%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET103-6 : TPTP v9.0.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n003.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 05:25:14 AM UTC 2025
% Result : Unsatisfiable 0.43s 0.58s
% Output : Proof 0.43s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.13 % Problem : SET103-6 : TPTP v9.0.0. Bugfixed v2.1.0.
% 0.03/0.13 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n003.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Jun 20 04:18:39 EDT 2025
% 0.13/0.34 % CPUTime :
% 0.43/0.58 % SZS status Unsatisfiable
% 0.43/0.58 % SZS output start Proof
% 0.43/0.58 tff(ordered_pair_type, type, (
% 0.43/0.58 ordered_pair: ( $i * $i ) > $i)).
% 0.43/0.58 tff(y_type, type, (
% 0.43/0.58 y: $i)).
% 0.43/0.58 tff(x_type, type, (
% 0.43/0.58 x: $i)).
% 0.43/0.58 tff(unordered_pair_type, type, (
% 0.43/0.58 unordered_pair: ( $i * $i ) > $i)).
% 0.43/0.58 tff(null_class_type, type, (
% 0.43/0.58 null_class: $i)).
% 0.43/0.58 tff(singleton_type, type, (
% 0.43/0.58 singleton: $i > $i)).
% 0.43/0.58 tff(member_type, type, (
% 0.43/0.58 member: ( $i * $i ) > $o)).
% 0.43/0.58 tff(apply_type, type, (
% 0.43/0.58 apply: ( $i * $i ) > $i)).
% 0.43/0.58 tff(choice_type, type, (
% 0.43/0.58 choice: $i)).
% 0.43/0.58 tff(universal_class_type, type, (
% 0.43/0.58 universal_class: $i)).
% 0.43/0.58 tff(image_type, type, (
% 0.43/0.58 image: ( $i * $i ) > $i)).
% 0.43/0.58 tff(sum_class_type, type, (
% 0.43/0.58 sum_class: $i > $i)).
% 0.43/0.58 tff(function_type, type, (
% 0.43/0.58 function: $i > $o)).
% 0.43/0.58 tff(1,plain,
% 0.43/0.58 (^[Y: $i, X: $i] : refl((unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y)) <=> (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y)))),
% 0.43/0.58 inference(bind,[status(th)],[])).
% 0.43/0.58 tff(2,plain,
% 0.43/0.58 (![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y)) <=> ![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))),
% 0.43/0.58 inference(quant_intro,[status(thm)],[1])).
% 0.43/0.58 tff(3,plain,
% 0.43/0.58 (![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y)) <=> ![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))),
% 0.43/0.58 inference(rewrite,[status(thm)],[])).
% 0.43/0.58 tff(4,axiom,(![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','ordered_pair')).
% 0.43/0.58 tff(5,plain,
% 0.43/0.58 (![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))),
% 0.43/0.58 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.43/0.58 tff(6,plain,(
% 0.43/0.58 ![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))),
% 0.43/0.58 inference(skolemize,[status(sab)],[5])).
% 0.43/0.58 tff(7,plain,
% 0.43/0.58 (![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))),
% 0.43/0.58 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.43/0.58 tff(8,plain,
% 0.43/0.58 ((~![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))) | (unordered_pair(singleton(x), unordered_pair(x, singleton(y))) = ordered_pair(x, y))),
% 0.43/0.58 inference(quant_inst,[status(thm)],[])).
% 0.43/0.58 tff(9,plain,
% 0.43/0.58 (unordered_pair(singleton(x), unordered_pair(x, singleton(y))) = ordered_pair(x, y)),
% 0.43/0.58 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.43/0.58 tff(10,plain,
% 0.43/0.58 (^[X: $i] : refl((unordered_pair(X, X) = singleton(X)) <=> (unordered_pair(X, X) = singleton(X)))),
% 0.43/0.58 inference(bind,[status(th)],[])).
% 0.43/0.58 tff(11,plain,
% 0.43/0.58 (![X: $i] : (unordered_pair(X, X) = singleton(X)) <=> ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.43/0.58 inference(quant_intro,[status(thm)],[10])).
% 0.43/0.58 tff(12,plain,
% 0.43/0.58 (![X: $i] : (unordered_pair(X, X) = singleton(X)) <=> ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.43/0.58 inference(rewrite,[status(thm)],[])).
% 0.43/0.58 tff(13,axiom,(![X: $i] : (unordered_pair(X, X) = singleton(X))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','singleton_set')).
% 0.43/0.58 tff(14,plain,
% 0.43/0.58 (![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.43/0.58 inference(modus_ponens,[status(thm)],[13, 12])).
% 0.43/0.58 tff(15,plain,(
% 0.43/0.58 ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.43/0.58 inference(skolemize,[status(sab)],[14])).
% 0.43/0.58 tff(16,plain,
% 0.43/0.58 (![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.43/0.58 inference(modus_ponens,[status(thm)],[15, 11])).
% 0.43/0.58 tff(17,plain,
% 0.43/0.58 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(singleton(y), singleton(y)) = singleton(singleton(y)))),
% 0.43/0.58 inference(quant_inst,[status(thm)],[])).
% 0.43/0.58 tff(18,plain,
% 0.43/0.58 (unordered_pair(singleton(y), singleton(y)) = singleton(singleton(y))),
% 0.43/0.58 inference(unit_resolution,[status(thm)],[17, 16])).
% 0.43/0.58 tff(19,plain,
% 0.43/0.58 (member(unordered_pair(singleton(y), singleton(y)), universal_class) <=> member(singleton(singleton(y)), universal_class)),
% 0.43/0.58 inference(monotonicity,[status(thm)],[18])).
% 0.43/0.58 tff(20,plain,
% 0.43/0.58 (member(singleton(singleton(y)), universal_class) <=> member(unordered_pair(singleton(y), singleton(y)), universal_class)),
% 0.43/0.58 inference(symmetry,[status(thm)],[19])).
% 0.43/0.58 tff(21,plain,
% 0.43/0.58 ((~member(singleton(singleton(y)), universal_class)) <=> (~member(unordered_pair(singleton(y), singleton(y)), universal_class))),
% 0.43/0.58 inference(monotonicity,[status(thm)],[20])).
% 0.43/0.58 tff(22,plain,
% 0.43/0.58 (^[Y: $i, Xf: $i] : refl((sum_class(image(Xf, singleton(Y))) = apply(Xf, Y)) <=> (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y)))),
% 0.43/0.58 inference(bind,[status(th)],[])).
% 0.43/0.58 tff(23,plain,
% 0.43/0.58 (![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y)) <=> ![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))),
% 0.43/0.58 inference(quant_intro,[status(thm)],[22])).
% 0.43/0.58 tff(24,plain,
% 0.43/0.58 (![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y)) <=> ![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))),
% 0.43/0.58 inference(rewrite,[status(thm)],[])).
% 0.43/0.58 tff(25,axiom,(![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','apply')).
% 0.43/0.58 tff(26,plain,
% 0.43/0.58 (![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))),
% 0.43/0.58 inference(modus_ponens,[status(thm)],[25, 24])).
% 0.43/0.58 tff(27,plain,(
% 0.43/0.58 ![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))),
% 0.43/0.58 inference(skolemize,[status(sab)],[26])).
% 0.43/0.58 tff(28,plain,
% 0.43/0.58 (![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))),
% 0.43/0.58 inference(modus_ponens,[status(thm)],[27, 23])).
% 0.43/0.58 tff(29,plain,
% 0.43/0.58 ((~![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))) | (sum_class(image(choice, singleton(singleton(y)))) = apply(choice, singleton(y)))),
% 0.43/0.58 inference(quant_inst,[status(thm)],[])).
% 0.43/0.58 tff(30,plain,
% 0.43/0.58 (sum_class(image(choice, singleton(singleton(y)))) = apply(choice, singleton(y))),
% 0.43/0.58 inference(unit_resolution,[status(thm)],[29, 28])).
% 0.43/0.58 tff(31,plain,
% 0.43/0.58 (apply(choice, singleton(y)) = sum_class(image(choice, singleton(singleton(y))))),
% 0.43/0.58 inference(symmetry,[status(thm)],[30])).
% 0.43/0.58 tff(32,plain,
% 0.43/0.58 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(y, y) = singleton(y))),
% 0.43/0.58 inference(quant_inst,[status(thm)],[])).
% 0.43/0.58 tff(33,plain,
% 0.43/0.58 (unordered_pair(y, y) = singleton(y)),
% 0.43/0.58 inference(unit_resolution,[status(thm)],[32, 16])).
% 0.43/0.58 tff(34,plain,
% 0.43/0.58 (member(apply(choice, singleton(y)), unordered_pair(y, y)) <=> member(apply(choice, singleton(y)), singleton(y))),
% 0.43/0.58 inference(monotonicity,[status(thm)],[33])).
% 0.43/0.58 tff(35,plain,
% 0.43/0.58 (member(apply(choice, singleton(y)), singleton(y)) <=> member(apply(choice, singleton(y)), unordered_pair(y, y))),
% 0.43/0.58 inference(symmetry,[status(thm)],[34])).
% 0.43/0.58 tff(36,assumption,(member(apply(choice, singleton(y)), singleton(y))), introduced(assumption)).
% 0.43/0.58 tff(37,plain,
% 0.43/0.58 (member(apply(choice, singleton(y)), unordered_pair(y, y))),
% 0.43/0.58 inference(modus_ponens,[status(thm)],[36, 35])).
% 0.43/0.58 tff(38,plain,
% 0.43/0.58 (^[Y: $i, U: $i, X: $i] : refl(((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y)))) <=> ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y)))))),
% 0.43/0.58 inference(bind,[status(th)],[])).
% 0.43/0.58 tff(39,plain,
% 0.43/0.58 (![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y)))) <=> ![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 0.43/0.58 inference(quant_intro,[status(thm)],[38])).
% 0.43/0.58 tff(40,plain,
% 0.43/0.58 (![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y)))) <=> ![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 0.43/0.58 inference(rewrite,[status(thm)],[])).
% 0.43/0.58 tff(41,plain,
% 0.43/0.58 (^[Y: $i, U: $i, X: $i] : rewrite((((~member(U, unordered_pair(X, Y))) | (U = X)) | (U = Y)) <=> ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y)))))),
% 0.43/0.58 inference(bind,[status(th)],[])).
% 0.43/0.58 tff(42,plain,
% 0.43/0.58 (![Y: $i, U: $i, X: $i] : (((~member(U, unordered_pair(X, Y))) | (U = X)) | (U = Y)) <=> ![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 0.43/0.59 inference(quant_intro,[status(thm)],[41])).
% 0.43/0.59 tff(43,axiom,(![Y: $i, U: $i, X: $i] : (((~member(U, unordered_pair(X, Y))) | (U = X)) | (U = Y))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','unordered_pair_member')).
% 0.43/0.59 tff(44,plain,
% 0.43/0.59 (![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[43, 42])).
% 0.43/0.59 tff(45,plain,
% 0.43/0.59 (![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[44, 40])).
% 0.43/0.59 tff(46,plain,(
% 0.43/0.59 ![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 0.43/0.59 inference(skolemize,[status(sab)],[45])).
% 0.43/0.59 tff(47,plain,
% 0.43/0.59 (![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[46, 39])).
% 0.43/0.59 tff(48,plain,
% 0.43/0.59 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, singleton(y)) = y) | (~member(apply(choice, singleton(y)), unordered_pair(y, y))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, singleton(y)) = y) | (~member(apply(choice, singleton(y)), unordered_pair(y, y))))),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(49,plain,
% 0.43/0.59 (((apply(choice, singleton(y)) = y) | (apply(choice, singleton(y)) = y) | (~member(apply(choice, singleton(y)), unordered_pair(y, y)))) <=> ((apply(choice, singleton(y)) = y) | (~member(apply(choice, singleton(y)), unordered_pair(y, y))))),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(50,plain,
% 0.43/0.59 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, singleton(y)) = y) | (apply(choice, singleton(y)) = y) | (~member(apply(choice, singleton(y)), unordered_pair(y, y))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, singleton(y)) = y) | (~member(apply(choice, singleton(y)), unordered_pair(y, y)))))),
% 0.43/0.59 inference(monotonicity,[status(thm)],[49])).
% 0.43/0.59 tff(51,plain,
% 0.43/0.59 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, singleton(y)) = y) | (apply(choice, singleton(y)) = y) | (~member(apply(choice, singleton(y)), unordered_pair(y, y))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, singleton(y)) = y) | (~member(apply(choice, singleton(y)), unordered_pair(y, y))))),
% 0.43/0.59 inference(transitivity,[status(thm)],[50, 48])).
% 0.43/0.59 tff(52,plain,
% 0.43/0.59 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, singleton(y)) = y) | (apply(choice, singleton(y)) = y) | (~member(apply(choice, singleton(y)), unordered_pair(y, y))))),
% 0.43/0.59 inference(quant_inst,[status(thm)],[])).
% 0.43/0.59 tff(53,plain,
% 0.43/0.59 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, singleton(y)) = y) | (~member(apply(choice, singleton(y)), unordered_pair(y, y)))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[52, 51])).
% 0.43/0.59 tff(54,plain,
% 0.43/0.59 ((apply(choice, singleton(y)) = y) | (~member(apply(choice, singleton(y)), unordered_pair(y, y)))),
% 0.43/0.59 inference(unit_resolution,[status(thm)],[53, 47])).
% 0.43/0.59 tff(55,plain,
% 0.43/0.59 (apply(choice, singleton(y)) = y),
% 0.43/0.59 inference(unit_resolution,[status(thm)],[54, 37])).
% 0.43/0.59 tff(56,plain,
% 0.43/0.59 (y = apply(choice, singleton(y))),
% 0.43/0.59 inference(symmetry,[status(thm)],[55])).
% 0.43/0.59 tff(57,plain,
% 0.43/0.59 (y = sum_class(image(choice, singleton(singleton(y))))),
% 0.43/0.59 inference(transitivity,[status(thm)],[56, 31])).
% 0.43/0.59 tff(58,plain,
% 0.43/0.59 (member(y, universal_class) <=> member(sum_class(image(choice, singleton(singleton(y)))), universal_class)),
% 0.43/0.59 inference(monotonicity,[status(thm)],[57])).
% 0.43/0.59 tff(59,plain,
% 0.43/0.59 ((~member(y, universal_class)) <=> (~member(sum_class(image(choice, singleton(singleton(y)))), universal_class))),
% 0.43/0.59 inference(monotonicity,[status(thm)],[58])).
% 0.43/0.59 tff(60,plain,
% 0.43/0.59 ((~member(y, universal_class)) <=> (~member(y, universal_class))),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(61,axiom,(~member(y, universal_class)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_property_1_of_ordered_pair_2')).
% 0.43/0.59 tff(62,plain,
% 0.43/0.59 (~member(y, universal_class)),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[61, 60])).
% 0.43/0.59 tff(63,plain,
% 0.43/0.59 (~member(sum_class(image(choice, singleton(singleton(y)))), universal_class)),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[62, 59])).
% 0.43/0.59 tff(64,plain,
% 0.43/0.59 (^[X: $i] : refl(((~member(X, universal_class)) | member(sum_class(X), universal_class)) <=> ((~member(X, universal_class)) | member(sum_class(X), universal_class)))),
% 0.43/0.59 inference(bind,[status(th)],[])).
% 0.43/0.59 tff(65,plain,
% 0.43/0.59 (![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class)) <=> ![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))),
% 0.43/0.59 inference(quant_intro,[status(thm)],[64])).
% 0.43/0.59 tff(66,plain,
% 0.43/0.59 (![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class)) <=> ![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(67,axiom,(![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','sum_class2')).
% 0.43/0.59 tff(68,plain,
% 0.43/0.59 (![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[67, 66])).
% 0.43/0.59 tff(69,plain,(
% 0.43/0.59 ![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))),
% 0.43/0.59 inference(skolemize,[status(sab)],[68])).
% 0.43/0.59 tff(70,plain,
% 0.43/0.59 (![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[69, 65])).
% 0.43/0.59 tff(71,plain,
% 0.43/0.59 (((~![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))) | ((~member(image(choice, singleton(singleton(y))), universal_class)) | member(sum_class(image(choice, singleton(singleton(y)))), universal_class))) <=> ((~![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))) | (~member(image(choice, singleton(singleton(y))), universal_class)) | member(sum_class(image(choice, singleton(singleton(y)))), universal_class))),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(72,plain,
% 0.43/0.59 ((~![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))) | ((~member(image(choice, singleton(singleton(y))), universal_class)) | member(sum_class(image(choice, singleton(singleton(y)))), universal_class))),
% 0.43/0.59 inference(quant_inst,[status(thm)],[])).
% 0.43/0.59 tff(73,plain,
% 0.43/0.59 ((~![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))) | (~member(image(choice, singleton(singleton(y))), universal_class)) | member(sum_class(image(choice, singleton(singleton(y)))), universal_class)),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[72, 71])).
% 0.43/0.59 tff(74,plain,
% 0.43/0.59 ((~member(image(choice, singleton(singleton(y))), universal_class)) | member(sum_class(image(choice, singleton(singleton(y)))), universal_class)),
% 0.43/0.59 inference(unit_resolution,[status(thm)],[73, 70])).
% 0.43/0.59 tff(75,plain,
% 0.43/0.59 (~member(image(choice, singleton(singleton(y))), universal_class)),
% 0.43/0.59 inference(unit_resolution,[status(thm)],[74, 63])).
% 0.43/0.59 tff(76,plain,
% 0.43/0.59 (function(choice) <=> function(choice)),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(77,axiom,(function(choice)), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','choice1')).
% 0.43/0.59 tff(78,plain,
% 0.43/0.59 (function(choice)),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[77, 76])).
% 0.43/0.59 tff(79,plain,
% 0.43/0.59 (^[Xf: $i, X: $i] : refl(((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf))) <=> ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf))))),
% 0.43/0.59 inference(bind,[status(th)],[])).
% 0.43/0.59 tff(80,plain,
% 0.43/0.59 (![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf))) <=> ![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))),
% 0.43/0.59 inference(quant_intro,[status(thm)],[79])).
% 0.43/0.59 tff(81,plain,
% 0.43/0.59 (![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf))) <=> ![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(82,plain,
% 0.43/0.59 (^[Xf: $i, X: $i] : rewrite((((~function(Xf)) | (~member(X, universal_class))) | member(image(Xf, X), universal_class)) <=> ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf))))),
% 0.43/0.59 inference(bind,[status(th)],[])).
% 0.43/0.59 tff(83,plain,
% 0.43/0.59 (![Xf: $i, X: $i] : (((~function(Xf)) | (~member(X, universal_class))) | member(image(Xf, X), universal_class)) <=> ![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))),
% 0.43/0.59 inference(quant_intro,[status(thm)],[82])).
% 0.43/0.59 tff(84,axiom,(![Xf: $i, X: $i] : (((~function(Xf)) | (~member(X, universal_class))) | member(image(Xf, X), universal_class))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','replacement')).
% 0.43/0.59 tff(85,plain,
% 0.43/0.59 (![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[84, 83])).
% 0.43/0.59 tff(86,plain,
% 0.43/0.59 (![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[85, 81])).
% 0.43/0.59 tff(87,plain,(
% 0.43/0.59 ![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))),
% 0.43/0.59 inference(skolemize,[status(sab)],[86])).
% 0.43/0.59 tff(88,plain,
% 0.43/0.59 (![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[87, 80])).
% 0.43/0.59 tff(89,plain,
% 0.43/0.59 (((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | ((~function(choice)) | member(image(choice, singleton(singleton(y))), universal_class) | (~member(singleton(singleton(y)), universal_class)))) <=> ((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | (~function(choice)) | member(image(choice, singleton(singleton(y))), universal_class) | (~member(singleton(singleton(y)), universal_class)))),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(90,plain,
% 0.43/0.59 (((~member(singleton(singleton(y)), universal_class)) | member(image(choice, singleton(singleton(y))), universal_class) | (~function(choice))) <=> ((~function(choice)) | member(image(choice, singleton(singleton(y))), universal_class) | (~member(singleton(singleton(y)), universal_class)))),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(91,plain,
% 0.43/0.59 (((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | ((~member(singleton(singleton(y)), universal_class)) | member(image(choice, singleton(singleton(y))), universal_class) | (~function(choice)))) <=> ((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | ((~function(choice)) | member(image(choice, singleton(singleton(y))), universal_class) | (~member(singleton(singleton(y)), universal_class))))),
% 0.43/0.59 inference(monotonicity,[status(thm)],[90])).
% 0.43/0.59 tff(92,plain,
% 0.43/0.59 (((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | ((~member(singleton(singleton(y)), universal_class)) | member(image(choice, singleton(singleton(y))), universal_class) | (~function(choice)))) <=> ((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | (~function(choice)) | member(image(choice, singleton(singleton(y))), universal_class) | (~member(singleton(singleton(y)), universal_class)))),
% 0.43/0.59 inference(transitivity,[status(thm)],[91, 89])).
% 0.43/0.59 tff(93,plain,
% 0.43/0.59 ((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | ((~member(singleton(singleton(y)), universal_class)) | member(image(choice, singleton(singleton(y))), universal_class) | (~function(choice)))),
% 0.43/0.59 inference(quant_inst,[status(thm)],[])).
% 0.43/0.59 tff(94,plain,
% 0.43/0.59 ((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | (~function(choice)) | member(image(choice, singleton(singleton(y))), universal_class) | (~member(singleton(singleton(y)), universal_class))),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[93, 92])).
% 0.43/0.59 tff(95,plain,
% 0.43/0.59 (member(image(choice, singleton(singleton(y))), universal_class) | (~member(singleton(singleton(y)), universal_class))),
% 0.43/0.59 inference(unit_resolution,[status(thm)],[94, 88, 78])).
% 0.43/0.59 tff(96,plain,
% 0.43/0.59 (~member(singleton(singleton(y)), universal_class)),
% 0.43/0.59 inference(unit_resolution,[status(thm)],[95, 75])).
% 0.43/0.59 tff(97,plain,
% 0.43/0.59 (~member(unordered_pair(singleton(y), singleton(y)), universal_class)),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[96, 21])).
% 0.43/0.59 tff(98,plain,
% 0.43/0.59 (^[Y: $i, X: $i] : refl(member(unordered_pair(X, Y), universal_class) <=> member(unordered_pair(X, Y), universal_class))),
% 0.43/0.59 inference(bind,[status(th)],[])).
% 0.43/0.59 tff(99,plain,
% 0.43/0.59 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class) <=> ![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.43/0.59 inference(quant_intro,[status(thm)],[98])).
% 0.43/0.59 tff(100,plain,
% 0.43/0.59 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class) <=> ![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(101,axiom,(![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','unordered_pairs_in_universal')).
% 0.43/0.59 tff(102,plain,
% 0.43/0.59 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[101, 100])).
% 0.43/0.59 tff(103,plain,(
% 0.43/0.59 ![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.43/0.59 inference(skolemize,[status(sab)],[102])).
% 0.43/0.59 tff(104,plain,
% 0.43/0.59 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[103, 99])).
% 0.43/0.59 tff(105,plain,
% 0.43/0.59 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(singleton(y), singleton(y)), universal_class)),
% 0.43/0.59 inference(quant_inst,[status(thm)],[])).
% 0.43/0.59 tff(106,plain,
% 0.43/0.59 (member(unordered_pair(singleton(y), singleton(y)), universal_class)),
% 0.43/0.59 inference(unit_resolution,[status(thm)],[105, 104])).
% 0.43/0.59 tff(107,plain,
% 0.43/0.59 ($false),
% 0.43/0.59 inference(unit_resolution,[status(thm)],[106, 97])).
% 0.43/0.59 tff(108,plain,(~member(apply(choice, singleton(y)), singleton(y))), inference(lemma,lemma(discharge,[]))).
% 0.43/0.59 tff(109,plain,
% 0.43/0.59 (member(unordered_pair(y, y), universal_class) <=> member(singleton(y), universal_class)),
% 0.43/0.59 inference(monotonicity,[status(thm)],[33])).
% 0.43/0.59 tff(110,plain,
% 0.43/0.59 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(y, y), universal_class)),
% 0.43/0.59 inference(quant_inst,[status(thm)],[])).
% 0.43/0.59 tff(111,plain,
% 0.43/0.59 (member(unordered_pair(y, y), universal_class)),
% 0.43/0.59 inference(unit_resolution,[status(thm)],[110, 104])).
% 0.43/0.59 tff(112,plain,
% 0.43/0.59 (member(singleton(y), universal_class)),
% 0.43/0.59 inference(modus_ponens,[status(thm)],[111, 109])).
% 0.43/0.59 tff(113,plain,
% 0.43/0.59 (^[Y: $i] : refl(((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y)) <=> ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y)))),
% 0.43/0.59 inference(bind,[status(th)],[])).
% 0.43/0.59 tff(114,plain,
% 0.43/0.59 (![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y)) <=> ![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))),
% 0.43/0.59 inference(quant_intro,[status(thm)],[113])).
% 0.43/0.59 tff(115,plain,
% 0.43/0.59 (![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y)) <=> ![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))),
% 0.43/0.59 inference(rewrite,[status(thm)],[])).
% 0.43/0.59 tff(116,plain,
% 0.43/0.59 (^[Y: $i] : rewrite((((~member(Y, universal_class)) | (Y = null_class)) | member(apply(choice, Y), Y)) <=> ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y)))),
% 0.43/0.60 inference(bind,[status(th)],[])).
% 0.43/0.60 tff(117,plain,
% 0.43/0.60 (![Y: $i] : (((~member(Y, universal_class)) | (Y = null_class)) | member(apply(choice, Y), Y)) <=> ![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))),
% 0.43/0.60 inference(quant_intro,[status(thm)],[116])).
% 0.43/0.60 tff(118,axiom,(![Y: $i] : (((~member(Y, universal_class)) | (Y = null_class)) | member(apply(choice, Y), Y))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','choice2')).
% 0.43/0.60 tff(119,plain,
% 0.43/0.60 (![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))),
% 0.43/0.60 inference(modus_ponens,[status(thm)],[118, 117])).
% 0.43/0.60 tff(120,plain,
% 0.43/0.60 (![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))),
% 0.43/0.60 inference(modus_ponens,[status(thm)],[119, 115])).
% 0.43/0.60 tff(121,plain,(
% 0.43/0.60 ![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))),
% 0.43/0.60 inference(skolemize,[status(sab)],[120])).
% 0.43/0.60 tff(122,plain,
% 0.43/0.60 (![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))),
% 0.43/0.60 inference(modus_ponens,[status(thm)],[121, 114])).
% 0.43/0.60 tff(123,plain,
% 0.43/0.60 (((~![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))) | ((~member(singleton(y), universal_class)) | (singleton(y) = null_class) | member(apply(choice, singleton(y)), singleton(y)))) <=> ((~![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))) | (~member(singleton(y), universal_class)) | (singleton(y) = null_class) | member(apply(choice, singleton(y)), singleton(y)))),
% 0.43/0.60 inference(rewrite,[status(thm)],[])).
% 0.43/0.60 tff(124,plain,
% 0.43/0.60 ((~![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))) | ((~member(singleton(y), universal_class)) | (singleton(y) = null_class) | member(apply(choice, singleton(y)), singleton(y)))),
% 0.43/0.60 inference(quant_inst,[status(thm)],[])).
% 0.43/0.60 tff(125,plain,
% 0.43/0.60 ((~![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))) | (~member(singleton(y), universal_class)) | (singleton(y) = null_class) | member(apply(choice, singleton(y)), singleton(y))),
% 0.43/0.60 inference(modus_ponens,[status(thm)],[124, 123])).
% 0.43/0.60 tff(126,plain,
% 0.43/0.60 ((singleton(y) = null_class) | member(apply(choice, singleton(y)), singleton(y))),
% 0.43/0.60 inference(unit_resolution,[status(thm)],[125, 122, 112])).
% 0.43/0.60 tff(127,plain,
% 0.43/0.60 (singleton(y) = null_class),
% 0.43/0.60 inference(unit_resolution,[status(thm)],[126, 108])).
% 0.43/0.60 tff(128,plain,
% 0.43/0.60 (null_class = singleton(y)),
% 0.43/0.60 inference(symmetry,[status(thm)],[127])).
% 0.43/0.60 tff(129,plain,
% 0.43/0.60 (unordered_pair(x, null_class) = unordered_pair(x, singleton(y))),
% 0.43/0.60 inference(monotonicity,[status(thm)],[128])).
% 0.43/0.60 tff(130,plain,
% 0.43/0.60 (unordered_pair(singleton(x), unordered_pair(x, null_class)) = unordered_pair(singleton(x), unordered_pair(x, singleton(y)))),
% 0.43/0.60 inference(monotonicity,[status(thm)],[129])).
% 0.43/0.60 tff(131,plain,
% 0.43/0.60 (unordered_pair(singleton(x), unordered_pair(x, null_class)) = ordered_pair(x, y)),
% 0.43/0.60 inference(transitivity,[status(thm)],[130, 9])).
% 0.43/0.60 tff(132,plain,
% 0.43/0.60 ((~(unordered_pair(singleton(x), unordered_pair(x, null_class)) = ordered_pair(x, y))) <=> (~(unordered_pair(singleton(x), unordered_pair(x, null_class)) = ordered_pair(x, y)))),
% 0.43/0.60 inference(rewrite,[status(thm)],[])).
% 0.43/0.60 tff(133,axiom,(~(unordered_pair(singleton(x), unordered_pair(x, null_class)) = ordered_pair(x, y))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_property_1_of_ordered_pair_1')).
% 0.43/0.60 tff(134,plain,
% 0.43/0.60 (~(unordered_pair(singleton(x), unordered_pair(x, null_class)) = ordered_pair(x, y))),
% 0.43/0.60 inference(modus_ponens,[status(thm)],[133, 132])).
% 0.43/0.60 tff(135,plain,
% 0.43/0.60 ($false),
% 0.43/0.60 inference(unit_resolution,[status(thm)],[134, 131])).
% 0.43/0.60 % SZS output end Proof
% 0.43/0.60 % E exiting
%------------------------------------------------------------------------------