1: $o). 135.41/58.48 thf(powersetTI1_def, definition, (powersetTI1 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => ((! [D:$i]: ((in @ D @ A) => ((in @ D @ B) => (in @ D @ C)))) => (in @ B @ (powerset @ C))))))))). 135.41/58.48 thf(powersetTE1_type, type, powersetTE1: $o). 135.41/58.48 thf(powersetTE1_def, definition, (powersetTE1 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((in @ B @ (powerset @ C)) => ((in @ D @ B) => (in @ D @ C))))))))))). 135.41/58.48 thf(complementTI1_type, type, complementTI1: $o). 135.41/58.48 thf(complementTI1_def, definition, (complementTI1 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ A) => ((in @ C @ B) => (~ (in @ C @ (setminus @ A @ B)))))))))). 135.41/58.48 thf(complementTE1_type, type, complementTE1: $o). 135.41/58.48 thf(complementTE1_def, definition, (complementTE1 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ A) => ((~ (in @ C @ (setminus @ A @ B))) => (in @ C @ B)))))))). 135.41/58.48 thf(binintersectTELcontra_type, type, binintersectTELcontra: $o). 135.41/58.48 thf(binintersectTELcontra_def, definition, (binintersectTELcontra = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((~ (in @ D @ B)) => (~ (in @ D @ (binintersect @ B @ C)))))))))))). 135.41/58.48 thf(binintersectTERcontra_type, type, binintersectTERcontra: $o). 135.41/58.48 thf(binintersectTERcontra_def, definition, (binintersectTERcontra = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((~ (in @ D @ C)) => (~ (in @ D @ (binintersect @ B @ C)))))))))))). 135.41/58.48 thf(contrasubsetT_type, type, contrasubsetT: $o). 135.41/58.48 thf(contrasubsetT_def, definition, (contrasubsetT = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((subset @ B @ (setminus @ A @ C)) => ((in @ D @ C) => (~ (in @ D @ B)))))))))))). 135.41/58.48 thf(contrasubsetT1_type, type, contrasubsetT1: $o). 135.41/58.48 thf(contrasubsetT1_def, definition, (contrasubsetT1 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((subset @ B @ C) => ((~ (in @ D @ C)) => (~ (in @ D @ B)))))))))))). 135.41/58.48 thf(contrasubsetT2_type, type, contrasubsetT2: $o). 135.41/58.48 thf(contrasubsetT2_def, definition, (contrasubsetT2 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => ((subset @ B @ C) => (subset @ (setminus @ A @ C) @ (setminus @ A @ B))))))))). 135.41/58.48 thf(contrasubsetT3_type, type, contrasubsetT3: $o). 135.41/58.48 thf(contrasubsetT3_def, definition, (contrasubsetT3 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => ((subset @ (setminus @ A @ C) @ (setminus @ A @ B)) => (subset @ B @ C)))))))). 135.41/58.48 thf(doubleComplementI1_type, type, doubleComplementI1: $o). 135.41/58.48 thf(doubleComplementI1_def, definition, (doubleComplementI1 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ A) => ((in @ C @ B) => (in @ C @ (setminus @ A @ (setminus @ A @ B)))))))))). 135.41/58.48 thf(doubleComplementE1_type, type, doubleComplementE1: $o). 135.41/58.48 thf(doubleComplementE1_def, definition, (doubleComplementE1 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ A) => ((in @ C @ (setminus @ A @ (setminus @ A @ B))) => (in @ C @ B)))))))). 135.41/58.48 thf(doubleComplementSub1_type, type, doubleComplementSub1: $o). 135.41/58.48 thf(doubleComplementSub1_def, definition, (doubleComplementSub1 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (subset @ B @ (setminus @ A @ (setminus @ A @ B))))))). 135.41/58.48 thf(doubleComplementSub2_type, type, doubleComplementSub2: $o). 135.41/58.48 thf(doubleComplementSub2_def, definition, (doubleComplementSub2 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (subset @ (setminus @ A @ (setminus @ A @ B)) @ B))))). 135.41/58.48 thf(doubleComplementEq_type, type, doubleComplementEq: $o). 135.41/58.48 thf(doubleComplementEq_def, definition, (doubleComplementEq = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (B = (setminus @ A @ (setminus @ A @ B))))))). 135.41/58.48 thf(complementTnotintersectT_type, type, complementTnotintersectT: $o). 135.41/58.48 thf(complementTnotintersectT_def, definition, (complementTnotintersectT = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((in @ D @ (setminus @ A @ B)) => (~ (in @ D @ (binintersect @ B @ C)))))))))))). 135.41/58.48 thf(complementImpComplementIntersect_type, type, complementImpComplementIntersect: $o). 135.41/58.48 thf(complementImpComplementIntersect_def, definition, (complementImpComplementIntersect = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((in @ D @ (setminus @ A @ B)) => (in @ D @ (setminus @ A @ (binintersect @ B @ C)))))))))))). 135.41/58.48 thf(complementSubsetComplementIntersect_type, type, complementSubsetComplementIntersect: $o). 135.41/58.48 thf(complementSubsetComplementIntersect_def, definition, (complementSubsetComplementIntersect = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (subset @ (setminus @ A @ B) @ (setminus @ A @ (binintersect @ B @ C))))))))). 135.41/58.48 thf(complementInPowersetComplementIntersect_type, type, complementInPowersetComplementIntersect: $o). 135.41/58.48 thf(complementInPowersetComplementIntersect_def, definition, (complementInPowersetComplementIntersect = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (in @ (setminus @ A @ B) @ (powerset @ (setminus @ A @ (binintersect @ B @ C)))))))))). 135.41/58.48 thf(contraSubsetComplement_type, type, contraSubsetComplement: $o). 135.41/58.48 thf(contraSubsetComplement_def, definition, (contraSubsetComplement = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => ((subset @ B @ (setminus @ A @ C)) => (! [D:$i]: ((in @ D @ A) => ((in @ D @ C) => (in @ D @ (setminus @ A @ B)))))))))))). 135.41/58.48 thf(complementTcontraSubset_type, type, complementTcontraSubset: $o). 135.41/58.48 thf(complementTcontraSubset_def, definition, (complementTcontraSubset = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => ((subset @ B @ (setminus @ A @ C)) => (subset @ C @ (setminus @ A @ B))))))))). 135.41/58.48 thf(binunionTILcontra_type, type, binunionTILcontra: $o). 135.41/58.48 thf(binunionTILcontra_def, definition, (binunionTILcontra = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((~ (in @ D @ (binunion @ B @ C))) => (~ (in @ D @ B))))))))))). 135.41/58.48 thf(binunionTIRcontra_type, type, binunionTIRcontra: $o). 135.41/58.48 thf(binunionTIRcontra_def, definition, (binunionTIRcontra = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((~ (in @ D @ (binunion @ B @ C))) => (~ (in @ D @ C))))))))))). 135.41/58.48 thf(inIntersectImpInUnion_type, type, inIntersectImpInUnion: $o). 135.41/58.48 thf(inIntersectImpInUnion_def, definition, (inIntersectImpInUnion = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ (powerset @ A)) => (! [E:$i]: ((in @ E @ A) => ((in @ E @ (binintersect @ B @ C)) => (in @ E @ (binunion @ B @ D))))))))))))). 135.41/58.48 thf(inIntersectImpInUnion2_type, type, inIntersectImpInUnion2: $o). 135.41/58.48 thf(inIntersectImpInUnion2_def, definition, (inIntersectImpInUnion2 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ (powerset @ A)) => (! [E:$i]: ((in @ E @ A) => ((in @ E @ (binintersect @ B @ C)) => (in @ E @ (binunion @ C @ D))))))))))))). 135.41/58.48 thf(inIntersectImpInIntersectUnions_type, type, inIntersectImpInIntersectUnions: $o). 135.41/58.48 thf(inIntersectImpInIntersectUnions_def, definition, (inIntersectImpInIntersectUnions = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ (powerset @ A)) => (! [E:$i]: ((in @ E @ A) => ((in @ E @ (binintersect @ B @ C)) => (in @ E @ (binintersect @ (binunion @ B @ D) @ (binunion @ C @ D)))))))))))))). 135.41/58.48 thf(intersectInPowersetIntersectUnions_type, type, intersectInPowersetIntersectUnions: $o). 135.41/58.48 thf(intersectInPowersetIntersectUnions_def, definition, (intersectInPowersetIntersectUnions = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ (powerset @ A)) => (in @ (binintersect @ B @ C) @ (powerset @ (binintersect @ (binunion @ B @ D) @ (binunion @ C @ D)))))))))))). 135.41/58.48 thf(inComplementUnionImpNotIn1_type, type, inComplementUnionImpNotIn1: $o). 135.41/58.48 thf(inComplementUnionImpNotIn1_def, definition, (inComplementUnionImpNotIn1 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((in @ D @ (setminus @ A @ (binunion @ B @ C))) => (~ (in @ D @ B))))))))))). 135.41/58.48 thf(inComplementUnionImpInComplement1_type, type, inComplementUnionImpInComplement1: $o). 135.41/58.48 thf(inComplementUnionImpInComplement1_def, definition, (inComplementUnionImpInComplement1 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((in @ D @ (setminus @ A @ (binunion @ B @ C))) => (in @ D @ (setminus @ A @ B))))))))))). 135.41/58.48 thf(binunionTE_type, type, binunionTE: $o). 135.41/58.48 thf(binunionTE_def, definition, (binunionTE = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$o,E:$i]: ((in @ E @ A) => ((in @ E @ (binunion @ B @ C)) => (((in @ E @ B) => (D)) => (((in @ E @ C) => (D)) => (D)))))))))))). 135.41/58.48 thf(binunionTEcontra_type, type, binunionTEcontra: $o). 135.41/58.48 thf(binunionTEcontra_def, definition, (binunionTEcontra = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((~ (in @ D @ B)) => ((~ (in @ D @ C)) => (~ (in @ D @ (binunion @ B @ C))))))))))))). 135.41/58.48 thf(demorgan2a1_type, type, demorgan2a1: $o). 135.41/58.48 thf(demorgan2a1_def, definition, (demorgan2a1 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((in @ D @ (setminus @ A @ (binunion @ B @ C))) => (in @ D @ (setminus @ A @ B))))))))))). 135.41/58.48 thf(complementUnionInPowersetComplement_type, type, complementUnionInPowersetComplement: $o). 135.41/58.48 thf(complementUnionInPowersetComplement_def, definition, (complementUnionInPowersetComplement = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (in @ (setminus @ A @ (binunion @ B @ C)) @ (powerset @ (setminus @ A @ B))))))))). 135.41/58.48 thf(demorgan2a2_type, type, demorgan2a2: $o). 135.41/58.48 thf(demorgan2a2_def, definition, (demorgan2a2 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((in @ D @ (setminus @ A @ (binunion @ B @ C))) => (in @ D @ (setminus @ A @ C))))))))))). 135.41/58.48 thf(demorgan1a_type, type, demorgan1a: $o). 135.41/58.48 thf(demorgan1a_def, definition, (demorgan1a = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((in @ D @ (setminus @ A @ (binintersect @ B @ C))) => (in @ D @ (binunion @ (setminus @ A @ B) @ (setminus @ A @ C)))))))))))). 135.41/58.48 thf(demorgan1b_type, type, demorgan1b: $o). 135.41/58.48 thf(demorgan1b_def, definition, (demorgan1b = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((in @ D @ (binunion @ (setminus @ A @ B) @ (setminus @ A @ C))) => (in @ D @ (setminus @ A @ (binintersect @ B @ C)))))))))))). 135.41/58.48 thf(demorgan1_type, type, demorgan1: $o). 135.41/58.48 thf(demorgan1_def, definition, (demorgan1 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => ((setminus @ A @ (binintersect @ B @ C)) = (binunion @ (setminus @ A @ B) @ (setminus @ A @ C))))))))). 135.41/58.48 thf(demorgan2a_type, type, demorgan2a: $o). 135.41/58.48 thf(demorgan2a_def, definition, (demorgan2a = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((in @ D @ (setminus @ A @ (binunion @ B @ C))) => (in @ D @ (binintersect @ (setminus @ A @ B) @ (setminus @ A @ C)))))))))))). 135.41/58.48 thf(demorgan2b2_type, type, demorgan2b2: $o). 135.41/58.48 thf(demorgan2b2_def, definition, (demorgan2b2 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((in @ D @ (setminus @ A @ B)) => ((in @ D @ (setminus @ A @ C)) => (in @ D @ (setminus @ A @ (binunion @ B @ C))))))))))))). 135.41/58.48 thf(demorgan2b_type, type, demorgan2b: $o). 135.41/58.48 thf(demorgan2b_def, definition, (demorgan2b = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((in @ D @ (binintersect @ (setminus @ A @ B) @ (setminus @ A @ C))) => (in @ D @ (setminus @ A @ (binunion @ B @ C)))))))))))). 135.41/58.48 thf(demorgan2_type, type, demorgan2: $o). 135.41/58.48 thf(demorgan2_def, definition, (demorgan2 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => ((setminus @ A @ (binunion @ B @ C)) = (binintersect @ (setminus @ A @ B) @ (setminus @ A @ C))))))))). 135.41/58.48 thf(woz13rule0_type, type, woz13rule0: $o). 135.41/58.48 thf(woz13rule0_def, definition, (woz13rule0 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ (binintersect @ B @ C)) => (in @ D @ A))))))))). 135.41/58.48 thf(woz13rule1_type, type, woz13rule1: $o). 135.41/58.48 thf(woz13rule1_def, definition, (woz13rule1 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ (powerset @ A)) => ((subset @ B @ D) => (subset @ (binintersect @ B @ C) @ D)))))))))). 135.41/58.48 thf(woz13rule2_type, type, woz13rule2: $o). 135.41/58.48 thf(woz13rule2_def, definition, (woz13rule2 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ (powerset @ A)) => ((subset @ C @ D) => (subset @ (binintersect @ B @ C) @ D)))))))))). 135.41/58.48 thf(woz13rule3_type, type, woz13rule3: $o). 135.41/58.48 thf(woz13rule3_def, definition, (woz13rule3 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ (powerset @ A)) => ((subset @ B @ C) => ((subset @ B @ D) => (subset @ B @ (binintersect @ C @ D)))))))))))). 135.41/58.48 thf(woz13rule4_type, type, woz13rule4: $o). 135.41/58.48 thf(woz13rule4_def, definition, (woz13rule4 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ (powerset @ A)) => (! [E:$i]: ((in @ E @ (powerset @ A)) => ((subset @ B @ D) => ((subset @ C @ E) => (subset @ (binintersect @ B @ C) @ (binintersect @ D @ E)))))))))))))). 135.41/58.48 thf(woz1_1_type, type, woz1_1: $o). 135.41/58.48 thf(woz1_1_def, definition, (woz1_1 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (in @ (setminus @ A @ B) @ (powerset @ (setminus @ A @ (binintersect @ B @ C)))))))))). 135.41/58.48 thf(woz1_2_type, type, woz1_2: $o). 135.41/58.48 thf(woz1_2_def, definition, (woz1_2 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ (powerset @ A)) => (! [E:$i]: ((in @ E @ (powerset @ A)) => ((setminus @ A @ (binintersect @ (binunion @ B @ C) @ (binunion @ D @ E))) = (binunion @ (binintersect @ (setminus @ A @ B) @ (setminus @ A @ C)) @ (binintersect @ (setminus @ A @ D) @ (setminus @ A @ E)))))))))))))). 135.41/58.48 thf(woz1_3_type, type, woz1_3: $o). 135.41/58.48 thf(woz1_3_def, definition, (woz1_3 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ (powerset @ A)) => (in @ (binintersect @ B @ C) @ (powerset @ (binintersect @ (binunion @ B @ D) @ (binunion @ C @ D)))))))))))). 135.41/58.48 thf(woz1_4_type, type, woz1_4: $o). 135.41/58.48 thf(woz1_4_def, definition, (woz1_4 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => ((subset @ B @ (setminus @ A @ C)) => (subset @ C @ (setminus @ A @ B))))))))). 135.41/58.48 thf(woz1_5_type, type, woz1_5: $o). 135.41/58.48 thf(woz1_5_def, definition, (woz1_5 = (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (in @ (setminus @ A @ (binunion @ B @ C)) @ (powerset @ (setminus @ A @ B))))))))). 135.41/58.48 thf(breln1_type, type, breln1: ($i > ($i > $o))). 135.41/58.48 thf(breln1all2_type, type, breln1all2: $o). 135.41/58.48 thf(breln1all2_def, definition, (breln1all2 = (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:($i > $o)]: ((! [D:$i]: ((in @ D @ A) => (! [E:$i]: ((in @ E @ A) => ((in @ (kpair @ D @ E) @ B) => (C @ (kpair @ D @ E))))))) => (! [D:$i]: ((in @ D @ B) => (C @ D))))))))). 135.41/58.48 thf(breln1Set_type, type, breln1Set: ($i > $i)). 135.41/58.48 thf(breln1SetBreln1_type, type, breln1SetBreln1: $o). 135.41/58.48 thf(breln1SetBreln1_def, definition, (breln1SetBreln1 = (! [A:$i,B:$i]: ((in @ B @ (breln1Set @ A)) => (breln1 @ A @ B))))). 135.41/58.48 thf(reflwellordering_type, type, reflwellordering: ($i > ($i > $o))). 135.41/58.48 thf(choice2fnsingleton_type, type, choice2fnsingleton: $o). 135.41/58.48 thf(choice2fnsingleton_def, definition, (choice2fnsingleton = (! [A:$i,B:$i,C:($i > ($i > $o))]: ((! [D:$i]: ((in @ D @ A) => (? [E:$i]: ((in @ E @ B) & (C @ D @ E))))) => (! [D:$i]: ((in @ D @ (breln1Set @ B)) => ((reflwellordering @ B @ D) => (! [E:$i]: ((in @ E @ A) => (singleton @ (dsetconstr @ B @ (^ [F:$i]: ((C @ E @ F) & ! [G:$i]: ((in @ G @ B) => ((C @ E @ G) => (in @ (kpair @ F @ G) @ D)))))))))))))))). 135.41/58.48 thf(setOfPairsIsBReln1_type, type, setOfPairsIsBReln1: $o). 135.41/58.48 thf(setOfPairsIsBReln1_def, definition, (setOfPairsIsBReln1 = (! [A:$i,B:($i > ($i > $o))]: (breln1 @ A @ (dpsetconstr @ A @ A @ (B)))))). 135.41/58.48 thf(breln1all1_type, type, breln1all1: $o). 135.41/58.48 thf(breln1all1_def, definition, (breln1all1 = (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:($i > $o)]: ((! [D:$i]: ((in @ D @ A) => (! [E:$i]: ((in @ E @ A) => ((in @ (kpair @ D @ E) @ B) => (C @ (kpair @ D @ E))))))) => (! [D:$i]: ((in @ D @ B) => (C @ D))))))))). 135.41/58.48 thf(subbreln1_type, type, subbreln1: $o). 135.41/58.48 thf(subbreln1_def, definition, (subbreln1 = (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:$i]: ((breln1 @ A @ C) => ((! [D:$i]: ((in @ D @ A) => (! [E:$i]: ((in @ E @ A) => ((in @ (kpair @ D @ E) @ B) => (in @ (kpair @ D @ E) @ C)))))) => (subset @ B @ C)))))))). 135.41/58.48 thf(eqbreln1_type, type, eqbreln1: $o). 135.41/58.48 thf(eqbreln1_def, definition, (eqbreln1 = (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:$i]: ((breln1 @ A @ C) => ((! [D:$i]: ((in @ D @ A) => (! [E:$i]: ((in @ E @ A) => ((in @ (kpair @ D @ E) @ B) => (in @ (kpair @ D @ E) @ C)))))) => ((! [D:$i]: ((in @ D @ A) => (! [E:$i]: ((in @ E @ A) => ((in @ (kpair @ D @ E) @ C) => (in @ (kpair @ D @ E) @ B)))))) => (B = C))))))))). 135.41/58.48 thf(breln1invset_type, type, breln1invset: ($i > ($i > $i))). 135.41/58.48 thf(breln1invprop_type, type, breln1invprop: $o). 135.41/58.48 thf(breln1invprop_def, definition, (breln1invprop = (! [A:$i,B:$i]: ((breln1 @ A @ B) => (breln1 @ A @ (breln1invset @ A @ B)))))). 135.41/58.48 thf(breln1invI_type, type, breln1invI: $o). 135.41/58.48 thf(breln1invI_def, definition, (breln1invI = (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ A) => ((in @ (kpair @ C @ D) @ B) => (in @ (kpair @ D @ C) @ (breln1invset @ A @ B))))))))))). 135.41/58.48 thf(breln1invE_type, type, breln1invE: $o). 135.41/58.48 thf(breln1invE_def, definition, (breln1invE = (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ A) => ((in @ (kpair @ D @ C) @ (breln1invset @ A @ B)) => (in @ (kpair @ C @ D) @ B)))))))))). 135.41/58.48 thf(breln1compset_type, type, breln1compset: ($i > ($i > ($i > $i)))). 135.41/58.48 thf(breln1compprop_type, type, breln1compprop: $o). 135.41/58.48 thf(breln1compprop_def, definition, (breln1compprop = (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:$i]: ((breln1 @ A @ C) => (breln1 @ A @ (breln1compset @ A @ B @ C)))))))). 135.41/58.48 thf(breln1compI_type, type, breln1compI: $o). 135.41/58.48 thf(breln1compI_def, definition, (breln1compI = (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:$i]: ((breln1 @ A @ C) => (! [D:$i]: ((in @ D @ A) => (! [E:$i]: ((in @ E @ A) => (! [F:$i]: ((in @ F @ A) => ((in @ (kpair @ D @ F) @ B) => ((in @ (kpair @ F @ E) @ C) => (in @ (kpair @ D @ E) @ (breln1compset @ A @ B @ C)))))))))))))))). 135.41/58.48 thf(breln1compE_type, type, breln1compE: $o). 135.41/58.48 thf(breln1compE_def, definition, (breln1compE = (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:$i]: ((breln1 @ A @ C) => (! [D:$i]: ((in @ D @ A) => (! [E:$i]: ((in @ E @ A) => ((in @ (kpair @ D @ E) @ (breln1compset @ A @ B @ C)) => (? [F:$i]: ((in @ F @ A) & (in @ (kpair @ D @ F) @ B) & (in @ (kpair @ F @ E) @ C)))))))))))))). 135.41/58.48 thf(breln1compEex_type, type, breln1compEex: $o). 135.41/58.48 thf(breln1compEex_def, definition, (breln1compEex = (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:$i]: ((breln1 @ A @ C) => (! [D:$i]: ((in @ D @ A) => (! [E:$i]: ((in @ E @ A) => ((in @ (kpair @ D @ E) @ (breln1compset @ A @ B @ C)) => (! [F:$o]: ((! [G:$i]: ((in @ G @ A) => ((in @ (kpair @ D @ G) @ B) => ((in @ (kpair @ G @ E) @ C) => (F))))) => (F)))))))))))))). 135.41/58.48 thf(breln1unionprop_type, type, breln1unionprop: $o). 135.41/58.48 thf(breln1unionprop_def, definition, (breln1unionprop = (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:$i]: ((breln1 @ A @ C) => (breln1 @ A @ (binunion @ B @ C)))))))). 135.41/58.48 thf(breln1unionIL_type, type, breln1unionIL: $o). 135.41/58.48 thf(breln1unionIL_def, definition, (breln1unionIL = (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:$i]: ((breln1 @ A @ C) => (! [D:$i]: ((in @ D @ A) => (! [E:$i]: ((in @ E @ A) => ((in @ (kpair @ D @ E) @ B) => (in @ (kpair @ D @ E) @ (binunion @ B @ C))))))))))))). 135.41/58.48 thf(breln1unionIR_type, type, breln1unionIR: $o). 135.41/58.48 thf(breln1unionIR_def, definition, (breln1unionIR = (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:$i]: ((breln1 @ A @ C) => (! [D:$i]: ((in @ D @ A) => (! [E:$i]: ((in @ E @ A) => ((in @ (kpair @ D @ E) @ C) => (in @ (kpair @ D @ E) @ (binunion @ B @ C))))))))))))). 135.41/58.48 thf(breln1unionI_type, type, breln1unionI: $o). 135.41/58.48 thf(breln1unionI_def, definition, (breln1unionI = (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:$i]: ((breln1 @ A @ C) => (! [D:$i]: ((in @ D @ A) => (! [E:$i]: ((in @ E @ A) => (((in @ (kpair @ D @ E) @ B) | (in @ (kpair @ D @ E) @ C)) => (in @ (kpair @ D @ E) @ (binunion @ B @ C))))))))))))). 135.41/58.48 thf(breln1unionE_type, type, breln1unionE: $o). 135.41/58.48 thf(breln1unionE_def, definition, (breln1unionE = (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:$i]: ((breln1 @ A @ C) => (! [D:$i]: ((in @ D @ A) => (! [E:$i]: ((in @ E @ A) => ((in @ (kpair @ D @ E) @ (binunion @ B @ C)) => ((in @ (kpair @ D @ E) @ B) | (in @ (kpair @ D @ E) @ C))))))))))))). 135.41/58.48 thf(breln1unionEcases_type, type, breln1unionEcases: $o). 135.41/58.48 thf(breln1unionEcases_def, definition, (breln1unionEcases = (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:$i]: ((breln1 @ A @ C) => (! [D:$i]: ((in @ D @ A) => (! [E:$i]: ((in @ E @ A) => ((in @ (kpair @ D @ E) @ (binunion @ B @ C)) => (! [F:$o]: (((in @ (kpair @ D @ E) @ B) => (F)) => (((in @ (kpair @ D @ E) @ C) => (F)) => (F))))))))))))))). 135.41/58.48 thf(breln1unionCommutes_type, type, breln1unionCommutes: $o). 135.41/58.48 thf(breln1unionCommutes_def, definition, (breln1unionCommutes = (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:$i]: ((breln1 @ A @ C) => ((binunion @ B @ C) = (binunion @ C @ B)))))))). 135.41/58.48 thf(woz2Ex_type, type, woz2Ex: $o). 135.41/58.48 thf(woz2Ex_def, definition, (woz2Ex = (! [A:$i,B:$i]: ((breln1 @ A @ B) => (B = (breln1invset @ A @ (breln1invset @ A @ B))))))). 135.41/58.48 thf(woz2W_type, type, woz2W: $o). 135.41/58.48 thf(woz2W_def, definition, (woz2W = (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:$i]: ((breln1 @ A @ C) => ((breln1invset @ A @ (breln1compset @ A @ B @ C)) = (breln1compset @ A @ (breln1invset @ A @ C) @ (breln1invset @ A @ B))))))))). 135.41/58.48 thf(woz2A_type, type, woz2A: $o). 135.41/58.48 thf(woz2A_def, definition, (woz2A = (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:$i]: ((breln1 @ A @ C) => (! [D:$i]: ((breln1 @ A @ D) => ((breln1compset @ A @ (binunion @ B @ C) @ D) = (binunion @ (breln1compset @ A @ B @ D) @ (breln1compset @ A @ C @ D))))))))))). 135.41/58.48 thf(woz2B_type, type, woz2B: $o). 135.41/58.48 thf(woz2B_def, definition, (woz2B = (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:$i]: ((breln1 @ A @ C) => (! [D:$i]: ((breln1 @ A @ D) => ((breln1compset @ A @ (binunion @ B @ C) @ D) = (binunion @ (breln1invset @ A @ (breln1compset @ A @ (breln1invset @ A @ D) @ (breln1invset @ A @ C))) @ (breln1invset @ A @ (breln1compset @ A @ (breln1invset @ A @ D) @ (breln1invset @ A @ B))))))))))))). 135.41/58.48 thf(image1Ex_type, type, image1Ex: $o). 135.41/58.48 thf(image1Ex_def, definition, (image1Ex = (! [A:$i,B:($i > $i)]: ? [C:$i]: ! [D:$i]: ((in @ D @ C) = (? [E:$i]: ((in @ E @ A) & (D = (B @ E)))))))). 135.41/58.48 thf(image1Ex1_type, type, image1Ex1: $o). 135.41/58.48 thf(image1Ex1_def, definition, (image1Ex1 = (! [A:$i,B:($i > $i)]: (exu @ (^ [C:$i]: ! [D:$i]: ((in @ D @ C) = (? [E:$i]: ((in @ E @ A) & (D = (B @ E)))))))))). 135.41/58.48 thf(image1_type, type, image1: ($i > (($i > $i) > $i))). 135.41/58.48 thf(image1Equiv_type, type, image1Equiv: $o). 135.41/58.48 thf(image1Equiv_def, definition, (image1Equiv = (! [A:$i,B:($i > $i),C:$i]: ((in @ C @ (image1 @ A @ (B))) = (? [D:$i]: ((in @ D @ A) & (C = (B @ D)))))))). 135.41/58.48 thf(image1E_type, type, image1E: $o). 135.41/58.48 thf(image1E_def, definition, (image1E = (! [A:$i,B:($i > $i),C:$i]: ((in @ C @ (image1 @ A @ (B))) => (? [D:$i]: ((in @ D @ A) & (C = (B @ D)))))))). 135.41/58.48 thf(image1I_type, type, image1I: $o). 135.41/58.48 thf(image1I_def, definition, (image1I = (! [A:$i,B:($i > $i),C:$i]: ((? [D:$i]: ((in @ D @ A) & (C = (B @ D)))) => (in @ C @ (image1 @ A @ (B))))))). 135.41/58.48 thf(injective_type, type, injective: ($i > ($i > ($i > $o)))). 135.41/58.48 thf(injFuncSet_type, type, injFuncSet: ($i > ($i > $i))). 135.41/58.48 thf(injFuncInInjFuncSet_type, type, injFuncInInjFuncSet: $o). 135.41/58.48 thf(injFuncInInjFuncSet_def, definition, (injFuncInInjFuncSet = (! [A:$i,B:$i,C:$i]: ((in @ C @ (funcSet @ A @ B)) => ((injective @ A @ B @ C) => (in @ C @ (injFuncSet @ A @ B))))))). 135.41/58.48 thf(injFuncSetFuncIn_type, type, injFuncSetFuncIn: $o). 135.41/58.48 thf(injFuncSetFuncIn_def, definition, (injFuncSetFuncIn = (! [A:$i,B:$i,C:$i]: ((in @ C @ (injFuncSet @ A @ B)) => (in @ C @ (funcSet @ A @ B)))))). 135.41/58.48 thf(injFuncSetFuncInj_type, type, injFuncSetFuncInj: $o). 135.41/58.48 thf(injFuncSetFuncInj_def, definition, (injFuncSetFuncInj = (! [A:$i,B:$i,C:$i]: ((in @ C @ (injFuncSet @ A @ B)) => (injective @ A @ B @ C))))). 135.41/58.48 thf(surjective_type, type, surjective: ($i > ($i > ($i > $o)))). 135.41/58.48 thf(surjFuncSet_type, type, surjFuncSet: ($i > ($i > $i))). 135.41/58.48 thf(surjFuncSetFuncIn_type, type, surjFuncSetFuncIn: $o). 135.41/58.48 thf(surjFuncSetFuncIn_def, definition, (surjFuncSetFuncIn = (! [A:$i,B:$i,C:$i]: ((in @ C @ (surjFuncSet @ A @ B)) => (in @ C @ (funcSet @ A @ B)))))). 135.41/58.48 thf(surjFuncSetFuncSurj_type, type, surjFuncSetFuncSurj: $o). 135.41/58.48 thf(surjFuncSetFuncSurj_def, definition, (surjFuncSetFuncSurj = (! [A:$i,B:$i,C:$i]: ((in @ C @ (surjFuncSet @ A @ B)) => (surjective @ A @ B @ C))))). 135.41/58.48 thf(leftInvIsSurj_type, type, leftInvIsSurj: $o). 135.41/58.48 thf(leftInvIsSurj_def, definition, (leftInvIsSurj = (! [A:$i,B:$i,C:($i > $i)]: ((! [D:$i]: ((in @ D @ A) => (in @ (C @ D) @ B))) => (! [D:$i]: ((in @ D @ (funcSet @ B @ A)) => ((! [E:$i]: ((in @ E @ A) => ((ap @ B @ A @ D @ (C @ E)) = E))) => (surjective @ B @ A @ D)))))))). 135.41/58.48 thf(surjCantorThm_type, type, surjCantorThm: $o). 135.41/58.48 thf(surjCantorThm_def, definition, (surjCantorThm = (! [A:$i,B:$i]: ((in @ B @ (funcSet @ A @ (powerset @ A))) => (~ (surjective @ A @ (powerset @ A) @ B)))))). 135.41/58.48 thf(foundation2_type, type, foundation2: $o). 135.41/58.48 thf(foundation2_def, definition, (foundation2 = (! [A:$i]: ((nonempty @ A) => (? [B:$i]: ((in @ B @ A) & ((binintersect @ B @ A) = emptyset))))))). 135.41/58.48 thf(notinself_type, type, notinself: $o). 135.41/58.48 thf(notinself_def, definition, (notinself = (! [A:$i]: ~ (in @ A @ A)))). 135.41/58.48 thf(notinself2_type, type, notinself2: $o). 135.41/58.48 thf(notinself2_def, definition, (notinself2 = (! [A:$i,B:$i]: ((in @ A @ B) => (~ (in @ B @ A)))))). 135.41/58.48 thf(omegaS_type, type, omegaS: ($i > $i)). 135.41/58.48 thf(omegaSp_type, type, omegaSp: $o). 135.41/58.48 thf(omegaSp_def, definition, (omegaSp = (! [A:$i]: ((in @ A @ omega) => (in @ (omegaS @ A) @ omega))))). 135.41/58.48 thf(omegaSclos_type, type, omegaSclos: $o). 135.41/58.48 thf(omegaSclos_def, definition, (omegaSclos = (! [A:$i]: ((in @ A @ omega) => (in @ (setadjoin @ A @ A) @ omega))))). 135.41/58.48 thf(peano0notS_type, type, peano0notS: $o). 135.41/58.48 thf(peano0notS_def, definition, (peano0notS = (! [A:$i]: ((in @ A @ omega) => ((omegaS @ A) != emptyset))))). 135.41/58.48 thf(peanoSinj_type, type, peanoSinj: $o). 135.41/58.48 thf(peanoSinj_def, definition, (peanoSinj = (! [A:$i]: ((in @ A @ omega) => (! [B:$i]: ((in @ B @ omega) => (((omegaS @ A) = (omegaS @ B)) => (A = B)))))))). 135.41/58.48 thf(transitivesetOp1_type, type, transitivesetOp1: $o). 135.41/58.48 thf(transitivesetOp1_def, definition, (transitivesetOp1 = (! [A:$i]: ((transitiveset @ A) => (! [B:$i]: ((in @ B @ A) => (subset @ B @ A))))))). 135.41/58.48 thf(binintTransitive_type, type, binintTransitive: $o). 135.41/58.48 thf(binintTransitive_def, definition, (binintTransitive = (! [A:$i]: ((transitiveset @ A) => (! [B:$i]: ((transitiveset @ B) => (transitiveset @ (binintersect @ A @ B)))))))). 135.41/58.48 thf(transitivesetOp2_type, type, transitivesetOp2: $o). 135.41/58.48 thf(transitivesetOp2_def, definition, (transitivesetOp2 = (! [A:$i]: ((transitiveset @ A) => (! [B:$i,C:$i]: ((in @ B @ A) => ((in @ C @ B) => (in @ C @ A)))))))). 135.41/58.48 thf(setunionTransitive_type, type, setunionTransitive: $o). 135.41/58.48 thf(setunionTransitive_def, definition, (setunionTransitive = (! [A:$i]: ((! [B:$i]: ((in @ B @ A) => (transitiveset @ B))) => (transitiveset @ (setunion @ A)))))). 135.41/58.48 thf(ordinal_type, type, ordinal: ($i > $o)). 135.41/58.48 thf(ordinal_def, definition, (ordinal = (^ [A:$i]: ((transitiveset @ A) & (wellorderedByIn @ A))))). 135.41/58.48 thf(ordinalMinLem1_type, type, ordinalMinLem1: $o). 135.41/58.48 thf(ordinalMinLem1_def, definition, (ordinalMinLem1 = (! [A:$i]: ((ordinal @ A) => (! [B:$i]: ((ordinal @ B) => (transitiveset @ (binintersect @ A @ B)))))))). 135.41/58.48 thf(ordinalTransSet_type, type, ordinalTransSet: $o). 135.41/58.48 thf(ordinalTransSet_def, definition, (ordinalTransSet = (! [A:$i]: ((ordinal @ A) => (! [B:$i,C:$i]: ((in @ C @ A) => ((in @ B @ C) => (in @ B @ A)))))))). 135.41/58.48 thf(1,conjecture,((((emptysetAx) => (((powersetAx) => ((((omegaSAx) => (((((wellorderingAx) => ((descrp) => (((dsetconstrEL) => ((dsetconstrER) => ((exuE1) => (((emptysetE) => (((notinemptyset) => ((exuE3e) => ((setext) => ((emptyI) => ((((nonemptyE1) => (((nonemptyI1) => ((setadjoinIL) => ((emptyinunitempty) => ((setadjoinIR) => ((setadjoinE) => ((setadjoinOr) => ((setoftrueEq) => ((powersetI) => ((emptyinPowerset) => ((emptyInPowerset) => ((((((exuE2) => (((((eqinunit) => ((singletonsswitch) => (((upairsetIL) => (((emptyE1) => ((vacuousDall) => ((quantDeMorgan1) => ((quantDeMorgan2) => ((quantDeMorgan3) => (((prop2setI) => (((notdexE) => (((exuI1) => ((exuI3) => ((exuI2) => ((inCongP) => ((((exu__Cong) => ((((((omega__Cong) => (((((subsetI1) => ((eqimpsubset2) => ((((((subsetE2) => ((notsubsetI) => (((notequalI2) => (((subsetTrans) => ((setadjoinSub) => (((subset2powerset) => ((((powersetI1) => ((powersetE1) => ((inPowerset) => ((powersetsubset) => ((sepInPowerset) => (((binunionIL) => (((binunionIR) => ((binunionEcases) => ((((binunionRsub) => ((binintersectI) => ((((binintersectLsub) => (((((disjointsetsI1) => (((((bs114d) => ((setminusI) => ((((setminusSubset2) => ((((setminusILneg) => ((setminusIRneg) => (((setminusSubset1) => ((symdiffE) => ((symdiffI1) => (((symdiffIneg1) => (((secondinupair) => (((setukpairIR) => (((kpairp) => ((singletonsubset) => ((singletoninpowerset) => ((singletoninpowunion) => ((((upairinpowunion) => ((ubforcartprodlem1) => ((((cartprodpairin) => ((cartprodmempair1) => ((cartprodmempair) => ((setunionE2) => ((setunionsingleton1) => (((setunionsingleton) => ((singletonprop) => ((ex1E1) => ((ex1I) => ((ex1I2) => ((singletonsuniq) => ((setukpairinjL1) => ((((((setukpairinjL2) => (((setukpairinjR11) => (((((setukpairinjR2) => ((setukpairinjR) => ((ksndsingleton) => ((ksndpairEq) => (((cartprodsndin) => ((((cartprodmempaircEq) => ((cartprodfstpairEq) => ((cartprodsndpairEq) => (((((setOfPairsIsBReln) => ((dpsetconstrERa) => ((dpsetconstrEL1) => ((dpsetconstrEL2) => ((((apProp) => ((((ap2p) => ((((((((ex1E2) => ((funcGraphProp1) => ((funcGraphProp3) => ((funcGraphProp2) => (((funcGraphProp4) => (((eqbreln) => ((funcext) => ((((((((beta2) => ((eta2) => ((((iftrueProp1) => ((iftrueProp2) => ((ifSingleton) => ((ifp) => ((theeq) => (((iffalse) => ((iftrueorfalse) => ((binintersectT_lem) => ((binunionT_lem) => ((powersetT_lem) => (((complementT_lem) => (((subsetTI) => (((powersetTE1) => ((complementTI1) => ((complementTE1) => ((((contrasubsetT) => (((contrasubsetT2) => ((contrasubsetT3) => (((doubleComplementE1) => (((doubleComplementSub2) => ((((complementImpComplementIntersect) => ((complementSubsetComplementIntersect) => ((((complementTcontraSubset) => (((binunionTIRcontra) => ((((inIntersectImpInIntersectUnions) => ((((((binunionTEcontra) => ((demorgan2a1) => ((((demorgan1a) => ((demorgan1b) => ((demorgan1) => (((demorgan2b2) => ((demorgan2b) => (((woz13rule0) => ((((woz13rule3) => (((woz1_1) => ((woz1_2) => ((woz1_3) => ((woz1_4) => (((breln1all2) => ((breln1SetBreln1) => (((setOfPairsIsBReln1) => ((breln1all1) => (((eqbreln1) => (((breln1invI) => ((breln1invE) => (((breln1compI) => (((breln1compEex) => ((((breln1unionIR) => ((breln1unionI) => ((breln1unionE) => (((breln1unionCommutes) => ((woz2Ex) => ((woz2W) => (((woz2B) => (((image1Ex1) => ((((image1I) => ((injFuncInInjFuncSet) => ((injFuncSetFuncIn) => ((injFuncSetFuncInj) => ((surjFuncSetFuncIn) => ((surjFuncSetFuncSurj) => ((((foundation2) => ((notinself) => ((notinself2) => (((omegaSclos) => ((((transitivesetOp1) => ((((setunionTransitive) => ((ordinalMinLem1) => ((ordinalTransSet) => (! [A:$i]: ((! [B:$i]: ((subset @ B @ A) <= (in @ B @ A))) <= (ordinal @ A)))))) <= (transitivesetOp2)) <= (binintTransitive))) <= (peanoSinj)) <= (peano0notS))) <= (omegaSp))))) <= (surjCantorThm)) <= (leftInvIsSurj)))))))) <= (image1E)) <= (image1Equiv))) <= (image1Ex))) <= (woz2A))))) <= (breln1unionEcases))))) <= (breln1unionIL)) <= (breln1unionprop))) <= (breln1compE))) <= (breln1compprop)))) <= (breln1invprop))) <= (subbreln1)))) <= (choice2fnsingleton)))) <= (woz1_5)))))) <= (woz13rule4))) <= (woz13rule2)) <= (woz13rule1))) <= (demorgan2)))) <= (demorgan2a))))) <= (demorgan2a2)) <= (complementUnionInPowersetComplement)))) <= (binunionTE)) <= (inComplementUnionImpInComplement1)) <= (inComplementUnionImpNotIn1)) <= (intersectInPowersetIntersectUnions))) <= (inIntersectImpInUnion2)) <= (inIntersectImpInUnion))) <= (binunionTILcontra))) <= (contraSubsetComplement)) <= (complementInPowersetComplementIntersect)))) <= (complementTnotintersectT)) <= (doubleComplementEq))) <= (doubleComplementSub1))) <= (doubleComplementI1)))) <= (contrasubsetT1))) <= (binintersectTERcontra)) <= (binintersectTELcontra))))) <= (powersetTI1))) <= (setextT))) <= (setminusT_lem))))))) <= (iftrue))))))) <= (iffalseProp2)) <= (iffalseProp1)))) <= (lam2lamEq)) <= (eta1)) <= (beta1)) <= (ap2apEq2)) <= (ap2apEq1)) <= (funcext2)))) <= (subbreln))) <= (funcextLem)))))) <= (brelnall2)) <= (brelnall1)) <= (lam2p)) <= (lamp)) <= (lamProp)) <= (funcinfuncset))) <= (infuncsetfunc)) <= (app))) <= (funcImageSingleton)) <= (dpsetconstrER)))))) <= (dpsetconstrSub)) <= (dpsetconstrI)) <= (cartprodpairsurjEq))))) <= (cartprodpairmemER)) <= (cartprodpairmemEL))) <= (kpairsurjEq)))))) <= (upairequniteq)) <= (setukpairinjR1)) <= (setukpairinjR12))) <= (setukpairinjL))) <= (cartprodfstin)) <= (kfstpairEq)) <= (theprop)) <= (kfstsingleton))))))))) <= (setunionsingleton2))))))) <= (ubforcartprodlem3)) <= (ubforcartprodlem2)))) <= (upairsubunion)) <= (upairset2E)))))) <= (kpairiskpair))) <= (setukpairIL))) <= (symdiffIneg2))) <= (symdiffI2))))) <= (setminusLsub)))) <= (setminusELneg)) <= (setminusERneg))) <= (setminusER)) <= (setminusEL)))) <= (binintersectSubset1)) <= (binintersectSubset4)) <= (binintersectRsub))) <= (binintersectER)) <= (binintersectSubset3)) <= (binintersectSubset2))) <= (binintersectEL)) <= (binintersectSubset5)))) <= (binunionLsub)) <= (binunionE)))) <= (upairset2IR))) <= (sepSubset))))))) <= (subsetemptysetimpeq)) <= (setextsub))) <= (setadjoinSub2)))) <= (subsetRefl))) <= (notequalI1)))) <= (subsetE)) <= (emptysetsubset)) <= (subsetI2)) <= (eqimpsubset1)))) <= (dsetconstr__Cong)) <= (descr__Cong)) <= (exuEu))) <= (setunion__Cong)) <= (powerset__Cong)) <= (setadjoin__Cong)) <= (emptyset__Cong))) <= (exuE3u)) <= (in__Cong)))))) <= (notdallE))) <= (prop2set2propI))) <= (quantDeMorgan4))))))) <= (upairsetIR))) <= (upairsetE)))) <= (notinsingleton)) <= (uniqinunit)) <= (nonemptyImpWitness))) <= (subPowSU)) <= (setunionE)) <= (setunionI)) <= (powersetE)))))))))))) <= (nonemptyI))) <= (setbeta)) <= (noeltsimpempty)))))) <= (emptysetimpfalse))) <= (prop2setE))))) <= (dsetconstrI)))) <= (foundationAx)) <= (replAx)) <= (omegaIndAx))) <= (omega0Ax)) <= (setunionAx))) <= (setadjoinAx))) <= (setextAx))),file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordinalTransSet1)). 135.41/58.48 thf(2,negated_conjecture,((~ (((emptysetAx) => (((powersetAx) => ((((omegaSAx) => (((((wellorderingAx) => ((descrp) => (((dsetconstrEL) => ((dsetconstrER) => ((exuE1) => (((emptysetE) => (((notinemptyset) => ((exuE3e) => ((setext) => ((emptyI) => ((((nonemptyE1) => (((nonemptyI1) => ((setadjoinIL) => ((emptyinunitempty) => ((setadjoinIR) => ((setadjoinE) => ((setadjoinOr) => ((setoftrueEq) => ((powersetI) => ((emptyinPowerset) => ((emptyInPowerset) => ((((((exuE2) => (((((eqinunit) => ((singletonsswitch) => (((upairsetIL) => (((emptyE1) => ((vacuousDall) => ((quantDeMorgan1) => ((quantDeMorgan2) => ((quantDeMorgan3) => (((prop2setI) => (((notdexE) => (((exuI1) => ((exuI3) => ((exuI2) => ((inCongP) => ((((exu__Cong) => ((((((omega__Cong) => (((((subsetI1) => ((eqimpsubset2) => ((((((subsetE2) => ((notsubsetI) => (((notequalI2) => (((subsetTrans) => ((setadjoinSub) => (((subset2powerset) => ((((powersetI1) => ((powersetE1) => ((inPowerset) => ((powersetsubset) => ((sepInPowerset) => (((binunionIL) => (((binunionIR) => ((binunionEcases) => ((((binunionRsub) => ((binintersectI) => ((((binintersectLsub) => (((((disjointsetsI1) => (((((bs114d) => ((setminusI) => ((((setminusSubset2) => ((((setminusILneg) => ((setminusIRneg) => (((setminusSubset1) => ((symdiffE) => ((symdiffI1) => (((symdiffIneg1) => (((secondinupair) => (((setukpairIR) => (((kpairp) => ((singletonsubset) => ((singletoninpowerset) => ((singletoninpowunion) => ((((upairinpowunion) => ((ubforcartprodlem1) => ((((cartprodpairin) => ((cartprodmempair1) => ((cartprodmempair) => ((setunionE2) => ((setunionsingleton1) => (((setunionsingleton) => ((singletonprop) => ((ex1E1) => ((ex1I) => ((ex1I2) => ((singletonsuniq) => ((setukpairinjL1) => ((((((setukpairinjL2) => (((setukpairinjR11) => (((((setukpairinjR2) => ((setukpairinjR) => ((ksndsingleton) => ((ksndpairEq) => (((cartprodsndin) => ((((cartprodmempaircEq) => ((cartprodfstpairEq) => ((cartprodsndpairEq) => (((((setOfPairsIsBReln) => ((dpsetconstrERa) => ((dpsetconstrEL1) => ((dpsetconstrEL2) => ((((apProp) => ((((ap2p) => ((((((((ex1E2) => ((funcGraphProp1) => ((funcGraphProp3) => ((funcGraphProp2) => (((funcGraphProp4) => (((eqbreln) => ((funcext) => ((((((((beta2) => ((eta2) => ((((iftrueProp1) => ((iftrueProp2) => ((ifSingleton) => ((ifp) => ((theeq) => (((iffalse) => ((iftrueorfalse) => ((binintersectT_lem) => ((binunionT_lem) => ((powersetT_lem) => (((complementT_lem) => (((subsetTI) => (((powersetTE1) => ((complementTI1) => ((complementTE1) => ((((contrasubsetT) => (((contrasubsetT2) => ((contrasubsetT3) => (((doubleComplementE1) => (((doubleComplementSub2) => ((((complementImpComplementIntersect) => ((complementSubsetComplementIntersect) => ((((complementTcontraSubset) => (((binunionTIRcontra) => ((((inIntersectImpInIntersectUnions) => ((((((binunionTEcontra) => ((demorgan2a1) => ((((demorgan1a) => ((demorgan1b) => ((demorgan1) => (((demorgan2b2) => ((demorgan2b) => (((woz13rule0) => ((((woz13rule3) => (((woz1_1) => ((woz1_2) => ((woz1_3) => ((woz1_4) => (((breln1all2) => ((breln1SetBreln1) => (((setOfPairsIsBReln1) => ((breln1all1) => (((eqbreln1) => (((breln1invI) => ((breln1invE) => (((breln1compI) => (((breln1compEex) => ((((breln1unionIR) => ((breln1unionI) => ((breln1unionE) => (((breln1unionCommutes) => ((woz2Ex) => ((woz2W) => (((woz2B) => (((image1Ex1) => ((((image1I) => ((injFuncInInjFuncSet) => ((injFuncSetFuncIn) => ((injFuncSetFuncInj) => ((surjFuncSetFuncIn) => ((surjFuncSetFuncSurj) => ((((foundation2) => ((notinself) => ((notinself2) => (((omegaSclos) => ((((transitivesetOp1) => ((((setunionTransitive) => ((ordinalMinLem1) => ((ordinalTransSet) => (! [A:$i]: ((! [B:$i]: ((subset @ B @ A) <= (in @ B @ A))) <= (ordinal @ A)))))) <= (transitivesetOp2)) <= (binintTransitive))) <= (peanoSinj)) <= (peano0notS))) <= (omegaSp))))) <= (surjCantorThm)) <= (leftInvIsSurj)))))))) <= (image1E)) <= (image1Equiv))) <= (image1Ex))) <= (woz2A))))) <= (breln1unionEcases))))) <= (breln1unionIL)) <= (breln1unionprop))) <= (breln1compE))) <= (breln1compprop)))) <= (breln1invprop))) <= (subbreln1)))) <= (choice2fnsingleton)))) <= (woz1_5)))))) <= (woz13rule4))) <= (woz13rule2)) <= (woz13rule1))) <= (demorgan2)))) <= (demorgan2a))))) <= (demorgan2a2)) <= (complementUnionInPowersetComplement)))) <= (binunionTE)) <= (inComplementUnionImpInComplement1)) <= (inComplementUnionImpNotIn1)) <= (intersectInPowersetIntersectUnions))) <= (inIntersectImpInUnion2)) <= (inIntersectImpInUnion))) <= (binunionTILcontra))) <= (contraSubsetComplement)) <= (complementInPowersetComplementIntersect)))) <= (complementTnotintersectT)) <= (doubleComplementEq))) <= (doubleComplementSub1))) <= (doubleComplementI1)))) <= (contrasubsetT1))) <= (binintersectTERcontra)) <= (binintersectTELcontra))))) <= (powersetTI1))) <= (setextT))) <= (setminusT_lem))))))) <= (iftrue))))))) <= (iffalseProp2)) <= (iffalseProp1)))) <= (lam2lamEq)) <= (eta1)) <= (beta1)) <= (ap2apEq2)) <= (ap2apEq1)) <= (funcext2)))) <= (subbreln))) <= (funcextLem)))))) <= (brelnall2)) <= (brelnall1)) <= (lam2p)) <= (lamp)) <= (lamProp)) <= (funcinfuncset))) <= (infuncsetfunc)) <= (app))) <= (funcImageSingleton)) <= (dpsetconstrER)))))) <= (dpsetconstrSub)) <= (dpsetconstrI)) <= (cartprodpairsurjEq))))) <= (cartprodpairmemER)) <= (cartprodpairmemEL))) <= (kpairsurjEq)))))) <= (upairequniteq)) <= (setukpairinjR1)) <= (setukpairinjR12))) <= (setukpairinjL))) <= (cartprodfstin)) <= (kfstpairEq)) <= (theprop)) <= (kfstsingleton))))))))) <= (setunionsingleton2))))))) <= (ubforcartprodlem3)) <= (ubforcartprodlem2)))) <= (upairsubunion)) <= (upairset2E)))))) <= (kpairiskpair))) <= (setukpairIL))) <= (symdiffIneg2))) <= (symdiffI2))))) <= (setminusLsub)))) <= (setminusELneg)) <= (setminusERneg))) <= (setminusER)) <= (setminusEL)))) <= (binintersectSubset1)) <= (binintersectSubset4)) <= (binintersectRsub))) <= (binintersectER)) <= (binintersectSubset3)) <= (binintersectSubset2))) <= (binintersectEL)) <= (binintersectSubset5)))) <= (binunionLsub)) <= (binunionE)))) <= (upairset2IR))) <= (sepSubset))))))) <= (subsetemptysetimpeq)) <= (setextsub))) <= (setadjoinSub2)))) <= (subsetRefl))) <= (notequalI1)))) <= (subsetE)) <= (emptysetsubset)) <= (subsetI2)) <= (eqimpsubset1)))) <= (dsetconstr__Cong)) <= (descr__Cong)) <= (exuEu))) <= (setunion__Cong)) <= (powerset__Cong)) <= (setadjoin__Cong)) <= (emptyset__Cong))) <= (exuE3u)) <= (in__Cong)))))) <= (notdallE))) <= (prop2set2propI))) <= (quantDeMorgan4))))))) <= (upairsetIR))) <= (upairsetE)))) <= (notinsingleton)) <= (uniqinunit)) <= (nonemptyImpWitness))) <= (subPowSU)) <= (setunionE)) <= (setunionI)) <= (powersetE)))))))))))) <= (nonemptyI))) <= (setbeta)) <= (noeltsimpempty)))))) <= (emptysetimpfalse))) <= (prop2setE))))) <= (dsetconstrI)))) <= (foundationAx)) <= (replAx)) <= (omegaIndAx))) <= (omega0Ax)) <= (setunionAx))) <= (setadjoinAx))) <= (setextAx)))),inference(neg_conjecture,[status(cth)],[1])). 135.41/58.48 thf(3,plain,((~ (((! [A:$i]: ~ (in @ A @ emptyset)) => (((! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) = (! [C:$i]: ((in @ C @ B) => (in @ C @ A))))) => (((! [A:$i]: ((in @ A @ omega) => (in @ (setadjoin @ A @ A) @ omega))) => (((! [A:$i]: ? [B:$i]: (! [C:$i]: ((in @ C @ B) => (! [D:$i]: ((in @ D @ C) => (in @ D @ A)))) & ! [C:$i,D:$i]: (((in @ C @ A) & (in @ D @ A)) => ((! [E:$i]: ((in @ E @ B) => ((in @ C @ E) = (in @ D @ E)))) => (C = D))) & ! [C:$i,D:$i]: (((in @ C @ B) & (in @ D @ B)) => (! [E:$i]: ((in @ E @ C) => (in @ E @ D)) | ! [E:$i]: ((in @ E @ D) => (in @ E @ C)))) & ! [C:$i]: ((! [D:$i]: ((in @ D @ C) => (in @ D @ A)) & ? [D:$i]: (in @ D @ C)) => (? [D:$i,E:$i]: ((in @ D @ B) & (in @ E @ C) & ~ (? [F:$i]: ((in @ F @ D) & (in @ F @ C))) & ! [F:$i]: ((in @ F @ B) => (! [G:$i]: ((in @ G @ F) => (in @ G @ D)) | (in @ E @ F)))))))) => ((! [A:($i > $o)]: ((? [B:$i]: ((A @ B) & ! [C:$i]: ((A @ C) => (B = C)))) => (A @ (descr @ (A))))) => (((! [A:$i,B:($i > $o),C:$i]: ((in @ C @ (dsetconstr @ A @ (B))) => (in @ C @ A))) => ((! [A:$i,B:($i > $o),C:$i]: ((in @ C @ (dsetconstr @ A @ (B))) => (B @ C))) => (((! [A:$i]: ((in @ A @ emptyset) => (! [B:$o]: (B)))) => (((! [A:$i]: ~ (in @ A @ emptyset)) => ((! [A:($i > $o)]: ((? [B:$i]: ((A @ B) & ! [C:$i]: ((A @ C) => (B = C)))) => (? [B:$i]: (A @ B)))) => ((! [A:$i,B:$i]: ((! [C:$i]: ((in @ C @ A) => (in @ C @ B))) => ((! [C:$i]: ((in @ C @ B) => (in @ C @ A))) => (A = B)))) => ((! [A:$i]: ((! [B:$i]: ~ (in @ B @ A)) => (A = emptyset))) => (((! [A:$i]: ((~ (A = emptyset)) => (? [B:$i]: (in @ B @ A)))) => (((! [A:$i]: ((? [B:$i]: (in @ B @ A)) => (~ (A = emptyset)))) => ((! [A:$i,B:$i]: (in @ A @ (setadjoin @ A @ B))) => ((in @ emptyset @ (setadjoin @ emptyset @ emptyset)) => ((! [A:$i,B:$i,C:$i]: ((in @ C @ B) => (in @ C @ (setadjoin @ A @ B)))) => ((! [A:$i,B:$i,C:$i]: ((in @ C @ (setadjoin @ A @ B)) => (! [D:$o]: (((C = A) => (D)) => (((in @ C @ B) => (D)) => (D)))))) => ((! [A:$i,B:$i,C:$i]: ((in @ C @ (setadjoin @ A @ B)) => ((C = A) | (in @ C @ B)))) => ((! [A:$i]: ((dsetconstr @ A @ (^ [B:$i]: ($true))) = A)) => ((! [A:$i,B:$i]: ((! [C:$i]: ((in @ C @ B) => (in @ C @ A))) => (in @ B @ (powerset @ A)))) => ((! [A:$i]: (in @ emptyset @ (powerset @ A))) => ((! [A:$i]: (in @ emptyset @ (powerset @ A))) => (((! [A:($i > $o)]: ((? [B:$i]: ((A @ B) & ! [C:$i]: ((A @ C) => (B = C)))) => (? [B:$i]: ! [C:$i]: ((A @ C) = (C = B))))) => (((! [A:$i,B:$i]: ((A = B) => (in @ A @ (setadjoin @ B @ emptyset)))) => ((! [A:$i,B:$i]: ((in @ A @ (setadjoin @ B @ emptyset)) => (in @ B @ (setadjoin @ A @ emptyset)))) => (((! [A:$i,B:$i]: (in @ A @ (setadjoin @ A @ (setadjoin @ B @ emptyset)))) => (((! [A:$i,B:($i > $o)]: ((? [C:$i]: ((in @ C @ A) & (B @ C))) => (~ ((dsetconstr @ A @ (B)) = emptyset)))) => ((! [A:($i > $o),B:$i]: ((in @ B @ emptyset) => (A @ B))) => ((! [A:$i,B:($i > $o)]: ((~ (! [C:$i]: ((in @ C @ A) => (B @ C)))) => (? [C:$i]: ((in @ C @ A) & ~ (B @ C))))) => ((! [A:$i,B:($i > $o)]: ((! [C:$i]: ((in @ C @ A) => (~ (B @ C)))) => (~ (? [C:$i]: ((in @ C @ A) & (B @ C)))))) => ((! [A:$i,B:($i > $o)]: ((~ (? [C:$i]: ((in @ C @ A) & (B @ C)))) => (! [C:$i]: ((in @ C @ A) => (~ (B @ C)))))) => (((! [A:$o]: ((A) => (in @ emptyset @ (prop2set @ A)))) => (((! [A:$i,B:($i > $o)]: ((~ (? [C:$i]: ((in @ C @ A) & (B @ C)))) => (! [C:$i]: ((in @ C @ A) => (~ (B @ C)))))) => (((! [A:($i > $o)]: ((? [B:$i]: (A @ B)) => ((! [B:$i,C:$i]: ((A @ B) => ((A @ C) => (B = C)))) => (? [B:$i]: ((A @ B) & ! [C:$i]: ((A @ C) => (B = C))))))) => ((! [A:($i > $o)]: ((? [B:$i]: ! [C:$i]: ((A @ C) = (C = B))) => (? [B:$i]: ((A @ B) & ! [C:$i]: ((A @ C) => (B = C)))))) => ((! [A:$i,B:$i]: ((A = B) => (! [C:$i,D:$i]: ((C = D) => ((in @ C @ A) => (in @ D @ B)))))) => (((! [A:($i > $o),B:($i > $o)]: ((! [C:$i,D:$i]: ((C = D) => ((A @ C) = (B @ D)))) => ((? [C:$i]: ((A @ C) & ! [D:$i]: ((A @ D) => (C = D)))) = (? [C:$i]: ((B @ C) & ! [D:$i]: ((B @ D) => (C = D))))))) => (((! [A:$i,B:$i]: ((! [C:$i]: ((in @ C @ A) => (in @ C @ B))) => (subset @ A @ B))) => ((! [A:$i,B:$i]: ((A = B) => (subset @ B @ A))) => (((! [A:$i,B:$i,C:$i]: ((subset @ A @ B) => ((~ (in @ C @ B)) => (~ (in @ C @ A))))) => ((! [A:$i,B:$i,C:$i]: ((in @ C @ A) => ((~ (in @ C @ B)) => (~ (subset @ A @ B))))) => (((! [A:$i,B:$i,C:$i]: ((in @ C @ A) => ((~ (in @ C @ B)) => (~ (A = B))))) => (((! [A:$i,B:$i,C:$i]: ((subset @ A @ B) => ((subset @ B @ C) => (subset @ A @ C)))) => ((! [A:$i,B:$i]: (subset @ B @ (setadjoin @ A @ B))) => (((! [A:$i,B:$i]: ((subset @ A @ B) => (in @ A @ (powerset @ B)))) => (((! [A:$i,B:$i]: ((subset @ B @ A) => (in @ B @ (powerset @ A)))) => ((! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (subset @ B @ A))) => ((! [A:$i]: (in @ A @ (powerset @ A))) => ((! [A:$i,B:$i]: ((subset @ A @ B) => (subset @ (powerset @ A) @ (powerset @ B)))) => ((! [A:$i,B:($i > $o)]: (in @ (dsetconstr @ A @ (B)) @ (powerset @ A))) => (((! [A:$i,B:$i,C:$i]: ((in @ C @ A) => (in @ C @ (binunion @ A @ B)))) => (((! [A:$i,B:$i,C:$i]: ((in @ C @ B) => (in @ C @ (binunion @ A @ B)))) => ((! [A:$i,B:$i,C:$i,D:$o]: ((in @ C @ (binunion @ A @ B)) => (((in @ C @ A) => (D)) => (((in @ C @ B) => (D)) => (D))))) => (((! [A:$i,B:$i]: (subset @ B @ (binunion @ A @ B))) => ((! [A:$i,B:$i,C:$i]: ((in @ C @ A) => ((in @ C @ B) => (in @ C @ (binintersect @ A @ B))))) => (((! [A:$i,B:$i]: (subset @ (binintersect @ A @ B) @ A)) => (((! [A:$i,B:$i]: ((~ (? [C:$i]: ((in @ C @ A) & (in @ C @ B)))) => ((binintersect @ A @ B) = emptyset))) => (((! [A:$i,B:$i,C:$i]: ((binintersect @ A @ (binunion @ B @ C)) = (binunion @ (binintersect @ A @ B) @ (binintersect @ A @ C)))) => ((! [A:$i,B:$i,C:$i]: ((in @ C @ A) => ((~ (in @ C @ B)) => (in @ C @ (setminus @ A @ B))))) => (((! [A:$i,B:$i]: ((subset @ A @ B) => ((setminus @ A @ B) = emptyset))) => (((! [A:$i,B:$i,C:$i]: ((~ (in @ C @ A)) => (~ (in @ C @ (setminus @ A @ B))))) => ((! [A:$i,B:$i,C:$i]: ((in @ C @ B) => (~ (in @ C @ (setminus @ A @ B))))) => (((! [A:$i,B:$i]: (((setminus @ A @ B) = emptyset) => (subset @ A @ B))) => ((! [A:$i,B:$i,C:$i]: ((in @ C @ (symdiff @ A @ B)) => (! [D:$o]: (((in @ C @ A) => ((~ (in @ C @ B)) => (D))) => (((~ (in @ C @ A)) => ((in @ C @ B) => (D))) => (D)))))) => ((! [A:$i,B:$i,C:$i]: ((in @ C @ A) => ((~ (in @ C @ B)) => (in @ C @ (symdiff @ A @ B))))) => (((! [A:$i,B:$i,C:$i]: ((in @ C @ A) => ((in @ C @ B) => (~ (in @ C @ (symdiff @ A @ B)))))) => (((! [A:$i,B:$i]: (in @ B @ (setadjoin @ A @ (setadjoin @ B @ emptyset)))) => (((! [A:$i,B:$i]: (in @ B @ (setunion @ (setadjoin @ (setadjoin @ A @ emptyset) @ (setadjoin @ (setadjoin @ A @ (setadjoin @ B @ emptyset)) @ emptyset))))) => (((! [A:$i,B:$i]: (iskpair @ (kpair @ A @ B))) => ((! [A:$i,B:$i]: ((in @ B @ A) => (subset @ (setadjoin @ B @ emptyset) @ A))) => ((! [A:$i,B:$i]: ((in @ B @ A) => (in @ (setadjoin @ B @ emptyset) @ (powerset @ A)))) => ((! [A:$i,B:$i,C:$i]: ((in @ C @ A) => (in @ (setadjoin @ C @ emptyset) @ (powerset @ (binunion @ A @ B))))) => (((! [A:$i,B:$i,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ B) => (in @ (setadjoin @ C @ (setadjoin @ D @ emptyset)) @ (powerset @ (binunion @ A @ B))))))) => ((! [A:$i,B:$i,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ B) => (subset @ (setadjoin @ (setadjoin @ C @ emptyset) @ (setadjoin @ (setadjoin @ C @ (setadjoin @ D @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ A @ B))))))) => (((! [A:$i,B:$i,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ B) => (in @ (kpair @ C @ D) @ (cartprod @ A @ B)))))) => ((! [A:$i,B:$i,C:$i]: ((in @ C @ (cartprod @ A @ B)) => (? [D:$i]: ((in @ D @ A) & ? [E:$i]: ((in @ E @ B) & (C = (kpair @ D @ E))))))) => ((! [A:$i,B:$i,C:$i]: ((in @ C @ (cartprod @ A @ B)) => (iskpair @ C))) => ((! [A:$i,B:$i]: ((in @ B @ (setunion @ A)) => (? [C:$i]: ((in @ C @ A) & (in @ B @ C))))) => ((! 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[B:$i]: ((in @ B @ A) => (! [C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ A) => (((in @ B @ C) & (in @ C @ D)) => (in @ B @ D))))))) & ! [B:$i]: ((in @ B @ A) => (! [C:$i]: ((in @ C @ A) => ((B = C) | (in @ B @ C) | (in @ C @ B))))) & ! [B:$i]: ((in @ B @ A) => (~ (in @ B @ B))) & ! [B:$i]: ((in @ B @ (powerset @ A)) => ((~ (B = emptyset)) => (? [C:$i]: ((in @ C @ B) & ! [D:$i]: ((in @ D @ B) => ((C = D) | (in @ C @ D)))))))) => (! [B:$i]: ((! [C:$i]: ((in @ C @ B) => (subset @ C @ B)) & ! [C:$i]: ((in @ C @ B) => (! [D:$i]: ((in @ D @ B) => (! [E:$i]: ((in @ E @ B) => (((in @ C @ D) & (in @ D @ E)) => (in @ C @ E))))))) & ! [C:$i]: ((in @ C @ B) => (! [D:$i]: ((in @ D @ B) => ((C = D) | (in @ C @ D) | (in @ D @ C))))) & ! [C:$i]: ((in @ C @ B) => (~ (in @ C @ C))) & ! [C:$i]: ((in @ C @ (powerset @ B)) => ((~ (C = emptyset)) => (? [D:$i]: ((in @ D @ C) & ! [E:$i]: ((in @ E @ C) => ((D = E) | (in @ D @ E)))))))) => (! [C:$i]: ((in @ C @ (binintersect @ A @ B)) => (subset @ C @ (binintersect @ A @ B)))))))) => ((! [A:$i]: ((! [B:$i]: ((in @ B @ A) => (subset @ B @ A)) & ! [B:$i]: ((in @ B @ A) => (! [C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ A) => (((in @ B @ C) & (in @ C @ D)) => (in @ B @ D))))))) & ! [B:$i]: ((in @ B @ A) => (! [C:$i]: ((in @ C @ A) => ((B = C) | (in @ B @ C) | (in @ C @ B))))) & ! [B:$i]: ((in @ B @ A) => (~ (in @ B @ B))) & ! [B:$i]: ((in @ B @ (powerset @ A)) => ((~ (B = emptyset)) => (? [C:$i]: ((in @ C @ B) & ! [D:$i]: ((in @ D @ B) => ((C = D) | (in @ C @ D)))))))) => (! [B:$i,C:$i]: ((in @ C @ A) => ((in @ B @ C) => (in @ B @ A)))))) => (! [A:$i]: (! [B:$i]: ((subset @ B @ A) | ~ (in @ B @ A)) | ~ (! [B:$i]: ((in @ B @ A) => (subset @ B @ A)) & ! [B:$i]: ((in @ B @ A) => (! [C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ A) => (((in @ B @ C) & (in @ C @ D)) => (in @ B @ D))))))) & ! [B:$i]: ((in @ B @ A) => (! [C:$i]: ((in @ C @ A) => ((B = C) | (in @ B @ C) | (in @ C @ B))))) & ! [B:$i]: ((in @ B @ A) => (~ (in @ B @ B))) & ! [B:$i]: ((in @ B @ (powerset @ A)) => ((~ (B = emptyset)) => (? [C:$i]: ((in @ C @ B) & ! [D:$i]: ((in @ D @ B) => ((C = D) | (in @ C @ D))))))))))))) | ~ (! [A:$i]: ((! [B:$i]: ((in @ B @ A) => (subset @ B @ A))) => (! [B:$i,C:$i]: ((in @ B @ A) => ((in @ C @ B) => (in @ C @ A)))))) | ~ (! [A:$i]: ((! [B:$i]: ((in @ B @ A) => (subset @ B @ A))) => (! [B:$i]: ((! [C:$i]: ((in @ C @ B) => (subset @ C @ B))) => (! [C:$i]: ((in @ C @ (binintersect @ A @ B)) => (subset @ C @ (binintersect @ A @ B)))))))) | ~ (! [A:$i]: ((in @ A @ omega) => (! [B:$i]: ((in @ B @ omega) => (((omegaS @ A) = (omegaS @ B)) => (A = B)))))) | ~ (! [A:$i]: ((in @ A @ omega) => (~ ((omegaS @ A) = emptyset)))))) | ~ (! [A:$i]: ((in @ A @ omega) => (in @ (omegaS @ A) @ omega))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (funcSet @ A @ (powerset @ A))) => (~ (surjective @ A @ (powerset @ A) @ B)))) | ~ (! [A:$i,B:$i,C:($i > $i)]: ((! [D:$i]: ((in @ D @ A) => (in @ (C @ D) @ B))) => (! [D:$i]: ((in @ D @ (funcSet @ B @ A)) => ((! [E:$i]: ((in @ E @ A) => ((ap @ B @ A @ D @ (C @ E)) = E))) => (surjective @ B @ A @ D))))))))))))) | ~ (! [A:$i,B:($i > $i),C:$i]: ((in @ C @ (image1 @ A @ (B))) => (? [D:$i]: ((in @ D @ A) & (C = (B @ D)))))) | ~ (! [A:$i,B:($i > $i),C:$i]: ((in @ C @ (image1 @ A @ (B))) = (? [D:$i]: ((in @ D @ A) & (C = (B @ D)))))))) | ~ (! [A:$i,B:($i > $i)]: ? [C:$i]: ! [D:$i]: ((in @ D @ C) = (? [E:$i]: ((in @ E @ A) & (D = (B @ E)))))))) | ~ (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:$i]: ((breln1 @ A @ C) => (! [D:$i]: ((breln1 @ A @ D) => ((breln1compset @ A @ (binunion @ B @ C) @ D) = (binunion @ (breln1compset @ A @ B @ D) @ (breln1compset @ A @ C @ D))))))))))))) | ~ (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:$i]: ((breln1 @ A @ C) => (! [D:$i]: ((in @ D @ A) => (! [E:$i]: ((in @ E @ A) => ((in @ (kpair @ D @ E) @ (binunion @ B @ C)) => (! [F:$o]: (((in @ (kpair @ D @ E) @ B) => (F)) => (((in @ (kpair @ D @ E) @ C) => (F)) => (F))))))))))))))))) | ~ (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:$i]: ((breln1 @ A @ C) => (! [D:$i]: ((in @ D @ A) => (! [E:$i]: ((in @ E @ A) => ((in @ (kpair @ D @ E) @ B) => (in @ (kpair @ D @ E) @ (binunion @ B @ C))))))))))) | ~ (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:$i]: ((breln1 @ A @ C) => (breln1 @ A @ (binunion @ B @ C)))))))) | ~ (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:$i]: ((breln1 @ A @ C) => (! [D:$i]: ((in @ D @ A) => (! [E:$i]: ((in @ E @ A) => ((in @ (kpair @ D @ E) @ (breln1compset @ A @ B @ C)) => (? [F:$i]: ((in @ F @ A) & (in @ (kpair @ D @ F) @ B) & (in @ (kpair @ F @ E) @ C)))))))))))))) | ~ (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:$i]: ((breln1 @ A @ C) => (breln1 @ A @ (breln1compset @ A @ B @ C))))))))) | ~ (! [A:$i,B:$i]: ((breln1 @ A @ B) => (breln1 @ A @ (breln1invset @ A @ B)))))) | ~ (! [A:$i,B:$i]: ((breln1 @ A @ B) => (! [C:$i]: ((breln1 @ A @ C) => ((! [D:$i]: ((in @ D @ A) => (! [E:$i]: ((in @ E @ A) => ((in @ (kpair @ D @ E) @ B) => (in @ (kpair @ D @ E) @ C)))))) => (subset @ B @ C))))))))) | ~ (! [A:$i,B:$i,C:($i > ($i > $o))]: ((! [D:$i]: ((in @ D @ A) => (? [E:$i]: ((in @ E @ B) & (C @ D @ E))))) => (! [D:$i]: ((in @ D @ (breln1Set @ B)) => ((reflwellordering @ B @ D) => (! [E:$i]: ((in @ E @ A) => (singleton @ (dsetconstr @ B @ (^ [F:$i]: ((C @ E @ F) & ! [G:$i]: ((in @ G @ B) => ((C @ E @ G) => (in @ (kpair @ F @ G) @ D))))))))))))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (in @ (setminus @ A @ (binunion @ B @ C)) @ (powerset @ (setminus @ A @ B)))))))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ (powerset @ A)) => (! [E:$i]: ((in @ E @ (powerset @ A)) => ((subset @ B @ D) => ((subset @ C @ E) => (subset @ (binintersect @ B @ C) @ (binintersect @ D @ E)))))))))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ (powerset @ A)) => ((subset @ C @ D) => (subset @ (binintersect @ B @ C) @ D)))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ (powerset @ A)) => ((subset @ B @ D) => (subset @ (binintersect @ B @ C) @ D)))))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => ((setminus @ A @ (binunion @ B @ C)) = (binintersect @ (setminus @ A @ B) @ (setminus @ A @ C)))))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((in @ D @ (setminus @ A @ (binunion @ B @ C))) => (in @ D @ (binintersect @ (setminus @ A @ B) @ (setminus @ A @ C)))))))))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((in @ D @ (setminus @ A @ (binunion @ B @ C))) => (in @ D @ (setminus @ A @ C))))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (in @ (setminus @ A @ (binunion @ B @ C)) @ (powerset @ (setminus @ A @ B)))))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$o,E:$i]: ((in @ E @ A) => ((in @ E @ (binunion @ B @ C)) => (((in @ E @ B) => (D)) => (((in @ E @ C) => (D)) => (D)))))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((in @ D @ (setminus @ A @ (binunion @ B @ C))) => (in @ D @ (setminus @ A @ B))))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((in @ D @ (setminus @ A @ (binunion @ B @ C))) => (~ (in @ D @ B))))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ (powerset @ A)) => (in @ (binintersect @ B @ C) @ (powerset @ (binintersect @ (binunion @ B @ D) @ (binunion @ C @ D)))))))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ (powerset @ A)) => (! [E:$i]: ((in @ E @ A) => ((in @ E @ (binintersect @ B @ C)) => (in @ E @ (binunion @ C @ D))))))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ (powerset @ A)) => (! [E:$i]: ((in @ E @ A) => ((in @ E @ (binintersect @ B @ C)) => (in @ E @ (binunion @ B @ D))))))))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((~ (in @ D @ (binunion @ B @ C))) => (~ (in @ D @ B))))))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => ((subset @ B @ (setminus @ A @ C)) => (! [D:$i]: ((in @ D @ A) => ((in @ D @ C) => (in @ D @ (setminus @ A @ B)))))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (in @ (setminus @ A @ B) @ (powerset @ (setminus @ A @ (binintersect @ B @ C))))))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((in @ D @ (setminus @ A @ B)) => (~ (in @ D @ (binintersect @ B @ C)))))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (B = (setminus @ A @ (setminus @ A @ B))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (subset @ B @ (setminus @ A @ (setminus @ A @ B))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ A) => ((in @ C @ B) => (in @ C @ (setminus @ A @ (setminus @ A @ B))))))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((subset @ B @ C) => ((~ (in @ D @ C)) => (~ (in @ D @ B)))))))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((~ (in @ D @ C)) => (~ (in @ D @ (binintersect @ B @ C)))))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (! [D:$i]: ((in @ D @ A) => ((~ (in @ D @ B)) => (~ (in @ D @ (binintersect @ B @ C)))))))))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => ((! [D:$i]: ((in @ D @ A) => ((in @ D @ B) => (in @ D @ C)))) => (in @ B @ (powerset @ C))))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => ((! [D:$i]: ((in @ D @ A) => ((in @ D @ B) => (in @ D @ C)))) => ((! [D:$i]: ((in @ D @ A) => ((in @ D @ C) => (in @ D @ B)))) => (B = C))))))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (powerset @ A)) => (! [C:$i]: ((in @ C @ (powerset @ A)) => (in @ (setminus @ B @ C) @ (powerset @ A)))))))))))) | ~ (! [A:$i,B:$o,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ A) => ((B) => ((if @ A @ B @ C @ D) = C)))))))))))) | ~ (! [A:$i,B:$o,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ A) => ((~ (B)) => ((dsetconstr @ A @ (^ [E:$i]: (((B) & (E = C)) | (~ (B) & (E = D))))) = (setadjoin @ D @ emptyset))))))) | ~ (! [A:$i,B:$o,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ A) => ((~ (B)) => (in @ D @ (dsetconstr @ A @ (^ [E:$i]: (((B) & (E = C)) | (~ (B) & (E = D)))))))))))))) | ~ (! [A:$i,B:$i,C:$i]: ((func @ A @ B @ C) => ((lam @ A @ B @ (ap @ A @ B @ C)) = C))) | ~ (! [A:$i,B:$i,C:($i > $i)]: ((! [D:$i]: ((in @ D @ A) => (in @ (C @ D) @ B))) => (! [D:$i]: ((in @ D @ A) => ((ap @ A @ B @ (lam @ A @ B @ (C)) @ D) = (C @ D)))))) | ~ (! [A:$i,B:$i,C:$i]: ((in @ C @ (funcSet @ A @ B)) => (! [D:$i]: ((in @ D @ (funcSet @ A @ B)) => ((! [E:$i]: ((in @ E @ A) => ((ap @ A @ B @ C @ E) = (ap @ A @ B @ D @ E)))) => (C = D))))))))) | ~ (! [A:$i,B:$i,C:$i]: ((breln @ A @ B @ C) => (! [D:$i]: ((breln @ A @ B @ D) => ((! [E:$i]: ((in @ E @ A) => (! [F:$i]: ((in @ F @ B) => ((in @ (kpair @ E @ F) @ C) => (in @ (kpair @ E @ F) @ D)))))) => (subset @ C @ D)))))))) | ~ (! [A:$i,B:$i,C:$i]: ((func @ A @ B @ C) => (! [D:$i]: ((func @ A @ B @ D) => ((! [E:$i]: ((in @ E @ A) => ((ap @ A @ B @ C @ E) = (ap @ A @ B @ D @ E)))) => (! [E:$i]: ((in @ E @ A) => (! [F:$i]: ((in @ F @ B) => ((in @ (kpair @ E @ F) @ D) => (in @ (kpair @ E @ F) @ C)))))))))))))))) | ~ (! [A:$i,B:$i,C:$i]: ((breln @ A @ B @ C) => (! [D:($i > $o)]: ((! [E:$i]: ((in @ E @ A) => (! [F:$i]: ((in @ F @ B) => ((in @ (kpair @ E @ F) @ C) => (D @ (kpair @ E @ F))))))) => (! [E:$i]: ((in @ E @ C) => (D @ E))))))) | ~ (! [A:$i,B:$i,C:$i]: ((breln @ A @ B @ C) => (! [D:($i > $o)]: ((! [E:$i]: ((in @ E @ A) => (! [F:$i]: ((in @ F @ B) => ((in @ (kpair @ E @ F) @ C) => (D @ (kpair @ E @ F))))))) => (! [E:$i]: ((in @ E @ C) => (D @ E))))))) | ~ (! [A:$i,B:$i,C:($i > $i)]: ((! [D:$i]: ((in @ D @ A) => (in @ (C @ D) @ B))) => (in @ (lam @ A @ B @ (C)) @ (funcSet @ A @ B)))) | ~ (! [A:$i,B:$i,C:($i > $i)]: ((! [D:$i]: ((in @ D @ A) => (in @ (C @ D) @ B))) => (func @ A @ B @ (lam @ A @ B @ (C))))) | ~ (! [A:$i,B:$i,C:($i > $i)]: ((! [D:$i]: ((in @ D @ A) => (in @ (C @ D) @ B))) => (func @ A @ B @ (dpsetconstr @ A @ B @ (^ [D:$i]: (= @ $i @ (C @ D))))))) | ~ (! [A:$i,B:$i,C:$i]: ((func @ A @ B @ C) => (in @ C @ (funcSet @ A @ B)))))) | ~ (! [A:$i,B:$i,C:$i]: ((in @ C @ (funcSet @ A @ B)) => (func @ A @ B @ C))) | ~ (! [A:$i,B:$i,C:$i]: ((func @ A @ B @ C) => (! [D:$i]: ((in @ D @ A) => (in @ (ap @ A @ B @ C @ D) @ B))))))) | ~ (! [A:$i,B:$i,C:$i]: ((func @ A @ B @ C) => (! [D:$i]: ((in @ D @ A) => (singleton @ (dsetconstr @ B @ (^ [E:$i]: (in @ (kpair @ D @ E) @ C)))))))) | ~ (! [A:$i,B:$i,C:($i > ($i > $o)),D:$i,E:$i]: ((in @ (kpair @ D @ E) @ (dpsetconstr @ A @ B @ (C))) => (C @ D @ E)))))))) | ~ (! [A:$i,B:$i,C:($i > ($i > $o))]: (subset @ (dpsetconstr @ A @ B @ (C)) @ (cartprod @ A @ B))) | ~ (! [A:$i,B:$i,C:($i > ($i > $o)),D:$i]: ((in @ D @ A) => (! [E:$i]: ((in @ E @ B) => ((C @ D @ E) => (in @ (kpair @ D @ E) @ (dpsetconstr @ A @ B @ (C)))))))) | ~ (! [A:$i,B:$i,C:$i]: ((in @ C @ (cartprod @ A @ B)) => ((kpair @ (kfst @ C) @ (ksnd @ C)) = C)))))) | ~ (! [A:$i,B:$i,C:$i,D:$i]: ((in @ (kpair @ C @ D) @ (cartprod @ A @ B)) => (in @ D @ B))) | ~ (! [A:$i,B:$i,C:$i,D:$i]: ((in @ (kpair @ C @ D) @ (cartprod @ A @ B)) => (in @ C @ A))))) | ~ (! [A:$i]: ((iskpair @ A) => ((kpair @ (kfst @ A) @ (ksnd @ A)) = A)))))))) | ~ (! [A:$i,B:$i,C:$i]: (((setadjoin @ A @ (setadjoin @ B @ emptyset)) = (setadjoin @ C @ emptyset)) => (A = B))) | ~ (! [A:$i,B:$i,C:$i,D:$i]: (((setadjoin @ (setadjoin @ A @ emptyset) @ (setadjoin @ (setadjoin @ A @ (setadjoin @ B @ emptyset)) @ emptyset)) = (setadjoin @ (setadjoin @ C @ emptyset) @ (setadjoin @ (setadjoin @ C @ (setadjoin @ D @ emptyset)) @ emptyset))) => ((C = D) => (B = D)))) | ~ (! [A:$i,B:$i]: ((A = B) => ((setadjoin @ (setadjoin @ A @ emptyset) @ (setadjoin @ (setadjoin @ A @ (setadjoin @ B @ emptyset)) @ emptyset)) = (setadjoin @ (setadjoin @ A @ emptyset) @ emptyset)))))) | ~ (! [A:$i,B:$i,C:$i,D:$i]: (((kpair @ A @ B) = (kpair @ C @ D)) => (A = C))))) | ~ (! [A:$i,B:$i,C:$i]: ((in @ C @ (cartprod @ A @ B)) => (in @ (kfst @ C) @ A))) | ~ (! [A:$i,B:$i]: ((kfst @ (kpair @ A @ B)) = A)) | ~ (! [A:$i]: ((singleton @ A) => (in @ (setunion @ A) @ A))) | ~ (! [A:$i]: ((iskpair @ A) => (singleton @ (dsetconstr @ (setunion @ A) @ (^ [B:$i]: (in @ (setadjoin @ B @ emptyset) @ A)))))))))))))) | ~ (! [A:$i]: (subset @ A @ (setunion @ (setadjoin @ A @ emptyset)))))))))) | ~ (! [A:$i,B:$i,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ B) => (in @ (kpair @ C @ D) @ (powerset @ (powerset @ (binunion @ A @ B)))))))) | ~ (! [A:$i,B:$i,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ B) => (in @ (setadjoin @ (setadjoin @ C @ emptyset) @ (setadjoin @ (setadjoin @ C @ (setadjoin @ D @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ A @ B))))))))))) | ~ (! [A:$i,B:$i,C:$i]: ((in @ C @ A) => (! [D:$i]: ((in @ D @ B) => (subset @ (setadjoin @ C @ (setadjoin @ D @ emptyset)) @ (binunion @ A @ B)))))) | ~ (! [A:$i,B:$i,C:$i]: ((in @ C @ (setadjoin @ A @ (setadjoin @ B @ emptyset))) => ((C = A) | (C = B))))))))) | ~ (! [A:$i,B:$i]: (iskpair @ (setadjoin @ (setadjoin @ A @ emptyset) @ (setadjoin @ (setadjoin @ A @ (setadjoin @ B @ emptyset)) @ emptyset)))))) | ~ (! [A:$i,B:$i]: (in @ A @ (setunion @ (setadjoin @ (setadjoin @ A @ emptyset) @ (setadjoin @ (setadjoin @ A @ (setadjoin @ B @ emptyset)) @ emptyset))))))) | ~ (! [A:$i,B:$i,C:$i]: ((~ (in @ C @ A)) => ((~ (in @ C @ B)) => (~ (in @ C @ (symdiff @ A @ B)))))))) | ~ (! [A:$i,B:$i,C:$i]: ((~ (in @ C @ A)) => ((in @ C @ B) => (in @ C @ (symdiff @ A @ B))))))))) | ~ (! [A:$i,B:$i]: (subset @ (setminus @ A @ B) @ A))))) | ~ (! [A:$i,B:$i,C:$i]: ((~ (in @ C @ (setminus @ A @ B))) => ((~ (in @ C @ B)) => (~ (in @ C @ A))))) | ~ (! [A:$i,B:$i,C:$i]: ((~ (in @ C @ (setminus @ A @ B))) => ((in @ C @ A) => (in @ C @ B)))))) | ~ (! [A:$i,B:$i,C:$i]: ((in @ C @ (setminus @ A @ B)) => (~ (in @ C @ B)))) | ~ (! [A:$i,B:$i,C:$i]: ((in @ C @ (setminus @ A @ B)) => (in @ C @ A)))))) | ~ (! [A:$i,B:$i]: (((binintersect @ A @ B) = A) => (subset @ A @ B))) | ~ (! [A:$i,B:$i]: ((subset @ B @ A) => ((binintersect @ A @ B) = B))) | ~ (! [A:$i,B:$i]: (subset @ (binintersect @ A @ B) @ B)))) | ~ (! [A:$i,B:$i,C:$i]: ((in @ C @ (binintersect @ A @ B)) => (in @ C @ B))) | ~ (! [A:$i,B:$i]: (((binintersect @ A @ B) = B) => (subset @ B @ A))) | ~ (! [A:$i,B:$i]: ((subset @ A @ B) => ((binintersect @ A @ B) = A))))) | ~ (! [A:$i,B:$i,C:$i]: ((in @ C @ (binintersect @ A @ B)) => (in @ C @ A))) | ~ (! [A:$i,B:$i,C:$i]: ((subset @ C @ A) => ((subset @ C @ B) => (subset @ C @ (binintersect @ A @ B)))))))) | ~ (! [A:$i,B:$i]: (subset @ A @ (binunion @ A @ B))) | ~ (! [A:$i,B:$i,C:$i]: ((in @ C @ (binunion @ A @ B)) => ((in @ C @ A) | (in @ C @ B))))))) | ~ (! [A:$i,B:$i]: (in @ B @ (setadjoin @ A @ (setadjoin @ B @ emptyset)))))) | ~ (! [A:$i,B:($i > $o)]: (subset @ (dsetconstr @ A @ (B)) @ A)))))))) | ~ (! [A:$i]: ((subset @ A @ emptyset) => (A = emptyset))) | ~ (! [A:$i,B:$i]: ((subset @ A @ B) => ((subset @ B @ A) => (A = B)))))) | ~ (! [A:$i,B:$i,C:$i]: ((subset @ A @ C) => (subset @ A @ (setadjoin @ B @ C))))))) | ~ (! [A:$i]: (subset @ A @ A)))) | ~ (! [A:$i,B:$i]: ((~ (subset @ A @ B)) => (~ (A = B))))))) | ~ (! [A:$i,B:$i,C:$i]: ((subset @ A @ B) => ((in @ C @ A) => (in @ C @ B)))) | ~ (! [A:$i]: (subset @ emptyset @ A)) | ~ (! [A:$i,B:$i]: ((! [C:$i]: ((in @ C @ A) => (in @ C @ B))) => (subset @ A @ B))) | ~ (! [A:$i,B:$i]: ((A = B) => (subset @ A @ B)))))) | ~ (! [A:$i,B:$i]: ((A = B) => (! [C:($i > $o),D:($i > $o)]: ((! [E:$i]: ((in @ E @ A) => (! [F:$i]: ((in @ F @ B) => ((E = F) => ((C @ E) = (D @ F))))))) => ((dsetconstr @ A @ (C)) = (dsetconstr @ B @ (D))))))) | ~ (! [A:($i > $o),B:($i > $o)]: ((! [C:$i,D:$i]: ((C = D) => ((A @ C) = (B @ D)))) => ((? [C:$i]: ((A @ C) & ! [D:$i]: ((A @ D) => (C = D)))) => ((? [C:$i]: ((B @ C) & ! [D:$i]: ((B @ D) => (C = D)))) => ((descr @ (A)) = (descr @ (B))))))) | ~ (! [A:($i > $o)]: ((? [B:$i]: ((A @ B) & ! [C:$i]: ((A @ C) => (B = C)))) => (! [B:$i,C:$i]: ((A @ B) => ((A @ C) => (B = C)))))) | ~ (! [A:$i,B:$i]: ((A = B) => ((setunion @ A) = (setunion @ B)))) | ~ (! [A:$i,B:$i]: ((A = B) => ((powerset @ A) = (powerset @ B)))) | ~ (! [A:$i,B:$i]: ((A = B) => (! [C:$i,D:$i]: ((C = D) => ((setadjoin @ A @ C) = (setadjoin @ B @ D)))))))) | ~ (! [A:($i > $o)]: ((? [B:$i]: ((A @ B) & ! [C:$i]: ((A @ C) => (B = C)))) => (! [B:$i,C:$i]: ((A @ B) => ((A @ C) => (B = C)))))) | ~ (! [A:$i,B:$i]: ((A = B) => (! [C:$i,D:$i]: ((C = D) => ((in @ C @ A) = (in @ D @ B)))))))))) | ~ (! [A:$i,B:($i > $o)]: ((~ (! [C:$i]: ((in @ C @ A) => (B @ C)))) => (? [C:$i]: ((in @ C @ A) & ~ (B @ C))))))) | ~ (! [A:$o]: ((A) => (set2prop @ (prop2set @ A)))))) | ~ (! [A:$i,B:($i > $o)]: ((? [C:$i]: ((in @ C @ A) & ~ (B @ C))) => (~ (! [C:$i]: ((in @ C @ A) => (B @ C)))))))))))) | ~ (! [A:$i,B:$i]: (in @ B @ (setadjoin @ A @ (setadjoin @ B @ emptyset)))))) | ~ (! [A:$i,B:$i,C:$i]: ((in @ C @ (setadjoin @ A @ (setadjoin @ B @ emptyset))) => ((C = A) | (C = B))))))) | ~ (! [A:$i,B:$i]: ((~ (A = B)) => (~ (in @ B @ (setadjoin @ A @ emptyset))))) | ~ (! [A:$i,B:$i]: ((in @ A @ (setadjoin @ B @ emptyset)) => (A = B))) | ~ (! [A:$i]: ((~ (A = emptyset)) => (? [B:$i]: (in @ B @ A)))))) | ~ (! [A:$i,B:$i]: ((in @ B @ A) => (in @ B @ (powerset @ (setunion @ A))))) | ~ (! [A:$i,B:$i]: ((in @ B @ (setunion @ A)) => (! [C:$o]: ((! [D:$i]: ((in @ B @ D) => ((in @ D @ A) => (C)))) => (C))))) | ~ (! [A:$i,B:$i,C:$i]: ((in @ B @ C) => ((in @ C @ A) => (in @ B @ (setunion @ A))))) | ~ (! [A:$i,B:$i,C:$i]: ((in @ B @ (powerset @ A)) => ((in @ C @ B) => (in @ C @ A))))))))))))))) | ~ (! [A:$i,B:($i > $o),C:$i]: ((in @ C @ A) => ((B @ C) => (~ ((dsetconstr @ A @ (B)) = emptyset))))))) | ~ (! [A:$i,B:($i > $o),C:$i]: ((in @ C @ A) => ((in @ C @ (dsetconstr @ A @ (B))) = (B @ C)))) | ~ (! [A:$i]: ((! [B:$i]: ~ (in @ B @ A)) => (A = emptyset)))))))) | ~ (! [A:$i]: ~ (in @ A @ emptyset)))) | ~ (! [A:$o,B:$i]: ((in @ B @ (prop2set @ A)) => (A)))))) | ~ (! [A:$i,B:($i > $o),C:$i]: ((in @ C @ A) => ((B @ C) => (in @ C @ (dsetconstr @ A @ (B))))))))) | ~ (! [A:$i]: ((? [B:$i]: (in @ B @ A)) => (? [B:$i]: ((in @ B @ A) & ~ (? [C:$i]: ((in @ C @ B) & (in @ C @ A))))))) | ~ (! [A:($i > ($i > $o)),B:$i]: ((! [C:$i]: ((in @ C @ B) => (? [D:$i]: ((A @ C @ D) & ! [E:$i]: ((A @ C @ E) => (D = E)))))) => (? [C:$i]: ! [D:$i]: ((in @ D @ C) = (? [E:$i]: ((in @ E @ B) & (A @ E @ D))))))) | ~ (! [A:$i]: (((in @ emptyset @ A) & ! [B:$i]: (((in @ B @ omega) & (in @ B @ A)) => (in @ (setadjoin @ B @ B) @ A))) => (! [B:$i]: ((in @ B @ omega) => (in @ B @ A))))))) | ~ (in @ emptyset @ omega) | ~ (! [A:$i,B:$i]: ((in @ B @ (setunion @ A)) = (? [C:$i]: ((in @ B @ C) & (in @ C @ A))))))) | ~ (! [A:$i,B:$i,C:$i]: ((in @ C @ (setadjoin @ A @ B)) = ((C = A) | (in @ C @ B)))))) | ~ (! [A:$i,B:$i]: ((! [C:$i]: ((in @ C @ A) = (in @ C @ B))) => (A = B)))))),inference(defexp_and_simp_and_etaexpand,[status(thm)],[2])). 135.41/58.48 thf(1693,plain,($false),inference(cvc4,[status(thm)],[3])). 135.41/58.48 % SZS output end CNFRefutation for /export/starexec/sandbox/benchmark/theBenchmark.p 135.41/58.48 % [INFO] Killing All external provers ... 135.41/58.48 FINAL WATCH: 135.1 CPU 58.1 WC 135.41/58.48 EOF