TSTP Solution File: SET103+1 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET103+1 : TPTP v8.1.0. Bugfixed v5.4.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n013.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 : Theorem 0.77s 0.72s
% Output : Proof 0.77s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : SET103+1 : TPTP v8.1.0. Bugfixed v5.4.0.
% 0.07/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.14/0.35 % Computer : n013.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Sat Sep 3 02:21:47 EDT 2022
% 0.14/0.35 % CPUTime :
% 0.14/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.14/0.35 Usage: tptp [options] [-file:]file
% 0.14/0.35 -h, -? prints this message.
% 0.14/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.14/0.35 -m, -model generate model.
% 0.14/0.35 -p, -proof generate proof.
% 0.14/0.35 -c, -core generate unsat core of named formulas.
% 0.14/0.35 -st, -statistics display statistics.
% 0.14/0.35 -t:timeout set timeout (in second).
% 0.14/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.14/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.14/0.35 -<param>:<value> configuration parameter and value.
% 0.14/0.35 -o:<output-file> file to place output in.
% 0.77/0.72 % SZS status Theorem
% 0.77/0.72 % SZS output start Proof
% 0.77/0.72 tff(ordered_pair_type, type, (
% 0.77/0.72 ordered_pair: ( $i * $i ) > $i)).
% 0.77/0.72 tff(tptp_fun_Y_7_type, type, (
% 0.77/0.72 tptp_fun_Y_7: $i)).
% 0.77/0.72 tff(tptp_fun_X_8_type, type, (
% 0.77/0.72 tptp_fun_X_8: $i)).
% 0.77/0.72 tff(unordered_pair_type, type, (
% 0.77/0.72 unordered_pair: ( $i * $i ) > $i)).
% 0.77/0.72 tff(null_class_type, type, (
% 0.77/0.72 null_class: $i)).
% 0.77/0.72 tff(singleton_type, type, (
% 0.77/0.72 singleton: $i > $i)).
% 0.77/0.72 tff(member_type, type, (
% 0.77/0.72 member: ( $i * $i ) > $o)).
% 0.77/0.72 tff(apply_type, type, (
% 0.77/0.72 apply: ( $i * $i ) > $i)).
% 0.77/0.72 tff(tptp_fun_XF_6_type, type, (
% 0.77/0.72 tptp_fun_XF_6: $i)).
% 0.77/0.72 tff(universal_class_type, type, (
% 0.77/0.72 universal_class: $i)).
% 0.77/0.72 tff(function_type, type, (
% 0.77/0.72 function: $i > $o)).
% 0.77/0.72 tff(1,plain,
% 0.77/0.72 (^[X: $i, Y: $i] : refl((ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y)))) <=> (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y)))))),
% 0.77/0.72 inference(bind,[status(th)],[])).
% 0.77/0.72 tff(2,plain,
% 0.77/0.72 (![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y)))) <=> ![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))),
% 0.77/0.72 inference(quant_intro,[status(thm)],[1])).
% 0.77/0.72 tff(3,plain,
% 0.77/0.72 (![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y)))) <=> ![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))),
% 0.77/0.72 inference(rewrite,[status(thm)],[])).
% 0.77/0.72 tff(4,axiom,(![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))), file('/export/starexec/sandbox2/benchmark/Axioms/SET005+0.ax','ordered_pair_defn')).
% 0.77/0.72 tff(5,plain,
% 0.77/0.72 (![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))),
% 0.77/0.72 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.77/0.72 tff(6,plain,(
% 0.77/0.72 ![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))),
% 0.77/0.72 inference(skolemize,[status(sab)],[5])).
% 0.77/0.72 tff(7,plain,
% 0.77/0.72 (![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))),
% 0.77/0.72 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.77/0.72 tff(8,plain,
% 0.77/0.72 ((~![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))) | (ordered_pair(X!8, Y!7) = unordered_pair(singleton(X!8), unordered_pair(X!8, singleton(Y!7))))),
% 0.77/0.72 inference(quant_inst,[status(thm)],[])).
% 0.77/0.72 tff(9,plain,
% 0.77/0.72 (ordered_pair(X!8, Y!7) = unordered_pair(singleton(X!8), unordered_pair(X!8, singleton(Y!7)))),
% 0.77/0.72 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.77/0.72 tff(10,plain,
% 0.77/0.72 (unordered_pair(singleton(X!8), unordered_pair(X!8, singleton(Y!7))) = ordered_pair(X!8, Y!7)),
% 0.77/0.72 inference(symmetry,[status(thm)],[9])).
% 0.77/0.72 tff(11,plain,
% 0.77/0.72 (^[X: $i] : refl((singleton(X) = unordered_pair(X, X)) <=> (singleton(X) = unordered_pair(X, X)))),
% 0.77/0.72 inference(bind,[status(th)],[])).
% 0.77/0.72 tff(12,plain,
% 0.77/0.72 (![X: $i] : (singleton(X) = unordered_pair(X, X)) <=> ![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.77/0.72 inference(quant_intro,[status(thm)],[11])).
% 0.77/0.72 tff(13,plain,
% 0.77/0.72 (![X: $i] : (singleton(X) = unordered_pair(X, X)) <=> ![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.77/0.72 inference(rewrite,[status(thm)],[])).
% 0.77/0.72 tff(14,axiom,(![X: $i] : (singleton(X) = unordered_pair(X, X))), file('/export/starexec/sandbox2/benchmark/Axioms/SET005+0.ax','singleton_set_defn')).
% 0.77/0.72 tff(15,plain,
% 0.77/0.72 (![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.77/0.72 inference(modus_ponens,[status(thm)],[14, 13])).
% 0.77/0.72 tff(16,plain,(
% 0.77/0.72 ![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.77/0.72 inference(skolemize,[status(sab)],[15])).
% 0.77/0.72 tff(17,plain,
% 0.77/0.72 (![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.77/0.72 inference(modus_ponens,[status(thm)],[16, 12])).
% 0.77/0.72 tff(18,plain,
% 0.77/0.72 ((~![X: $i] : (singleton(X) = unordered_pair(X, X))) | (singleton(Y!7) = unordered_pair(Y!7, Y!7))),
% 0.77/0.72 inference(quant_inst,[status(thm)],[])).
% 0.77/0.72 tff(19,plain,
% 0.77/0.72 (singleton(Y!7) = unordered_pair(Y!7, Y!7)),
% 0.77/0.72 inference(unit_resolution,[status(thm)],[18, 17])).
% 0.77/0.72 tff(20,plain,
% 0.77/0.72 (unordered_pair(Y!7, Y!7) = singleton(Y!7)),
% 0.77/0.72 inference(symmetry,[status(thm)],[19])).
% 0.77/0.72 tff(21,assumption,(~(member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)) <=> (~((~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)) | (~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)))))), introduced(assumption)).
% 0.77/0.72 tff(22,plain,
% 0.77/0.72 (^[U: $i, X: $i, Y: $i] : refl((member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X)))))) <=> (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X)))))))),
% 0.77/0.72 inference(bind,[status(th)],[])).
% 0.77/0.72 tff(23,plain,
% 0.77/0.72 (![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X)))))) <=> ![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))),
% 0.77/0.72 inference(quant_intro,[status(thm)],[22])).
% 0.77/0.72 tff(24,plain,
% 0.77/0.72 (^[U: $i, X: $i, Y: $i] : rewrite((member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X)))) <=> (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X)))))))),
% 0.77/0.72 inference(bind,[status(th)],[])).
% 0.77/0.72 tff(25,plain,
% 0.77/0.72 (![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X)))) <=> ![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))),
% 0.77/0.72 inference(quant_intro,[status(thm)],[24])).
% 0.77/0.72 tff(26,plain,
% 0.77/0.72 (![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X)))) <=> ![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X))))),
% 0.77/0.72 inference(rewrite,[status(thm)],[])).
% 0.77/0.72 tff(27,plain,
% 0.77/0.72 (^[U: $i, X: $i, Y: $i] : rewrite((member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = X) | (U = Y)))) <=> (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X)))))),
% 0.77/0.72 inference(bind,[status(th)],[])).
% 0.77/0.72 tff(28,plain,
% 0.77/0.72 (![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = X) | (U = Y)))) <=> ![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X))))),
% 0.77/0.72 inference(quant_intro,[status(thm)],[27])).
% 0.77/0.72 tff(29,axiom,(![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = X) | (U = Y))))), file('/export/starexec/sandbox2/benchmark/Axioms/SET005+0.ax','unordered_pair_defn')).
% 0.77/0.72 tff(30,plain,
% 0.77/0.72 (![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X))))),
% 0.77/0.72 inference(modus_ponens,[status(thm)],[29, 28])).
% 0.77/0.72 tff(31,plain,
% 0.77/0.72 (![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X))))),
% 0.77/0.72 inference(modus_ponens,[status(thm)],[30, 26])).
% 0.77/0.72 tff(32,plain,(
% 0.77/0.72 ![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X))))),
% 0.77/0.72 inference(skolemize,[status(sab)],[31])).
% 0.77/0.72 tff(33,plain,
% 0.77/0.72 (![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))),
% 0.77/0.72 inference(modus_ponens,[status(thm)],[32, 25])).
% 0.77/0.72 tff(34,plain,
% 0.77/0.72 (![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))),
% 0.77/0.72 inference(modus_ponens,[status(thm)],[33, 23])).
% 0.77/0.72 tff(35,plain,
% 0.77/0.72 (((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)) <=> (~((~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)) | (~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)))))) <=> ((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)) <=> (~((~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)) | (~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7))))))),
% 0.77/0.72 inference(rewrite,[status(thm)],[])).
% 0.77/0.72 tff(36,plain,
% 0.77/0.72 ((member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)) <=> (~((~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)) | (~((apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7) | (apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)))))) <=> (member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)) <=> (~((~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)) | (~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)))))),
% 0.77/0.72 inference(rewrite,[status(thm)],[])).
% 0.77/0.72 tff(37,plain,
% 0.77/0.72 (((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)) <=> (~((~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)) | (~((apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7) | (apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7))))))) <=> ((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)) <=> (~((~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)) | (~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7))))))),
% 0.77/0.72 inference(monotonicity,[status(thm)],[36])).
% 0.77/0.72 tff(38,plain,
% 0.77/0.72 (((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)) <=> (~((~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)) | (~((apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7) | (apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7))))))) <=> ((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)) <=> (~((~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)) | (~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7))))))),
% 0.77/0.72 inference(transitivity,[status(thm)],[37, 35])).
% 0.77/0.72 tff(39,plain,
% 0.77/0.72 ((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)) <=> (~((~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)) | (~((apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7) | (apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7))))))),
% 0.77/0.72 inference(quant_inst,[status(thm)],[])).
% 0.77/0.72 tff(40,plain,
% 0.77/0.72 ((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)) <=> (~((~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)) | (~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)))))),
% 0.77/0.72 inference(modus_ponens,[status(thm)],[39, 38])).
% 0.77/0.72 tff(41,plain,
% 0.77/0.72 ($false),
% 0.77/0.72 inference(unit_resolution,[status(thm)],[40, 34, 21])).
% 0.77/0.72 tff(42,plain,(member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)) <=> (~((~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)) | (~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7))))), inference(lemma,lemma(discharge,[]))).
% 0.77/0.72 tff(43,assumption,(~((~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)) | (~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)))), introduced(assumption)).
% 0.77/0.72 tff(44,plain,
% 0.77/0.72 (((~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)) | (~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7))) | (apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)),
% 0.77/0.72 inference(tautology,[status(thm)],[])).
% 0.77/0.72 tff(45,plain,
% 0.77/0.72 (apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7),
% 0.77/0.72 inference(unit_resolution,[status(thm)],[44, 43])).
% 0.77/0.72 tff(46,plain,
% 0.77/0.72 (Y!7 = apply(XF!6, unordered_pair(Y!7, Y!7))),
% 0.77/0.72 inference(symmetry,[status(thm)],[45])).
% 0.77/0.72 tff(47,plain,
% 0.77/0.72 (member(Y!7, universal_class) <=> member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)),
% 0.77/0.72 inference(monotonicity,[status(thm)],[46])).
% 0.77/0.72 tff(48,plain,
% 0.77/0.72 (member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class) <=> member(Y!7, universal_class)),
% 0.77/0.72 inference(symmetry,[status(thm)],[47])).
% 0.77/0.72 tff(49,plain,
% 0.77/0.72 (((~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)) | (~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7))) | member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)),
% 0.77/0.72 inference(tautology,[status(thm)],[])).
% 0.77/0.72 tff(50,plain,
% 0.77/0.72 (member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)),
% 0.77/0.72 inference(unit_resolution,[status(thm)],[49, 43])).
% 0.77/0.72 tff(51,plain,
% 0.77/0.72 (member(Y!7, universal_class)),
% 0.77/0.72 inference(modus_ponens,[status(thm)],[50, 48])).
% 0.77/0.72 tff(52,plain,
% 0.77/0.72 ((~![X: $i, Y: $i] : (member(Y, universal_class) | (unordered_pair(singleton(X), unordered_pair(X, null_class)) = ordered_pair(X, Y)))) <=> (~![X: $i, Y: $i] : (member(Y, universal_class) | (unordered_pair(singleton(X), unordered_pair(X, null_class)) = ordered_pair(X, Y))))),
% 0.77/0.72 inference(rewrite,[status(thm)],[])).
% 0.77/0.72 tff(53,plain,
% 0.77/0.72 ((~![X: $i, Y: $i] : ((unordered_pair(singleton(X), unordered_pair(X, null_class)) = ordered_pair(X, Y)) | member(Y, universal_class))) <=> (~![X: $i, Y: $i] : (member(Y, universal_class) | (unordered_pair(singleton(X), unordered_pair(X, null_class)) = ordered_pair(X, Y))))),
% 0.77/0.72 inference(rewrite,[status(thm)],[])).
% 0.77/0.72 tff(54,axiom,(~![X: $i, Y: $i] : ((unordered_pair(singleton(X), unordered_pair(X, null_class)) = ordered_pair(X, Y)) | member(Y, universal_class))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','property_1_of_ordered_pair')).
% 0.77/0.72 tff(55,plain,
% 0.77/0.72 (~![X: $i, Y: $i] : (member(Y, universal_class) | (unordered_pair(singleton(X), unordered_pair(X, null_class)) = ordered_pair(X, Y)))),
% 0.77/0.72 inference(modus_ponens,[status(thm)],[54, 53])).
% 0.77/0.72 tff(56,plain,
% 0.77/0.72 (~![X: $i, Y: $i] : (member(Y, universal_class) | (unordered_pair(singleton(X), unordered_pair(X, null_class)) = ordered_pair(X, Y)))),
% 0.77/0.72 inference(modus_ponens,[status(thm)],[55, 52])).
% 0.77/0.72 tff(57,plain,
% 0.77/0.72 (~![X: $i, Y: $i] : (member(Y, universal_class) | (unordered_pair(singleton(X), unordered_pair(X, null_class)) = ordered_pair(X, Y)))),
% 0.77/0.72 inference(modus_ponens,[status(thm)],[56, 52])).
% 0.77/0.72 tff(58,plain,
% 0.77/0.72 (~![X: $i, Y: $i] : (member(Y, universal_class) | (unordered_pair(singleton(X), unordered_pair(X, null_class)) = ordered_pair(X, Y)))),
% 0.77/0.72 inference(modus_ponens,[status(thm)],[57, 52])).
% 0.77/0.72 tff(59,plain,
% 0.77/0.72 (~![X: $i, Y: $i] : (member(Y, universal_class) | (unordered_pair(singleton(X), unordered_pair(X, null_class)) = ordered_pair(X, Y)))),
% 0.77/0.72 inference(modus_ponens,[status(thm)],[58, 52])).
% 0.77/0.72 tff(60,plain,
% 0.77/0.72 (~![X: $i, Y: $i] : (member(Y, universal_class) | (unordered_pair(singleton(X), unordered_pair(X, null_class)) = ordered_pair(X, Y)))),
% 0.77/0.72 inference(modus_ponens,[status(thm)],[59, 52])).
% 0.77/0.72 tff(61,plain,
% 0.77/0.72 (~![X: $i, Y: $i] : (member(Y, universal_class) | (unordered_pair(singleton(X), unordered_pair(X, null_class)) = ordered_pair(X, Y)))),
% 0.77/0.72 inference(modus_ponens,[status(thm)],[60, 52])).
% 0.77/0.72 tff(62,plain,(
% 0.77/0.72 ~(member(Y!7, universal_class) | (unordered_pair(singleton(X!8), unordered_pair(X!8, null_class)) = ordered_pair(X!8, Y!7)))),
% 0.77/0.72 inference(skolemize,[status(sab)],[61])).
% 0.77/0.72 tff(63,plain,
% 0.77/0.72 (~member(Y!7, universal_class)),
% 0.77/0.72 inference(or_elim,[status(thm)],[62])).
% 0.77/0.72 tff(64,plain,
% 0.77/0.72 ($false),
% 0.77/0.72 inference(unit_resolution,[status(thm)],[63, 51])).
% 0.77/0.72 tff(65,plain,((~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)) | (~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7))), inference(lemma,lemma(discharge,[]))).
% 0.77/0.72 tff(66,plain,
% 0.77/0.72 ((~(member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)) <=> (~((~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)) | (~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)))))) | (~member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7))) | (~((~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)) | (~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7))))),
% 0.77/0.73 inference(tautology,[status(thm)],[])).
% 0.77/0.73 tff(67,plain,
% 0.77/0.73 ((~(member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)) <=> (~((~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)) | (~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)))))) | (~member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)))),
% 0.77/0.73 inference(unit_resolution,[status(thm)],[66, 65])).
% 0.77/0.73 tff(68,plain,
% 0.77/0.73 (~member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7))),
% 0.77/0.73 inference(unit_resolution,[status(thm)],[67, 42])).
% 0.77/0.73 tff(69,plain,
% 0.77/0.73 (^[X: $i, Y: $i] : refl(member(unordered_pair(X, Y), universal_class) <=> member(unordered_pair(X, Y), universal_class))),
% 0.77/0.73 inference(bind,[status(th)],[])).
% 0.77/0.73 tff(70,plain,
% 0.77/0.73 (![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class) <=> ![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.77/0.73 inference(quant_intro,[status(thm)],[69])).
% 0.77/0.73 tff(71,plain,
% 0.77/0.73 (![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class) <=> ![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.77/0.73 inference(rewrite,[status(thm)],[])).
% 0.77/0.73 tff(72,axiom,(![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)), file('/export/starexec/sandbox2/benchmark/Axioms/SET005+0.ax','unordered_pair')).
% 0.77/0.73 tff(73,plain,
% 0.77/0.73 (![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.77/0.73 inference(modus_ponens,[status(thm)],[72, 71])).
% 0.77/0.73 tff(74,plain,(
% 0.77/0.73 ![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.77/0.73 inference(skolemize,[status(sab)],[73])).
% 0.77/0.73 tff(75,plain,
% 0.77/0.73 (![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.77/0.73 inference(modus_ponens,[status(thm)],[74, 70])).
% 0.77/0.73 tff(76,plain,
% 0.77/0.73 ((~![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(Y!7, Y!7), universal_class)),
% 0.77/0.73 inference(quant_inst,[status(thm)],[])).
% 0.77/0.73 tff(77,plain,
% 0.77/0.73 (member(unordered_pair(Y!7, Y!7), universal_class)),
% 0.77/0.73 inference(unit_resolution,[status(thm)],[76, 75])).
% 0.77/0.73 tff(78,plain,
% 0.77/0.73 (^[Y: $i] : refl(((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y)) <=> ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y)))),
% 0.77/0.73 inference(bind,[status(th)],[])).
% 0.77/0.73 tff(79,plain,
% 0.77/0.73 (![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y)) <=> ![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))),
% 0.77/0.73 inference(quant_intro,[status(thm)],[78])).
% 0.77/0.73 tff(80,plain,
% 0.77/0.73 ((function(XF!6) & ![Y: $i] : (member(apply(XF!6, Y), Y) | (Y = null_class) | (~member(Y, universal_class)))) <=> (function(XF!6) & ![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y)))),
% 0.77/0.73 inference(rewrite,[status(thm)],[])).
% 0.77/0.73 tff(81,plain,
% 0.77/0.73 (?[XF: $i] : (function(XF) & ![Y: $i] : (member(apply(XF, Y), Y) | (Y = null_class) | (~member(Y, universal_class)))) <=> ?[XF: $i] : (function(XF) & ![Y: $i] : (member(apply(XF, Y), Y) | (Y = null_class) | (~member(Y, universal_class))))),
% 0.77/0.73 inference(rewrite,[status(thm)],[])).
% 0.77/0.73 tff(82,plain,
% 0.77/0.73 (^[XF: $i] : rewrite((function(XF) & ![Y: $i] : (member(Y, universal_class) => ((Y = null_class) | member(apply(XF, Y), Y)))) <=> (function(XF) & ![Y: $i] : (member(apply(XF, Y), Y) | (Y = null_class) | (~member(Y, universal_class)))))),
% 0.77/0.73 inference(bind,[status(th)],[])).
% 0.77/0.73 tff(83,plain,
% 0.77/0.73 (?[XF: $i] : (function(XF) & ![Y: $i] : (member(Y, universal_class) => ((Y = null_class) | member(apply(XF, Y), Y)))) <=> ?[XF: $i] : (function(XF) & ![Y: $i] : (member(apply(XF, Y), Y) | (Y = null_class) | (~member(Y, universal_class))))),
% 0.77/0.73 inference(quant_intro,[status(thm)],[82])).
% 0.77/0.73 tff(84,axiom,(?[XF: $i] : (function(XF) & ![Y: $i] : (member(Y, universal_class) => ((Y = null_class) | member(apply(XF, Y), Y))))), file('/export/starexec/sandbox2/benchmark/Axioms/SET005+0.ax','choice')).
% 0.77/0.73 tff(85,plain,
% 0.77/0.73 (?[XF: $i] : (function(XF) & ![Y: $i] : (member(apply(XF, Y), Y) | (Y = null_class) | (~member(Y, universal_class))))),
% 0.77/0.73 inference(modus_ponens,[status(thm)],[84, 83])).
% 0.77/0.73 tff(86,plain,
% 0.77/0.73 (?[XF: $i] : (function(XF) & ![Y: $i] : (member(apply(XF, Y), Y) | (Y = null_class) | (~member(Y, universal_class))))),
% 0.77/0.73 inference(modus_ponens,[status(thm)],[85, 81])).
% 0.77/0.73 tff(87,plain,
% 0.77/0.73 (function(XF!6) & ![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))),
% 0.77/0.73 inference(modus_ponens,[status(thm)],[86, 80])).
% 0.77/0.73 tff(88,plain,
% 0.77/0.73 (![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))),
% 0.77/0.73 inference(and_elim,[status(thm)],[87])).
% 0.77/0.73 tff(89,plain,
% 0.77/0.73 (![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))),
% 0.77/0.73 inference(modus_ponens,[status(thm)],[88, 79])).
% 0.77/0.73 tff(90,plain,
% 0.77/0.73 (((~![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))) | ((unordered_pair(Y!7, Y!7) = null_class) | (~member(unordered_pair(Y!7, Y!7), universal_class)) | member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)))) <=> ((~![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))) | (unordered_pair(Y!7, Y!7) = null_class) | (~member(unordered_pair(Y!7, Y!7), universal_class)) | member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)))),
% 0.77/0.73 inference(rewrite,[status(thm)],[])).
% 0.77/0.73 tff(91,plain,
% 0.77/0.73 ((~![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))) | ((unordered_pair(Y!7, Y!7) = null_class) | (~member(unordered_pair(Y!7, Y!7), universal_class)) | member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)))),
% 0.77/0.73 inference(quant_inst,[status(thm)],[])).
% 0.77/0.73 tff(92,plain,
% 0.77/0.73 ((~![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))) | (unordered_pair(Y!7, Y!7) = null_class) | (~member(unordered_pair(Y!7, Y!7), universal_class)) | member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7))),
% 0.77/0.73 inference(modus_ponens,[status(thm)],[91, 90])).
% 0.77/0.73 tff(93,plain,
% 0.77/0.73 ((unordered_pair(Y!7, Y!7) = null_class) | member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7))),
% 0.77/0.73 inference(unit_resolution,[status(thm)],[92, 89, 77])).
% 0.77/0.73 tff(94,plain,
% 0.77/0.73 (unordered_pair(Y!7, Y!7) = null_class),
% 0.77/0.73 inference(unit_resolution,[status(thm)],[93, 68])).
% 0.77/0.73 tff(95,plain,
% 0.77/0.73 (null_class = unordered_pair(Y!7, Y!7)),
% 0.77/0.73 inference(symmetry,[status(thm)],[94])).
% 0.77/0.73 tff(96,plain,
% 0.77/0.73 (null_class = singleton(Y!7)),
% 0.77/0.73 inference(transitivity,[status(thm)],[95, 20])).
% 0.77/0.73 tff(97,plain,
% 0.77/0.73 (unordered_pair(X!8, null_class) = unordered_pair(X!8, singleton(Y!7))),
% 0.77/0.73 inference(monotonicity,[status(thm)],[96])).
% 0.77/0.73 tff(98,plain,
% 0.77/0.73 (unordered_pair(singleton(X!8), unordered_pair(X!8, null_class)) = unordered_pair(singleton(X!8), unordered_pair(X!8, singleton(Y!7)))),
% 0.77/0.73 inference(monotonicity,[status(thm)],[97])).
% 0.77/0.73 tff(99,plain,
% 0.77/0.73 (unordered_pair(singleton(X!8), unordered_pair(X!8, null_class)) = ordered_pair(X!8, Y!7)),
% 0.77/0.73 inference(transitivity,[status(thm)],[98, 10])).
% 0.77/0.73 tff(100,plain,
% 0.77/0.73 (~(unordered_pair(singleton(X!8), unordered_pair(X!8, null_class)) = ordered_pair(X!8, Y!7))),
% 0.77/0.73 inference(or_elim,[status(thm)],[62])).
% 0.77/0.73 tff(101,plain,
% 0.77/0.73 ($false),
% 0.77/0.73 inference(unit_resolution,[status(thm)],[100, 99])).
% 0.77/0.73 % SZS output end Proof
%------------------------------------------------------------------------------