TSTP Solution File: SET090+1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET090+1 : TPTP v8.1.0. Bugfixed v7.3.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n019.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 05:05:33 EDT 2022
% Result : Theorem 0.16s 0.37s
% Output : Proof 0.16s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.05/0.10 % Problem : SET090+1 : TPTP v8.1.0. Bugfixed v7.3.0.
% 0.05/0.11 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.10/0.31 % Computer : n019.cluster.edu
% 0.10/0.31 % Model : x86_64 x86_64
% 0.10/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.31 % Memory : 8042.1875MB
% 0.10/0.31 % OS : Linux 3.10.0-693.el7.x86_64
% 0.10/0.31 % CPULimit : 300
% 0.10/0.31 % WCLimit : 300
% 0.10/0.31 % DateTime : Sat Sep 3 02:17:20 EDT 2022
% 0.15/0.31 % CPUTime :
% 0.15/0.31 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.15/0.31 Usage: tptp [options] [-file:]file
% 0.15/0.31 -h, -? prints this message.
% 0.15/0.31 -smt2 print SMT-LIB2 benchmark.
% 0.15/0.31 -m, -model generate model.
% 0.15/0.31 -p, -proof generate proof.
% 0.15/0.31 -c, -core generate unsat core of named formulas.
% 0.15/0.31 -st, -statistics display statistics.
% 0.15/0.31 -t:timeout set timeout (in second).
% 0.15/0.31 -smt2status display status in smt2 format instead of SZS.
% 0.15/0.31 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.15/0.31 -<param>:<value> configuration parameter and value.
% 0.15/0.31 -o:<output-file> file to place output in.
% 0.16/0.37 % SZS status Theorem
% 0.16/0.37 % SZS output start Proof
% 0.16/0.37 tff(tptp_fun_U_7_type, type, (
% 0.16/0.37 tptp_fun_U_7: $i)).
% 0.16/0.37 tff(member_of_type, type, (
% 0.16/0.37 member_of: $i > $i)).
% 0.16/0.37 tff(tptp_fun_X_8_type, type, (
% 0.16/0.37 tptp_fun_X_8: $i)).
% 0.16/0.37 tff(second_type, type, (
% 0.16/0.37 second: $i > $i)).
% 0.16/0.37 tff(ordered_pair_type, type, (
% 0.16/0.37 ordered_pair: ( $i * $i ) > $i)).
% 0.16/0.37 tff(singleton_type, type, (
% 0.16/0.37 singleton: $i > $i)).
% 0.16/0.37 tff(first_type, type, (
% 0.16/0.37 first: $i > $i)).
% 0.16/0.37 tff(member_type, type, (
% 0.16/0.37 member: ( $i * $i ) > $o)).
% 0.16/0.37 tff(universal_class_type, type, (
% 0.16/0.37 universal_class: $i)).
% 0.16/0.37 tff(unordered_pair_type, type, (
% 0.16/0.37 unordered_pair: ( $i * $i ) > $i)).
% 0.16/0.37 tff(1,plain,
% 0.16/0.37 (^[X: $i] : refl((singleton(X) = unordered_pair(X, X)) <=> (singleton(X) = unordered_pair(X, X)))),
% 0.16/0.37 inference(bind,[status(th)],[])).
% 0.16/0.37 tff(2,plain,
% 0.16/0.37 (![X: $i] : (singleton(X) = unordered_pair(X, X)) <=> ![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.16/0.37 inference(quant_intro,[status(thm)],[1])).
% 0.16/0.37 tff(3,plain,
% 0.16/0.37 (![X: $i] : (singleton(X) = unordered_pair(X, X)) <=> ![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.16/0.37 inference(rewrite,[status(thm)],[])).
% 0.16/0.37 tff(4,axiom,(![X: $i] : (singleton(X) = unordered_pair(X, X))), file('/export/starexec/sandbox/benchmark/Axioms/SET005+0.ax','singleton_set_defn')).
% 0.16/0.37 tff(5,plain,
% 0.16/0.37 (![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.16/0.37 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.16/0.37 tff(6,plain,(
% 0.16/0.37 ![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.16/0.37 inference(skolemize,[status(sab)],[5])).
% 0.16/0.37 tff(7,plain,
% 0.16/0.37 (![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.16/0.37 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.16/0.37 tff(8,plain,
% 0.16/0.37 ((~![X: $i] : (singleton(X) = unordered_pair(X, X))) | (singleton(U!7) = unordered_pair(U!7, U!7))),
% 0.16/0.37 inference(quant_inst,[status(thm)],[])).
% 0.16/0.37 tff(9,plain,
% 0.16/0.37 (singleton(U!7) = unordered_pair(U!7, U!7)),
% 0.16/0.37 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.16/0.37 tff(10,plain,
% 0.16/0.37 (member(singleton(U!7), universal_class) <=> member(unordered_pair(U!7, U!7), universal_class)),
% 0.16/0.37 inference(monotonicity,[status(thm)],[9])).
% 0.16/0.37 tff(11,plain,
% 0.16/0.37 (member(unordered_pair(U!7, U!7), universal_class) <=> member(singleton(U!7), universal_class)),
% 0.16/0.37 inference(symmetry,[status(thm)],[10])).
% 0.16/0.37 tff(12,plain,
% 0.16/0.37 (^[X: $i, Y: $i] : refl(member(unordered_pair(X, Y), universal_class) <=> member(unordered_pair(X, Y), universal_class))),
% 0.16/0.37 inference(bind,[status(th)],[])).
% 0.16/0.37 tff(13,plain,
% 0.16/0.37 (![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class) <=> ![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.16/0.37 inference(quant_intro,[status(thm)],[12])).
% 0.16/0.37 tff(14,plain,
% 0.16/0.37 (![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class) <=> ![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.16/0.37 inference(rewrite,[status(thm)],[])).
% 0.16/0.37 tff(15,axiom,(![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)), file('/export/starexec/sandbox/benchmark/Axioms/SET005+0.ax','unordered_pair')).
% 0.16/0.37 tff(16,plain,
% 0.16/0.37 (![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.16/0.37 inference(modus_ponens,[status(thm)],[15, 14])).
% 0.16/0.37 tff(17,plain,(
% 0.16/0.37 ![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.16/0.37 inference(skolemize,[status(sab)],[16])).
% 0.16/0.37 tff(18,plain,
% 0.16/0.37 (![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.16/0.37 inference(modus_ponens,[status(thm)],[17, 13])).
% 0.16/0.37 tff(19,plain,
% 0.16/0.37 ((~![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(U!7, U!7), universal_class)),
% 0.16/0.37 inference(quant_inst,[status(thm)],[])).
% 0.16/0.37 tff(20,plain,
% 0.16/0.37 (member(unordered_pair(U!7, U!7), universal_class)),
% 0.16/0.37 inference(unit_resolution,[status(thm)],[19, 18])).
% 0.16/0.37 tff(21,plain,
% 0.16/0.37 (member(singleton(U!7), universal_class)),
% 0.16/0.37 inference(modus_ponens,[status(thm)],[20, 11])).
% 0.16/0.37 tff(22,plain,
% 0.16/0.37 ((~![X: $i, U: $i] : ((~(member(U, universal_class) & (X = singleton(U)))) | (member_of(X) = U))) <=> (~![X: $i, U: $i] : ((~(member(U, universal_class) & (X = singleton(U)))) | (member_of(X) = U)))),
% 0.16/0.37 inference(rewrite,[status(thm)],[])).
% 0.16/0.37 tff(23,plain,
% 0.16/0.37 ((~![X: $i, U: $i] : ((member(U, universal_class) & (X = singleton(U))) => (member_of(X) = U))) <=> (~![X: $i, U: $i] : ((~(member(U, universal_class) & (X = singleton(U)))) | (member_of(X) = U)))),
% 0.16/0.38 inference(rewrite,[status(thm)],[])).
% 0.16/0.38 tff(24,axiom,(~![X: $i, U: $i] : ((member(U, universal_class) & (X = singleton(U))) => (member_of(X) = U))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','member_of_singleton')).
% 0.16/0.38 tff(25,plain,
% 0.16/0.38 (~![X: $i, U: $i] : ((~(member(U, universal_class) & (X = singleton(U)))) | (member_of(X) = U))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[24, 23])).
% 0.16/0.38 tff(26,plain,
% 0.16/0.38 (~![X: $i, U: $i] : ((~(member(U, universal_class) & (X = singleton(U)))) | (member_of(X) = U))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[25, 22])).
% 0.16/0.38 tff(27,plain,
% 0.16/0.38 (~![X: $i, U: $i] : ((~(member(U, universal_class) & (X = singleton(U)))) | (member_of(X) = U))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[26, 22])).
% 0.16/0.38 tff(28,plain,
% 0.16/0.38 (~![X: $i, U: $i] : ((~(member(U, universal_class) & (X = singleton(U)))) | (member_of(X) = U))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[27, 22])).
% 0.16/0.38 tff(29,plain,
% 0.16/0.38 (~![X: $i, U: $i] : ((~(member(U, universal_class) & (X = singleton(U)))) | (member_of(X) = U))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[28, 22])).
% 0.16/0.38 tff(30,plain,
% 0.16/0.38 (~![X: $i, U: $i] : ((~(member(U, universal_class) & (X = singleton(U)))) | (member_of(X) = U))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[29, 22])).
% 0.16/0.38 tff(31,plain,
% 0.16/0.38 (~![X: $i, U: $i] : ((~(member(U, universal_class) & (X = singleton(U)))) | (member_of(X) = U))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[30, 22])).
% 0.16/0.38 tff(32,plain,(
% 0.16/0.38 ~((~(member(U!7, universal_class) & (X!8 = singleton(U!7)))) | (member_of(X!8) = U!7))),
% 0.16/0.38 inference(skolemize,[status(sab)],[31])).
% 0.16/0.38 tff(33,plain,
% 0.16/0.38 (member(U!7, universal_class) & (X!8 = singleton(U!7))),
% 0.16/0.38 inference(or_elim,[status(thm)],[32])).
% 0.16/0.38 tff(34,plain,
% 0.16/0.38 (member(U!7, universal_class)),
% 0.16/0.38 inference(and_elim,[status(thm)],[33])).
% 0.16/0.38 tff(35,plain,
% 0.16/0.38 (^[X: $i, Y: $i] : refl(((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y))))) <=> ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y))))))),
% 0.16/0.38 inference(bind,[status(th)],[])).
% 0.16/0.38 tff(36,plain,
% 0.16/0.38 (![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y))))) <=> ![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))),
% 0.16/0.38 inference(quant_intro,[status(thm)],[35])).
% 0.16/0.38 tff(37,plain,
% 0.16/0.38 (^[X: $i, Y: $i] : trans(monotonicity(trans(monotonicity(rewrite((member(X, universal_class) & member(Y, universal_class)) <=> (~((~member(Y, universal_class)) | (~member(X, universal_class))))), ((~(member(X, universal_class) & member(Y, universal_class))) <=> (~(~((~member(Y, universal_class)) | (~member(X, universal_class))))))), rewrite((~(~((~member(Y, universal_class)) | (~member(X, universal_class))))) <=> ((~member(Y, universal_class)) | (~member(X, universal_class)))), ((~(member(X, universal_class) & member(Y, universal_class))) <=> ((~member(Y, universal_class)) | (~member(X, universal_class))))), rewrite(((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y)) <=> (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y))))), (((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y))) <=> (((~member(Y, universal_class)) | (~member(X, universal_class))) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y))))))), rewrite((((~member(Y, universal_class)) | (~member(X, universal_class))) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y))))) <=> ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))), (((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y))) <=> ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))))),
% 0.16/0.38 inference(bind,[status(th)],[])).
% 0.16/0.38 tff(38,plain,
% 0.16/0.38 (![X: $i, Y: $i] : ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y))) <=> ![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))),
% 0.16/0.38 inference(quant_intro,[status(thm)],[37])).
% 0.16/0.38 tff(39,plain,
% 0.16/0.38 (![X: $i, Y: $i] : ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y))) <=> ![X: $i, Y: $i] : ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y)))),
% 0.16/0.38 inference(rewrite,[status(thm)],[])).
% 0.16/0.38 tff(40,plain,
% 0.16/0.38 (^[X: $i, Y: $i] : rewrite(((member(X, universal_class) & member(Y, universal_class)) => ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y))) <=> ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y))))),
% 0.16/0.38 inference(bind,[status(th)],[])).
% 0.16/0.38 tff(41,plain,
% 0.16/0.38 (![X: $i, Y: $i] : ((member(X, universal_class) & member(Y, universal_class)) => ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y))) <=> ![X: $i, Y: $i] : ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y)))),
% 0.16/0.38 inference(quant_intro,[status(thm)],[40])).
% 0.16/0.38 tff(42,axiom,(![X: $i, Y: $i] : ((member(X, universal_class) & member(Y, universal_class)) => ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y)))), file('/export/starexec/sandbox/benchmark/Axioms/SET005+0.ax','first_second')).
% 0.16/0.38 tff(43,plain,
% 0.16/0.38 (![X: $i, Y: $i] : ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y)))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[42, 41])).
% 0.16/0.38 tff(44,plain,
% 0.16/0.38 (![X: $i, Y: $i] : ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y)))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[43, 39])).
% 0.16/0.38 tff(45,plain,(
% 0.16/0.38 ![X: $i, Y: $i] : ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y)))),
% 0.16/0.38 inference(skolemize,[status(sab)],[44])).
% 0.16/0.38 tff(46,plain,
% 0.16/0.38 (![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[45, 38])).
% 0.16/0.38 tff(47,plain,
% 0.16/0.38 (![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[46, 36])).
% 0.16/0.38 tff(48,plain,
% 0.16/0.38 (((~![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))) | ((~member(U!7, universal_class)) | (~member(singleton(U!7), universal_class)) | (~((~(first(ordered_pair(singleton(U!7), U!7)) = singleton(U!7))) | (~(second(ordered_pair(singleton(U!7), U!7)) = U!7)))))) <=> ((~![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))) | (~member(U!7, universal_class)) | (~member(singleton(U!7), universal_class)) | (~((~(first(ordered_pair(singleton(U!7), U!7)) = singleton(U!7))) | (~(second(ordered_pair(singleton(U!7), U!7)) = U!7)))))),
% 0.16/0.38 inference(rewrite,[status(thm)],[])).
% 0.16/0.38 tff(49,plain,
% 0.16/0.38 ((~![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))) | ((~member(U!7, universal_class)) | (~member(singleton(U!7), universal_class)) | (~((~(first(ordered_pair(singleton(U!7), U!7)) = singleton(U!7))) | (~(second(ordered_pair(singleton(U!7), U!7)) = U!7)))))),
% 0.16/0.38 inference(quant_inst,[status(thm)],[])).
% 0.16/0.38 tff(50,plain,
% 0.16/0.38 ((~![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))) | (~member(U!7, universal_class)) | (~member(singleton(U!7), universal_class)) | (~((~(first(ordered_pair(singleton(U!7), U!7)) = singleton(U!7))) | (~(second(ordered_pair(singleton(U!7), U!7)) = U!7))))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[49, 48])).
% 0.16/0.38 tff(51,plain,
% 0.16/0.38 (~((~(first(ordered_pair(singleton(U!7), U!7)) = singleton(U!7))) | (~(second(ordered_pair(singleton(U!7), U!7)) = U!7)))),
% 0.16/0.38 inference(unit_resolution,[status(thm)],[50, 47, 34, 21])).
% 0.16/0.38 tff(52,plain,
% 0.16/0.38 (((~(first(ordered_pair(singleton(U!7), U!7)) = singleton(U!7))) | (~(second(ordered_pair(singleton(U!7), U!7)) = U!7))) | (second(ordered_pair(singleton(U!7), U!7)) = U!7)),
% 0.16/0.38 inference(tautology,[status(thm)],[])).
% 0.16/0.38 tff(53,plain,
% 0.16/0.38 (second(ordered_pair(singleton(U!7), U!7)) = U!7),
% 0.16/0.38 inference(unit_resolution,[status(thm)],[52, 51])).
% 0.16/0.38 tff(54,plain,
% 0.16/0.38 (^[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)))))),
% 0.16/0.38 inference(bind,[status(th)],[])).
% 0.16/0.38 tff(55,plain,
% 0.16/0.38 (![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))))),
% 0.16/0.38 inference(quant_intro,[status(thm)],[54])).
% 0.16/0.38 tff(56,plain,
% 0.16/0.38 (![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))))),
% 0.16/0.38 inference(rewrite,[status(thm)],[])).
% 0.16/0.38 tff(57,axiom,(![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))), file('/export/starexec/sandbox/benchmark/Axioms/SET005+0.ax','ordered_pair_defn')).
% 0.16/0.38 tff(58,plain,
% 0.16/0.38 (![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[57, 56])).
% 0.16/0.38 tff(59,plain,(
% 0.16/0.38 ![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))),
% 0.16/0.38 inference(skolemize,[status(sab)],[58])).
% 0.16/0.38 tff(60,plain,
% 0.16/0.38 (![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[59, 55])).
% 0.16/0.38 tff(61,plain,
% 0.16/0.38 ((~![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))) | (ordered_pair(singleton(U!7), member_of(X!8)) = unordered_pair(singleton(singleton(U!7)), unordered_pair(singleton(U!7), singleton(member_of(X!8)))))),
% 0.16/0.38 inference(quant_inst,[status(thm)],[])).
% 0.16/0.38 tff(62,plain,
% 0.16/0.38 (ordered_pair(singleton(U!7), member_of(X!8)) = unordered_pair(singleton(singleton(U!7)), unordered_pair(singleton(U!7), singleton(member_of(X!8))))),
% 0.16/0.38 inference(unit_resolution,[status(thm)],[61, 60])).
% 0.16/0.38 tff(63,plain,
% 0.16/0.38 (unordered_pair(singleton(singleton(U!7)), unordered_pair(singleton(U!7), singleton(member_of(X!8)))) = ordered_pair(singleton(U!7), member_of(X!8))),
% 0.16/0.38 inference(symmetry,[status(thm)],[62])).
% 0.16/0.38 tff(64,plain,
% 0.16/0.38 (^[Y: $i] : refl(((~member(Y, universal_class)) | (singleton(member_of(singleton(Y))) = singleton(Y))) <=> ((~member(Y, universal_class)) | (singleton(member_of(singleton(Y))) = singleton(Y))))),
% 0.16/0.38 inference(bind,[status(th)],[])).
% 0.16/0.38 tff(65,plain,
% 0.16/0.38 (![Y: $i] : ((~member(Y, universal_class)) | (singleton(member_of(singleton(Y))) = singleton(Y))) <=> ![Y: $i] : ((~member(Y, universal_class)) | (singleton(member_of(singleton(Y))) = singleton(Y)))),
% 0.16/0.38 inference(quant_intro,[status(thm)],[64])).
% 0.16/0.38 tff(66,plain,
% 0.16/0.38 (![Y: $i] : ((~member(Y, universal_class)) | (singleton(member_of(singleton(Y))) = singleton(Y))) <=> ![Y: $i] : ((~member(Y, universal_class)) | (singleton(member_of(singleton(Y))) = singleton(Y)))),
% 0.16/0.38 inference(rewrite,[status(thm)],[])).
% 0.16/0.38 tff(67,plain,
% 0.16/0.38 (^[Y: $i] : rewrite((member(Y, universal_class) => (singleton(member_of(singleton(Y))) = singleton(Y))) <=> ((~member(Y, universal_class)) | (singleton(member_of(singleton(Y))) = singleton(Y))))),
% 0.16/0.38 inference(bind,[status(th)],[])).
% 0.16/0.38 tff(68,plain,
% 0.16/0.38 (![Y: $i] : (member(Y, universal_class) => (singleton(member_of(singleton(Y))) = singleton(Y))) <=> ![Y: $i] : ((~member(Y, universal_class)) | (singleton(member_of(singleton(Y))) = singleton(Y)))),
% 0.16/0.38 inference(quant_intro,[status(thm)],[67])).
% 0.16/0.38 tff(69,axiom,(![Y: $i] : (member(Y, universal_class) => (singleton(member_of(singleton(Y))) = singleton(Y)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','member_singleton_singleton')).
% 0.16/0.38 tff(70,plain,
% 0.16/0.38 (![Y: $i] : ((~member(Y, universal_class)) | (singleton(member_of(singleton(Y))) = singleton(Y)))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[69, 68])).
% 0.16/0.38 tff(71,plain,
% 0.16/0.38 (![Y: $i] : ((~member(Y, universal_class)) | (singleton(member_of(singleton(Y))) = singleton(Y)))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[70, 66])).
% 0.16/0.38 tff(72,plain,(
% 0.16/0.38 ![Y: $i] : ((~member(Y, universal_class)) | (singleton(member_of(singleton(Y))) = singleton(Y)))),
% 0.16/0.38 inference(skolemize,[status(sab)],[71])).
% 0.16/0.38 tff(73,plain,
% 0.16/0.38 (![Y: $i] : ((~member(Y, universal_class)) | (singleton(member_of(singleton(Y))) = singleton(Y)))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[72, 65])).
% 0.16/0.38 tff(74,plain,
% 0.16/0.38 (((~![Y: $i] : ((~member(Y, universal_class)) | (singleton(member_of(singleton(Y))) = singleton(Y)))) | ((~member(U!7, universal_class)) | (singleton(member_of(singleton(U!7))) = singleton(U!7)))) <=> ((~![Y: $i] : ((~member(Y, universal_class)) | (singleton(member_of(singleton(Y))) = singleton(Y)))) | (~member(U!7, universal_class)) | (singleton(member_of(singleton(U!7))) = singleton(U!7)))),
% 0.16/0.38 inference(rewrite,[status(thm)],[])).
% 0.16/0.38 tff(75,plain,
% 0.16/0.38 ((~![Y: $i] : ((~member(Y, universal_class)) | (singleton(member_of(singleton(Y))) = singleton(Y)))) | ((~member(U!7, universal_class)) | (singleton(member_of(singleton(U!7))) = singleton(U!7)))),
% 0.16/0.38 inference(quant_inst,[status(thm)],[])).
% 0.16/0.38 tff(76,plain,
% 0.16/0.38 ((~![Y: $i] : ((~member(Y, universal_class)) | (singleton(member_of(singleton(Y))) = singleton(Y)))) | (~member(U!7, universal_class)) | (singleton(member_of(singleton(U!7))) = singleton(U!7))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[75, 74])).
% 0.16/0.38 tff(77,plain,
% 0.16/0.38 (singleton(member_of(singleton(U!7))) = singleton(U!7)),
% 0.16/0.38 inference(unit_resolution,[status(thm)],[76, 73, 34])).
% 0.16/0.38 tff(78,plain,
% 0.16/0.38 (X!8 = singleton(U!7)),
% 0.16/0.38 inference(and_elim,[status(thm)],[33])).
% 0.16/0.38 tff(79,plain,
% 0.16/0.38 (singleton(U!7) = X!8),
% 0.16/0.38 inference(symmetry,[status(thm)],[78])).
% 0.16/0.38 tff(80,plain,
% 0.16/0.38 (member_of(singleton(U!7)) = member_of(X!8)),
% 0.16/0.38 inference(monotonicity,[status(thm)],[79])).
% 0.16/0.38 tff(81,plain,
% 0.16/0.38 (member_of(X!8) = member_of(singleton(U!7))),
% 0.16/0.38 inference(symmetry,[status(thm)],[80])).
% 0.16/0.38 tff(82,plain,
% 0.16/0.38 (singleton(member_of(X!8)) = singleton(member_of(singleton(U!7)))),
% 0.16/0.38 inference(monotonicity,[status(thm)],[81])).
% 0.16/0.38 tff(83,plain,
% 0.16/0.38 (singleton(member_of(X!8)) = singleton(U!7)),
% 0.16/0.38 inference(transitivity,[status(thm)],[82, 77])).
% 0.16/0.38 tff(84,plain,
% 0.16/0.38 (unordered_pair(singleton(U!7), singleton(member_of(X!8))) = unordered_pair(singleton(U!7), singleton(U!7))),
% 0.16/0.38 inference(monotonicity,[status(thm)],[83])).
% 0.16/0.38 tff(85,plain,
% 0.16/0.38 (unordered_pair(singleton(singleton(U!7)), unordered_pair(singleton(U!7), singleton(member_of(X!8)))) = unordered_pair(singleton(singleton(U!7)), unordered_pair(singleton(U!7), singleton(U!7)))),
% 0.16/0.38 inference(monotonicity,[status(thm)],[84])).
% 0.16/0.38 tff(86,plain,
% 0.16/0.38 (unordered_pair(singleton(singleton(U!7)), unordered_pair(singleton(U!7), singleton(U!7))) = unordered_pair(singleton(singleton(U!7)), unordered_pair(singleton(U!7), singleton(member_of(X!8))))),
% 0.16/0.38 inference(symmetry,[status(thm)],[85])).
% 0.16/0.38 tff(87,plain,
% 0.16/0.38 ((~![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))) | (ordered_pair(singleton(U!7), U!7) = unordered_pair(singleton(singleton(U!7)), unordered_pair(singleton(U!7), singleton(U!7))))),
% 0.16/0.38 inference(quant_inst,[status(thm)],[])).
% 0.16/0.38 tff(88,plain,
% 0.16/0.38 (ordered_pair(singleton(U!7), U!7) = unordered_pair(singleton(singleton(U!7)), unordered_pair(singleton(U!7), singleton(U!7)))),
% 0.16/0.38 inference(unit_resolution,[status(thm)],[87, 60])).
% 0.16/0.38 tff(89,plain,
% 0.16/0.38 (ordered_pair(singleton(U!7), U!7) = ordered_pair(singleton(U!7), member_of(X!8))),
% 0.16/0.38 inference(transitivity,[status(thm)],[88, 86, 63])).
% 0.16/0.38 tff(90,plain,
% 0.16/0.38 (second(ordered_pair(singleton(U!7), U!7)) = second(ordered_pair(singleton(U!7), member_of(X!8)))),
% 0.16/0.38 inference(monotonicity,[status(thm)],[89])).
% 0.16/0.38 tff(91,plain,
% 0.16/0.38 (second(ordered_pair(singleton(U!7), member_of(X!8))) = second(ordered_pair(singleton(U!7), U!7))),
% 0.16/0.38 inference(symmetry,[status(thm)],[90])).
% 0.16/0.38 tff(92,plain,
% 0.16/0.38 (member(member_of(X!8), universal_class) <=> member(member_of(singleton(U!7)), universal_class)),
% 0.16/0.38 inference(monotonicity,[status(thm)],[81])).
% 0.16/0.38 tff(93,plain,
% 0.16/0.38 (member(member_of(singleton(U!7)), universal_class) <=> member(member_of(X!8), universal_class)),
% 0.16/0.38 inference(symmetry,[status(thm)],[92])).
% 0.16/0.38 tff(94,plain,
% 0.16/0.38 (^[Y: $i] : refl(((~member(Y, universal_class)) | member(member_of(singleton(Y)), universal_class)) <=> ((~member(Y, universal_class)) | member(member_of(singleton(Y)), universal_class)))),
% 0.16/0.38 inference(bind,[status(th)],[])).
% 0.16/0.38 tff(95,plain,
% 0.16/0.38 (![Y: $i] : ((~member(Y, universal_class)) | member(member_of(singleton(Y)), universal_class)) <=> ![Y: $i] : ((~member(Y, universal_class)) | member(member_of(singleton(Y)), universal_class))),
% 0.16/0.38 inference(quant_intro,[status(thm)],[94])).
% 0.16/0.38 tff(96,plain,
% 0.16/0.38 (![Y: $i] : ((~member(Y, universal_class)) | member(member_of(singleton(Y)), universal_class)) <=> ![Y: $i] : ((~member(Y, universal_class)) | member(member_of(singleton(Y)), universal_class))),
% 0.16/0.38 inference(rewrite,[status(thm)],[])).
% 0.16/0.38 tff(97,plain,
% 0.16/0.38 (^[Y: $i] : rewrite((member(Y, universal_class) => member(member_of(singleton(Y)), universal_class)) <=> ((~member(Y, universal_class)) | member(member_of(singleton(Y)), universal_class)))),
% 0.16/0.38 inference(bind,[status(th)],[])).
% 0.16/0.38 tff(98,plain,
% 0.16/0.38 (![Y: $i] : (member(Y, universal_class) => member(member_of(singleton(Y)), universal_class)) <=> ![Y: $i] : ((~member(Y, universal_class)) | member(member_of(singleton(Y)), universal_class))),
% 0.16/0.38 inference(quant_intro,[status(thm)],[97])).
% 0.16/0.38 tff(99,axiom,(![Y: $i] : (member(Y, universal_class) => member(member_of(singleton(Y)), universal_class))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','member_singleton_universal')).
% 0.16/0.38 tff(100,plain,
% 0.16/0.38 (![Y: $i] : ((~member(Y, universal_class)) | member(member_of(singleton(Y)), universal_class))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[99, 98])).
% 0.16/0.38 tff(101,plain,
% 0.16/0.38 (![Y: $i] : ((~member(Y, universal_class)) | member(member_of(singleton(Y)), universal_class))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[100, 96])).
% 0.16/0.38 tff(102,plain,(
% 0.16/0.38 ![Y: $i] : ((~member(Y, universal_class)) | member(member_of(singleton(Y)), universal_class))),
% 0.16/0.38 inference(skolemize,[status(sab)],[101])).
% 0.16/0.38 tff(103,plain,
% 0.16/0.38 (![Y: $i] : ((~member(Y, universal_class)) | member(member_of(singleton(Y)), universal_class))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[102, 95])).
% 0.16/0.38 tff(104,plain,
% 0.16/0.38 (((~![Y: $i] : ((~member(Y, universal_class)) | member(member_of(singleton(Y)), universal_class))) | ((~member(U!7, universal_class)) | member(member_of(singleton(U!7)), universal_class))) <=> ((~![Y: $i] : ((~member(Y, universal_class)) | member(member_of(singleton(Y)), universal_class))) | (~member(U!7, universal_class)) | member(member_of(singleton(U!7)), universal_class))),
% 0.16/0.38 inference(rewrite,[status(thm)],[])).
% 0.16/0.38 tff(105,plain,
% 0.16/0.38 ((~![Y: $i] : ((~member(Y, universal_class)) | member(member_of(singleton(Y)), universal_class))) | ((~member(U!7, universal_class)) | member(member_of(singleton(U!7)), universal_class))),
% 0.16/0.39 inference(quant_inst,[status(thm)],[])).
% 0.16/0.39 tff(106,plain,
% 0.16/0.39 ((~![Y: $i] : ((~member(Y, universal_class)) | member(member_of(singleton(Y)), universal_class))) | (~member(U!7, universal_class)) | member(member_of(singleton(U!7)), universal_class)),
% 0.16/0.39 inference(modus_ponens,[status(thm)],[105, 104])).
% 0.16/0.39 tff(107,plain,
% 0.16/0.39 (member(member_of(singleton(U!7)), universal_class)),
% 0.16/0.39 inference(unit_resolution,[status(thm)],[106, 103, 34])).
% 0.16/0.39 tff(108,plain,
% 0.16/0.39 (member(member_of(X!8), universal_class)),
% 0.16/0.39 inference(modus_ponens,[status(thm)],[107, 93])).
% 0.16/0.39 tff(109,plain,
% 0.16/0.39 (((~![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))) | ((~member(member_of(X!8), universal_class)) | (~member(singleton(U!7), universal_class)) | (~((~(first(ordered_pair(singleton(U!7), member_of(X!8))) = singleton(U!7))) | (~(second(ordered_pair(singleton(U!7), member_of(X!8))) = member_of(X!8))))))) <=> ((~![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))) | (~member(member_of(X!8), universal_class)) | (~member(singleton(U!7), universal_class)) | (~((~(first(ordered_pair(singleton(U!7), member_of(X!8))) = singleton(U!7))) | (~(second(ordered_pair(singleton(U!7), member_of(X!8))) = member_of(X!8))))))),
% 0.16/0.39 inference(rewrite,[status(thm)],[])).
% 0.16/0.39 tff(110,plain,
% 0.16/0.39 ((~![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))) | ((~member(member_of(X!8), universal_class)) | (~member(singleton(U!7), universal_class)) | (~((~(first(ordered_pair(singleton(U!7), member_of(X!8))) = singleton(U!7))) | (~(second(ordered_pair(singleton(U!7), member_of(X!8))) = member_of(X!8))))))),
% 0.16/0.39 inference(quant_inst,[status(thm)],[])).
% 0.16/0.39 tff(111,plain,
% 0.16/0.39 ((~![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))) | (~member(member_of(X!8), universal_class)) | (~member(singleton(U!7), universal_class)) | (~((~(first(ordered_pair(singleton(U!7), member_of(X!8))) = singleton(U!7))) | (~(second(ordered_pair(singleton(U!7), member_of(X!8))) = member_of(X!8)))))),
% 0.16/0.39 inference(modus_ponens,[status(thm)],[110, 109])).
% 0.16/0.39 tff(112,plain,
% 0.16/0.39 (~((~(first(ordered_pair(singleton(U!7), member_of(X!8))) = singleton(U!7))) | (~(second(ordered_pair(singleton(U!7), member_of(X!8))) = member_of(X!8))))),
% 0.16/0.39 inference(unit_resolution,[status(thm)],[111, 47, 108, 21])).
% 0.16/0.39 tff(113,plain,
% 0.16/0.39 (((~(first(ordered_pair(singleton(U!7), member_of(X!8))) = singleton(U!7))) | (~(second(ordered_pair(singleton(U!7), member_of(X!8))) = member_of(X!8)))) | (second(ordered_pair(singleton(U!7), member_of(X!8))) = member_of(X!8))),
% 0.16/0.39 inference(tautology,[status(thm)],[])).
% 0.16/0.39 tff(114,plain,
% 0.16/0.39 (second(ordered_pair(singleton(U!7), member_of(X!8))) = member_of(X!8)),
% 0.16/0.39 inference(unit_resolution,[status(thm)],[113, 112])).
% 0.16/0.39 tff(115,plain,
% 0.16/0.39 (member_of(X!8) = second(ordered_pair(singleton(U!7), member_of(X!8)))),
% 0.16/0.39 inference(symmetry,[status(thm)],[114])).
% 0.16/0.39 tff(116,plain,
% 0.16/0.39 (member_of(X!8) = U!7),
% 0.16/0.39 inference(transitivity,[status(thm)],[115, 91, 53])).
% 0.16/0.39 tff(117,plain,
% 0.16/0.39 (~(member_of(X!8) = U!7)),
% 0.16/0.39 inference(or_elim,[status(thm)],[32])).
% 0.16/0.39 tff(118,plain,
% 0.16/0.39 ($false),
% 0.16/0.39 inference(unit_resolution,[status(thm)],[117, 116])).
% 0.16/0.39 % SZS output end Proof
%------------------------------------------------------------------------------