%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET084+1 : TPTP v9.0.0. Bugfixed v5.4.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n007.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 05:25:09 AM UTC 2025
% Result : Theorem 0.18s 0.40s
% Output : Proof 0.18s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SET084+1 : TPTP v9.0.0. Bugfixed v5.4.0.
% 0.00/0.11 % Command : run_E %s %d THM
% 0.10/0.32 % Computer : n007.cluster.edu
% 0.10/0.32 % Model : x86_64 x86_64
% 0.10/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.32 % Memory : 8042.1875MB
% 0.10/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.10/0.32 % CPULimit : 300
% 0.10/0.32 % WCLimit : 300
% 0.10/0.32 % DateTime : Fri Jun 20 04:15:05 EDT 2025
% 0.10/0.32 % CPUTime :
% 0.18/0.40 % SZS status Theorem
% 0.18/0.40 % SZS output start Proof
% 0.18/0.40 tff(tptp_fun_Y_7_type, type, (
% 0.18/0.40 tptp_fun_Y_7: $i)).
% 0.18/0.40 tff(second_type, type, (
% 0.18/0.40 second: $i > $i)).
% 0.18/0.40 tff(ordered_pair_type, type, (
% 0.18/0.40 ordered_pair: ( $i * $i ) > $i)).
% 0.18/0.40 tff(first_type, type, (
% 0.18/0.40 first: $i > $i)).
% 0.18/0.40 tff(tptp_fun_X_8_type, type, (
% 0.18/0.40 tptp_fun_X_8: $i)).
% 0.18/0.40 tff(unordered_pair_type, type, (
% 0.18/0.40 unordered_pair: ( $i * $i ) > $i)).
% 0.18/0.40 tff(singleton_type, type, (
% 0.18/0.40 singleton: $i > $i)).
% 0.18/0.40 tff(member_type, type, (
% 0.18/0.40 member: ( $i * $i ) > $o)).
% 0.18/0.40 tff(universal_class_type, type, (
% 0.18/0.40 universal_class: $i)).
% 0.18/0.40 tff(cross_product_type, type, (
% 0.18/0.40 cross_product: ( $i * $i ) > $i)).
% 0.18/0.40 tff(1,plain,
% 0.18/0.40 ((Y!7 = X!8) <=> (X!8 = Y!7)),
% 0.18/0.40 inference(commutativity,[status(thm)],[])).
% 0.18/0.40 tff(2,assumption,(~((~(first(ordered_pair(Y!7, Y!7)) = Y!7)) | (~(second(ordered_pair(Y!7, Y!7)) = Y!7)))), introduced(assumption)).
% 0.18/0.40 tff(3,plain,
% 0.18/0.40 (((~(first(ordered_pair(Y!7, Y!7)) = Y!7)) | (~(second(ordered_pair(Y!7, Y!7)) = Y!7))) | (second(ordered_pair(Y!7, Y!7)) = Y!7)),
% 0.18/0.40 inference(tautology,[status(thm)],[])).
% 0.18/0.40 tff(4,plain,
% 0.18/0.40 (second(ordered_pair(Y!7, Y!7)) = Y!7),
% 0.18/0.40 inference(unit_resolution,[status(thm)],[3, 2])).
% 0.18/0.40 tff(5,plain,
% 0.18/0.40 (^[X: $i, Y: $i] : refl((ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y)))) <=> (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y)))))),
% 0.18/0.40 inference(bind,[status(th)],[])).
% 0.18/0.40 tff(6,plain,
% 0.18/0.40 (![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y)))) <=> ![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))),
% 0.18/0.40 inference(quant_intro,[status(thm)],[5])).
% 0.18/0.40 tff(7,plain,
% 0.18/0.40 (![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y)))) <=> ![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))),
% 0.18/0.40 inference(rewrite,[status(thm)],[])).
% 0.18/0.40 tff(8,axiom,(![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))), file('/export/starexec/sandbox/benchmark/Axioms/SET005+0.ax','ordered_pair_defn')).
% 0.18/0.40 tff(9,plain,
% 0.18/0.40 (![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))),
% 0.18/0.40 inference(modus_ponens,[status(thm)],[8, 7])).
% 0.18/0.40 tff(10,plain,(
% 0.18/0.40 ![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))),
% 0.18/0.40 inference(skolemize,[status(sab)],[9])).
% 0.18/0.40 tff(11,plain,
% 0.18/0.40 (![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))),
% 0.18/0.40 inference(modus_ponens,[status(thm)],[10, 6])).
% 0.18/0.40 tff(12,plain,
% 0.18/0.40 ((~![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))) | (ordered_pair(Y!7, Y!7) = unordered_pair(singleton(Y!7), unordered_pair(Y!7, singleton(Y!7))))),
% 0.18/0.40 inference(quant_inst,[status(thm)],[])).
% 0.18/0.40 tff(13,plain,
% 0.18/0.40 (ordered_pair(Y!7, Y!7) = unordered_pair(singleton(Y!7), unordered_pair(Y!7, singleton(Y!7)))),
% 0.18/0.40 inference(unit_resolution,[status(thm)],[12, 11])).
% 0.18/0.40 tff(14,plain,
% 0.18/0.40 (unordered_pair(singleton(Y!7), unordered_pair(Y!7, singleton(Y!7))) = ordered_pair(Y!7, Y!7)),
% 0.18/0.40 inference(symmetry,[status(thm)],[13])).
% 0.18/0.40 tff(15,plain,
% 0.18/0.40 ((~![X: $i, Y: $i] : ((~((singleton(X) = singleton(Y)) & member(Y, universal_class))) | (X = Y))) <=> (~![X: $i, Y: $i] : ((~((singleton(X) = singleton(Y)) & member(Y, universal_class))) | (X = Y)))),
% 0.18/0.40 inference(rewrite,[status(thm)],[])).
% 0.18/0.40 tff(16,plain,
% 0.18/0.40 ((~![X: $i, Y: $i] : (((singleton(X) = singleton(Y)) & member(Y, universal_class)) => (X = Y))) <=> (~![X: $i, Y: $i] : ((~((singleton(X) = singleton(Y)) & member(Y, universal_class))) | (X = Y)))),
% 0.18/0.40 inference(rewrite,[status(thm)],[])).
% 0.18/0.40 tff(17,axiom,(~![X: $i, Y: $i] : (((singleton(X) = singleton(Y)) & member(Y, universal_class)) => (X = Y))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','singleton_identified_by_element2')).
% 0.18/0.40 tff(18,plain,
% 0.18/0.40 (~![X: $i, Y: $i] : ((~((singleton(X) = singleton(Y)) & member(Y, universal_class))) | (X = Y))),
% 0.18/0.41 inference(modus_ponens,[status(thm)],[17, 16])).
% 0.18/0.41 tff(19,plain,
% 0.18/0.41 (~![X: $i, Y: $i] : ((~((singleton(X) = singleton(Y)) & member(Y, universal_class))) | (X = Y))),
% 0.18/0.41 inference(modus_ponens,[status(thm)],[18, 15])).
% 0.18/0.41 tff(20,plain,
% 0.18/0.41 (~![X: $i, Y: $i] : ((~((singleton(X) = singleton(Y)) & member(Y, universal_class))) | (X = Y))),
% 0.18/0.41 inference(modus_ponens,[status(thm)],[19, 15])).
% 0.18/0.41 tff(21,plain,
% 0.18/0.41 (~![X: $i, Y: $i] : ((~((singleton(X) = singleton(Y)) & member(Y, universal_class))) | (X = Y))),
% 0.18/0.41 inference(modus_ponens,[status(thm)],[20, 15])).
% 0.18/0.41 tff(22,plain,(
% 0.18/0.41 ~((~((singleton(X!8) = singleton(Y!7)) & member(Y!7, universal_class))) | (X!8 = Y!7))),
% 0.18/0.41 inference(skolemize,[status(sab)],[21])).
% 0.18/0.41 tff(23,plain,
% 0.18/0.41 ((singleton(X!8) = singleton(Y!7)) & member(Y!7, universal_class)),
% 0.18/0.41 inference(or_elim,[status(thm)],[22])).
% 0.18/0.41 tff(24,plain,
% 0.18/0.41 (singleton(X!8) = singleton(Y!7)),
% 0.18/0.41 inference(and_elim,[status(thm)],[23])).
% 0.18/0.41 tff(25,plain,
% 0.18/0.41 (unordered_pair(Y!7, singleton(X!8)) = unordered_pair(Y!7, singleton(Y!7))),
% 0.18/0.41 inference(monotonicity,[status(thm)],[24])).
% 0.18/0.41 tff(26,plain,
% 0.18/0.41 (unordered_pair(singleton(Y!7), unordered_pair(Y!7, singleton(X!8))) = unordered_pair(singleton(Y!7), unordered_pair(Y!7, singleton(Y!7)))),
% 0.18/0.41 inference(monotonicity,[status(thm)],[25])).
% 0.18/0.41 tff(27,plain,
% 0.18/0.41 ((~![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))) | (ordered_pair(Y!7, X!8) = unordered_pair(singleton(Y!7), unordered_pair(Y!7, singleton(X!8))))),
% 0.18/0.41 inference(quant_inst,[status(thm)],[])).
% 0.18/0.41 tff(28,plain,
% 0.18/0.41 (ordered_pair(Y!7, X!8) = unordered_pair(singleton(Y!7), unordered_pair(Y!7, singleton(X!8)))),
% 0.18/0.41 inference(unit_resolution,[status(thm)],[27, 11])).
% 0.18/0.41 tff(29,plain,
% 0.18/0.41 (ordered_pair(Y!7, X!8) = ordered_pair(Y!7, Y!7)),
% 0.18/0.41 inference(transitivity,[status(thm)],[28, 26, 14])).
% 0.18/0.41 tff(30,plain,
% 0.18/0.41 (second(ordered_pair(Y!7, X!8)) = second(ordered_pair(Y!7, Y!7))),
% 0.18/0.41 inference(monotonicity,[status(thm)],[29])).
% 0.18/0.41 tff(31,plain,
% 0.18/0.41 (second(ordered_pair(Y!7, X!8)) = Y!7),
% 0.18/0.41 inference(transitivity,[status(thm)],[30, 4])).
% 0.18/0.41 tff(32,plain,
% 0.18/0.41 ((second(ordered_pair(Y!7, X!8)) = X!8) <=> (Y!7 = X!8)),
% 0.18/0.41 inference(monotonicity,[status(thm)],[31])).
% 0.18/0.41 tff(33,plain,
% 0.18/0.41 ((second(ordered_pair(Y!7, X!8)) = X!8) <=> (X!8 = Y!7)),
% 0.18/0.41 inference(transitivity,[status(thm)],[32, 1])).
% 0.18/0.41 tff(34,plain,
% 0.18/0.41 ((X!8 = Y!7) <=> (second(ordered_pair(Y!7, X!8)) = X!8)),
% 0.18/0.41 inference(symmetry,[status(thm)],[33])).
% 0.18/0.41 tff(35,plain,
% 0.18/0.41 ((~(X!8 = Y!7)) <=> (~(second(ordered_pair(Y!7, X!8)) = X!8))),
% 0.18/0.41 inference(monotonicity,[status(thm)],[34])).
% 0.18/0.41 tff(36,plain,
% 0.18/0.41 (~(X!8 = Y!7)),
% 0.18/0.41 inference(or_elim,[status(thm)],[22])).
% 0.18/0.41 tff(37,plain,
% 0.18/0.41 (~(second(ordered_pair(Y!7, X!8)) = X!8)),
% 0.18/0.41 inference(modus_ponens,[status(thm)],[36, 35])).
% 0.18/0.41 tff(38,plain,
% 0.18/0.41 (((~(first(ordered_pair(Y!7, X!8)) = Y!7)) | (~(second(ordered_pair(Y!7, X!8)) = X!8))) | (second(ordered_pair(Y!7, X!8)) = X!8)),
% 0.18/0.41 inference(tautology,[status(thm)],[])).
% 0.18/0.41 tff(39,plain,
% 0.18/0.41 ((~(first(ordered_pair(Y!7, X!8)) = Y!7)) | (~(second(ordered_pair(Y!7, X!8)) = X!8))),
% 0.18/0.41 inference(unit_resolution,[status(thm)],[38, 37])).
% 0.18/0.41 tff(40,plain,
% 0.18/0.41 (^[U: $i, V: $i, X: $i, Y: $i] : refl((~(((~member(U, X)) | (~member(V, Y))) <=> member(ordered_pair(U, V), cross_product(X, Y)))) <=> (~(((~member(U, X)) | (~member(V, Y))) <=> member(ordered_pair(U, V), cross_product(X, Y)))))),
% 0.18/0.41 inference(bind,[status(th)],[])).
% 0.18/0.41 tff(41,plain,
% 0.18/0.41 (![U: $i, V: $i, X: $i, Y: $i] : (~(((~member(U, X)) | (~member(V, Y))) <=> member(ordered_pair(U, V), cross_product(X, Y)))) <=> ![U: $i, V: $i, X: $i, Y: $i] : (~(((~member(U, X)) | (~member(V, Y))) <=> member(ordered_pair(U, V), cross_product(X, Y))))),
% 0.18/0.41 inference(quant_intro,[status(thm)],[40])).
% 0.18/0.41 tff(42,plain,
% 0.18/0.41 (^[U: $i, V: $i, X: $i, Y: $i] : trans(trans(monotonicity(rewrite((member(U, X) & member(V, Y)) <=> (~((~member(U, X)) | (~member(V, Y))))), ((member(ordered_pair(U, V), cross_product(X, Y)) <=> (member(U, X) & member(V, Y))) <=> (member(ordered_pair(U, V), cross_product(X, Y)) <=> (~((~member(U, X)) | (~member(V, Y))))))), rewrite((member(ordered_pair(U, V), cross_product(X, Y)) <=> (~((~member(U, X)) | (~member(V, Y))))) <=> (~(((~member(U, X)) | (~member(V, Y))) <=> member(ordered_pair(U, V), cross_product(X, Y))))), ((member(ordered_pair(U, V), cross_product(X, Y)) <=> (member(U, X) & member(V, Y))) <=> (~(((~member(U, X)) | (~member(V, Y))) <=> member(ordered_pair(U, V), cross_product(X, Y)))))), rewrite((~(((~member(U, X)) | (~member(V, Y))) <=> member(ordered_pair(U, V), cross_product(X, Y)))) <=> (~(((~member(U, X)) | (~member(V, Y))) <=> member(ordered_pair(U, V), cross_product(X, Y))))), ((member(ordered_pair(U, V), cross_product(X, Y)) <=> (member(U, X) & member(V, Y))) <=> (~(((~member(U, X)) | (~member(V, Y))) <=> member(ordered_pair(U, V), cross_product(X, Y))))))),
% 0.18/0.41 inference(bind,[status(th)],[])).
% 0.18/0.41 tff(43,plain,
% 0.18/0.41 (![U: $i, V: $i, X: $i, Y: $i] : (member(ordered_pair(U, V), cross_product(X, Y)) <=> (member(U, X) & member(V, Y))) <=> ![U: $i, V: $i, X: $i, Y: $i] : (~(((~member(U, X)) | (~member(V, Y))) <=> member(ordered_pair(U, V), cross_product(X, Y))))),
% 0.18/0.41 inference(quant_intro,[status(thm)],[42])).
% 0.18/0.41 tff(44,plain,
% 0.18/0.41 (![U: $i, V: $i, X: $i, Y: $i] : (member(ordered_pair(U, V), cross_product(X, Y)) <=> (member(U, X) & member(V, Y))) <=> ![U: $i, V: $i, X: $i, Y: $i] : (member(ordered_pair(U, V), cross_product(X, Y)) <=> (member(U, X) & member(V, Y)))),
% 0.18/0.41 inference(rewrite,[status(thm)],[])).
% 0.18/0.41 tff(45,axiom,(![U: $i, V: $i, X: $i, Y: $i] : (member(ordered_pair(U, V), cross_product(X, Y)) <=> (member(U, X) & member(V, Y)))), file('/export/starexec/sandbox/benchmark/Axioms/SET005+0.ax','cross_product_defn')).
% 0.18/0.41 tff(46,plain,
% 0.18/0.41 (![U: $i, V: $i, X: $i, Y: $i] : (member(ordered_pair(U, V), cross_product(X, Y)) <=> (member(U, X) & member(V, Y)))),
% 0.18/0.41 inference(modus_ponens,[status(thm)],[45, 44])).
% 0.18/0.41 tff(47,plain,(
% 0.18/0.41 ![U: $i, V: $i, X: $i, Y: $i] : (member(ordered_pair(U, V), cross_product(X, Y)) <=> (member(U, X) & member(V, Y)))),
% 0.18/0.41 inference(skolemize,[status(sab)],[46])).
% 0.18/0.41 tff(48,plain,
% 0.18/0.41 (![U: $i, V: $i, X: $i, Y: $i] : (~(((~member(U, X)) | (~member(V, Y))) <=> member(ordered_pair(U, V), cross_product(X, Y))))),
% 0.18/0.41 inference(modus_ponens,[status(thm)],[47, 43])).
% 0.18/0.41 tff(49,plain,
% 0.18/0.41 (![U: $i, V: $i, X: $i, Y: $i] : (~(((~member(U, X)) | (~member(V, Y))) <=> member(ordered_pair(U, V), cross_product(X, Y))))),
% 0.18/0.41 inference(modus_ponens,[status(thm)],[48, 41])).
% 0.18/0.41 tff(50,plain,
% 0.18/0.41 ((~![U: $i, V: $i, X: $i, Y: $i] : (~(((~member(U, X)) | (~member(V, Y))) <=> member(ordered_pair(U, V), cross_product(X, Y))))) | (~(((~member(Y!7, universal_class)) | (~member(X!8, universal_class))) <=> member(ordered_pair(Y!7, X!8), cross_product(universal_class, universal_class))))),
% 0.18/0.41 inference(quant_inst,[status(thm)],[])).
% 0.18/0.41 tff(51,plain,
% 0.18/0.41 (~(((~member(Y!7, universal_class)) | (~member(X!8, universal_class))) <=> member(ordered_pair(Y!7, X!8), cross_product(universal_class, universal_class)))),
% 0.18/0.41 inference(unit_resolution,[status(thm)],[50, 49])).
% 0.18/0.41 tff(52,plain,
% 0.18/0.41 (member(ordered_pair(Y!7, X!8), cross_product(universal_class, universal_class)) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class))),
% 0.18/0.41 inference(monotonicity,[status(thm)],[29])).
% 0.18/0.41 tff(53,plain,
% 0.18/0.41 (member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class)) <=> member(ordered_pair(Y!7, X!8), cross_product(universal_class, universal_class))),
% 0.18/0.41 inference(symmetry,[status(thm)],[52])).
% 0.18/0.41 tff(54,plain,
% 0.18/0.41 (((~![U: $i, V: $i, X: $i, Y: $i] : (~(((~member(U, X)) | (~member(V, Y))) <=> member(ordered_pair(U, V), cross_product(X, Y))))) | (member(Y!7, universal_class) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class)))) <=> ((~![U: $i, V: $i, X: $i, Y: $i] : (~(((~member(U, X)) | (~member(V, Y))) <=> member(ordered_pair(U, V), cross_product(X, Y))))) | (member(Y!7, universal_class) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class))))),
% 0.18/0.41 inference(rewrite,[status(thm)],[])).
% 0.18/0.41 tff(55,plain,
% 0.18/0.41 ((~(~(member(Y!7, universal_class) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class))))) <=> (member(Y!7, universal_class) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class)))),
% 0.18/0.41 inference(rewrite,[status(thm)],[])).
% 0.18/0.41 tff(56,plain,
% 0.18/0.41 ((~(member(Y!7, universal_class) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class)))) <=> (~(member(Y!7, universal_class) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class))))),
% 0.18/0.41 inference(rewrite,[status(thm)],[])).
% 0.18/0.41 tff(57,plain,
% 0.18/0.41 (((~member(Y!7, universal_class)) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class))) <=> (~(member(Y!7, universal_class) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class))))),
% 0.18/0.41 inference(rewrite,[status(thm)],[])).
% 0.18/0.41 tff(58,plain,
% 0.18/0.41 (((~member(Y!7, universal_class)) | (~member(Y!7, universal_class))) <=> (~member(Y!7, universal_class))),
% 0.18/0.41 inference(rewrite,[status(thm)],[])).
% 0.18/0.41 tff(59,plain,
% 0.18/0.41 ((((~member(Y!7, universal_class)) | (~member(Y!7, universal_class))) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class))) <=> ((~member(Y!7, universal_class)) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class)))),
% 0.18/0.41 inference(monotonicity,[status(thm)],[58])).
% 0.18/0.41 tff(60,plain,
% 0.18/0.41 ((((~member(Y!7, universal_class)) | (~member(Y!7, universal_class))) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class))) <=> (~(member(Y!7, universal_class) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class))))),
% 0.18/0.41 inference(transitivity,[status(thm)],[59, 57])).
% 0.18/0.41 tff(61,plain,
% 0.18/0.41 ((((~member(Y!7, universal_class)) | (~member(Y!7, universal_class))) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class))) <=> (~(member(Y!7, universal_class) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class))))),
% 0.18/0.41 inference(transitivity,[status(thm)],[60, 56])).
% 0.18/0.41 tff(62,plain,
% 0.18/0.41 ((~(((~member(Y!7, universal_class)) | (~member(Y!7, universal_class))) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class)))) <=> (~(~(member(Y!7, universal_class) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class)))))),
% 0.18/0.41 inference(monotonicity,[status(thm)],[61])).
% 0.18/0.41 tff(63,plain,
% 0.18/0.41 ((~(((~member(Y!7, universal_class)) | (~member(Y!7, universal_class))) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class)))) <=> (member(Y!7, universal_class) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class)))),
% 0.18/0.41 inference(transitivity,[status(thm)],[62, 55])).
% 0.18/0.41 tff(64,plain,
% 0.18/0.41 (((~![U: $i, V: $i, X: $i, Y: $i] : (~(((~member(U, X)) | (~member(V, Y))) <=> member(ordered_pair(U, V), cross_product(X, Y))))) | (~(((~member(Y!7, universal_class)) | (~member(Y!7, universal_class))) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class))))) <=> ((~![U: $i, V: $i, X: $i, Y: $i] : (~(((~member(U, X)) | (~member(V, Y))) <=> member(ordered_pair(U, V), cross_product(X, Y))))) | (member(Y!7, universal_class) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class))))),
% 0.18/0.41 inference(monotonicity,[status(thm)],[63])).
% 0.18/0.41 tff(65,plain,
% 0.18/0.41 (((~![U: $i, V: $i, X: $i, Y: $i] : (~(((~member(U, X)) | (~member(V, Y))) <=> member(ordered_pair(U, V), cross_product(X, Y))))) | (~(((~member(Y!7, universal_class)) | (~member(Y!7, universal_class))) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class))))) <=> ((~![U: $i, V: $i, X: $i, Y: $i] : (~(((~member(U, X)) | (~member(V, Y))) <=> member(ordered_pair(U, V), cross_product(X, Y))))) | (member(Y!7, universal_class) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class))))),
% 0.18/0.41 inference(transitivity,[status(thm)],[64, 54])).
% 0.18/0.41 tff(66,plain,
% 0.18/0.41 ((~![U: $i, V: $i, X: $i, Y: $i] : (~(((~member(U, X)) | (~member(V, Y))) <=> member(ordered_pair(U, V), cross_product(X, Y))))) | (~(((~member(Y!7, universal_class)) | (~member(Y!7, universal_class))) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class))))),
% 0.18/0.41 inference(quant_inst,[status(thm)],[])).
% 0.18/0.41 tff(67,plain,
% 0.18/0.41 ((~![U: $i, V: $i, X: $i, Y: $i] : (~(((~member(U, X)) | (~member(V, Y))) <=> member(ordered_pair(U, V), cross_product(X, Y))))) | (member(Y!7, universal_class) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class)))),
% 0.18/0.41 inference(modus_ponens,[status(thm)],[66, 65])).
% 0.18/0.41 tff(68,plain,
% 0.18/0.41 (member(Y!7, universal_class) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class))),
% 0.18/0.41 inference(unit_resolution,[status(thm)],[67, 49])).
% 0.18/0.41 tff(69,plain,
% 0.18/0.41 (member(Y!7, universal_class)),
% 0.18/0.41 inference(and_elim,[status(thm)],[23])).
% 0.18/0.41 tff(70,plain,
% 0.18/0.41 ((~(member(Y!7, universal_class) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class)))) | (~member(Y!7, universal_class)) | member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class))),
% 0.18/0.41 inference(tautology,[status(thm)],[])).
% 0.18/0.41 tff(71,plain,
% 0.18/0.41 ((~(member(Y!7, universal_class) <=> member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class)))) | member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class))),
% 0.18/0.41 inference(unit_resolution,[status(thm)],[70, 69])).
% 0.18/0.41 tff(72,plain,
% 0.18/0.41 (member(ordered_pair(Y!7, Y!7), cross_product(universal_class, universal_class))),
% 0.18/0.41 inference(unit_resolution,[status(thm)],[71, 68])).
% 0.18/0.41 tff(73,plain,
% 0.18/0.41 (member(ordered_pair(Y!7, X!8), cross_product(universal_class, universal_class))),
% 0.18/0.41 inference(modus_ponens,[status(thm)],[72, 53])).
% 0.18/0.41 tff(74,plain,
% 0.18/0.41 ((((~member(Y!7, universal_class)) | (~member(X!8, universal_class))) <=> member(ordered_pair(Y!7, X!8), cross_product(universal_class, universal_class))) | (~((~member(Y!7, universal_class)) | (~member(X!8, universal_class)))) | (~member(ordered_pair(Y!7, X!8), cross_product(universal_class, universal_class)))),
% 0.18/0.41 inference(tautology,[status(thm)],[])).
% 0.18/0.41 tff(75,plain,
% 0.18/0.41 (~((~member(Y!7, universal_class)) | (~member(X!8, universal_class)))),
% 0.18/0.41 inference(unit_resolution,[status(thm)],[74, 73, 51])).
% 0.18/0.41 tff(76,plain,
% 0.18/0.41 (((~member(Y!7, universal_class)) | (~member(X!8, universal_class))) | member(X!8, universal_class)),
% 0.18/0.41 inference(tautology,[status(thm)],[])).
% 0.18/0.41 tff(77,plain,
% 0.18/0.41 (member(X!8, universal_class)),
% 0.18/0.41 inference(unit_resolution,[status(thm)],[76, 75])).
% 0.18/0.41 tff(78,plain,
% 0.18/0.41 (^[X: $i, Y: $i] : refl(((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y))))) <=> ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y))))))),
% 0.18/0.41 inference(bind,[status(th)],[])).
% 0.18/0.41 tff(79,plain,
% 0.18/0.41 (![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y))))) <=> ![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))),
% 0.18/0.41 inference(quant_intro,[status(thm)],[78])).
% 0.18/0.41 tff(80,plain,
% 0.18/0.41 (^[X: $i, Y: $i] : trans(monotonicity(trans(monotonicity(rewrite((member(X, universal_class) & member(Y, universal_class)) <=> (~((~member(Y, universal_class)) | (~member(X, universal_class))))), ((~(member(X, universal_class) & member(Y, universal_class))) <=> (~(~((~member(Y, universal_class)) | (~member(X, universal_class))))))), rewrite((~(~((~member(Y, universal_class)) | (~member(X, universal_class))))) <=> ((~member(Y, universal_class)) | (~member(X, universal_class)))), ((~(member(X, universal_class) & member(Y, universal_class))) <=> ((~member(Y, universal_class)) | (~member(X, universal_class))))), rewrite(((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y)) <=> (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y))))), (((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y))) <=> (((~member(Y, universal_class)) | (~member(X, universal_class))) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y))))))), rewrite((((~member(Y, universal_class)) | (~member(X, universal_class))) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y))))) <=> ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))), (((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y))) <=> ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))))),
% 0.18/0.42 inference(bind,[status(th)],[])).
% 0.18/0.42 tff(81,plain,
% 0.18/0.42 (![X: $i, Y: $i] : ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y))) <=> ![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))),
% 0.18/0.42 inference(quant_intro,[status(thm)],[80])).
% 0.18/0.42 tff(82,plain,
% 0.18/0.42 (![X: $i, Y: $i] : ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y))) <=> ![X: $i, Y: $i] : ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y)))),
% 0.18/0.42 inference(rewrite,[status(thm)],[])).
% 0.18/0.42 tff(83,plain,
% 0.18/0.42 (^[X: $i, Y: $i] : rewrite(((member(X, universal_class) & member(Y, universal_class)) => ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y))) <=> ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y))))),
% 0.18/0.42 inference(bind,[status(th)],[])).
% 0.18/0.42 tff(84,plain,
% 0.18/0.42 (![X: $i, Y: $i] : ((member(X, universal_class) & member(Y, universal_class)) => ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y))) <=> ![X: $i, Y: $i] : ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y)))),
% 0.18/0.42 inference(quant_intro,[status(thm)],[83])).
% 0.18/0.42 tff(85,axiom,(![X: $i, Y: $i] : ((member(X, universal_class) & member(Y, universal_class)) => ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y)))), file('/export/starexec/sandbox/benchmark/Axioms/SET005+0.ax','first_second')).
% 0.18/0.42 tff(86,plain,
% 0.18/0.42 (![X: $i, Y: $i] : ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y)))),
% 0.18/0.42 inference(modus_ponens,[status(thm)],[85, 84])).
% 0.18/0.42 tff(87,plain,
% 0.18/0.42 (![X: $i, Y: $i] : ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y)))),
% 0.18/0.42 inference(modus_ponens,[status(thm)],[86, 82])).
% 0.18/0.42 tff(88,plain,(
% 0.18/0.42 ![X: $i, Y: $i] : ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y)))),
% 0.18/0.42 inference(skolemize,[status(sab)],[87])).
% 0.18/0.42 tff(89,plain,
% 0.18/0.42 (![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))),
% 0.18/0.42 inference(modus_ponens,[status(thm)],[88, 81])).
% 0.18/0.42 tff(90,plain,
% 0.18/0.42 (![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))),
% 0.18/0.42 inference(modus_ponens,[status(thm)],[89, 79])).
% 0.18/0.42 tff(91,plain,
% 0.18/0.42 (((~![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))) | ((~member(X!8, universal_class)) | (~member(Y!7, universal_class)) | (~((~(first(ordered_pair(Y!7, X!8)) = Y!7)) | (~(second(ordered_pair(Y!7, X!8)) = X!8)))))) <=> ((~![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))) | (~member(X!8, universal_class)) | (~member(Y!7, universal_class)) | (~((~(first(ordered_pair(Y!7, X!8)) = Y!7)) | (~(second(ordered_pair(Y!7, X!8)) = X!8)))))),
% 0.18/0.42 inference(rewrite,[status(thm)],[])).
% 0.18/0.42 tff(92,plain,
% 0.18/0.42 ((~![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))) | ((~member(X!8, universal_class)) | (~member(Y!7, universal_class)) | (~((~(first(ordered_pair(Y!7, X!8)) = Y!7)) | (~(second(ordered_pair(Y!7, X!8)) = X!8)))))),
% 0.18/0.42 inference(quant_inst,[status(thm)],[])).
% 0.18/0.42 tff(93,plain,
% 0.18/0.42 ((~![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))) | (~member(X!8, universal_class)) | (~member(Y!7, universal_class)) | (~((~(first(ordered_pair(Y!7, X!8)) = Y!7)) | (~(second(ordered_pair(Y!7, X!8)) = X!8))))),
% 0.18/0.42 inference(modus_ponens,[status(thm)],[92, 91])).
% 0.18/0.42 tff(94,plain,
% 0.18/0.42 ((~member(X!8, universal_class)) | (~((~(first(ordered_pair(Y!7, X!8)) = Y!7)) | (~(second(ordered_pair(Y!7, X!8)) = X!8))))),
% 0.18/0.42 inference(unit_resolution,[status(thm)],[93, 90, 69])).
% 0.18/0.42 tff(95,plain,
% 0.18/0.42 ($false),
% 0.18/0.42 inference(unit_resolution,[status(thm)],[94, 77, 39])).
% 0.18/0.42 tff(96,plain,((~(first(ordered_pair(Y!7, Y!7)) = Y!7)) | (~(second(ordered_pair(Y!7, Y!7)) = Y!7))), inference(lemma,lemma(discharge,[]))).
% 0.18/0.42 tff(97,plain,
% 0.18/0.42 (((~![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))) | ((~member(Y!7, universal_class)) | (~((~(first(ordered_pair(Y!7, Y!7)) = Y!7)) | (~(second(ordered_pair(Y!7, Y!7)) = Y!7)))))) <=> ((~![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))) | (~member(Y!7, universal_class)) | (~((~(first(ordered_pair(Y!7, Y!7)) = Y!7)) | (~(second(ordered_pair(Y!7, Y!7)) = Y!7)))))),
% 0.18/0.42 inference(rewrite,[status(thm)],[])).
% 0.18/0.42 tff(98,plain,
% 0.18/0.42 (((~member(Y!7, universal_class)) | (~member(Y!7, universal_class)) | (~((~(first(ordered_pair(Y!7, Y!7)) = Y!7)) | (~(second(ordered_pair(Y!7, Y!7)) = Y!7))))) <=> ((~member(Y!7, universal_class)) | (~((~(first(ordered_pair(Y!7, Y!7)) = Y!7)) | (~(second(ordered_pair(Y!7, Y!7)) = Y!7)))))),
% 0.18/0.42 inference(rewrite,[status(thm)],[])).
% 0.18/0.42 tff(99,plain,
% 0.18/0.42 (((~![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))) | ((~member(Y!7, universal_class)) | (~member(Y!7, universal_class)) | (~((~(first(ordered_pair(Y!7, Y!7)) = Y!7)) | (~(second(ordered_pair(Y!7, Y!7)) = Y!7)))))) <=> ((~![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))) | ((~member(Y!7, universal_class)) | (~((~(first(ordered_pair(Y!7, Y!7)) = Y!7)) | (~(second(ordered_pair(Y!7, Y!7)) = Y!7))))))),
% 0.18/0.42 inference(monotonicity,[status(thm)],[98])).
% 0.18/0.42 tff(100,plain,
% 0.18/0.42 (((~![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))) | ((~member(Y!7, universal_class)) | (~member(Y!7, universal_class)) | (~((~(first(ordered_pair(Y!7, Y!7)) = Y!7)) | (~(second(ordered_pair(Y!7, Y!7)) = Y!7)))))) <=> ((~![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))) | (~member(Y!7, universal_class)) | (~((~(first(ordered_pair(Y!7, Y!7)) = Y!7)) | (~(second(ordered_pair(Y!7, Y!7)) = Y!7)))))),
% 0.18/0.42 inference(transitivity,[status(thm)],[99, 97])).
% 0.18/0.42 tff(101,plain,
% 0.18/0.42 ((~![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))) | ((~member(Y!7, universal_class)) | (~member(Y!7, universal_class)) | (~((~(first(ordered_pair(Y!7, Y!7)) = Y!7)) | (~(second(ordered_pair(Y!7, Y!7)) = Y!7)))))),
% 0.18/0.42 inference(quant_inst,[status(thm)],[])).
% 0.18/0.42 tff(102,plain,
% 0.18/0.42 ((~![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))) | (~member(Y!7, universal_class)) | (~((~(first(ordered_pair(Y!7, Y!7)) = Y!7)) | (~(second(ordered_pair(Y!7, Y!7)) = Y!7))))),
% 0.18/0.42 inference(modus_ponens,[status(thm)],[101, 100])).
% 0.18/0.42 tff(103,plain,
% 0.18/0.42 ($false),
% 0.18/0.42 inference(unit_resolution,[status(thm)],[102, 90, 69, 96])).
% 0.18/0.42 % SZS output end Proof
% 0.18/0.43 % E exiting
%------------------------------------------------------------------------------