%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET090+1 : TPTP v9.0.0. Bugfixed v7.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n005.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:25:10 AM UTC 2025
% Result : Theorem 0.19s 0.38s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SET090+1 : TPTP v9.0.0. Bugfixed v7.3.0.
% 0.03/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n005.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Jun 20 04:15:05 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.19/0.38 % SZS status Theorem
% 0.19/0.38 % SZS output start Proof
% 0.19/0.38 tff(tptp_fun_U_7_type, type, (
% 0.19/0.38 tptp_fun_U_7: $i)).
% 0.19/0.38 tff(member_of_type, type, (
% 0.19/0.38 member_of: $i > $i)).
% 0.19/0.38 tff(singleton_type, type, (
% 0.19/0.38 singleton: $i > $i)).
% 0.19/0.38 tff(second_type, type, (
% 0.19/0.38 second: $i > $i)).
% 0.19/0.38 tff(ordered_pair_type, type, (
% 0.19/0.38 ordered_pair: ( $i * $i ) > $i)).
% 0.19/0.38 tff(first_type, type, (
% 0.19/0.38 first: $i > $i)).
% 0.19/0.38 tff(member_type, type, (
% 0.19/0.38 member: ( $i * $i ) > $o)).
% 0.19/0.38 tff(universal_class_type, type, (
% 0.19/0.38 universal_class: $i)).
% 0.19/0.38 tff(unordered_pair_type, type, (
% 0.19/0.38 unordered_pair: ( $i * $i ) > $i)).
% 0.19/0.38 tff(1,plain,
% 0.19/0.38 (^[X: $i] : refl((singleton(X) = unordered_pair(X, X)) <=> (singleton(X) = unordered_pair(X, X)))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(2,plain,
% 0.19/0.38 (![X: $i] : (singleton(X) = unordered_pair(X, X)) <=> ![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[1])).
% 0.19/0.38 tff(3,plain,
% 0.19/0.38 (![X: $i] : (singleton(X) = unordered_pair(X, X)) <=> ![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(4,axiom,(![X: $i] : (singleton(X) = unordered_pair(X, X))), file('/export/starexec/sandbox2/benchmark/Axioms/SET005+0.ax','singleton_set_defn')).
% 0.19/0.39 tff(5,plain,
% 0.19/0.39 (![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.19/0.39 tff(6,plain,(
% 0.19/0.39 ![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.19/0.39 inference(skolemize,[status(sab)],[5])).
% 0.19/0.39 tff(7,plain,
% 0.19/0.39 (![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.19/0.39 tff(8,plain,
% 0.19/0.39 ((~![X: $i] : (singleton(X) = unordered_pair(X, X))) | (singleton(U!7) = unordered_pair(U!7, U!7))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(9,plain,
% 0.19/0.39 (singleton(U!7) = unordered_pair(U!7, U!7)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.19/0.39 tff(10,plain,
% 0.19/0.39 (member(singleton(U!7), universal_class) <=> member(unordered_pair(U!7, U!7), universal_class)),
% 0.19/0.39 inference(monotonicity,[status(thm)],[9])).
% 0.19/0.39 tff(11,plain,
% 0.19/0.39 (member(unordered_pair(U!7, U!7), universal_class) <=> member(singleton(U!7), universal_class)),
% 0.19/0.39 inference(symmetry,[status(thm)],[10])).
% 0.19/0.39 tff(12,plain,
% 0.19/0.39 (^[X: $i, Y: $i] : refl(member(unordered_pair(X, Y), universal_class) <=> member(unordered_pair(X, Y), universal_class))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(13,plain,
% 0.19/0.39 (![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class) <=> ![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.19/0.39 inference(quant_intro,[status(thm)],[12])).
% 0.19/0.39 tff(14,plain,
% 0.19/0.39 (![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class) <=> ![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(15,axiom,(![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)), file('/export/starexec/sandbox2/benchmark/Axioms/SET005+0.ax','unordered_pair')).
% 0.19/0.39 tff(16,plain,
% 0.19/0.39 (![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[15, 14])).
% 0.19/0.39 tff(17,plain,(
% 0.19/0.39 ![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.19/0.39 inference(skolemize,[status(sab)],[16])).
% 0.19/0.39 tff(18,plain,
% 0.19/0.39 (![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[17, 13])).
% 0.19/0.39 tff(19,plain,
% 0.19/0.39 ((~![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(U!7, U!7), universal_class)),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(20,plain,
% 0.19/0.39 (member(unordered_pair(U!7, U!7), universal_class)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[19, 18])).
% 0.19/0.39 tff(21,plain,
% 0.19/0.39 (member(singleton(U!7), universal_class)),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[20, 11])).
% 0.19/0.39 tff(22,plain,
% 0.19/0.39 ((~![U: $i] : ((~member(U, universal_class)) | (member_of(singleton(U)) = U))) <=> (~![U: $i] : ((~member(U, universal_class)) | (member_of(singleton(U)) = U)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(23,plain,
% 0.19/0.39 ((~![X: $i, U: $i] : ((member(U, universal_class) & (X = singleton(U))) => (member_of(X) = U))) <=> (~![U: $i] : ((~member(U, universal_class)) | (member_of(singleton(U)) = U)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(24,axiom,(~![X: $i, U: $i] : ((member(U, universal_class) & (X = singleton(U))) => (member_of(X) = U))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','member_of_singleton')).
% 0.19/0.39 tff(25,plain,
% 0.19/0.39 (~![U: $i] : ((~member(U, universal_class)) | (member_of(singleton(U)) = U))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[24, 23])).
% 0.19/0.39 tff(26,plain,
% 0.19/0.39 (~![U: $i] : ((~member(U, universal_class)) | (member_of(singleton(U)) = U))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[25, 22])).
% 0.19/0.39 tff(27,plain,
% 0.19/0.39 (~![U: $i] : ((~member(U, universal_class)) | (member_of(singleton(U)) = U))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[26, 22])).
% 0.19/0.39 tff(28,plain,
% 0.19/0.39 (~![U: $i] : ((~member(U, universal_class)) | (member_of(singleton(U)) = U))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[27, 22])).
% 0.19/0.39 tff(29,plain,(
% 0.19/0.39 ~((~member(U!7, universal_class)) | (member_of(singleton(U!7)) = U!7))),
% 0.19/0.39 inference(skolemize,[status(sab)],[28])).
% 0.19/0.39 tff(30,plain,
% 0.19/0.39 (member(U!7, universal_class)),
% 0.19/0.39 inference(or_elim,[status(thm)],[29])).
% 0.19/0.39 tff(31,plain,
% 0.19/0.39 (^[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))))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(32,plain,
% 0.19/0.39 (![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y))))) <=> ![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[31])).
% 0.19/0.39 tff(33,plain,
% 0.19/0.39 (^[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)))))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(34,plain,
% 0.19/0.39 (![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)))))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[33])).
% 0.19/0.39 tff(35,plain,
% 0.19/0.39 (![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)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(36,plain,
% 0.19/0.39 (^[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))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(37,plain,
% 0.19/0.39 (![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)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[36])).
% 0.19/0.39 tff(38,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/sandbox2/benchmark/Axioms/SET005+0.ax','first_second')).
% 0.19/0.39 tff(39,plain,
% 0.19/0.39 (![X: $i, Y: $i] : ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[38, 37])).
% 0.19/0.39 tff(40,plain,
% 0.19/0.39 (![X: $i, Y: $i] : ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[39, 35])).
% 0.19/0.39 tff(41,plain,(
% 0.19/0.39 ![X: $i, Y: $i] : ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y)))),
% 0.19/0.39 inference(skolemize,[status(sab)],[40])).
% 0.19/0.39 tff(42,plain,
% 0.19/0.39 (![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[41, 34])).
% 0.19/0.39 tff(43,plain,
% 0.19/0.39 (![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[42, 32])).
% 0.19/0.39 tff(44,plain,
% 0.19/0.39 (((~![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(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)))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(45,plain,
% 0.19/0.39 ((~![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(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)))))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(46,plain,
% 0.19/0.39 ((~![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(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))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[45, 44])).
% 0.19/0.39 tff(47,plain,
% 0.19/0.39 (~((~(first(ordered_pair(singleton(U!7), U!7)) = singleton(U!7))) | (~(second(ordered_pair(singleton(U!7), U!7)) = U!7)))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[46, 43, 30, 21])).
% 0.19/0.39 tff(48,plain,
% 0.19/0.39 (((~(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)),
% 0.19/0.39 inference(tautology,[status(thm)],[])).
% 0.19/0.39 tff(49,plain,
% 0.19/0.39 (second(ordered_pair(singleton(U!7), U!7)) = U!7),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[48, 47])).
% 0.19/0.39 tff(50,plain,
% 0.19/0.39 (^[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)))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(51,plain,
% 0.19/0.39 (![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))))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[50])).
% 0.19/0.39 tff(52,plain,
% 0.19/0.39 (![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))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(53,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')).
% 0.19/0.39 tff(54,plain,
% 0.19/0.39 (![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[53, 52])).
% 0.19/0.39 tff(55,plain,(
% 0.19/0.39 ![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))),
% 0.19/0.39 inference(skolemize,[status(sab)],[54])).
% 0.19/0.39 tff(56,plain,
% 0.19/0.39 (![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[55, 51])).
% 0.19/0.39 tff(57,plain,
% 0.19/0.39 ((~![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))))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(58,plain,
% 0.19/0.39 (ordered_pair(singleton(U!7), U!7) = unordered_pair(singleton(singleton(U!7)), unordered_pair(singleton(U!7), singleton(U!7)))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[57, 56])).
% 0.19/0.39 tff(59,plain,
% 0.19/0.39 (unordered_pair(singleton(singleton(U!7)), unordered_pair(singleton(U!7), singleton(U!7))) = ordered_pair(singleton(U!7), U!7)),
% 0.19/0.39 inference(symmetry,[status(thm)],[58])).
% 0.19/0.39 tff(60,plain,
% 0.19/0.39 (^[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))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(61,plain,
% 0.19/0.39 (![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)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[60])).
% 0.19/0.39 tff(62,plain,
% 0.19/0.39 (![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)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(63,plain,
% 0.19/0.39 (^[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))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(64,plain,
% 0.19/0.39 (![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)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[63])).
% 0.19/0.39 tff(65,axiom,(![Y: $i] : (member(Y, universal_class) => (singleton(member_of(singleton(Y))) = singleton(Y)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','member_singleton_singleton')).
% 0.19/0.39 tff(66,plain,
% 0.19/0.39 (![Y: $i] : ((~member(Y, universal_class)) | (singleton(member_of(singleton(Y))) = singleton(Y)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[65, 64])).
% 0.19/0.39 tff(67,plain,
% 0.19/0.39 (![Y: $i] : ((~member(Y, universal_class)) | (singleton(member_of(singleton(Y))) = singleton(Y)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[66, 62])).
% 0.19/0.39 tff(68,plain,(
% 0.19/0.39 ![Y: $i] : ((~member(Y, universal_class)) | (singleton(member_of(singleton(Y))) = singleton(Y)))),
% 0.19/0.39 inference(skolemize,[status(sab)],[67])).
% 0.19/0.39 tff(69,plain,
% 0.19/0.39 (![Y: $i] : ((~member(Y, universal_class)) | (singleton(member_of(singleton(Y))) = singleton(Y)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[68, 61])).
% 0.19/0.39 tff(70,plain,
% 0.19/0.39 (((~![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)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(71,plain,
% 0.19/0.39 ((~![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)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(72,plain,
% 0.19/0.39 ((~![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))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[71, 70])).
% 0.19/0.39 tff(73,plain,
% 0.19/0.39 (singleton(member_of(singleton(U!7))) = singleton(U!7)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[72, 69, 30])).
% 0.19/0.39 tff(74,plain,
% 0.19/0.39 (unordered_pair(singleton(U!7), singleton(member_of(singleton(U!7)))) = unordered_pair(singleton(U!7), singleton(U!7))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[73])).
% 0.19/0.39 tff(75,plain,
% 0.19/0.39 (unordered_pair(singleton(singleton(U!7)), unordered_pair(singleton(U!7), singleton(member_of(singleton(U!7))))) = unordered_pair(singleton(singleton(U!7)), unordered_pair(singleton(U!7), singleton(U!7)))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[74])).
% 0.19/0.39 tff(76,plain,
% 0.19/0.39 ((~![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))) | (ordered_pair(singleton(U!7), member_of(singleton(U!7))) = unordered_pair(singleton(singleton(U!7)), unordered_pair(singleton(U!7), singleton(member_of(singleton(U!7))))))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(77,plain,
% 0.19/0.39 (ordered_pair(singleton(U!7), member_of(singleton(U!7))) = unordered_pair(singleton(singleton(U!7)), unordered_pair(singleton(U!7), singleton(member_of(singleton(U!7)))))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[76, 56])).
% 0.19/0.39 tff(78,plain,
% 0.19/0.39 (ordered_pair(singleton(U!7), member_of(singleton(U!7))) = ordered_pair(singleton(U!7), U!7)),
% 0.19/0.39 inference(transitivity,[status(thm)],[77, 75, 59])).
% 0.19/0.39 tff(79,plain,
% 0.19/0.39 (second(ordered_pair(singleton(U!7), member_of(singleton(U!7)))) = second(ordered_pair(singleton(U!7), U!7))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[78])).
% 0.19/0.39 tff(80,plain,
% 0.19/0.39 (^[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)))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(81,plain,
% 0.19/0.39 (![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))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[80])).
% 0.19/0.39 tff(82,plain,
% 0.19/0.39 (![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))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(83,plain,
% 0.19/0.40 (^[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)))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(84,plain,
% 0.19/0.40 (![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))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[83])).
% 0.19/0.40 tff(85,axiom,(![Y: $i] : (member(Y, universal_class) => member(member_of(singleton(Y)), universal_class))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','member_singleton_universal')).
% 0.19/0.40 tff(86,plain,
% 0.19/0.40 (![Y: $i] : ((~member(Y, universal_class)) | member(member_of(singleton(Y)), universal_class))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[85, 84])).
% 0.19/0.40 tff(87,plain,
% 0.19/0.40 (![Y: $i] : ((~member(Y, universal_class)) | member(member_of(singleton(Y)), universal_class))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[86, 82])).
% 0.19/0.40 tff(88,plain,(
% 0.19/0.40 ![Y: $i] : ((~member(Y, universal_class)) | member(member_of(singleton(Y)), universal_class))),
% 0.19/0.40 inference(skolemize,[status(sab)],[87])).
% 0.19/0.40 tff(89,plain,
% 0.19/0.40 (![Y: $i] : ((~member(Y, universal_class)) | member(member_of(singleton(Y)), universal_class))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[88, 81])).
% 0.19/0.40 tff(90,plain,
% 0.19/0.40 (((~![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))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(91,plain,
% 0.19/0.40 ((~![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))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(92,plain,
% 0.19/0.40 ((~![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)),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[91, 90])).
% 0.19/0.40 tff(93,plain,
% 0.19/0.40 (member(member_of(singleton(U!7)), universal_class)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[92, 89, 30])).
% 0.19/0.40 tff(94,plain,
% 0.19/0.40 (((~![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(singleton(U!7)), universal_class)) | (~member(singleton(U!7), universal_class)) | (~((~(first(ordered_pair(singleton(U!7), member_of(singleton(U!7)))) = singleton(U!7))) | (~(second(ordered_pair(singleton(U!7), member_of(singleton(U!7)))) = member_of(singleton(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(member_of(singleton(U!7)), universal_class)) | (~member(singleton(U!7), universal_class)) | (~((~(first(ordered_pair(singleton(U!7), member_of(singleton(U!7)))) = singleton(U!7))) | (~(second(ordered_pair(singleton(U!7), member_of(singleton(U!7)))) = member_of(singleton(U!7)))))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(95,plain,
% 0.19/0.40 ((~![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(singleton(U!7)), universal_class)) | (~member(singleton(U!7), universal_class)) | (~((~(first(ordered_pair(singleton(U!7), member_of(singleton(U!7)))) = singleton(U!7))) | (~(second(ordered_pair(singleton(U!7), member_of(singleton(U!7)))) = member_of(singleton(U!7)))))))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(96,plain,
% 0.19/0.40 ((~![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))) | (~member(member_of(singleton(U!7)), universal_class)) | (~member(singleton(U!7), universal_class)) | (~((~(first(ordered_pair(singleton(U!7), member_of(singleton(U!7)))) = singleton(U!7))) | (~(second(ordered_pair(singleton(U!7), member_of(singleton(U!7)))) = member_of(singleton(U!7))))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[95, 94])).
% 0.19/0.40 tff(97,plain,
% 0.19/0.40 (~((~(first(ordered_pair(singleton(U!7), member_of(singleton(U!7)))) = singleton(U!7))) | (~(second(ordered_pair(singleton(U!7), member_of(singleton(U!7)))) = member_of(singleton(U!7)))))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[96, 43, 93, 21])).
% 0.19/0.40 tff(98,plain,
% 0.19/0.40 (((~(first(ordered_pair(singleton(U!7), member_of(singleton(U!7)))) = singleton(U!7))) | (~(second(ordered_pair(singleton(U!7), member_of(singleton(U!7)))) = member_of(singleton(U!7))))) | (second(ordered_pair(singleton(U!7), member_of(singleton(U!7)))) = member_of(singleton(U!7)))),
% 0.19/0.40 inference(tautology,[status(thm)],[])).
% 0.19/0.40 tff(99,plain,
% 0.19/0.40 (second(ordered_pair(singleton(U!7), member_of(singleton(U!7)))) = member_of(singleton(U!7))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[98, 97])).
% 0.19/0.40 tff(100,plain,
% 0.19/0.40 (member_of(singleton(U!7)) = second(ordered_pair(singleton(U!7), member_of(singleton(U!7))))),
% 0.19/0.40 inference(symmetry,[status(thm)],[99])).
% 0.19/0.40 tff(101,plain,
% 0.19/0.40 (member_of(singleton(U!7)) = U!7),
% 0.19/0.40 inference(transitivity,[status(thm)],[100, 79, 49])).
% 0.19/0.40 tff(102,plain,
% 0.19/0.40 (~(member_of(singleton(U!7)) = U!7)),
% 0.19/0.40 inference(or_elim,[status(thm)],[29])).
% 0.19/0.40 tff(103,plain,
% 0.19/0.40 ($false),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[102, 101])).
% 0.19/0.40 % SZS output end Proof
% 0.19/0.40 % E exiting
%------------------------------------------------------------------------------