%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET071-6 : TPTP v9.0.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n012.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 05:25:05 AM UTC 2025
% Result : Unsatisfiable 0.73s 0.91s
% Output : Proof 0.73s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.13 % Problem : SET071-6 : TPTP v9.0.0. Bugfixed v2.1.0.
% 0.12/0.13 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n012.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 04:11:40 EDT 2025
% 0.13/0.35 % CPUTime :
% 0.73/0.91 % SZS status Unsatisfiable
% 0.73/0.91 % SZS output start Proof
% 0.73/0.91 tff(member_type, type, (
% 0.73/0.91 member: ( $i * $i ) > $o)).
% 0.73/0.91 tff(universal_class_type, type, (
% 0.73/0.91 universal_class: $i)).
% 0.73/0.91 tff(y_type, type, (
% 0.73/0.91 y: $i)).
% 0.73/0.91 tff(sum_class_type, type, (
% 0.73/0.91 sum_class: $i > $i)).
% 0.73/0.91 tff(image_type, type, (
% 0.73/0.91 image: ( $i * $i ) > $i)).
% 0.73/0.91 tff(singleton_type, type, (
% 0.73/0.91 singleton: $i > $i)).
% 0.73/0.91 tff(unordered_pair_type, type, (
% 0.73/0.91 unordered_pair: ( $i * $i ) > $i)).
% 0.73/0.91 tff(x_type, type, (
% 0.73/0.91 x: $i)).
% 0.73/0.91 tff(choice_type, type, (
% 0.73/0.91 choice: $i)).
% 0.73/0.91 tff(apply_type, type, (
% 0.73/0.91 apply: ( $i * $i ) > $i)).
% 0.73/0.91 tff(function_type, type, (
% 0.73/0.91 function: $i > $o)).
% 0.73/0.91 tff(null_class_type, type, (
% 0.73/0.91 null_class: $i)).
% 0.73/0.91 tff(1,plain,
% 0.73/0.91 (^[Y: $i, Xf: $i] : refl((sum_class(image(Xf, singleton(Y))) = apply(Xf, Y)) <=> (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y)))),
% 0.73/0.91 inference(bind,[status(th)],[])).
% 0.73/0.91 tff(2,plain,
% 0.73/0.91 (![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))),
% 0.73/0.91 inference(quant_intro,[status(thm)],[1])).
% 0.73/0.91 tff(3,plain,
% 0.73/0.91 (![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))),
% 0.73/0.91 inference(rewrite,[status(thm)],[])).
% 0.73/0.91 tff(4,axiom,(![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','apply')).
% 0.73/0.91 tff(5,plain,
% 0.73/0.91 (![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))),
% 0.73/0.91 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.73/0.91 tff(6,plain,(
% 0.73/0.91 ![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))),
% 0.73/0.91 inference(skolemize,[status(sab)],[5])).
% 0.73/0.91 tff(7,plain,
% 0.73/0.91 (![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))),
% 0.73/0.91 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.73/0.91 tff(8,plain,
% 0.73/0.91 ((~![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))) | (sum_class(image(choice, singleton(unordered_pair(x, y)))) = apply(choice, unordered_pair(x, y)))),
% 0.73/0.91 inference(quant_inst,[status(thm)],[])).
% 0.73/0.91 tff(9,plain,
% 0.73/0.91 (sum_class(image(choice, singleton(unordered_pair(x, y)))) = apply(choice, unordered_pair(x, y))),
% 0.73/0.91 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.73/0.91 tff(10,plain,
% 0.73/0.91 (apply(choice, unordered_pair(x, y)) = sum_class(image(choice, singleton(unordered_pair(x, y))))),
% 0.73/0.91 inference(symmetry,[status(thm)],[9])).
% 0.73/0.91 tff(11,assumption,(apply(choice, unordered_pair(x, y)) = x), introduced(assumption)).
% 0.73/0.91 tff(12,plain,
% 0.73/0.91 (x = apply(choice, unordered_pair(x, y))),
% 0.73/0.91 inference(symmetry,[status(thm)],[11])).
% 0.73/0.91 tff(13,plain,
% 0.73/0.91 (x = sum_class(image(choice, singleton(unordered_pair(x, y))))),
% 0.73/0.91 inference(transitivity,[status(thm)],[12, 10])).
% 0.73/0.91 tff(14,plain,
% 0.73/0.91 (member(x, universal_class) <=> member(sum_class(image(choice, singleton(unordered_pair(x, y)))), universal_class)),
% 0.73/0.91 inference(monotonicity,[status(thm)],[13])).
% 0.73/0.91 tff(15,plain,
% 0.73/0.91 (member(sum_class(image(choice, singleton(unordered_pair(x, y)))), universal_class) <=> member(x, universal_class)),
% 0.73/0.91 inference(symmetry,[status(thm)],[14])).
% 0.73/0.91 tff(16,plain,
% 0.73/0.91 (^[X: $i] : refl((unordered_pair(X, X) = singleton(X)) <=> (unordered_pair(X, X) = singleton(X)))),
% 0.73/0.91 inference(bind,[status(th)],[])).
% 0.73/0.91 tff(17,plain,
% 0.73/0.91 (![X: $i] : (unordered_pair(X, X) = singleton(X)) <=> ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.73/0.91 inference(quant_intro,[status(thm)],[16])).
% 0.73/0.91 tff(18,plain,
% 0.73/0.91 (![X: $i] : (unordered_pair(X, X) = singleton(X)) <=> ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.73/0.91 inference(rewrite,[status(thm)],[])).
% 0.73/0.91 tff(19,axiom,(![X: $i] : (unordered_pair(X, X) = singleton(X))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','singleton_set')).
% 0.73/0.91 tff(20,plain,
% 0.73/0.91 (![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.73/0.91 inference(modus_ponens,[status(thm)],[19, 18])).
% 0.73/0.91 tff(21,plain,(
% 0.73/0.91 ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.73/0.91 inference(skolemize,[status(sab)],[20])).
% 0.73/0.91 tff(22,plain,
% 0.73/0.91 (![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.73/0.91 inference(modus_ponens,[status(thm)],[21, 17])).
% 0.73/0.91 tff(23,plain,
% 0.73/0.91 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(unordered_pair(x, y), unordered_pair(x, y)) = singleton(unordered_pair(x, y)))),
% 0.73/0.91 inference(quant_inst,[status(thm)],[])).
% 0.73/0.91 tff(24,plain,
% 0.73/0.91 (unordered_pair(unordered_pair(x, y), unordered_pair(x, y)) = singleton(unordered_pair(x, y))),
% 0.73/0.91 inference(unit_resolution,[status(thm)],[23, 22])).
% 0.73/0.91 tff(25,plain,
% 0.73/0.91 (member(unordered_pair(unordered_pair(x, y), unordered_pair(x, y)), universal_class) <=> member(singleton(unordered_pair(x, y)), universal_class)),
% 0.73/0.91 inference(monotonicity,[status(thm)],[24])).
% 0.73/0.91 tff(26,plain,
% 0.73/0.91 (^[Y: $i, X: $i] : refl(member(unordered_pair(X, Y), universal_class) <=> member(unordered_pair(X, Y), universal_class))),
% 0.73/0.91 inference(bind,[status(th)],[])).
% 0.73/0.91 tff(27,plain,
% 0.73/0.91 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class) <=> ![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.73/0.91 inference(quant_intro,[status(thm)],[26])).
% 0.73/0.91 tff(28,plain,
% 0.73/0.91 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class) <=> ![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.73/0.91 inference(rewrite,[status(thm)],[])).
% 0.73/0.91 tff(29,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')).
% 0.73/0.91 tff(30,plain,
% 0.73/0.91 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.73/0.91 inference(modus_ponens,[status(thm)],[29, 28])).
% 0.73/0.91 tff(31,plain,(
% 0.73/0.91 ![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.73/0.91 inference(skolemize,[status(sab)],[30])).
% 0.73/0.91 tff(32,plain,
% 0.73/0.91 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.73/0.91 inference(modus_ponens,[status(thm)],[31, 27])).
% 0.73/0.91 tff(33,plain,
% 0.73/0.91 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(unordered_pair(x, y), unordered_pair(x, y)), universal_class)),
% 0.73/0.91 inference(quant_inst,[status(thm)],[])).
% 0.73/0.91 tff(34,plain,
% 0.73/0.91 (member(unordered_pair(unordered_pair(x, y), unordered_pair(x, y)), universal_class)),
% 0.73/0.91 inference(unit_resolution,[status(thm)],[33, 32])).
% 0.73/0.91 tff(35,plain,
% 0.73/0.91 (member(singleton(unordered_pair(x, y)), universal_class)),
% 0.73/0.91 inference(modus_ponens,[status(thm)],[34, 25])).
% 0.73/0.91 tff(36,plain,
% 0.73/0.91 (function(choice) <=> function(choice)),
% 0.73/0.91 inference(rewrite,[status(thm)],[])).
% 0.73/0.91 tff(37,axiom,(function(choice)), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','choice1')).
% 0.73/0.91 tff(38,plain,
% 0.73/0.91 (function(choice)),
% 0.73/0.91 inference(modus_ponens,[status(thm)],[37, 36])).
% 0.73/0.91 tff(39,plain,
% 0.73/0.91 (^[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))))),
% 0.73/0.91 inference(bind,[status(th)],[])).
% 0.73/0.91 tff(40,plain,
% 0.73/0.91 (![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)))),
% 0.73/0.91 inference(quant_intro,[status(thm)],[39])).
% 0.73/0.91 tff(41,plain,
% 0.73/0.91 (![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)))),
% 0.73/0.91 inference(rewrite,[status(thm)],[])).
% 0.73/0.91 tff(42,plain,
% 0.73/0.91 (^[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))))),
% 0.73/0.91 inference(bind,[status(th)],[])).
% 0.73/0.91 tff(43,plain,
% 0.73/0.91 (![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)))),
% 0.73/0.91 inference(quant_intro,[status(thm)],[42])).
% 0.73/0.91 tff(44,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')).
% 0.73/0.91 tff(45,plain,
% 0.73/0.91 (![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))),
% 0.73/0.91 inference(modus_ponens,[status(thm)],[44, 43])).
% 0.73/0.91 tff(46,plain,
% 0.73/0.91 (![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))),
% 0.73/0.91 inference(modus_ponens,[status(thm)],[45, 41])).
% 0.73/0.91 tff(47,plain,(
% 0.73/0.91 ![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))),
% 0.73/0.91 inference(skolemize,[status(sab)],[46])).
% 0.73/0.91 tff(48,plain,
% 0.73/0.91 (![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))),
% 0.73/0.91 inference(modus_ponens,[status(thm)],[47, 40])).
% 0.73/0.91 tff(49,plain,
% 0.73/0.91 (((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | ((~function(choice)) | member(image(choice, singleton(unordered_pair(x, y))), universal_class) | (~member(singleton(unordered_pair(x, 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(unordered_pair(x, y))), universal_class) | (~member(singleton(unordered_pair(x, y)), universal_class)))),
% 0.73/0.91 inference(rewrite,[status(thm)],[])).
% 0.73/0.91 tff(50,plain,
% 0.73/0.91 (((~member(singleton(unordered_pair(x, y)), universal_class)) | member(image(choice, singleton(unordered_pair(x, y))), universal_class) | (~function(choice))) <=> ((~function(choice)) | member(image(choice, singleton(unordered_pair(x, y))), universal_class) | (~member(singleton(unordered_pair(x, y)), universal_class)))),
% 0.73/0.91 inference(rewrite,[status(thm)],[])).
% 0.73/0.91 tff(51,plain,
% 0.73/0.91 (((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | ((~member(singleton(unordered_pair(x, y)), universal_class)) | member(image(choice, singleton(unordered_pair(x, 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(unordered_pair(x, y))), universal_class) | (~member(singleton(unordered_pair(x, y)), universal_class))))),
% 0.73/0.91 inference(monotonicity,[status(thm)],[50])).
% 0.73/0.91 tff(52,plain,
% 0.73/0.91 (((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | ((~member(singleton(unordered_pair(x, y)), universal_class)) | member(image(choice, singleton(unordered_pair(x, 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(unordered_pair(x, y))), universal_class) | (~member(singleton(unordered_pair(x, y)), universal_class)))),
% 0.73/0.91 inference(transitivity,[status(thm)],[51, 49])).
% 0.73/0.91 tff(53,plain,
% 0.73/0.91 ((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | ((~member(singleton(unordered_pair(x, y)), universal_class)) | member(image(choice, singleton(unordered_pair(x, y))), universal_class) | (~function(choice)))),
% 0.73/0.91 inference(quant_inst,[status(thm)],[])).
% 0.73/0.91 tff(54,plain,
% 0.73/0.91 ((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | (~function(choice)) | member(image(choice, singleton(unordered_pair(x, y))), universal_class) | (~member(singleton(unordered_pair(x, y)), universal_class))),
% 0.73/0.91 inference(modus_ponens,[status(thm)],[53, 52])).
% 0.73/0.91 tff(55,plain,
% 0.73/0.91 (member(image(choice, singleton(unordered_pair(x, y))), universal_class) | (~member(singleton(unordered_pair(x, y)), universal_class))),
% 0.73/0.91 inference(unit_resolution,[status(thm)],[54, 48, 38])).
% 0.73/0.91 tff(56,plain,
% 0.73/0.91 (member(image(choice, singleton(unordered_pair(x, y))), universal_class)),
% 0.73/0.91 inference(unit_resolution,[status(thm)],[55, 35])).
% 0.73/0.91 tff(57,plain,
% 0.73/0.91 (^[X: $i] : refl(((~member(X, universal_class)) | member(sum_class(X), universal_class)) <=> ((~member(X, universal_class)) | member(sum_class(X), universal_class)))),
% 0.73/0.91 inference(bind,[status(th)],[])).
% 0.73/0.91 tff(58,plain,
% 0.73/0.91 (![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class)) <=> ![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))),
% 0.73/0.91 inference(quant_intro,[status(thm)],[57])).
% 0.73/0.91 tff(59,plain,
% 0.73/0.91 (![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class)) <=> ![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))),
% 0.73/0.91 inference(rewrite,[status(thm)],[])).
% 0.73/0.91 tff(60,axiom,(![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','sum_class2')).
% 0.73/0.91 tff(61,plain,
% 0.73/0.91 (![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))),
% 0.73/0.91 inference(modus_ponens,[status(thm)],[60, 59])).
% 0.73/0.91 tff(62,plain,(
% 0.73/0.91 ![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))),
% 0.73/0.91 inference(skolemize,[status(sab)],[61])).
% 0.73/0.91 tff(63,plain,
% 0.73/0.91 (![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))),
% 0.73/0.91 inference(modus_ponens,[status(thm)],[62, 58])).
% 0.73/0.91 tff(64,plain,
% 0.73/0.91 (((~![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))) | ((~member(image(choice, singleton(unordered_pair(x, y))), universal_class)) | member(sum_class(image(choice, singleton(unordered_pair(x, y)))), universal_class))) <=> ((~![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))) | (~member(image(choice, singleton(unordered_pair(x, y))), universal_class)) | member(sum_class(image(choice, singleton(unordered_pair(x, y)))), universal_class))),
% 0.73/0.91 inference(rewrite,[status(thm)],[])).
% 0.73/0.91 tff(65,plain,
% 0.73/0.91 ((~![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))) | ((~member(image(choice, singleton(unordered_pair(x, y))), universal_class)) | member(sum_class(image(choice, singleton(unordered_pair(x, y)))), universal_class))),
% 0.73/0.91 inference(quant_inst,[status(thm)],[])).
% 0.73/0.91 tff(66,plain,
% 0.73/0.91 ((~![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))) | (~member(image(choice, singleton(unordered_pair(x, y))), universal_class)) | member(sum_class(image(choice, singleton(unordered_pair(x, y)))), universal_class)),
% 0.73/0.91 inference(modus_ponens,[status(thm)],[65, 64])).
% 0.73/0.91 tff(67,plain,
% 0.73/0.91 ((~member(image(choice, singleton(unordered_pair(x, y))), universal_class)) | member(sum_class(image(choice, singleton(unordered_pair(x, y)))), universal_class)),
% 0.73/0.91 inference(unit_resolution,[status(thm)],[66, 63])).
% 0.73/0.91 tff(68,plain,
% 0.73/0.91 (member(sum_class(image(choice, singleton(unordered_pair(x, y)))), universal_class)),
% 0.73/0.91 inference(unit_resolution,[status(thm)],[67, 56])).
% 0.73/0.91 tff(69,plain,
% 0.73/0.91 (member(x, universal_class)),
% 0.73/0.91 inference(modus_ponens,[status(thm)],[68, 15])).
% 0.73/0.91 tff(70,plain,
% 0.73/0.91 ((~member(x, universal_class)) <=> (~member(x, universal_class))),
% 0.73/0.91 inference(rewrite,[status(thm)],[])).
% 0.73/0.91 tff(71,axiom,(~member(x, universal_class)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_null_unordered_pair_2')).
% 0.73/0.91 tff(72,plain,
% 0.73/0.91 (~member(x, universal_class)),
% 0.73/0.91 inference(modus_ponens,[status(thm)],[71, 70])).
% 0.73/0.91 tff(73,plain,
% 0.73/0.91 ($false),
% 0.73/0.91 inference(unit_resolution,[status(thm)],[72, 69])).
% 0.73/0.91 tff(74,plain,(~(apply(choice, unordered_pair(x, y)) = x)), inference(lemma,lemma(discharge,[]))).
% 0.73/0.91 tff(75,plain,
% 0.73/0.91 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(x, y), universal_class)),
% 0.73/0.91 inference(quant_inst,[status(thm)],[])).
% 0.73/0.91 tff(76,plain,
% 0.73/0.91 (member(unordered_pair(x, y), universal_class)),
% 0.73/0.91 inference(unit_resolution,[status(thm)],[75, 32])).
% 0.73/0.91 tff(77,plain,
% 0.73/0.91 ((~(unordered_pair(x, y) = null_class)) <=> (~(unordered_pair(x, y) = null_class))),
% 0.73/0.91 inference(rewrite,[status(thm)],[])).
% 0.73/0.91 tff(78,axiom,(~(unordered_pair(x, y) = null_class)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_null_unordered_pair_1')).
% 0.73/0.91 tff(79,plain,
% 0.73/0.91 (~(unordered_pair(x, y) = null_class)),
% 0.73/0.91 inference(modus_ponens,[status(thm)],[78, 77])).
% 0.73/0.91 tff(80,plain,
% 0.73/0.91 (^[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)))),
% 0.73/0.91 inference(bind,[status(th)],[])).
% 0.73/0.91 tff(81,plain,
% 0.73/0.91 (![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))),
% 0.73/0.91 inference(quant_intro,[status(thm)],[80])).
% 0.73/0.91 tff(82,plain,
% 0.73/0.91 (![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))),
% 0.73/0.91 inference(rewrite,[status(thm)],[])).
% 0.73/0.91 tff(83,plain,
% 0.73/0.91 (^[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)))),
% 0.73/0.91 inference(bind,[status(th)],[])).
% 0.73/0.91 tff(84,plain,
% 0.73/0.91 (![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))),
% 0.73/0.91 inference(quant_intro,[status(thm)],[83])).
% 0.73/0.91 tff(85,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')).
% 0.73/0.91 tff(86,plain,
% 0.73/0.91 (![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))),
% 0.73/0.91 inference(modus_ponens,[status(thm)],[85, 84])).
% 0.73/0.91 tff(87,plain,
% 0.73/0.91 (![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))),
% 0.73/0.91 inference(modus_ponens,[status(thm)],[86, 82])).
% 0.73/0.91 tff(88,plain,(
% 0.73/0.91 ![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))),
% 0.73/0.91 inference(skolemize,[status(sab)],[87])).
% 0.73/0.91 tff(89,plain,
% 0.73/0.91 (![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))),
% 0.73/0.91 inference(modus_ponens,[status(thm)],[88, 81])).
% 0.73/0.91 tff(90,plain,
% 0.73/0.91 (((~![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))) | ((~member(unordered_pair(x, y), universal_class)) | (unordered_pair(x, y) = null_class) | member(apply(choice, unordered_pair(x, y)), unordered_pair(x, y)))) <=> ((~![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))) | (~member(unordered_pair(x, y), universal_class)) | (unordered_pair(x, y) = null_class) | member(apply(choice, unordered_pair(x, y)), unordered_pair(x, y)))),
% 0.73/0.91 inference(rewrite,[status(thm)],[])).
% 0.73/0.91 tff(91,plain,
% 0.73/0.91 ((~![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))) | ((~member(unordered_pair(x, y), universal_class)) | (unordered_pair(x, y) = null_class) | member(apply(choice, unordered_pair(x, y)), unordered_pair(x, y)))),
% 0.73/0.91 inference(quant_inst,[status(thm)],[])).
% 0.73/0.91 tff(92,plain,
% 0.73/0.91 ((~![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))) | (~member(unordered_pair(x, y), universal_class)) | (unordered_pair(x, y) = null_class) | member(apply(choice, unordered_pair(x, y)), unordered_pair(x, y))),
% 0.73/0.91 inference(modus_ponens,[status(thm)],[91, 90])).
% 0.73/0.91 tff(93,plain,
% 0.73/0.91 (member(apply(choice, unordered_pair(x, y)), unordered_pair(x, y))),
% 0.73/0.91 inference(unit_resolution,[status(thm)],[92, 89, 79, 76])).
% 0.73/0.91 tff(94,plain,
% 0.73/0.91 (^[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)))))),
% 0.73/0.91 inference(bind,[status(th)],[])).
% 0.73/0.91 tff(95,plain,
% 0.73/0.91 (![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))))),
% 0.73/0.91 inference(quant_intro,[status(thm)],[94])).
% 0.73/0.92 tff(96,plain,
% 0.73/0.92 (![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))))),
% 0.73/0.92 inference(rewrite,[status(thm)],[])).
% 0.73/0.92 tff(97,plain,
% 0.73/0.92 (^[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)))))),
% 0.73/0.92 inference(bind,[status(th)],[])).
% 0.73/0.92 tff(98,plain,
% 0.73/0.92 (![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))))),
% 0.73/0.92 inference(quant_intro,[status(thm)],[97])).
% 0.73/0.92 tff(99,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')).
% 0.73/0.92 tff(100,plain,
% 0.73/0.92 (![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 0.73/0.92 inference(modus_ponens,[status(thm)],[99, 98])).
% 0.73/0.92 tff(101,plain,
% 0.73/0.92 (![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 0.73/0.92 inference(modus_ponens,[status(thm)],[100, 96])).
% 0.73/0.92 tff(102,plain,(
% 0.73/0.92 ![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 0.73/0.92 inference(skolemize,[status(sab)],[101])).
% 0.73/0.92 tff(103,plain,
% 0.73/0.92 (![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 0.73/0.92 inference(modus_ponens,[status(thm)],[102, 95])).
% 0.73/0.92 tff(104,plain,
% 0.73/0.92 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, unordered_pair(x, y)) = y) | (apply(choice, unordered_pair(x, y)) = x) | (~member(apply(choice, unordered_pair(x, y)), unordered_pair(x, y))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, unordered_pair(x, y)) = y) | (apply(choice, unordered_pair(x, y)) = x) | (~member(apply(choice, unordered_pair(x, y)), unordered_pair(x, y))))),
% 0.73/0.92 inference(rewrite,[status(thm)],[])).
% 0.73/0.92 tff(105,plain,
% 0.73/0.92 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, unordered_pair(x, y)) = y) | (apply(choice, unordered_pair(x, y)) = x) | (~member(apply(choice, unordered_pair(x, y)), unordered_pair(x, y))))),
% 0.73/0.92 inference(quant_inst,[status(thm)],[])).
% 0.73/0.92 tff(106,plain,
% 0.73/0.92 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, unordered_pair(x, y)) = y) | (apply(choice, unordered_pair(x, y)) = x) | (~member(apply(choice, unordered_pair(x, y)), unordered_pair(x, y)))),
% 0.73/0.92 inference(modus_ponens,[status(thm)],[105, 104])).
% 0.73/0.92 tff(107,plain,
% 0.73/0.92 ((apply(choice, unordered_pair(x, y)) = y) | (apply(choice, unordered_pair(x, y)) = x)),
% 0.73/0.92 inference(unit_resolution,[status(thm)],[106, 103, 93])).
% 0.73/0.92 tff(108,plain,
% 0.73/0.92 (apply(choice, unordered_pair(x, y)) = y),
% 0.73/0.92 inference(unit_resolution,[status(thm)],[107, 74])).
% 0.73/0.92 tff(109,plain,
% 0.73/0.92 (y = apply(choice, unordered_pair(x, y))),
% 0.73/0.92 inference(symmetry,[status(thm)],[108])).
% 0.73/0.92 tff(110,plain,
% 0.73/0.92 (y = sum_class(image(choice, singleton(unordered_pair(x, y))))),
% 0.73/0.92 inference(transitivity,[status(thm)],[109, 10])).
% 0.73/0.92 tff(111,plain,
% 0.73/0.92 (member(y, universal_class) <=> member(sum_class(image(choice, singleton(unordered_pair(x, y)))), universal_class)),
% 0.73/0.92 inference(monotonicity,[status(thm)],[110])).
% 0.73/0.92 tff(112,plain,
% 0.73/0.92 (member(sum_class(image(choice, singleton(unordered_pair(x, y)))), universal_class) <=> member(y, universal_class)),
% 0.73/0.92 inference(symmetry,[status(thm)],[111])).
% 0.73/0.92 tff(113,plain,
% 0.73/0.92 (member(y, universal_class)),
% 0.73/0.92 inference(modus_ponens,[status(thm)],[68, 112])).
% 0.73/0.92 tff(114,plain,
% 0.73/0.92 ((~member(y, universal_class)) <=> (~member(y, universal_class))),
% 0.73/0.92 inference(rewrite,[status(thm)],[])).
% 0.73/0.92 tff(115,axiom,(~member(y, universal_class)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_null_unordered_pair_3')).
% 0.73/0.92 tff(116,plain,
% 0.73/0.92 (~member(y, universal_class)),
% 0.73/0.92 inference(modus_ponens,[status(thm)],[115, 114])).
% 0.73/0.92 tff(117,plain,
% 0.73/0.92 ($false),
% 0.73/0.92 inference(unit_resolution,[status(thm)],[116, 113])).
% 0.73/0.92 % SZS output end Proof
% 0.73/0.92 % E exiting
%------------------------------------------------------------------------------