%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET024+1 : TPTP v9.0.0. Bugfixed v5.4.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n001.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:24:52 AM UTC 2025
% Result : Theorem 0.13s 0.40s
% Output : Proof 0.13s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.13/0.13 % Problem : SET024+1 : TPTP v9.0.0. Bugfixed v5.4.0.
% 0.13/0.13 % Command : run_E %s %d THM
% 0.13/0.35 % Computer : n001.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Fri Jun 20 04:07:04 EDT 2025
% 0.13/0.35 % CPUTime :
% 0.13/0.40 % SZS status Theorem
% 0.13/0.40 % SZS output start Proof
% 0.13/0.40 tff(member_type, type, (
% 0.13/0.40 member: ( $i * $i ) > $o)).
% 0.13/0.40 tff(unordered_pair_type, type, (
% 0.13/0.40 unordered_pair: ( $i * $i ) > $i)).
% 0.13/0.40 tff(tptp_fun_X_7_type, type, (
% 0.13/0.40 tptp_fun_X_7: $i)).
% 0.13/0.40 tff(singleton_type, type, (
% 0.13/0.40 singleton: $i > $i)).
% 0.13/0.40 tff(universal_class_type, type, (
% 0.13/0.40 universal_class: $i)).
% 0.13/0.40 tff(1,plain,
% 0.13/0.40 (^[X: $i] : refl((singleton(X) = unordered_pair(X, X)) <=> (singleton(X) = unordered_pair(X, X)))),
% 0.13/0.40 inference(bind,[status(th)],[])).
% 0.13/0.40 tff(2,plain,
% 0.13/0.40 (![X: $i] : (singleton(X) = unordered_pair(X, X)) <=> ![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.13/0.40 inference(quant_intro,[status(thm)],[1])).
% 0.13/0.40 tff(3,plain,
% 0.13/0.40 (![X: $i] : (singleton(X) = unordered_pair(X, X)) <=> ![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(4,axiom,(![X: $i] : (singleton(X) = unordered_pair(X, X))), file('/export/starexec/sandbox2/benchmark/Axioms/SET005+0.ax','singleton_set_defn')).
% 0.13/0.40 tff(5,plain,
% 0.13/0.40 (![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.13/0.40 tff(6,plain,(
% 0.13/0.40 ![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.13/0.40 inference(skolemize,[status(sab)],[5])).
% 0.13/0.40 tff(7,plain,
% 0.13/0.40 (![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.13/0.40 tff(8,plain,
% 0.13/0.40 ((~![X: $i] : (singleton(X) = unordered_pair(X, X))) | (singleton(X!7) = unordered_pair(X!7, X!7))),
% 0.13/0.40 inference(quant_inst,[status(thm)],[])).
% 0.13/0.40 tff(9,plain,
% 0.13/0.40 (singleton(X!7) = unordered_pair(X!7, X!7)),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.13/0.40 tff(10,plain,
% 0.13/0.40 (unordered_pair(X!7, X!7) = singleton(X!7)),
% 0.13/0.40 inference(symmetry,[status(thm)],[9])).
% 0.13/0.40 tff(11,plain,
% 0.13/0.40 (member(X!7, unordered_pair(X!7, X!7)) <=> member(X!7, singleton(X!7))),
% 0.13/0.40 inference(monotonicity,[status(thm)],[10])).
% 0.13/0.40 tff(12,plain,
% 0.13/0.40 (member(X!7, singleton(X!7)) <=> member(X!7, unordered_pair(X!7, X!7))),
% 0.13/0.40 inference(symmetry,[status(thm)],[11])).
% 0.13/0.40 tff(13,plain,
% 0.13/0.40 ((~member(X!7, singleton(X!7))) <=> (~member(X!7, unordered_pair(X!7, X!7)))),
% 0.13/0.40 inference(monotonicity,[status(thm)],[12])).
% 0.13/0.40 tff(14,plain,
% 0.13/0.40 ((~![X: $i] : ((~member(X, universal_class)) | member(X, singleton(X)))) <=> (~![X: $i] : ((~member(X, universal_class)) | member(X, singleton(X))))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(15,plain,
% 0.13/0.40 ((~![X: $i] : (member(X, universal_class) => member(X, singleton(X)))) <=> (~![X: $i] : ((~member(X, universal_class)) | member(X, singleton(X))))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(16,axiom,(~![X: $i] : (member(X, universal_class) => member(X, singleton(X)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','set_in_its_singleton')).
% 0.13/0.40 tff(17,plain,
% 0.13/0.40 (~![X: $i] : ((~member(X, universal_class)) | member(X, singleton(X)))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[16, 15])).
% 0.13/0.40 tff(18,plain,
% 0.13/0.40 (~![X: $i] : ((~member(X, universal_class)) | member(X, singleton(X)))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[17, 14])).
% 0.13/0.40 tff(19,plain,
% 0.13/0.40 (~![X: $i] : ((~member(X, universal_class)) | member(X, singleton(X)))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[18, 14])).
% 0.13/0.40 tff(20,plain,
% 0.13/0.40 (~![X: $i] : ((~member(X, universal_class)) | member(X, singleton(X)))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[19, 14])).
% 0.13/0.40 tff(21,plain,(
% 0.13/0.40 ~((~member(X!7, universal_class)) | member(X!7, singleton(X!7)))),
% 0.13/0.40 inference(skolemize,[status(sab)],[20])).
% 0.13/0.40 tff(22,plain,
% 0.13/0.40 (~member(X!7, singleton(X!7))),
% 0.13/0.40 inference(or_elim,[status(thm)],[21])).
% 0.13/0.40 tff(23,plain,
% 0.13/0.40 (~member(X!7, unordered_pair(X!7, X!7))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[22, 13])).
% 0.13/0.40 tff(24,plain,
% 0.13/0.40 (^[U: $i, X: $i, Y: $i] : refl((~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y)))) <=> (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y)))))),
% 0.13/0.40 inference(bind,[status(th)],[])).
% 0.13/0.40 tff(25,plain,
% 0.13/0.40 (![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y)))) <=> ![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))),
% 0.13/0.40 inference(quant_intro,[status(thm)],[24])).
% 0.13/0.40 tff(26,plain,
% 0.13/0.40 (^[U: $i, X: $i, Y: $i] : trans(trans(monotonicity(rewrite((member(U, universal_class) & ((U = Y) | (U = X))) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X)))))), ((member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X)))) <=> (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X)))))))), rewrite((member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X)))))) <=> (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))), ((member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X)))) <=> (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y)))))), rewrite((~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y)))) <=> (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))), ((member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X)))) <=> (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))))),
% 0.13/0.40 inference(bind,[status(th)],[])).
% 0.13/0.40 tff(27,plain,
% 0.13/0.40 (![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X)))) <=> ![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))),
% 0.13/0.40 inference(quant_intro,[status(thm)],[26])).
% 0.13/0.40 tff(28,plain,
% 0.13/0.40 (![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X)))) <=> ![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X))))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(29,plain,
% 0.13/0.40 (^[U: $i, X: $i, Y: $i] : rewrite((member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = X) | (U = Y)))) <=> (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X)))))),
% 0.13/0.40 inference(bind,[status(th)],[])).
% 0.13/0.40 tff(30,plain,
% 0.13/0.40 (![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = X) | (U = Y)))) <=> ![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X))))),
% 0.13/0.40 inference(quant_intro,[status(thm)],[29])).
% 0.13/0.40 tff(31,axiom,(![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = X) | (U = Y))))), file('/export/starexec/sandbox2/benchmark/Axioms/SET005+0.ax','unordered_pair_defn')).
% 0.13/0.40 tff(32,plain,
% 0.13/0.40 (![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X))))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[31, 30])).
% 0.13/0.40 tff(33,plain,
% 0.13/0.40 (![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X))))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[32, 28])).
% 0.13/0.40 tff(34,plain,(
% 0.13/0.40 ![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X))))),
% 0.13/0.40 inference(skolemize,[status(sab)],[33])).
% 0.13/0.40 tff(35,plain,
% 0.13/0.40 (![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[34, 27])).
% 0.13/0.40 tff(36,plain,
% 0.13/0.40 (![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[35, 25])).
% 0.13/0.40 tff(37,plain,
% 0.13/0.40 (((~![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))) | (member(X!7, universal_class) <=> member(X!7, unordered_pair(X!7, X!7)))) <=> ((~![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))) | (member(X!7, universal_class) <=> member(X!7, unordered_pair(X!7, X!7))))),
% 0.13/0.41 inference(rewrite,[status(thm)],[])).
% 0.13/0.41 tff(38,plain,
% 0.13/0.41 ((~(~(member(X!7, universal_class) <=> member(X!7, unordered_pair(X!7, X!7))))) <=> (member(X!7, universal_class) <=> member(X!7, unordered_pair(X!7, X!7)))),
% 0.13/0.41 inference(rewrite,[status(thm)],[])).
% 0.13/0.41 tff(39,plain,
% 0.13/0.41 ((~(member(X!7, universal_class) <=> member(X!7, unordered_pair(X!7, X!7)))) <=> (~(member(X!7, universal_class) <=> member(X!7, unordered_pair(X!7, X!7))))),
% 0.13/0.41 inference(rewrite,[status(thm)],[])).
% 0.13/0.41 tff(40,plain,
% 0.13/0.41 (((~member(X!7, universal_class)) <=> member(X!7, unordered_pair(X!7, X!7))) <=> (~(member(X!7, universal_class) <=> member(X!7, unordered_pair(X!7, X!7))))),
% 0.13/0.41 inference(rewrite,[status(thm)],[])).
% 0.13/0.41 tff(41,plain,
% 0.13/0.41 (((~member(X!7, universal_class)) | $false) <=> (~member(X!7, universal_class))),
% 0.13/0.41 inference(rewrite,[status(thm)],[])).
% 0.13/0.41 tff(42,plain,
% 0.13/0.41 ((~$true) <=> $false),
% 0.13/0.41 inference(rewrite,[status(thm)],[])).
% 0.13/0.41 tff(43,plain,
% 0.13/0.41 (($true | $true) <=> $true),
% 0.13/0.41 inference(rewrite,[status(thm)],[])).
% 0.13/0.41 tff(44,plain,
% 0.13/0.41 ((X!7 = X!7) <=> $true),
% 0.13/0.41 inference(rewrite,[status(thm)],[])).
% 0.13/0.41 tff(45,plain,
% 0.13/0.41 (((X!7 = X!7) | (X!7 = X!7)) <=> ($true | $true)),
% 0.13/0.41 inference(monotonicity,[status(thm)],[44, 44])).
% 0.13/0.41 tff(46,plain,
% 0.13/0.41 (((X!7 = X!7) | (X!7 = X!7)) <=> $true),
% 0.13/0.41 inference(transitivity,[status(thm)],[45, 43])).
% 0.13/0.41 tff(47,plain,
% 0.13/0.41 ((~((X!7 = X!7) | (X!7 = X!7))) <=> (~$true)),
% 0.13/0.41 inference(monotonicity,[status(thm)],[46])).
% 0.13/0.41 tff(48,plain,
% 0.13/0.41 ((~((X!7 = X!7) | (X!7 = X!7))) <=> $false),
% 0.13/0.41 inference(transitivity,[status(thm)],[47, 42])).
% 0.13/0.41 tff(49,plain,
% 0.13/0.41 (((~member(X!7, universal_class)) | (~((X!7 = X!7) | (X!7 = X!7)))) <=> ((~member(X!7, universal_class)) | $false)),
% 0.13/0.41 inference(monotonicity,[status(thm)],[48])).
% 0.13/0.41 tff(50,plain,
% 0.13/0.41 (((~member(X!7, universal_class)) | (~((X!7 = X!7) | (X!7 = X!7)))) <=> (~member(X!7, universal_class))),
% 0.13/0.41 inference(transitivity,[status(thm)],[49, 41])).
% 0.13/0.41 tff(51,plain,
% 0.13/0.41 ((((~member(X!7, universal_class)) | (~((X!7 = X!7) | (X!7 = X!7)))) <=> member(X!7, unordered_pair(X!7, X!7))) <=> ((~member(X!7, universal_class)) <=> member(X!7, unordered_pair(X!7, X!7)))),
% 0.13/0.41 inference(monotonicity,[status(thm)],[50])).
% 0.13/0.41 tff(52,plain,
% 0.13/0.41 ((((~member(X!7, universal_class)) | (~((X!7 = X!7) | (X!7 = X!7)))) <=> member(X!7, unordered_pair(X!7, X!7))) <=> (~(member(X!7, universal_class) <=> member(X!7, unordered_pair(X!7, X!7))))),
% 0.13/0.41 inference(transitivity,[status(thm)],[51, 40])).
% 0.13/0.41 tff(53,plain,
% 0.13/0.41 ((((~member(X!7, universal_class)) | (~((X!7 = X!7) | (X!7 = X!7)))) <=> member(X!7, unordered_pair(X!7, X!7))) <=> (~(member(X!7, universal_class) <=> member(X!7, unordered_pair(X!7, X!7))))),
% 0.13/0.41 inference(transitivity,[status(thm)],[52, 39])).
% 0.13/0.41 tff(54,plain,
% 0.13/0.41 ((~(((~member(X!7, universal_class)) | (~((X!7 = X!7) | (X!7 = X!7)))) <=> member(X!7, unordered_pair(X!7, X!7)))) <=> (~(~(member(X!7, universal_class) <=> member(X!7, unordered_pair(X!7, X!7)))))),
% 0.13/0.41 inference(monotonicity,[status(thm)],[53])).
% 0.13/0.41 tff(55,plain,
% 0.13/0.41 ((~(((~member(X!7, universal_class)) | (~((X!7 = X!7) | (X!7 = X!7)))) <=> member(X!7, unordered_pair(X!7, X!7)))) <=> (member(X!7, universal_class) <=> member(X!7, unordered_pair(X!7, X!7)))),
% 0.13/0.41 inference(transitivity,[status(thm)],[54, 38])).
% 0.13/0.41 tff(56,plain,
% 0.13/0.41 (((~![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))) | (~(((~member(X!7, universal_class)) | (~((X!7 = X!7) | (X!7 = X!7)))) <=> member(X!7, unordered_pair(X!7, X!7))))) <=> ((~![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))) | (member(X!7, universal_class) <=> member(X!7, unordered_pair(X!7, X!7))))),
% 0.13/0.41 inference(monotonicity,[status(thm)],[55])).
% 0.13/0.41 tff(57,plain,
% 0.13/0.41 (((~![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))) | (~(((~member(X!7, universal_class)) | (~((X!7 = X!7) | (X!7 = X!7)))) <=> member(X!7, unordered_pair(X!7, X!7))))) <=> ((~![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))) | (member(X!7, universal_class) <=> member(X!7, unordered_pair(X!7, X!7))))),
% 0.13/0.41 inference(transitivity,[status(thm)],[56, 37])).
% 0.13/0.41 tff(58,plain,
% 0.13/0.41 ((~![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))) | (~(((~member(X!7, universal_class)) | (~((X!7 = X!7) | (X!7 = X!7)))) <=> member(X!7, unordered_pair(X!7, X!7))))),
% 0.13/0.41 inference(quant_inst,[status(thm)],[])).
% 0.13/0.41 tff(59,plain,
% 0.13/0.41 ((~![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))) | (member(X!7, universal_class) <=> member(X!7, unordered_pair(X!7, X!7)))),
% 0.13/0.41 inference(modus_ponens,[status(thm)],[58, 57])).
% 0.13/0.41 tff(60,plain,
% 0.13/0.41 (member(X!7, universal_class) <=> member(X!7, unordered_pair(X!7, X!7))),
% 0.13/0.41 inference(unit_resolution,[status(thm)],[59, 36])).
% 0.13/0.41 tff(61,plain,
% 0.13/0.41 (member(X!7, universal_class)),
% 0.13/0.41 inference(or_elim,[status(thm)],[21])).
% 0.13/0.41 tff(62,plain,
% 0.13/0.41 ((~(member(X!7, universal_class) <=> member(X!7, unordered_pair(X!7, X!7)))) | (~member(X!7, universal_class)) | member(X!7, unordered_pair(X!7, X!7))),
% 0.13/0.41 inference(tautology,[status(thm)],[])).
% 0.13/0.41 tff(63,plain,
% 0.13/0.41 ((~(member(X!7, universal_class) <=> member(X!7, unordered_pair(X!7, X!7)))) | member(X!7, unordered_pair(X!7, X!7))),
% 0.13/0.41 inference(unit_resolution,[status(thm)],[62, 61])).
% 0.13/0.41 tff(64,plain,
% 0.13/0.41 (member(X!7, unordered_pair(X!7, X!7))),
% 0.13/0.41 inference(unit_resolution,[status(thm)],[63, 60])).
% 0.13/0.41 tff(65,plain,
% 0.13/0.41 ($false),
% 0.13/0.41 inference(unit_resolution,[status(thm)],[64, 23])).
% 0.13/0.41 % SZS output end Proof
% 0.13/0.41 % E exiting
%------------------------------------------------------------------------------