TSTP Solution File: SET084+1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET084+1 : TPTP v8.1.0. Bugfixed v5.4.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n027.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 05:05:30 EDT 2022
% Result : Theorem 2.34s 1.74s
% Output : Proof 2.34s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.10 % Problem : SET084+1 : TPTP v8.1.0. Bugfixed v5.4.0.
% 0.00/0.11 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.11/0.32 % Computer : n027.cluster.edu
% 0.11/0.32 % Model : x86_64 x86_64
% 0.11/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.32 % Memory : 8042.1875MB
% 0.11/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.32 % CPULimit : 300
% 0.11/0.32 % WCLimit : 300
% 0.11/0.32 % DateTime : Sat Sep 3 02:23:50 EDT 2022
% 0.11/0.32 % CPUTime :
% 0.11/0.32 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.11/0.32 Usage: tptp [options] [-file:]file
% 0.11/0.32 -h, -? prints this message.
% 0.11/0.32 -smt2 print SMT-LIB2 benchmark.
% 0.11/0.32 -m, -model generate model.
% 0.11/0.32 -p, -proof generate proof.
% 0.11/0.32 -c, -core generate unsat core of named formulas.
% 0.11/0.32 -st, -statistics display statistics.
% 0.11/0.32 -t:timeout set timeout (in second).
% 0.11/0.32 -smt2status display status in smt2 format instead of SZS.
% 0.11/0.32 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.11/0.32 -<param>:<value> configuration parameter and value.
% 0.11/0.32 -o:<output-file> file to place output in.
% 2.34/1.74 % SZS status Theorem
% 2.34/1.74 % SZS output start Proof
% 2.34/1.74 tff(member_type, type, (
% 2.34/1.74 member: ( $i * $i ) > $o)).
% 2.34/1.74 tff(null_class_type, type, (
% 2.34/1.74 null_class: $i)).
% 2.34/1.74 tff(tptp_fun_Y_7_type, type, (
% 2.34/1.74 tptp_fun_Y_7: $i)).
% 2.34/1.74 tff(singleton_type, type, (
% 2.34/1.74 singleton: $i > $i)).
% 2.34/1.74 tff(apply_type, type, (
% 2.34/1.74 apply: ( $i * $i ) > $i)).
% 2.34/1.74 tff(tptp_fun_XF_6_type, type, (
% 2.34/1.74 tptp_fun_XF_6: $i)).
% 2.34/1.74 tff(tptp_fun_X_8_type, type, (
% 2.34/1.74 tptp_fun_X_8: $i)).
% 2.34/1.74 tff(universal_class_type, type, (
% 2.34/1.74 universal_class: $i)).
% 2.34/1.74 tff(unordered_pair_type, type, (
% 2.34/1.74 unordered_pair: ( $i * $i ) > $i)).
% 2.34/1.74 tff(function_type, type, (
% 2.34/1.74 function: $i > $o)).
% 2.34/1.74 tff(1,plain,
% 2.34/1.74 ((X!8 = apply(XF!6, singleton(Y!7))) <=> (apply(XF!6, singleton(Y!7)) = X!8)),
% 2.34/1.74 inference(commutativity,[status(thm)],[])).
% 2.34/1.74 tff(2,plain,
% 2.34/1.74 (^[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)))))))),
% 2.34/1.74 inference(bind,[status(th)],[])).
% 2.34/1.74 tff(3,plain,
% 2.34/1.74 (![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))))))),
% 2.34/1.74 inference(quant_intro,[status(thm)],[2])).
% 2.34/1.74 tff(4,plain,
% 2.34/1.74 (^[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)))))))),
% 2.34/1.74 inference(bind,[status(th)],[])).
% 2.34/1.74 tff(5,plain,
% 2.34/1.74 (![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))))))),
% 2.34/1.74 inference(quant_intro,[status(thm)],[4])).
% 2.34/1.74 tff(6,plain,
% 2.34/1.74 (![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))))),
% 2.34/1.74 inference(rewrite,[status(thm)],[])).
% 2.34/1.74 tff(7,plain,
% 2.34/1.74 (^[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)))))),
% 2.34/1.74 inference(bind,[status(th)],[])).
% 2.34/1.74 tff(8,plain,
% 2.34/1.74 (![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))))),
% 2.34/1.74 inference(quant_intro,[status(thm)],[7])).
% 2.34/1.74 tff(9,axiom,(![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = X) | (U = Y))))), file('/export/starexec/sandbox/benchmark/Axioms/SET005+0.ax','unordered_pair_defn')).
% 2.34/1.74 tff(10,plain,
% 2.34/1.74 (![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X))))),
% 2.34/1.74 inference(modus_ponens,[status(thm)],[9, 8])).
% 2.34/1.74 tff(11,plain,
% 2.34/1.74 (![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X))))),
% 2.34/1.74 inference(modus_ponens,[status(thm)],[10, 6])).
% 2.34/1.74 tff(12,plain,(
% 2.34/1.74 ![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X))))),
% 2.34/1.74 inference(skolemize,[status(sab)],[11])).
% 2.34/1.74 tff(13,plain,
% 2.34/1.74 (![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))),
% 2.34/1.74 inference(modus_ponens,[status(thm)],[12, 5])).
% 2.34/1.74 tff(14,plain,
% 2.34/1.74 (![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))),
% 2.34/1.74 inference(modus_ponens,[status(thm)],[13, 3])).
% 2.34/1.74 tff(15,plain,
% 2.34/1.74 (((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(apply(XF!6, singleton(Y!7)), unordered_pair(Y!7, Y!7)) <=> (~((~(apply(XF!6, singleton(Y!7)) = Y!7)) | (~member(apply(XF!6, singleton(Y!7)), universal_class)))))) <=> ((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(apply(XF!6, singleton(Y!7)), unordered_pair(Y!7, Y!7)) <=> (~((~(apply(XF!6, singleton(Y!7)) = Y!7)) | (~member(apply(XF!6, singleton(Y!7)), universal_class))))))),
% 2.34/1.75 inference(rewrite,[status(thm)],[])).
% 2.34/1.75 tff(16,plain,
% 2.34/1.75 ((member(apply(XF!6, singleton(Y!7)), unordered_pair(Y!7, Y!7)) <=> (~((~member(apply(XF!6, singleton(Y!7)), universal_class)) | (~((apply(XF!6, singleton(Y!7)) = Y!7) | (apply(XF!6, singleton(Y!7)) = Y!7)))))) <=> (member(apply(XF!6, singleton(Y!7)), unordered_pair(Y!7, Y!7)) <=> (~((~(apply(XF!6, singleton(Y!7)) = Y!7)) | (~member(apply(XF!6, singleton(Y!7)), universal_class)))))),
% 2.34/1.75 inference(rewrite,[status(thm)],[])).
% 2.34/1.75 tff(17,plain,
% 2.34/1.75 (((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(apply(XF!6, singleton(Y!7)), unordered_pair(Y!7, Y!7)) <=> (~((~member(apply(XF!6, singleton(Y!7)), universal_class)) | (~((apply(XF!6, singleton(Y!7)) = Y!7) | (apply(XF!6, singleton(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, singleton(Y!7)), unordered_pair(Y!7, Y!7)) <=> (~((~(apply(XF!6, singleton(Y!7)) = Y!7)) | (~member(apply(XF!6, singleton(Y!7)), universal_class))))))),
% 2.34/1.75 inference(monotonicity,[status(thm)],[16])).
% 2.34/1.75 tff(18,plain,
% 2.34/1.75 (((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(apply(XF!6, singleton(Y!7)), unordered_pair(Y!7, Y!7)) <=> (~((~member(apply(XF!6, singleton(Y!7)), universal_class)) | (~((apply(XF!6, singleton(Y!7)) = Y!7) | (apply(XF!6, singleton(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, singleton(Y!7)), unordered_pair(Y!7, Y!7)) <=> (~((~(apply(XF!6, singleton(Y!7)) = Y!7)) | (~member(apply(XF!6, singleton(Y!7)), universal_class))))))),
% 2.34/1.75 inference(transitivity,[status(thm)],[17, 15])).
% 2.34/1.75 tff(19,plain,
% 2.34/1.75 ((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(apply(XF!6, singleton(Y!7)), unordered_pair(Y!7, Y!7)) <=> (~((~member(apply(XF!6, singleton(Y!7)), universal_class)) | (~((apply(XF!6, singleton(Y!7)) = Y!7) | (apply(XF!6, singleton(Y!7)) = Y!7))))))),
% 2.34/1.75 inference(quant_inst,[status(thm)],[])).
% 2.34/1.75 tff(20,plain,
% 2.34/1.75 ((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(apply(XF!6, singleton(Y!7)), unordered_pair(Y!7, Y!7)) <=> (~((~(apply(XF!6, singleton(Y!7)) = Y!7)) | (~member(apply(XF!6, singleton(Y!7)), universal_class)))))),
% 2.34/1.75 inference(modus_ponens,[status(thm)],[19, 18])).
% 2.34/1.75 tff(21,plain,
% 2.34/1.75 (member(apply(XF!6, singleton(Y!7)), unordered_pair(Y!7, Y!7)) <=> (~((~(apply(XF!6, singleton(Y!7)) = Y!7)) | (~member(apply(XF!6, singleton(Y!7)), universal_class))))),
% 2.34/1.75 inference(unit_resolution,[status(thm)],[20, 14])).
% 2.34/1.75 tff(22,plain,
% 2.34/1.75 (^[X: $i] : refl((singleton(X) = unordered_pair(X, X)) <=> (singleton(X) = unordered_pair(X, X)))),
% 2.34/1.75 inference(bind,[status(th)],[])).
% 2.34/1.75 tff(23,plain,
% 2.34/1.75 (![X: $i] : (singleton(X) = unordered_pair(X, X)) <=> ![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 2.34/1.75 inference(quant_intro,[status(thm)],[22])).
% 2.34/1.75 tff(24,plain,
% 2.34/1.75 (![X: $i] : (singleton(X) = unordered_pair(X, X)) <=> ![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 2.34/1.75 inference(rewrite,[status(thm)],[])).
% 2.34/1.75 tff(25,axiom,(![X: $i] : (singleton(X) = unordered_pair(X, X))), file('/export/starexec/sandbox/benchmark/Axioms/SET005+0.ax','singleton_set_defn')).
% 2.34/1.75 tff(26,plain,
% 2.34/1.75 (![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 2.34/1.75 inference(modus_ponens,[status(thm)],[25, 24])).
% 2.34/1.75 tff(27,plain,(
% 2.34/1.75 ![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 2.34/1.75 inference(skolemize,[status(sab)],[26])).
% 2.34/1.75 tff(28,plain,
% 2.34/1.75 (![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 2.34/1.75 inference(modus_ponens,[status(thm)],[27, 23])).
% 2.34/1.75 tff(29,plain,
% 2.34/1.75 ((~![X: $i] : (singleton(X) = unordered_pair(X, X))) | (singleton(Y!7) = unordered_pair(Y!7, Y!7))),
% 2.34/1.75 inference(quant_inst,[status(thm)],[])).
% 2.34/1.75 tff(30,plain,
% 2.34/1.75 (singleton(Y!7) = unordered_pair(Y!7, Y!7)),
% 2.34/1.75 inference(unit_resolution,[status(thm)],[29, 28])).
% 2.34/1.75 tff(31,plain,
% 2.34/1.75 (unordered_pair(Y!7, Y!7) = singleton(Y!7)),
% 2.34/1.75 inference(symmetry,[status(thm)],[30])).
% 2.34/1.75 tff(32,plain,
% 2.34/1.75 (member(apply(XF!6, singleton(Y!7)), unordered_pair(Y!7, Y!7)) <=> member(apply(XF!6, singleton(Y!7)), singleton(Y!7))),
% 2.34/1.75 inference(monotonicity,[status(thm)],[31])).
% 2.34/1.75 tff(33,plain,
% 2.34/1.75 (member(apply(XF!6, singleton(Y!7)), singleton(Y!7)) <=> member(apply(XF!6, singleton(Y!7)), unordered_pair(Y!7, Y!7))),
% 2.34/1.75 inference(symmetry,[status(thm)],[32])).
% 2.34/1.75 tff(34,assumption,(member(apply(XF!6, singleton(Y!7)), singleton(Y!7))), introduced(assumption)).
% 2.34/1.75 tff(35,plain,
% 2.34/1.75 (member(apply(XF!6, singleton(Y!7)), unordered_pair(Y!7, Y!7))),
% 2.34/1.75 inference(modus_ponens,[status(thm)],[34, 33])).
% 2.34/1.75 tff(36,plain,
% 2.34/1.75 ((~(member(apply(XF!6, singleton(Y!7)), unordered_pair(Y!7, Y!7)) <=> (~((~(apply(XF!6, singleton(Y!7)) = Y!7)) | (~member(apply(XF!6, singleton(Y!7)), universal_class)))))) | (~member(apply(XF!6, singleton(Y!7)), unordered_pair(Y!7, Y!7))) | (~((~(apply(XF!6, singleton(Y!7)) = Y!7)) | (~member(apply(XF!6, singleton(Y!7)), universal_class))))),
% 2.34/1.75 inference(tautology,[status(thm)],[])).
% 2.34/1.75 tff(37,plain,
% 2.34/1.75 (~((~(apply(XF!6, singleton(Y!7)) = Y!7)) | (~member(apply(XF!6, singleton(Y!7)), universal_class)))),
% 2.34/1.75 inference(unit_resolution,[status(thm)],[36, 35, 21])).
% 2.34/1.75 tff(38,plain,
% 2.34/1.75 (((~(apply(XF!6, singleton(Y!7)) = Y!7)) | (~member(apply(XF!6, singleton(Y!7)), universal_class))) | (apply(XF!6, singleton(Y!7)) = Y!7)),
% 2.34/1.75 inference(tautology,[status(thm)],[])).
% 2.34/1.75 tff(39,plain,
% 2.34/1.75 (apply(XF!6, singleton(Y!7)) = Y!7),
% 2.34/1.75 inference(unit_resolution,[status(thm)],[38, 37])).
% 2.34/1.75 tff(40,plain,
% 2.34/1.75 (Y!7 = apply(XF!6, singleton(Y!7))),
% 2.34/1.75 inference(symmetry,[status(thm)],[39])).
% 2.34/1.75 tff(41,plain,
% 2.34/1.75 ((X!8 = Y!7) <=> (X!8 = apply(XF!6, singleton(Y!7)))),
% 2.34/1.75 inference(monotonicity,[status(thm)],[40])).
% 2.34/1.75 tff(42,plain,
% 2.34/1.75 ((X!8 = Y!7) <=> (apply(XF!6, singleton(Y!7)) = X!8)),
% 2.34/1.75 inference(transitivity,[status(thm)],[41, 1])).
% 2.34/1.75 tff(43,plain,
% 2.34/1.75 ((apply(XF!6, singleton(Y!7)) = X!8) <=> (X!8 = Y!7)),
% 2.34/1.75 inference(symmetry,[status(thm)],[42])).
% 2.34/1.75 tff(44,plain,
% 2.34/1.75 (((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(apply(XF!6, singleton(Y!7)), unordered_pair(X!8, X!8)) <=> (~((~member(apply(XF!6, singleton(Y!7)), universal_class)) | (~(apply(XF!6, singleton(Y!7)) = X!8)))))) <=> ((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(apply(XF!6, singleton(Y!7)), unordered_pair(X!8, X!8)) <=> (~((~member(apply(XF!6, singleton(Y!7)), universal_class)) | (~(apply(XF!6, singleton(Y!7)) = X!8))))))),
% 2.34/1.75 inference(rewrite,[status(thm)],[])).
% 2.34/1.75 tff(45,plain,
% 2.34/1.75 ((member(apply(XF!6, singleton(Y!7)), unordered_pair(X!8, X!8)) <=> (~((~member(apply(XF!6, singleton(Y!7)), universal_class)) | (~((apply(XF!6, singleton(Y!7)) = X!8) | (apply(XF!6, singleton(Y!7)) = X!8)))))) <=> (member(apply(XF!6, singleton(Y!7)), unordered_pair(X!8, X!8)) <=> (~((~member(apply(XF!6, singleton(Y!7)), universal_class)) | (~(apply(XF!6, singleton(Y!7)) = X!8)))))),
% 2.34/1.75 inference(rewrite,[status(thm)],[])).
% 2.34/1.75 tff(46,plain,
% 2.34/1.75 (((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(apply(XF!6, singleton(Y!7)), unordered_pair(X!8, X!8)) <=> (~((~member(apply(XF!6, singleton(Y!7)), universal_class)) | (~((apply(XF!6, singleton(Y!7)) = X!8) | (apply(XF!6, singleton(Y!7)) = X!8))))))) <=> ((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(apply(XF!6, singleton(Y!7)), unordered_pair(X!8, X!8)) <=> (~((~member(apply(XF!6, singleton(Y!7)), universal_class)) | (~(apply(XF!6, singleton(Y!7)) = X!8))))))),
% 2.34/1.75 inference(monotonicity,[status(thm)],[45])).
% 2.34/1.75 tff(47,plain,
% 2.34/1.75 (((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(apply(XF!6, singleton(Y!7)), unordered_pair(X!8, X!8)) <=> (~((~member(apply(XF!6, singleton(Y!7)), universal_class)) | (~((apply(XF!6, singleton(Y!7)) = X!8) | (apply(XF!6, singleton(Y!7)) = X!8))))))) <=> ((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(apply(XF!6, singleton(Y!7)), unordered_pair(X!8, X!8)) <=> (~((~member(apply(XF!6, singleton(Y!7)), universal_class)) | (~(apply(XF!6, singleton(Y!7)) = X!8))))))),
% 2.34/1.75 inference(transitivity,[status(thm)],[46, 44])).
% 2.34/1.75 tff(48,plain,
% 2.34/1.75 ((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(apply(XF!6, singleton(Y!7)), unordered_pair(X!8, X!8)) <=> (~((~member(apply(XF!6, singleton(Y!7)), universal_class)) | (~((apply(XF!6, singleton(Y!7)) = X!8) | (apply(XF!6, singleton(Y!7)) = X!8))))))),
% 2.34/1.75 inference(quant_inst,[status(thm)],[])).
% 2.34/1.75 tff(49,plain,
% 2.34/1.75 ((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(apply(XF!6, singleton(Y!7)), unordered_pair(X!8, X!8)) <=> (~((~member(apply(XF!6, singleton(Y!7)), universal_class)) | (~(apply(XF!6, singleton(Y!7)) = X!8)))))),
% 2.34/1.75 inference(modus_ponens,[status(thm)],[48, 47])).
% 2.34/1.75 tff(50,plain,
% 2.34/1.75 (member(apply(XF!6, singleton(Y!7)), unordered_pair(X!8, X!8)) <=> (~((~member(apply(XF!6, singleton(Y!7)), universal_class)) | (~(apply(XF!6, singleton(Y!7)) = X!8))))),
% 2.34/1.75 inference(unit_resolution,[status(thm)],[49, 14])).
% 2.34/1.75 tff(51,plain,
% 2.34/1.75 ((~![X: $i, Y: $i] : ((~((singleton(X) = singleton(Y)) & member(Y, universal_class))) | (X = Y))) <=> (~![X: $i, Y: $i] : ((~((singleton(X) = singleton(Y)) & member(Y, universal_class))) | (X = Y)))),
% 2.34/1.75 inference(rewrite,[status(thm)],[])).
% 2.34/1.75 tff(52,plain,
% 2.34/1.75 ((~![X: $i, Y: $i] : (((singleton(X) = singleton(Y)) & member(Y, universal_class)) => (X = Y))) <=> (~![X: $i, Y: $i] : ((~((singleton(X) = singleton(Y)) & member(Y, universal_class))) | (X = Y)))),
% 2.34/1.75 inference(rewrite,[status(thm)],[])).
% 2.34/1.75 tff(53,axiom,(~![X: $i, Y: $i] : (((singleton(X) = singleton(Y)) & member(Y, universal_class)) => (X = Y))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','singleton_identified_by_element2')).
% 2.34/1.75 tff(54,plain,
% 2.34/1.75 (~![X: $i, Y: $i] : ((~((singleton(X) = singleton(Y)) & member(Y, universal_class))) | (X = Y))),
% 2.34/1.75 inference(modus_ponens,[status(thm)],[53, 52])).
% 2.34/1.75 tff(55,plain,
% 2.34/1.75 (~![X: $i, Y: $i] : ((~((singleton(X) = singleton(Y)) & member(Y, universal_class))) | (X = Y))),
% 2.34/1.75 inference(modus_ponens,[status(thm)],[54, 51])).
% 2.34/1.75 tff(56,plain,
% 2.34/1.75 (~![X: $i, Y: $i] : ((~((singleton(X) = singleton(Y)) & member(Y, universal_class))) | (X = Y))),
% 2.34/1.75 inference(modus_ponens,[status(thm)],[55, 51])).
% 2.34/1.75 tff(57,plain,
% 2.34/1.75 (~![X: $i, Y: $i] : ((~((singleton(X) = singleton(Y)) & member(Y, universal_class))) | (X = Y))),
% 2.34/1.75 inference(modus_ponens,[status(thm)],[56, 51])).
% 2.34/1.75 tff(58,plain,
% 2.34/1.75 (~![X: $i, Y: $i] : ((~((singleton(X) = singleton(Y)) & member(Y, universal_class))) | (X = Y))),
% 2.34/1.75 inference(modus_ponens,[status(thm)],[57, 51])).
% 2.34/1.75 tff(59,plain,
% 2.34/1.75 (~![X: $i, Y: $i] : ((~((singleton(X) = singleton(Y)) & member(Y, universal_class))) | (X = Y))),
% 2.34/1.75 inference(modus_ponens,[status(thm)],[58, 51])).
% 2.34/1.75 tff(60,plain,
% 2.34/1.75 (~![X: $i, Y: $i] : ((~((singleton(X) = singleton(Y)) & member(Y, universal_class))) | (X = Y))),
% 2.34/1.75 inference(modus_ponens,[status(thm)],[59, 51])).
% 2.34/1.75 tff(61,plain,(
% 2.34/1.75 ~((~((singleton(X!8) = singleton(Y!7)) & member(Y!7, universal_class))) | (X!8 = Y!7))),
% 2.34/1.75 inference(skolemize,[status(sab)],[60])).
% 2.34/1.75 tff(62,plain,
% 2.34/1.75 ((singleton(X!8) = singleton(Y!7)) & member(Y!7, universal_class)),
% 2.34/1.75 inference(or_elim,[status(thm)],[61])).
% 2.34/1.75 tff(63,plain,
% 2.34/1.75 (singleton(X!8) = singleton(Y!7)),
% 2.34/1.75 inference(and_elim,[status(thm)],[62])).
% 2.34/1.75 tff(64,plain,
% 2.34/1.75 ((~![X: $i] : (singleton(X) = unordered_pair(X, X))) | (singleton(X!8) = unordered_pair(X!8, X!8))),
% 2.34/1.75 inference(quant_inst,[status(thm)],[])).
% 2.34/1.75 tff(65,plain,
% 2.34/1.75 (singleton(X!8) = unordered_pair(X!8, X!8)),
% 2.34/1.75 inference(unit_resolution,[status(thm)],[64, 28])).
% 2.34/1.75 tff(66,plain,
% 2.34/1.75 (unordered_pair(X!8, X!8) = singleton(X!8)),
% 2.34/1.75 inference(symmetry,[status(thm)],[65])).
% 2.34/1.75 tff(67,plain,
% 2.34/1.75 (unordered_pair(X!8, X!8) = singleton(Y!7)),
% 2.34/1.75 inference(transitivity,[status(thm)],[66, 63])).
% 2.34/1.75 tff(68,plain,
% 2.34/1.75 (member(apply(XF!6, singleton(Y!7)), unordered_pair(X!8, X!8)) <=> member(apply(XF!6, singleton(Y!7)), singleton(Y!7))),
% 2.34/1.76 inference(monotonicity,[status(thm)],[67])).
% 2.34/1.76 tff(69,plain,
% 2.34/1.76 (member(apply(XF!6, singleton(Y!7)), singleton(Y!7)) <=> member(apply(XF!6, singleton(Y!7)), unordered_pair(X!8, X!8))),
% 2.34/1.76 inference(symmetry,[status(thm)],[68])).
% 2.34/1.76 tff(70,plain,
% 2.34/1.76 (member(apply(XF!6, singleton(Y!7)), unordered_pair(X!8, X!8))),
% 2.34/1.76 inference(modus_ponens,[status(thm)],[34, 69])).
% 2.34/1.76 tff(71,plain,
% 2.34/1.76 ((~(member(apply(XF!6, singleton(Y!7)), unordered_pair(X!8, X!8)) <=> (~((~member(apply(XF!6, singleton(Y!7)), universal_class)) | (~(apply(XF!6, singleton(Y!7)) = X!8)))))) | (~member(apply(XF!6, singleton(Y!7)), unordered_pair(X!8, X!8))) | (~((~member(apply(XF!6, singleton(Y!7)), universal_class)) | (~(apply(XF!6, singleton(Y!7)) = X!8))))),
% 2.34/1.76 inference(tautology,[status(thm)],[])).
% 2.34/1.76 tff(72,plain,
% 2.34/1.76 (~((~member(apply(XF!6, singleton(Y!7)), universal_class)) | (~(apply(XF!6, singleton(Y!7)) = X!8)))),
% 2.34/1.76 inference(unit_resolution,[status(thm)],[71, 70, 50])).
% 2.34/1.76 tff(73,plain,
% 2.34/1.76 (((~member(apply(XF!6, singleton(Y!7)), universal_class)) | (~(apply(XF!6, singleton(Y!7)) = X!8))) | (apply(XF!6, singleton(Y!7)) = X!8)),
% 2.34/1.76 inference(tautology,[status(thm)],[])).
% 2.34/1.76 tff(74,plain,
% 2.34/1.76 (apply(XF!6, singleton(Y!7)) = X!8),
% 2.34/1.76 inference(unit_resolution,[status(thm)],[73, 72])).
% 2.34/1.76 tff(75,plain,
% 2.34/1.76 (X!8 = Y!7),
% 2.34/1.76 inference(modus_ponens,[status(thm)],[74, 43])).
% 2.34/1.76 tff(76,plain,
% 2.34/1.76 (~(X!8 = Y!7)),
% 2.34/1.76 inference(or_elim,[status(thm)],[61])).
% 2.34/1.76 tff(77,plain,
% 2.34/1.76 ($false),
% 2.34/1.76 inference(unit_resolution,[status(thm)],[76, 75])).
% 2.34/1.76 tff(78,plain,(~member(apply(XF!6, singleton(Y!7)), singleton(Y!7))), inference(lemma,lemma(discharge,[]))).
% 2.34/1.76 tff(79,plain,
% 2.34/1.76 (singleton(Y!7) = singleton(X!8)),
% 2.34/1.76 inference(symmetry,[status(thm)],[63])).
% 2.34/1.76 tff(80,plain,
% 2.34/1.76 (singleton(Y!7) = unordered_pair(X!8, X!8)),
% 2.34/1.76 inference(transitivity,[status(thm)],[79, 65])).
% 2.34/1.76 tff(81,plain,
% 2.34/1.76 (member(singleton(Y!7), universal_class) <=> member(unordered_pair(X!8, X!8), universal_class)),
% 2.34/1.76 inference(monotonicity,[status(thm)],[80])).
% 2.34/1.76 tff(82,plain,
% 2.34/1.76 (member(unordered_pair(X!8, X!8), universal_class) <=> member(singleton(Y!7), universal_class)),
% 2.34/1.76 inference(symmetry,[status(thm)],[81])).
% 2.34/1.76 tff(83,plain,
% 2.34/1.76 (^[X: $i, Y: $i] : refl(member(unordered_pair(X, Y), universal_class) <=> member(unordered_pair(X, Y), universal_class))),
% 2.34/1.76 inference(bind,[status(th)],[])).
% 2.34/1.76 tff(84,plain,
% 2.34/1.76 (![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class) <=> ![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 2.34/1.76 inference(quant_intro,[status(thm)],[83])).
% 2.34/1.76 tff(85,plain,
% 2.34/1.76 (![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class) <=> ![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 2.34/1.76 inference(rewrite,[status(thm)],[])).
% 2.34/1.76 tff(86,axiom,(![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)), file('/export/starexec/sandbox/benchmark/Axioms/SET005+0.ax','unordered_pair')).
% 2.34/1.76 tff(87,plain,
% 2.34/1.76 (![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 2.34/1.76 inference(modus_ponens,[status(thm)],[86, 85])).
% 2.34/1.76 tff(88,plain,(
% 2.34/1.76 ![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 2.34/1.76 inference(skolemize,[status(sab)],[87])).
% 2.34/1.76 tff(89,plain,
% 2.34/1.76 (![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 2.34/1.76 inference(modus_ponens,[status(thm)],[88, 84])).
% 2.34/1.76 tff(90,plain,
% 2.34/1.76 ((~![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(X!8, X!8), universal_class)),
% 2.34/1.76 inference(quant_inst,[status(thm)],[])).
% 2.34/1.76 tff(91,plain,
% 2.34/1.76 (member(unordered_pair(X!8, X!8), universal_class)),
% 2.34/1.76 inference(unit_resolution,[status(thm)],[90, 89])).
% 2.34/1.76 tff(92,plain,
% 2.34/1.76 (member(singleton(Y!7), universal_class)),
% 2.34/1.76 inference(modus_ponens,[status(thm)],[91, 82])).
% 2.34/1.76 tff(93,plain,
% 2.34/1.76 (^[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)))),
% 2.34/1.76 inference(bind,[status(th)],[])).
% 2.34/1.76 tff(94,plain,
% 2.34/1.76 (![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))),
% 2.34/1.76 inference(quant_intro,[status(thm)],[93])).
% 2.34/1.76 tff(95,plain,
% 2.34/1.76 ((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)))),
% 2.34/1.76 inference(rewrite,[status(thm)],[])).
% 2.34/1.76 tff(96,plain,
% 2.34/1.76 (?[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))))),
% 2.34/1.76 inference(rewrite,[status(thm)],[])).
% 2.34/1.76 tff(97,plain,
% 2.34/1.76 (^[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)))))),
% 2.34/1.76 inference(bind,[status(th)],[])).
% 2.34/1.76 tff(98,plain,
% 2.34/1.76 (?[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))))),
% 2.34/1.76 inference(quant_intro,[status(thm)],[97])).
% 2.34/1.76 tff(99,axiom,(?[XF: $i] : (function(XF) & ![Y: $i] : (member(Y, universal_class) => ((Y = null_class) | member(apply(XF, Y), Y))))), file('/export/starexec/sandbox/benchmark/Axioms/SET005+0.ax','choice')).
% 2.34/1.76 tff(100,plain,
% 2.34/1.76 (?[XF: $i] : (function(XF) & ![Y: $i] : (member(apply(XF, Y), Y) | (Y = null_class) | (~member(Y, universal_class))))),
% 2.34/1.76 inference(modus_ponens,[status(thm)],[99, 98])).
% 2.34/1.76 tff(101,plain,
% 2.34/1.76 (?[XF: $i] : (function(XF) & ![Y: $i] : (member(apply(XF, Y), Y) | (Y = null_class) | (~member(Y, universal_class))))),
% 2.34/1.76 inference(modus_ponens,[status(thm)],[100, 96])).
% 2.34/1.76 tff(102,plain,
% 2.34/1.76 (function(XF!6) & ![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))),
% 2.34/1.76 inference(modus_ponens,[status(thm)],[101, 95])).
% 2.34/1.76 tff(103,plain,
% 2.34/1.76 (![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))),
% 2.34/1.76 inference(and_elim,[status(thm)],[102])).
% 2.34/1.76 tff(104,plain,
% 2.34/1.76 (![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))),
% 2.34/1.76 inference(modus_ponens,[status(thm)],[103, 94])).
% 2.34/1.76 tff(105,plain,
% 2.34/1.76 (((~![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))) | ((singleton(Y!7) = null_class) | (~member(singleton(Y!7), universal_class)) | member(apply(XF!6, singleton(Y!7)), singleton(Y!7)))) <=> ((~![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))) | (singleton(Y!7) = null_class) | (~member(singleton(Y!7), universal_class)) | member(apply(XF!6, singleton(Y!7)), singleton(Y!7)))),
% 2.34/1.76 inference(rewrite,[status(thm)],[])).
% 2.34/1.76 tff(106,plain,
% 2.34/1.76 ((~![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))) | ((singleton(Y!7) = null_class) | (~member(singleton(Y!7), universal_class)) | member(apply(XF!6, singleton(Y!7)), singleton(Y!7)))),
% 2.34/1.76 inference(quant_inst,[status(thm)],[])).
% 2.34/1.76 tff(107,plain,
% 2.34/1.76 ((~![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))) | (singleton(Y!7) = null_class) | (~member(singleton(Y!7), universal_class)) | member(apply(XF!6, singleton(Y!7)), singleton(Y!7))),
% 2.34/1.76 inference(modus_ponens,[status(thm)],[106, 105])).
% 2.34/1.76 tff(108,plain,
% 2.34/1.76 ((singleton(Y!7) = null_class) | (~member(singleton(Y!7), universal_class)) | member(apply(XF!6, singleton(Y!7)), singleton(Y!7))),
% 2.34/1.76 inference(unit_resolution,[status(thm)],[107, 104])).
% 2.34/1.76 tff(109,plain,
% 2.34/1.76 ((singleton(Y!7) = null_class) | member(apply(XF!6, singleton(Y!7)), singleton(Y!7))),
% 2.34/1.76 inference(unit_resolution,[status(thm)],[108, 92])).
% 2.34/1.76 tff(110,plain,
% 2.34/1.76 (singleton(Y!7) = null_class),
% 2.34/1.76 inference(unit_resolution,[status(thm)],[109, 78])).
% 2.34/1.76 tff(111,plain,
% 2.34/1.76 (null_class = singleton(Y!7)),
% 2.34/1.76 inference(symmetry,[status(thm)],[110])).
% 2.34/1.76 tff(112,plain,
% 2.34/1.76 (member(Y!7, null_class) <=> member(Y!7, singleton(Y!7))),
% 2.34/1.76 inference(monotonicity,[status(thm)],[111])).
% 2.34/1.76 tff(113,plain,
% 2.34/1.76 (member(Y!7, singleton(Y!7)) <=> member(Y!7, null_class)),
% 2.34/1.76 inference(symmetry,[status(thm)],[112])).
% 2.34/1.76 tff(114,plain,
% 2.34/1.76 (member(Y!7, unordered_pair(Y!7, Y!7)) <=> member(Y!7, singleton(Y!7))),
% 2.34/1.76 inference(monotonicity,[status(thm)],[31])).
% 2.34/1.76 tff(115,plain,
% 2.34/1.76 (member(Y!7, singleton(Y!7)) <=> member(Y!7, unordered_pair(Y!7, Y!7))),
% 2.34/1.76 inference(symmetry,[status(thm)],[114])).
% 2.34/1.76 tff(116,plain,
% 2.34/1.76 ((~member(Y!7, singleton(Y!7))) <=> (~member(Y!7, unordered_pair(Y!7, Y!7)))),
% 2.34/1.76 inference(monotonicity,[status(thm)],[115])).
% 2.34/1.76 tff(117,assumption,(~member(Y!7, singleton(Y!7))), introduced(assumption)).
% 2.34/1.76 tff(118,plain,
% 2.34/1.76 (~member(Y!7, unordered_pair(Y!7, Y!7))),
% 2.34/1.76 inference(modus_ponens,[status(thm)],[117, 116])).
% 2.34/1.76 tff(119,plain,
% 2.34/1.76 (((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(Y!7, unordered_pair(Y!7, Y!7)) <=> member(Y!7, universal_class))) <=> ((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(Y!7, unordered_pair(Y!7, Y!7)) <=> member(Y!7, universal_class)))),
% 2.34/1.76 inference(rewrite,[status(thm)],[])).
% 2.34/1.76 tff(120,plain,
% 2.34/1.76 ((member(Y!7, unordered_pair(Y!7, Y!7)) <=> (~((~member(Y!7, universal_class)) | (~((Y!7 = Y!7) | (Y!7 = Y!7)))))) <=> (member(Y!7, unordered_pair(Y!7, Y!7)) <=> member(Y!7, universal_class))),
% 2.34/1.76 inference(rewrite,[status(thm)],[])).
% 2.34/1.76 tff(121,plain,
% 2.34/1.76 (((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(Y!7, unordered_pair(Y!7, Y!7)) <=> (~((~member(Y!7, universal_class)) | (~((Y!7 = 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(Y!7, unordered_pair(Y!7, Y!7)) <=> member(Y!7, universal_class)))),
% 2.34/1.76 inference(monotonicity,[status(thm)],[120])).
% 2.34/1.76 tff(122,plain,
% 2.34/1.76 (((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(Y!7, unordered_pair(Y!7, Y!7)) <=> (~((~member(Y!7, universal_class)) | (~((Y!7 = 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(Y!7, unordered_pair(Y!7, Y!7)) <=> member(Y!7, universal_class)))),
% 2.34/1.76 inference(transitivity,[status(thm)],[121, 119])).
% 2.34/1.76 tff(123,plain,
% 2.34/1.76 ((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(Y!7, unordered_pair(Y!7, Y!7)) <=> (~((~member(Y!7, universal_class)) | (~((Y!7 = Y!7) | (Y!7 = Y!7))))))),
% 2.34/1.76 inference(quant_inst,[status(thm)],[])).
% 2.34/1.76 tff(124,plain,
% 2.34/1.76 ((~![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X))))))) | (member(Y!7, unordered_pair(Y!7, Y!7)) <=> member(Y!7, universal_class))),
% 2.34/1.77 inference(modus_ponens,[status(thm)],[123, 122])).
% 2.34/1.77 tff(125,plain,
% 2.34/1.77 (member(Y!7, unordered_pair(Y!7, Y!7)) <=> member(Y!7, universal_class)),
% 2.34/1.77 inference(unit_resolution,[status(thm)],[124, 14])).
% 2.34/1.77 tff(126,plain,
% 2.34/1.77 (member(Y!7, universal_class)),
% 2.34/1.77 inference(and_elim,[status(thm)],[62])).
% 2.34/1.77 tff(127,plain,
% 2.34/1.77 ((~(member(Y!7, unordered_pair(Y!7, Y!7)) <=> member(Y!7, universal_class))) | member(Y!7, unordered_pair(Y!7, Y!7)) | (~member(Y!7, universal_class))),
% 2.34/1.77 inference(tautology,[status(thm)],[])).
% 2.34/1.77 tff(128,plain,
% 2.34/1.77 ((~(member(Y!7, unordered_pair(Y!7, Y!7)) <=> member(Y!7, universal_class))) | member(Y!7, unordered_pair(Y!7, Y!7))),
% 2.34/1.77 inference(unit_resolution,[status(thm)],[127, 126])).
% 2.34/1.77 tff(129,plain,
% 2.34/1.77 (member(Y!7, unordered_pair(Y!7, Y!7))),
% 2.34/1.77 inference(unit_resolution,[status(thm)],[128, 125])).
% 2.34/1.77 tff(130,plain,
% 2.34/1.77 ($false),
% 2.34/1.77 inference(unit_resolution,[status(thm)],[129, 118])).
% 2.34/1.77 tff(131,plain,(member(Y!7, singleton(Y!7))), inference(lemma,lemma(discharge,[]))).
% 2.34/1.77 tff(132,plain,
% 2.34/1.77 (member(Y!7, null_class)),
% 2.34/1.77 inference(modus_ponens,[status(thm)],[131, 113])).
% 2.34/1.77 tff(133,plain,
% 2.34/1.77 (^[X: $i] : refl((~member(X, null_class)) <=> (~member(X, null_class)))),
% 2.34/1.77 inference(bind,[status(th)],[])).
% 2.34/1.77 tff(134,plain,
% 2.34/1.77 (![X: $i] : (~member(X, null_class)) <=> ![X: $i] : (~member(X, null_class))),
% 2.34/1.77 inference(quant_intro,[status(thm)],[133])).
% 2.34/1.77 tff(135,plain,
% 2.34/1.77 (![X: $i] : (~member(X, null_class)) <=> ![X: $i] : (~member(X, null_class))),
% 2.34/1.77 inference(rewrite,[status(thm)],[])).
% 2.34/1.77 tff(136,axiom,(![X: $i] : (~member(X, null_class))), file('/export/starexec/sandbox/benchmark/Axioms/SET005+0.ax','null_class_defn')).
% 2.34/1.77 tff(137,plain,
% 2.34/1.77 (![X: $i] : (~member(X, null_class))),
% 2.34/1.77 inference(modus_ponens,[status(thm)],[136, 135])).
% 2.34/1.77 tff(138,plain,(
% 2.34/1.77 ![X: $i] : (~member(X, null_class))),
% 2.34/1.77 inference(skolemize,[status(sab)],[137])).
% 2.34/1.77 tff(139,plain,
% 2.34/1.77 (![X: $i] : (~member(X, null_class))),
% 2.34/1.77 inference(modus_ponens,[status(thm)],[138, 134])).
% 2.34/1.77 tff(140,plain,
% 2.34/1.77 ((~![X: $i] : (~member(X, null_class))) | (~member(Y!7, null_class))),
% 2.34/1.77 inference(quant_inst,[status(thm)],[])).
% 2.34/1.77 tff(141,plain,
% 2.34/1.77 (~member(Y!7, null_class)),
% 2.34/1.77 inference(unit_resolution,[status(thm)],[140, 139])).
% 2.34/1.77 tff(142,plain,
% 2.34/1.77 ($false),
% 2.34/1.77 inference(unit_resolution,[status(thm)],[141, 132])).
% 2.34/1.77 % SZS output end Proof
%------------------------------------------------------------------------------