TSTP Solution File: SET103-6 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET103-6 : TPTP v8.1.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n005.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 20 05:05:41 EDT 2022
% Result : Unsatisfiable 0.19s 0.44s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SET103-6 : TPTP v8.1.0. Bugfixed v2.1.0.
% 0.03/0.12 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.34 % Computer : n005.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Sat Sep 3 02:25:48 EDT 2022
% 0.12/0.34 % CPUTime :
% 0.12/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.34 Usage: tptp [options] [-file:]file
% 0.12/0.34 -h, -? prints this message.
% 0.12/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.34 -m, -model generate model.
% 0.12/0.34 -p, -proof generate proof.
% 0.12/0.34 -c, -core generate unsat core of named formulas.
% 0.12/0.34 -st, -statistics display statistics.
% 0.12/0.34 -t:timeout set timeout (in second).
% 0.12/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.34 -<param>:<value> configuration parameter and value.
% 0.12/0.34 -o:<output-file> file to place output in.
% 0.19/0.44 % SZS status Unsatisfiable
% 0.19/0.44 % SZS output start Proof
% 0.19/0.44 tff(member_type, type, (
% 0.19/0.44 member: ( $i * $i ) > $o)).
% 0.19/0.44 tff(universal_class_type, type, (
% 0.19/0.44 universal_class: $i)).
% 0.19/0.44 tff(unordered_pair_type, type, (
% 0.19/0.44 unordered_pair: ( $i * $i ) > $i)).
% 0.19/0.44 tff(singleton_type, type, (
% 0.19/0.44 singleton: $i > $i)).
% 0.19/0.44 tff(y_type, type, (
% 0.19/0.44 y: $i)).
% 0.19/0.44 tff(image_type, type, (
% 0.19/0.44 image: ( $i * $i ) > $i)).
% 0.19/0.44 tff(choice_type, type, (
% 0.19/0.44 choice: $i)).
% 0.19/0.44 tff(sum_class_type, type, (
% 0.19/0.44 sum_class: $i > $i)).
% 0.19/0.44 tff(apply_type, type, (
% 0.19/0.44 apply: ( $i * $i ) > $i)).
% 0.19/0.44 tff(null_class_type, type, (
% 0.19/0.44 null_class: $i)).
% 0.19/0.44 tff(ordered_pair_type, type, (
% 0.19/0.44 ordered_pair: ( $i * $i ) > $i)).
% 0.19/0.44 tff(x_type, type, (
% 0.19/0.44 x: $i)).
% 0.19/0.44 tff(function_type, type, (
% 0.19/0.44 function: $i > $o)).
% 0.19/0.44 tff(1,plain,
% 0.19/0.44 (^[X: $i] : refl((unordered_pair(X, X) = singleton(X)) <=> (unordered_pair(X, X) = singleton(X)))),
% 0.19/0.44 inference(bind,[status(th)],[])).
% 0.19/0.44 tff(2,plain,
% 0.19/0.44 (![X: $i] : (unordered_pair(X, X) = singleton(X)) <=> ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.19/0.44 inference(quant_intro,[status(thm)],[1])).
% 0.19/0.44 tff(3,plain,
% 0.19/0.44 (![X: $i] : (unordered_pair(X, X) = singleton(X)) <=> ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(4,axiom,(![X: $i] : (unordered_pair(X, X) = singleton(X))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','singleton_set')).
% 0.19/0.44 tff(5,plain,
% 0.19/0.44 (![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.19/0.44 tff(6,plain,(
% 0.19/0.44 ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.19/0.44 inference(skolemize,[status(sab)],[5])).
% 0.19/0.44 tff(7,plain,
% 0.19/0.44 (![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.19/0.44 tff(8,plain,
% 0.19/0.44 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(singleton(y), singleton(y)) = singleton(singleton(y)))),
% 0.19/0.44 inference(quant_inst,[status(thm)],[])).
% 0.19/0.44 tff(9,plain,
% 0.19/0.44 (unordered_pair(singleton(y), singleton(y)) = singleton(singleton(y))),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.19/0.44 tff(10,plain,
% 0.19/0.44 (member(unordered_pair(singleton(y), singleton(y)), universal_class) <=> member(singleton(singleton(y)), universal_class)),
% 0.19/0.44 inference(monotonicity,[status(thm)],[9])).
% 0.19/0.44 tff(11,plain,
% 0.19/0.44 (member(singleton(singleton(y)), universal_class) <=> member(unordered_pair(singleton(y), singleton(y)), universal_class)),
% 0.19/0.44 inference(symmetry,[status(thm)],[10])).
% 0.19/0.44 tff(12,plain,
% 0.19/0.44 ((~member(singleton(singleton(y)), universal_class)) <=> (~member(unordered_pair(singleton(y), singleton(y)), universal_class))),
% 0.19/0.44 inference(monotonicity,[status(thm)],[11])).
% 0.19/0.44 tff(13,plain,
% 0.19/0.44 (^[Y: $i, Xf: $i] : refl((sum_class(image(Xf, singleton(Y))) = apply(Xf, Y)) <=> (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y)))),
% 0.19/0.44 inference(bind,[status(th)],[])).
% 0.19/0.44 tff(14,plain,
% 0.19/0.44 (![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))),
% 0.19/0.44 inference(quant_intro,[status(thm)],[13])).
% 0.19/0.44 tff(15,plain,
% 0.19/0.44 (![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))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(16,axiom,(![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','apply')).
% 0.19/0.44 tff(17,plain,
% 0.19/0.44 (![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[16, 15])).
% 0.19/0.44 tff(18,plain,(
% 0.19/0.44 ![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))),
% 0.19/0.44 inference(skolemize,[status(sab)],[17])).
% 0.19/0.44 tff(19,plain,
% 0.19/0.44 (![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[18, 14])).
% 0.19/0.44 tff(20,plain,
% 0.19/0.44 ((~![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))) | (sum_class(image(choice, singleton(singleton(y)))) = apply(choice, singleton(y)))),
% 0.19/0.44 inference(quant_inst,[status(thm)],[])).
% 0.19/0.44 tff(21,plain,
% 0.19/0.44 (sum_class(image(choice, singleton(singleton(y)))) = apply(choice, singleton(y))),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[20, 19])).
% 0.19/0.44 tff(22,plain,
% 0.19/0.44 (apply(choice, singleton(y)) = sum_class(image(choice, singleton(singleton(y))))),
% 0.19/0.44 inference(symmetry,[status(thm)],[21])).
% 0.19/0.44 tff(23,plain,
% 0.19/0.44 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(y, y) = singleton(y))),
% 0.19/0.44 inference(quant_inst,[status(thm)],[])).
% 0.19/0.44 tff(24,plain,
% 0.19/0.44 (unordered_pair(y, y) = singleton(y)),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[23, 7])).
% 0.19/0.44 tff(25,plain,
% 0.19/0.44 (member(apply(choice, singleton(y)), unordered_pair(y, y)) <=> member(apply(choice, singleton(y)), singleton(y))),
% 0.19/0.44 inference(monotonicity,[status(thm)],[24])).
% 0.19/0.44 tff(26,plain,
% 0.19/0.44 (member(apply(choice, singleton(y)), singleton(y)) <=> member(apply(choice, singleton(y)), unordered_pair(y, y))),
% 0.19/0.44 inference(symmetry,[status(thm)],[25])).
% 0.19/0.44 tff(27,plain,
% 0.19/0.44 (^[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)))),
% 0.19/0.44 inference(bind,[status(th)],[])).
% 0.19/0.44 tff(28,plain,
% 0.19/0.44 (![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))),
% 0.19/0.44 inference(quant_intro,[status(thm)],[27])).
% 0.19/0.44 tff(29,plain,
% 0.19/0.44 (![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))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(30,axiom,(![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','ordered_pair')).
% 0.19/0.44 tff(31,plain,
% 0.19/0.44 (![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[30, 29])).
% 0.19/0.44 tff(32,plain,(
% 0.19/0.44 ![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))),
% 0.19/0.44 inference(skolemize,[status(sab)],[31])).
% 0.19/0.44 tff(33,plain,
% 0.19/0.44 (![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[32, 28])).
% 0.19/0.44 tff(34,plain,
% 0.19/0.44 ((~![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))) | (unordered_pair(singleton(x), unordered_pair(x, singleton(y))) = ordered_pair(x, y))),
% 0.19/0.44 inference(quant_inst,[status(thm)],[])).
% 0.19/0.44 tff(35,plain,
% 0.19/0.44 (unordered_pair(singleton(x), unordered_pair(x, singleton(y))) = ordered_pair(x, y)),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[34, 33])).
% 0.19/0.44 tff(36,assumption,(singleton(y) = null_class), introduced(assumption)).
% 0.19/0.44 tff(37,plain,
% 0.19/0.44 (null_class = singleton(y)),
% 0.19/0.44 inference(symmetry,[status(thm)],[36])).
% 0.19/0.44 tff(38,plain,
% 0.19/0.44 (unordered_pair(x, null_class) = unordered_pair(x, singleton(y))),
% 0.19/0.44 inference(monotonicity,[status(thm)],[37])).
% 0.19/0.44 tff(39,plain,
% 0.19/0.44 (unordered_pair(singleton(x), unordered_pair(x, null_class)) = unordered_pair(singleton(x), unordered_pair(x, singleton(y)))),
% 0.19/0.44 inference(monotonicity,[status(thm)],[38])).
% 0.19/0.44 tff(40,plain,
% 0.19/0.44 (unordered_pair(singleton(x), unordered_pair(x, null_class)) = ordered_pair(x, y)),
% 0.19/0.44 inference(transitivity,[status(thm)],[39, 35])).
% 0.19/0.44 tff(41,plain,
% 0.19/0.44 ((~(unordered_pair(singleton(x), unordered_pair(x, null_class)) = ordered_pair(x, y))) <=> (~(unordered_pair(singleton(x), unordered_pair(x, null_class)) = ordered_pair(x, y)))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(42,axiom,(~(unordered_pair(singleton(x), unordered_pair(x, null_class)) = ordered_pair(x, y))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_property_1_of_ordered_pair_1')).
% 0.19/0.44 tff(43,plain,
% 0.19/0.44 (~(unordered_pair(singleton(x), unordered_pair(x, null_class)) = ordered_pair(x, y))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[42, 41])).
% 0.19/0.44 tff(44,plain,
% 0.19/0.44 ($false),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[43, 40])).
% 0.19/0.44 tff(45,plain,(~(singleton(y) = null_class)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.44 tff(46,plain,
% 0.19/0.44 (member(unordered_pair(y, y), universal_class) <=> member(singleton(y), universal_class)),
% 0.19/0.44 inference(monotonicity,[status(thm)],[24])).
% 0.19/0.44 tff(47,plain,
% 0.19/0.44 (^[Y: $i, X: $i] : refl(member(unordered_pair(X, Y), universal_class) <=> member(unordered_pair(X, Y), universal_class))),
% 0.19/0.44 inference(bind,[status(th)],[])).
% 0.19/0.44 tff(48,plain,
% 0.19/0.44 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class) <=> ![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.19/0.44 inference(quant_intro,[status(thm)],[47])).
% 0.19/0.44 tff(49,plain,
% 0.19/0.44 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class) <=> ![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(50,axiom,(![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','unordered_pairs_in_universal')).
% 0.19/0.44 tff(51,plain,
% 0.19/0.44 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[50, 49])).
% 0.19/0.44 tff(52,plain,(
% 0.19/0.44 ![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.19/0.44 inference(skolemize,[status(sab)],[51])).
% 0.19/0.44 tff(53,plain,
% 0.19/0.44 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[52, 48])).
% 0.19/0.44 tff(54,plain,
% 0.19/0.44 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(y, y), universal_class)),
% 0.19/0.44 inference(quant_inst,[status(thm)],[])).
% 0.19/0.44 tff(55,plain,
% 0.19/0.44 (member(unordered_pair(y, y), universal_class)),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[54, 53])).
% 0.19/0.44 tff(56,plain,
% 0.19/0.44 (member(singleton(y), universal_class)),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[55, 46])).
% 0.19/0.44 tff(57,plain,
% 0.19/0.44 (^[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)))),
% 0.19/0.44 inference(bind,[status(th)],[])).
% 0.19/0.44 tff(58,plain,
% 0.19/0.44 (![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))),
% 0.19/0.44 inference(quant_intro,[status(thm)],[57])).
% 0.19/0.44 tff(59,plain,
% 0.19/0.44 (![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))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(60,plain,
% 0.19/0.44 (^[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)))),
% 0.19/0.44 inference(bind,[status(th)],[])).
% 0.19/0.44 tff(61,plain,
% 0.19/0.44 (![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))),
% 0.19/0.44 inference(quant_intro,[status(thm)],[60])).
% 0.19/0.44 tff(62,axiom,(![Y: $i] : (((~member(Y, universal_class)) | (Y = null_class)) | member(apply(choice, Y), Y))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','choice2')).
% 0.19/0.44 tff(63,plain,
% 0.19/0.44 (![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[62, 61])).
% 0.19/0.44 tff(64,plain,
% 0.19/0.44 (![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[63, 59])).
% 0.19/0.44 tff(65,plain,(
% 0.19/0.44 ![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))),
% 0.19/0.44 inference(skolemize,[status(sab)],[64])).
% 0.19/0.44 tff(66,plain,
% 0.19/0.44 (![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[65, 58])).
% 0.19/0.44 tff(67,plain,
% 0.19/0.44 (((~![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)))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(68,plain,
% 0.19/0.45 ((~![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)))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(69,plain,
% 0.19/0.45 ((~![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))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[68, 67])).
% 0.19/0.45 tff(70,plain,
% 0.19/0.45 ((singleton(y) = null_class) | member(apply(choice, singleton(y)), singleton(y))),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[69, 66, 56])).
% 0.19/0.45 tff(71,plain,
% 0.19/0.45 (member(apply(choice, singleton(y)), singleton(y))),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[70, 45])).
% 0.19/0.45 tff(72,plain,
% 0.19/0.45 (member(apply(choice, singleton(y)), unordered_pair(y, y))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[71, 26])).
% 0.19/0.45 tff(73,plain,
% 0.19/0.45 (^[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)))))),
% 0.19/0.45 inference(bind,[status(th)],[])).
% 0.19/0.45 tff(74,plain,
% 0.19/0.45 (![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))))),
% 0.19/0.45 inference(quant_intro,[status(thm)],[73])).
% 0.19/0.45 tff(75,plain,
% 0.19/0.45 (![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))))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(76,plain,
% 0.19/0.45 (^[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)))))),
% 0.19/0.45 inference(bind,[status(th)],[])).
% 0.19/0.45 tff(77,plain,
% 0.19/0.45 (![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))))),
% 0.19/0.45 inference(quant_intro,[status(thm)],[76])).
% 0.19/0.45 tff(78,axiom,(![Y: $i, U: $i, X: $i] : (((~member(U, unordered_pair(X, Y))) | (U = X)) | (U = Y))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','unordered_pair_member')).
% 0.19/0.45 tff(79,plain,
% 0.19/0.45 (![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[78, 77])).
% 0.19/0.45 tff(80,plain,
% 0.19/0.45 (![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[79, 75])).
% 0.19/0.45 tff(81,plain,(
% 0.19/0.45 ![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 0.19/0.45 inference(skolemize,[status(sab)],[80])).
% 0.19/0.45 tff(82,plain,
% 0.19/0.45 (![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[81, 74])).
% 0.19/0.45 tff(83,plain,
% 0.19/0.45 (((~![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))))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(84,plain,
% 0.19/0.45 (((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))))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(85,plain,
% 0.19/0.45 (((~![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)))))),
% 0.19/0.45 inference(monotonicity,[status(thm)],[84])).
% 0.19/0.45 tff(86,plain,
% 0.19/0.45 (((~![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))))),
% 0.19/0.45 inference(transitivity,[status(thm)],[85, 83])).
% 0.19/0.45 tff(87,plain,
% 0.19/0.45 ((~![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))))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(88,plain,
% 0.19/0.45 ((~![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)))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[87, 86])).
% 0.19/0.45 tff(89,plain,
% 0.19/0.45 ((apply(choice, singleton(y)) = y) | (~member(apply(choice, singleton(y)), unordered_pair(y, y)))),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[88, 82])).
% 0.19/0.45 tff(90,plain,
% 0.19/0.45 (apply(choice, singleton(y)) = y),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[89, 72])).
% 0.19/0.45 tff(91,plain,
% 0.19/0.45 (y = apply(choice, singleton(y))),
% 0.19/0.45 inference(symmetry,[status(thm)],[90])).
% 0.19/0.45 tff(92,plain,
% 0.19/0.45 (y = sum_class(image(choice, singleton(singleton(y))))),
% 0.19/0.45 inference(transitivity,[status(thm)],[91, 22])).
% 0.19/0.45 tff(93,plain,
% 0.19/0.45 (member(y, universal_class) <=> member(sum_class(image(choice, singleton(singleton(y)))), universal_class)),
% 0.19/0.45 inference(monotonicity,[status(thm)],[92])).
% 0.19/0.45 tff(94,plain,
% 0.19/0.45 ((~member(y, universal_class)) <=> (~member(sum_class(image(choice, singleton(singleton(y)))), universal_class))),
% 0.19/0.45 inference(monotonicity,[status(thm)],[93])).
% 0.19/0.45 tff(95,plain,
% 0.19/0.45 ((~member(y, universal_class)) <=> (~member(y, universal_class))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(96,axiom,(~member(y, universal_class)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_property_1_of_ordered_pair_2')).
% 0.19/0.45 tff(97,plain,
% 0.19/0.45 (~member(y, universal_class)),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[96, 95])).
% 0.19/0.45 tff(98,plain,
% 0.19/0.45 (~member(sum_class(image(choice, singleton(singleton(y)))), universal_class)),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[97, 94])).
% 0.19/0.45 tff(99,plain,
% 0.19/0.45 (^[X: $i] : refl(((~member(X, universal_class)) | member(sum_class(X), universal_class)) <=> ((~member(X, universal_class)) | member(sum_class(X), universal_class)))),
% 0.19/0.45 inference(bind,[status(th)],[])).
% 0.19/0.45 tff(100,plain,
% 0.19/0.45 (![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class)) <=> ![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))),
% 0.19/0.45 inference(quant_intro,[status(thm)],[99])).
% 0.19/0.45 tff(101,plain,
% 0.19/0.45 (![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class)) <=> ![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(102,axiom,(![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','sum_class2')).
% 0.19/0.45 tff(103,plain,
% 0.19/0.45 (![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[102, 101])).
% 0.19/0.45 tff(104,plain,(
% 0.19/0.45 ![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))),
% 0.19/0.45 inference(skolemize,[status(sab)],[103])).
% 0.19/0.45 tff(105,plain,
% 0.19/0.45 (![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[104, 100])).
% 0.19/0.45 tff(106,plain,
% 0.19/0.45 (((~![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))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(107,plain,
% 0.19/0.45 ((~![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))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(108,plain,
% 0.19/0.45 ((~![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)),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[107, 106])).
% 0.19/0.45 tff(109,plain,
% 0.19/0.45 ((~member(image(choice, singleton(singleton(y))), universal_class)) | member(sum_class(image(choice, singleton(singleton(y)))), universal_class)),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[108, 105])).
% 0.19/0.45 tff(110,plain,
% 0.19/0.45 (~member(image(choice, singleton(singleton(y))), universal_class)),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[109, 98])).
% 0.19/0.45 tff(111,plain,
% 0.19/0.45 (function(choice) <=> function(choice)),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(112,axiom,(function(choice)), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','choice1')).
% 0.19/0.45 tff(113,plain,
% 0.19/0.45 (function(choice)),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[112, 111])).
% 0.19/0.45 tff(114,plain,
% 0.19/0.45 (^[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))))),
% 0.19/0.45 inference(bind,[status(th)],[])).
% 0.19/0.45 tff(115,plain,
% 0.19/0.45 (![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)))),
% 0.19/0.45 inference(quant_intro,[status(thm)],[114])).
% 0.19/0.45 tff(116,plain,
% 0.19/0.45 (![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)))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(117,plain,
% 0.19/0.45 (^[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))))),
% 0.19/0.45 inference(bind,[status(th)],[])).
% 0.19/0.45 tff(118,plain,
% 0.19/0.45 (![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)))),
% 0.19/0.45 inference(quant_intro,[status(thm)],[117])).
% 0.19/0.45 tff(119,axiom,(![Xf: $i, X: $i] : (((~function(Xf)) | (~member(X, universal_class))) | member(image(Xf, X), universal_class))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','replacement')).
% 0.19/0.45 tff(120,plain,
% 0.19/0.45 (![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[119, 118])).
% 0.19/0.45 tff(121,plain,
% 0.19/0.45 (![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[120, 116])).
% 0.19/0.45 tff(122,plain,(
% 0.19/0.45 ![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))),
% 0.19/0.45 inference(skolemize,[status(sab)],[121])).
% 0.19/0.46 tff(123,plain,
% 0.19/0.46 (![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[122, 115])).
% 0.19/0.46 tff(124,plain,
% 0.19/0.46 (((~![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)))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(125,plain,
% 0.19/0.46 (((~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)))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(126,plain,
% 0.19/0.46 (((~![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))))),
% 0.19/0.46 inference(monotonicity,[status(thm)],[125])).
% 0.19/0.46 tff(127,plain,
% 0.19/0.46 (((~![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)))),
% 0.19/0.46 inference(transitivity,[status(thm)],[126, 124])).
% 0.19/0.46 tff(128,plain,
% 0.19/0.46 ((~![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)))),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(129,plain,
% 0.19/0.46 ((~![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))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[128, 127])).
% 0.19/0.46 tff(130,plain,
% 0.19/0.46 (member(image(choice, singleton(singleton(y))), universal_class) | (~member(singleton(singleton(y)), universal_class))),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[129, 123, 113])).
% 0.19/0.46 tff(131,plain,
% 0.19/0.46 (~member(singleton(singleton(y)), universal_class)),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[130, 110])).
% 0.19/0.46 tff(132,plain,
% 0.19/0.46 (~member(unordered_pair(singleton(y), singleton(y)), universal_class)),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[131, 12])).
% 0.19/0.46 tff(133,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(singleton(y), singleton(y)), universal_class)),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(134,plain,
% 0.19/0.46 (member(unordered_pair(singleton(y), singleton(y)), universal_class)),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[133, 53])).
% 0.19/0.46 tff(135,plain,
% 0.19/0.46 ($false),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[134, 132])).
% 0.19/0.46 % SZS output end Proof
%------------------------------------------------------------------------------