%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET094+1 : TPTP v9.0.0. Bugfixed v7.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n005.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 05:25:12 AM UTC 2025
% Result : Theorem 0.12s 0.38s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : SET094+1 : TPTP v9.0.0. Bugfixed v7.3.0.
% 0.06/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n005.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Jun 20 04:15:35 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.12/0.38 % SZS status Theorem
% 0.12/0.38 % SZS output start Proof
% 0.12/0.38 tff(member_type, type, (
% 0.12/0.38 member: ( $i * $i ) > $o)).
% 0.12/0.38 tff(unordered_pair_type, type, (
% 0.12/0.38 unordered_pair: ( $i * $i ) > $i)).
% 0.12/0.38 tff(member_of_type, type, (
% 0.12/0.38 member_of: $i > $i)).
% 0.12/0.38 tff(tptp_fun_X_8_type, type, (
% 0.12/0.38 tptp_fun_X_8: $i)).
% 0.12/0.38 tff(tptp_fun_Y_7_type, type, (
% 0.12/0.38 tptp_fun_Y_7: $i)).
% 0.12/0.38 tff(universal_class_type, type, (
% 0.12/0.38 universal_class: $i)).
% 0.12/0.38 tff(singleton_type, type, (
% 0.12/0.38 singleton: $i > $i)).
% 0.12/0.38 tff(1,plain,
% 0.12/0.38 (^[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)))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(2,plain,
% 0.12/0.38 (![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))))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[1])).
% 0.12/0.38 tff(3,plain,
% 0.12/0.38 (^[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))))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(4,plain,
% 0.12/0.38 (![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))))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[3])).
% 0.12/0.38 tff(5,plain,
% 0.12/0.38 (![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))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(6,plain,
% 0.12/0.38 (^[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)))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(7,plain,
% 0.12/0.38 (![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))))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[6])).
% 0.12/0.38 tff(8,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')).
% 0.12/0.38 tff(9,plain,
% 0.12/0.38 (![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[8, 7])).
% 0.12/0.38 tff(10,plain,
% 0.12/0.38 (![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[9, 5])).
% 0.12/0.38 tff(11,plain,(
% 0.12/0.38 ![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X))))),
% 0.12/0.38 inference(skolemize,[status(sab)],[10])).
% 0.12/0.38 tff(12,plain,
% 0.12/0.38 (![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[11, 4])).
% 0.12/0.38 tff(13,plain,
% 0.12/0.38 (![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[12, 2])).
% 0.12/0.38 tff(14,plain,
% 0.12/0.38 (((~![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))) | (~(((~member(Y!7, universal_class)) | (~(Y!7 = member_of(X!8)))) <=> member(Y!7, unordered_pair(member_of(X!8), member_of(X!8)))))) <=> ((~![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))) | (~(((~member(Y!7, universal_class)) | (~(Y!7 = member_of(X!8)))) <=> member(Y!7, unordered_pair(member_of(X!8), member_of(X!8))))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(15,plain,
% 0.12/0.38 ((~(((~member(Y!7, universal_class)) | (~(Y!7 = member_of(X!8)))) <=> member(Y!7, unordered_pair(member_of(X!8), member_of(X!8))))) <=> (~(((~member(Y!7, universal_class)) | (~(Y!7 = member_of(X!8)))) <=> member(Y!7, unordered_pair(member_of(X!8), member_of(X!8)))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(16,plain,
% 0.12/0.38 ((((~member(Y!7, universal_class)) | (~((Y!7 = member_of(X!8)) | (Y!7 = member_of(X!8))))) <=> member(Y!7, unordered_pair(member_of(X!8), member_of(X!8)))) <=> (((~member(Y!7, universal_class)) | (~(Y!7 = member_of(X!8)))) <=> member(Y!7, unordered_pair(member_of(X!8), member_of(X!8))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(17,plain,
% 0.12/0.38 ((~(((~member(Y!7, universal_class)) | (~((Y!7 = member_of(X!8)) | (Y!7 = member_of(X!8))))) <=> member(Y!7, unordered_pair(member_of(X!8), member_of(X!8))))) <=> (~(((~member(Y!7, universal_class)) | (~(Y!7 = member_of(X!8)))) <=> member(Y!7, unordered_pair(member_of(X!8), member_of(X!8)))))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[16])).
% 0.12/0.38 tff(18,plain,
% 0.12/0.38 ((~(((~member(Y!7, universal_class)) | (~((Y!7 = member_of(X!8)) | (Y!7 = member_of(X!8))))) <=> member(Y!7, unordered_pair(member_of(X!8), member_of(X!8))))) <=> (~(((~member(Y!7, universal_class)) | (~(Y!7 = member_of(X!8)))) <=> member(Y!7, unordered_pair(member_of(X!8), member_of(X!8)))))),
% 0.12/0.38 inference(transitivity,[status(thm)],[17, 15])).
% 0.12/0.38 tff(19,plain,
% 0.12/0.38 (((~![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))) | (~(((~member(Y!7, universal_class)) | (~((Y!7 = member_of(X!8)) | (Y!7 = member_of(X!8))))) <=> member(Y!7, unordered_pair(member_of(X!8), member_of(X!8)))))) <=> ((~![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))) | (~(((~member(Y!7, universal_class)) | (~(Y!7 = member_of(X!8)))) <=> member(Y!7, unordered_pair(member_of(X!8), member_of(X!8))))))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[18])).
% 0.12/0.38 tff(20,plain,
% 0.12/0.38 (((~![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))) | (~(((~member(Y!7, universal_class)) | (~((Y!7 = member_of(X!8)) | (Y!7 = member_of(X!8))))) <=> member(Y!7, unordered_pair(member_of(X!8), member_of(X!8)))))) <=> ((~![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))) | (~(((~member(Y!7, universal_class)) | (~(Y!7 = member_of(X!8)))) <=> member(Y!7, unordered_pair(member_of(X!8), member_of(X!8))))))),
% 0.12/0.38 inference(transitivity,[status(thm)],[19, 14])).
% 0.12/0.38 tff(21,plain,
% 0.12/0.38 ((~![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))) | (~(((~member(Y!7, universal_class)) | (~((Y!7 = member_of(X!8)) | (Y!7 = member_of(X!8))))) <=> member(Y!7, unordered_pair(member_of(X!8), member_of(X!8)))))),
% 0.12/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(22,plain,
% 0.19/0.38 ((~![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))) | (~(((~member(Y!7, universal_class)) | (~(Y!7 = member_of(X!8)))) <=> member(Y!7, unordered_pair(member_of(X!8), member_of(X!8)))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[21, 20])).
% 0.19/0.38 tff(23,plain,
% 0.19/0.38 (~(((~member(Y!7, universal_class)) | (~(Y!7 = member_of(X!8)))) <=> member(Y!7, unordered_pair(member_of(X!8), member_of(X!8))))),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[22, 13])).
% 0.19/0.38 tff(24,plain,
% 0.19/0.38 ((~![X: $i, Y: $i] : ((~((singleton(member_of(X)) = X) & member(Y, X))) | (member_of(X) = Y))) <=> (~![X: $i, Y: $i] : ((~((singleton(member_of(X)) = X) & member(Y, X))) | (member_of(X) = Y)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(25,plain,
% 0.19/0.38 ((~![X: $i, Y: $i] : (((singleton(member_of(X)) = X) & member(Y, X)) => (member_of(X) = Y))) <=> (~![X: $i, Y: $i] : ((~((singleton(member_of(X)) = X) & member(Y, X))) | (member_of(X) = Y)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(26,axiom,(~![X: $i, Y: $i] : (((singleton(member_of(X)) = X) & member(Y, X)) => (member_of(X) = Y))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','property_of_singletons1_1')).
% 0.19/0.38 tff(27,plain,
% 0.19/0.38 (~![X: $i, Y: $i] : ((~((singleton(member_of(X)) = X) & member(Y, X))) | (member_of(X) = Y))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[26, 25])).
% 0.19/0.38 tff(28,plain,
% 0.19/0.38 (~![X: $i, Y: $i] : ((~((singleton(member_of(X)) = X) & member(Y, X))) | (member_of(X) = Y))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[27, 24])).
% 0.19/0.38 tff(29,plain,
% 0.19/0.38 (~![X: $i, Y: $i] : ((~((singleton(member_of(X)) = X) & member(Y, X))) | (member_of(X) = Y))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[28, 24])).
% 0.19/0.38 tff(30,plain,
% 0.19/0.38 (~![X: $i, Y: $i] : ((~((singleton(member_of(X)) = X) & member(Y, X))) | (member_of(X) = Y))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[29, 24])).
% 0.19/0.38 tff(31,plain,(
% 0.19/0.38 ~((~((singleton(member_of(X!8)) = X!8) & member(Y!7, X!8))) | (member_of(X!8) = Y!7))),
% 0.19/0.38 inference(skolemize,[status(sab)],[30])).
% 0.19/0.38 tff(32,plain,
% 0.19/0.38 ((singleton(member_of(X!8)) = X!8) & member(Y!7, X!8)),
% 0.19/0.38 inference(or_elim,[status(thm)],[31])).
% 0.19/0.38 tff(33,plain,
% 0.19/0.38 (singleton(member_of(X!8)) = X!8),
% 0.19/0.38 inference(and_elim,[status(thm)],[32])).
% 0.19/0.38 tff(34,plain,
% 0.19/0.38 (^[X: $i] : refl((singleton(X) = unordered_pair(X, X)) <=> (singleton(X) = unordered_pair(X, X)))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(35,plain,
% 0.19/0.38 (![X: $i] : (singleton(X) = unordered_pair(X, X)) <=> ![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[34])).
% 0.19/0.38 tff(36,plain,
% 0.19/0.38 (![X: $i] : (singleton(X) = unordered_pair(X, X)) <=> ![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(37,axiom,(![X: $i] : (singleton(X) = unordered_pair(X, X))), file('/export/starexec/sandbox2/benchmark/Axioms/SET005+0.ax','singleton_set_defn')).
% 0.19/0.38 tff(38,plain,
% 0.19/0.38 (![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[37, 36])).
% 0.19/0.38 tff(39,plain,(
% 0.19/0.38 ![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.19/0.38 inference(skolemize,[status(sab)],[38])).
% 0.19/0.38 tff(40,plain,
% 0.19/0.38 (![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[39, 35])).
% 0.19/0.38 tff(41,plain,
% 0.19/0.38 ((~![X: $i] : (singleton(X) = unordered_pair(X, X))) | (singleton(member_of(X!8)) = unordered_pair(member_of(X!8), member_of(X!8)))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(42,plain,
% 0.19/0.38 (singleton(member_of(X!8)) = unordered_pair(member_of(X!8), member_of(X!8))),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[41, 40])).
% 0.19/0.38 tff(43,plain,
% 0.19/0.38 (unordered_pair(member_of(X!8), member_of(X!8)) = singleton(member_of(X!8))),
% 0.19/0.38 inference(symmetry,[status(thm)],[42])).
% 0.19/0.38 tff(44,plain,
% 0.19/0.38 (unordered_pair(member_of(X!8), member_of(X!8)) = X!8),
% 0.19/0.38 inference(transitivity,[status(thm)],[43, 33])).
% 0.19/0.38 tff(45,plain,
% 0.19/0.38 (member(Y!7, unordered_pair(member_of(X!8), member_of(X!8))) <=> member(Y!7, X!8)),
% 0.19/0.38 inference(monotonicity,[status(thm)],[44])).
% 0.19/0.38 tff(46,plain,
% 0.19/0.39 (member(Y!7, X!8) <=> member(Y!7, unordered_pair(member_of(X!8), member_of(X!8)))),
% 0.19/0.39 inference(symmetry,[status(thm)],[45])).
% 0.19/0.39 tff(47,plain,
% 0.19/0.39 (member(Y!7, X!8)),
% 0.19/0.39 inference(and_elim,[status(thm)],[32])).
% 0.19/0.39 tff(48,plain,
% 0.19/0.39 (member(Y!7, unordered_pair(member_of(X!8), member_of(X!8)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[47, 46])).
% 0.19/0.39 tff(49,plain,
% 0.19/0.39 ((Y!7 = member_of(X!8)) <=> (member_of(X!8) = Y!7)),
% 0.19/0.39 inference(commutativity,[status(thm)],[])).
% 0.19/0.39 tff(50,plain,
% 0.19/0.39 ((member_of(X!8) = Y!7) <=> (Y!7 = member_of(X!8))),
% 0.19/0.39 inference(symmetry,[status(thm)],[49])).
% 0.19/0.39 tff(51,plain,
% 0.19/0.39 ((~(member_of(X!8) = Y!7)) <=> (~(Y!7 = member_of(X!8)))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[50])).
% 0.19/0.39 tff(52,plain,
% 0.19/0.39 (~(member_of(X!8) = Y!7)),
% 0.19/0.39 inference(or_elim,[status(thm)],[31])).
% 0.19/0.39 tff(53,plain,
% 0.19/0.39 (~(Y!7 = member_of(X!8))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[52, 51])).
% 0.19/0.39 tff(54,plain,
% 0.19/0.39 (((~member(Y!7, universal_class)) | (~(Y!7 = member_of(X!8)))) | (Y!7 = member_of(X!8))),
% 0.19/0.39 inference(tautology,[status(thm)],[])).
% 0.19/0.39 tff(55,plain,
% 0.19/0.39 ((~member(Y!7, universal_class)) | (~(Y!7 = member_of(X!8)))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[54, 53])).
% 0.19/0.39 tff(56,plain,
% 0.19/0.39 ((((~member(Y!7, universal_class)) | (~(Y!7 = member_of(X!8)))) <=> member(Y!7, unordered_pair(member_of(X!8), member_of(X!8)))) | (~((~member(Y!7, universal_class)) | (~(Y!7 = member_of(X!8))))) | (~member(Y!7, unordered_pair(member_of(X!8), member_of(X!8))))),
% 0.19/0.39 inference(tautology,[status(thm)],[])).
% 0.19/0.39 tff(57,plain,
% 0.19/0.39 ($false),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[56, 55, 48, 23])).
% 0.19/0.39 % SZS output end Proof
% 0.19/0.39 % E exiting
%------------------------------------------------------------------------------