%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET103+1 : TPTP v9.0.0. Bugfixed v5.4.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n009.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 05:25:14 AM UTC 2025
% Result : Theorem 0.19s 0.46s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SET103+1 : TPTP v9.0.0. Bugfixed v5.4.0.
% 0.03/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n009.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 04:17:48 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.19/0.46 % SZS status Theorem
% 0.19/0.46 % SZS output start Proof
% 0.19/0.46 tff(member_type, type, (
% 0.19/0.46 member: ( $i * $i ) > $o)).
% 0.19/0.46 tff(universal_class_type, type, (
% 0.19/0.46 universal_class: $i)).
% 0.19/0.46 tff(tptp_fun_Y_7_type, type, (
% 0.19/0.46 tptp_fun_Y_7: $i)).
% 0.19/0.46 tff(apply_type, type, (
% 0.19/0.46 apply: ( $i * $i ) > $i)).
% 0.19/0.46 tff(unordered_pair_type, type, (
% 0.19/0.46 unordered_pair: ( $i * $i ) > $i)).
% 0.19/0.46 tff(tptp_fun_XF_6_type, type, (
% 0.19/0.46 tptp_fun_XF_6: $i)).
% 0.19/0.46 tff(null_class_type, type, (
% 0.19/0.46 null_class: $i)).
% 0.19/0.46 tff(ordered_pair_type, type, (
% 0.19/0.46 ordered_pair: ( $i * $i ) > $i)).
% 0.19/0.46 tff(tptp_fun_X_8_type, type, (
% 0.19/0.46 tptp_fun_X_8: $i)).
% 0.19/0.46 tff(singleton_type, type, (
% 0.19/0.46 singleton: $i > $i)).
% 0.19/0.46 tff(function_type, type, (
% 0.19/0.46 function: $i > $o)).
% 0.19/0.46 tff(1,plain,
% 0.19/0.46 (^[U: $i, X: $i, Y: $i] : refl((~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y)))) <=> (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y)))))),
% 0.19/0.46 inference(bind,[status(th)],[])).
% 0.19/0.46 tff(2,plain,
% 0.19/0.46 (![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y)))) <=> ![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))),
% 0.19/0.47 inference(quant_intro,[status(thm)],[1])).
% 0.19/0.47 tff(3,plain,
% 0.19/0.47 (^[U: $i, X: $i, Y: $i] : trans(trans(monotonicity(rewrite((member(U, universal_class) & ((U = Y) | (U = X))) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X)))))), ((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)))))))), rewrite((member(U, unordered_pair(X, Y)) <=> (~((~member(U, universal_class)) | (~((U = Y) | (U = X)))))) <=> (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))), ((member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X)))) <=> (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y)))))), rewrite((~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y)))) <=> (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))), ((member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X)))) <=> (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))))),
% 0.19/0.47 inference(bind,[status(th)],[])).
% 0.19/0.47 tff(4,plain,
% 0.19/0.47 (![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, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))),
% 0.19/0.47 inference(quant_intro,[status(thm)],[3])).
% 0.19/0.47 tff(5,plain,
% 0.19/0.47 (![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X)))) <=> ![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X))))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(6,plain,
% 0.19/0.47 (^[U: $i, X: $i, Y: $i] : rewrite((member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = X) | (U = Y)))) <=> (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X)))))),
% 0.19/0.47 inference(bind,[status(th)],[])).
% 0.19/0.47 tff(7,plain,
% 0.19/0.47 (![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = X) | (U = Y)))) <=> ![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X))))),
% 0.19/0.47 inference(quant_intro,[status(thm)],[6])).
% 0.19/0.47 tff(8,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')).
% 0.19/0.47 tff(9,plain,
% 0.19/0.47 (![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X))))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[8, 7])).
% 0.19/0.47 tff(10,plain,
% 0.19/0.47 (![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X))))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[9, 5])).
% 0.19/0.47 tff(11,plain,(
% 0.19/0.47 ![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X))))),
% 0.19/0.47 inference(skolemize,[status(sab)],[10])).
% 0.19/0.47 tff(12,plain,
% 0.19/0.47 (![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[11, 4])).
% 0.19/0.47 tff(13,plain,
% 0.19/0.47 (![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[12, 2])).
% 0.19/0.47 tff(14,plain,
% 0.19/0.47 (((~![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))) | (~(((~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)) | (~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class))) <=> member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7))))) <=> ((~![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))) | (~(((~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)) | (~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class))) <=> member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)))))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(15,plain,
% 0.19/0.47 ((~(((~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)) | (~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class))) <=> member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)))) <=> (~(((~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)) | (~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class))) <=> member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7))))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(16,plain,
% 0.19/0.47 ((((~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)) | (~((apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7) | (apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)))) <=> member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7))) <=> (((~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)) | (~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class))) <=> member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(17,plain,
% 0.19/0.47 ((~(((~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)) | (~((apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7) | (apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)))) <=> member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)))) <=> (~(((~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)) | (~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class))) <=> member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7))))),
% 0.19/0.47 inference(monotonicity,[status(thm)],[16])).
% 0.19/0.47 tff(18,plain,
% 0.19/0.47 ((~(((~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)) | (~((apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7) | (apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)))) <=> member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)))) <=> (~(((~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)) | (~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class))) <=> member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7))))),
% 0.19/0.47 inference(transitivity,[status(thm)],[17, 15])).
% 0.19/0.47 tff(19,plain,
% 0.19/0.47 (((~![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))) | (~(((~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)) | (~((apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7) | (apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)))) <=> member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7))))) <=> ((~![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))) | (~(((~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)) | (~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class))) <=> member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)))))),
% 0.19/0.47 inference(monotonicity,[status(thm)],[18])).
% 0.19/0.47 tff(20,plain,
% 0.19/0.47 (((~![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))) | (~(((~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)) | (~((apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7) | (apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)))) <=> member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7))))) <=> ((~![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))) | (~(((~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)) | (~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class))) <=> member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)))))),
% 0.19/0.47 inference(transitivity,[status(thm)],[19, 14])).
% 0.19/0.47 tff(21,plain,
% 0.19/0.47 ((~![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))) | (~(((~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)) | (~((apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7) | (apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)))) <=> member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7))))),
% 0.19/0.47 inference(quant_inst,[status(thm)],[])).
% 0.19/0.47 tff(22,plain,
% 0.19/0.47 ((~![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))) | (~(((~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)) | (~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class))) <=> member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7))))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[21, 20])).
% 0.19/0.47 tff(23,plain,
% 0.19/0.47 (~(((~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)) | (~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class))) <=> member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)))),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[22, 13])).
% 0.19/0.47 tff(24,plain,
% 0.19/0.47 (^[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)))))),
% 0.19/0.47 inference(bind,[status(th)],[])).
% 0.19/0.47 tff(25,plain,
% 0.19/0.47 (![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))))),
% 0.19/0.47 inference(quant_intro,[status(thm)],[24])).
% 0.19/0.47 tff(26,plain,
% 0.19/0.47 (![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))))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(27,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')).
% 0.19/0.47 tff(28,plain,
% 0.19/0.47 (![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[27, 26])).
% 0.19/0.47 tff(29,plain,(
% 0.19/0.47 ![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))),
% 0.19/0.47 inference(skolemize,[status(sab)],[28])).
% 0.19/0.47 tff(30,plain,
% 0.19/0.47 (![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[29, 25])).
% 0.19/0.47 tff(31,plain,
% 0.19/0.47 ((~![X: $i, Y: $i] : (ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))))) | (ordered_pair(X!8, Y!7) = unordered_pair(singleton(X!8), unordered_pair(X!8, singleton(Y!7))))),
% 0.19/0.47 inference(quant_inst,[status(thm)],[])).
% 0.19/0.47 tff(32,plain,
% 0.19/0.47 (ordered_pair(X!8, Y!7) = unordered_pair(singleton(X!8), unordered_pair(X!8, singleton(Y!7)))),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[31, 30])).
% 0.19/0.47 tff(33,plain,
% 0.19/0.47 (unordered_pair(singleton(X!8), unordered_pair(X!8, singleton(Y!7))) = ordered_pair(X!8, Y!7)),
% 0.19/0.47 inference(symmetry,[status(thm)],[32])).
% 0.19/0.47 tff(34,plain,
% 0.19/0.47 (^[X: $i] : refl((singleton(X) = unordered_pair(X, X)) <=> (singleton(X) = unordered_pair(X, X)))),
% 0.19/0.47 inference(bind,[status(th)],[])).
% 0.19/0.47 tff(35,plain,
% 0.19/0.47 (![X: $i] : (singleton(X) = unordered_pair(X, X)) <=> ![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.19/0.47 inference(quant_intro,[status(thm)],[34])).
% 0.19/0.47 tff(36,plain,
% 0.19/0.47 (![X: $i] : (singleton(X) = unordered_pair(X, X)) <=> ![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(37,axiom,(![X: $i] : (singleton(X) = unordered_pair(X, X))), file('/export/starexec/sandbox/benchmark/Axioms/SET005+0.ax','singleton_set_defn')).
% 0.19/0.47 tff(38,plain,
% 0.19/0.47 (![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[37, 36])).
% 0.19/0.47 tff(39,plain,(
% 0.19/0.47 ![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.19/0.47 inference(skolemize,[status(sab)],[38])).
% 0.19/0.47 tff(40,plain,
% 0.19/0.47 (![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[39, 35])).
% 0.19/0.47 tff(41,plain,
% 0.19/0.47 ((~![X: $i] : (singleton(X) = unordered_pair(X, X))) | (singleton(Y!7) = unordered_pair(Y!7, Y!7))),
% 0.19/0.47 inference(quant_inst,[status(thm)],[])).
% 0.19/0.47 tff(42,plain,
% 0.19/0.47 (singleton(Y!7) = unordered_pair(Y!7, Y!7)),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[41, 40])).
% 0.19/0.47 tff(43,plain,
% 0.19/0.47 (unordered_pair(Y!7, Y!7) = singleton(Y!7)),
% 0.19/0.47 inference(symmetry,[status(thm)],[42])).
% 0.19/0.47 tff(44,assumption,(unordered_pair(Y!7, Y!7) = null_class), introduced(assumption)).
% 0.19/0.47 tff(45,plain,
% 0.19/0.47 (null_class = unordered_pair(Y!7, Y!7)),
% 0.19/0.47 inference(symmetry,[status(thm)],[44])).
% 0.19/0.47 tff(46,plain,
% 0.19/0.47 (null_class = singleton(Y!7)),
% 0.19/0.47 inference(transitivity,[status(thm)],[45, 43])).
% 0.19/0.47 tff(47,plain,
% 0.19/0.47 (unordered_pair(X!8, null_class) = unordered_pair(X!8, singleton(Y!7))),
% 0.19/0.47 inference(monotonicity,[status(thm)],[46])).
% 0.19/0.47 tff(48,plain,
% 0.19/0.47 (unordered_pair(singleton(X!8), unordered_pair(X!8, null_class)) = unordered_pair(singleton(X!8), unordered_pair(X!8, singleton(Y!7)))),
% 0.19/0.47 inference(monotonicity,[status(thm)],[47])).
% 0.19/0.47 tff(49,plain,
% 0.19/0.47 (unordered_pair(singleton(X!8), unordered_pair(X!8, null_class)) = ordered_pair(X!8, Y!7)),
% 0.19/0.47 inference(transitivity,[status(thm)],[48, 33])).
% 0.19/0.47 tff(50,plain,
% 0.19/0.47 ((~![X: $i, Y: $i] : (member(Y, universal_class) | (unordered_pair(singleton(X), unordered_pair(X, null_class)) = ordered_pair(X, Y)))) <=> (~![X: $i, Y: $i] : (member(Y, universal_class) | (unordered_pair(singleton(X), unordered_pair(X, null_class)) = ordered_pair(X, Y))))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(51,plain,
% 0.19/0.47 ((~![X: $i, Y: $i] : ((unordered_pair(singleton(X), unordered_pair(X, null_class)) = ordered_pair(X, Y)) | member(Y, universal_class))) <=> (~![X: $i, Y: $i] : (member(Y, universal_class) | (unordered_pair(singleton(X), unordered_pair(X, null_class)) = ordered_pair(X, Y))))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(52,axiom,(~![X: $i, Y: $i] : ((unordered_pair(singleton(X), unordered_pair(X, null_class)) = ordered_pair(X, Y)) | member(Y, universal_class))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','property_1_of_ordered_pair')).
% 0.19/0.47 tff(53,plain,
% 0.19/0.47 (~![X: $i, Y: $i] : (member(Y, universal_class) | (unordered_pair(singleton(X), unordered_pair(X, null_class)) = ordered_pair(X, Y)))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[52, 51])).
% 0.19/0.47 tff(54,plain,
% 0.19/0.47 (~![X: $i, Y: $i] : (member(Y, universal_class) | (unordered_pair(singleton(X), unordered_pair(X, null_class)) = ordered_pair(X, Y)))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[53, 50])).
% 0.19/0.47 tff(55,plain,
% 0.19/0.47 (~![X: $i, Y: $i] : (member(Y, universal_class) | (unordered_pair(singleton(X), unordered_pair(X, null_class)) = ordered_pair(X, Y)))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[54, 50])).
% 0.19/0.47 tff(56,plain,
% 0.19/0.47 (~![X: $i, Y: $i] : (member(Y, universal_class) | (unordered_pair(singleton(X), unordered_pair(X, null_class)) = ordered_pair(X, Y)))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[55, 50])).
% 0.19/0.47 tff(57,plain,(
% 0.19/0.47 ~(member(Y!7, universal_class) | (unordered_pair(singleton(X!8), unordered_pair(X!8, null_class)) = ordered_pair(X!8, Y!7)))),
% 0.19/0.47 inference(skolemize,[status(sab)],[56])).
% 0.19/0.47 tff(58,plain,
% 0.19/0.47 (~(unordered_pair(singleton(X!8), unordered_pair(X!8, null_class)) = ordered_pair(X!8, Y!7))),
% 0.19/0.47 inference(or_elim,[status(thm)],[57])).
% 0.19/0.47 tff(59,plain,
% 0.19/0.47 ($false),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[58, 49])).
% 0.19/0.47 tff(60,plain,(~(unordered_pair(Y!7, Y!7) = null_class)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.47 tff(61,plain,
% 0.19/0.47 (^[X: $i, Y: $i] : refl(member(unordered_pair(X, Y), universal_class) <=> member(unordered_pair(X, Y), universal_class))),
% 0.19/0.47 inference(bind,[status(th)],[])).
% 0.19/0.47 tff(62,plain,
% 0.19/0.47 (![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class) <=> ![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.19/0.47 inference(quant_intro,[status(thm)],[61])).
% 0.19/0.47 tff(63,plain,
% 0.19/0.47 (![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class) <=> ![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(64,axiom,(![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)), file('/export/starexec/sandbox/benchmark/Axioms/SET005+0.ax','unordered_pair')).
% 0.19/0.47 tff(65,plain,
% 0.19/0.47 (![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[64, 63])).
% 0.19/0.47 tff(66,plain,(
% 0.19/0.47 ![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.19/0.47 inference(skolemize,[status(sab)],[65])).
% 0.19/0.47 tff(67,plain,
% 0.19/0.47 (![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[66, 62])).
% 0.19/0.47 tff(68,plain,
% 0.19/0.47 ((~![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(Y!7, Y!7), universal_class)),
% 0.19/0.47 inference(quant_inst,[status(thm)],[])).
% 0.19/0.47 tff(69,plain,
% 0.19/0.47 (member(unordered_pair(Y!7, Y!7), universal_class)),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[68, 67])).
% 0.19/0.47 tff(70,plain,
% 0.19/0.47 (^[Y: $i] : refl(((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y)) <=> ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y)))),
% 0.19/0.47 inference(bind,[status(th)],[])).
% 0.19/0.47 tff(71,plain,
% 0.19/0.47 (![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y)) <=> ![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))),
% 0.19/0.47 inference(quant_intro,[status(thm)],[70])).
% 0.19/0.47 tff(72,plain,
% 0.19/0.47 ((function(XF!6) & ![Y: $i] : (member(apply(XF!6, Y), Y) | (Y = null_class) | (~member(Y, universal_class)))) <=> (function(XF!6) & ![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y)))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(73,plain,
% 0.19/0.47 (?[XF: $i] : (function(XF) & ![Y: $i] : (member(apply(XF, Y), Y) | (Y = null_class) | (~member(Y, universal_class)))) <=> ?[XF: $i] : (function(XF) & ![Y: $i] : (member(apply(XF, Y), Y) | (Y = null_class) | (~member(Y, universal_class))))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(74,plain,
% 0.19/0.47 (^[XF: $i] : rewrite((function(XF) & ![Y: $i] : (member(Y, universal_class) => ((Y = null_class) | member(apply(XF, Y), Y)))) <=> (function(XF) & ![Y: $i] : (member(apply(XF, Y), Y) | (Y = null_class) | (~member(Y, universal_class)))))),
% 0.19/0.47 inference(bind,[status(th)],[])).
% 0.19/0.47 tff(75,plain,
% 0.19/0.47 (?[XF: $i] : (function(XF) & ![Y: $i] : (member(Y, universal_class) => ((Y = null_class) | member(apply(XF, Y), Y)))) <=> ?[XF: $i] : (function(XF) & ![Y: $i] : (member(apply(XF, Y), Y) | (Y = null_class) | (~member(Y, universal_class))))),
% 0.19/0.47 inference(quant_intro,[status(thm)],[74])).
% 0.19/0.47 tff(76,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')).
% 0.19/0.47 tff(77,plain,
% 0.19/0.47 (?[XF: $i] : (function(XF) & ![Y: $i] : (member(apply(XF, Y), Y) | (Y = null_class) | (~member(Y, universal_class))))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[76, 75])).
% 0.19/0.47 tff(78,plain,
% 0.19/0.47 (?[XF: $i] : (function(XF) & ![Y: $i] : (member(apply(XF, Y), Y) | (Y = null_class) | (~member(Y, universal_class))))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[77, 73])).
% 0.19/0.47 tff(79,plain,
% 0.19/0.47 (function(XF!6) & ![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[78, 72])).
% 0.19/0.47 tff(80,plain,
% 0.19/0.47 (![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))),
% 0.19/0.47 inference(and_elim,[status(thm)],[79])).
% 0.19/0.47 tff(81,plain,
% 0.19/0.47 (![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[80, 71])).
% 0.19/0.47 tff(82,plain,
% 0.19/0.47 (((~![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))) | ((unordered_pair(Y!7, Y!7) = null_class) | (~member(unordered_pair(Y!7, Y!7), universal_class)) | member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)))) <=> ((~![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))) | (unordered_pair(Y!7, Y!7) = null_class) | (~member(unordered_pair(Y!7, Y!7), universal_class)) | member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(83,plain,
% 0.19/0.47 ((~![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))) | ((unordered_pair(Y!7, Y!7) = null_class) | (~member(unordered_pair(Y!7, Y!7), universal_class)) | member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)))),
% 0.19/0.47 inference(quant_inst,[status(thm)],[])).
% 0.19/0.47 tff(84,plain,
% 0.19/0.47 ((~![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))) | (unordered_pair(Y!7, Y!7) = null_class) | (~member(unordered_pair(Y!7, Y!7), universal_class)) | member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[83, 82])).
% 0.19/0.47 tff(85,plain,
% 0.19/0.47 ((unordered_pair(Y!7, Y!7) = null_class) | member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7))),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[84, 81, 69])).
% 0.19/0.47 tff(86,plain,
% 0.19/0.47 (member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7))),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[85, 60])).
% 0.19/0.47 tff(87,plain,
% 0.19/0.47 ((((~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)) | (~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class))) <=> member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7))) | (~((~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)) | (~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)))) | (~member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7)))),
% 0.19/0.47 inference(tautology,[status(thm)],[])).
% 0.19/0.47 tff(88,plain,
% 0.19/0.47 ((((~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)) | (~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class))) <=> member(apply(XF!6, unordered_pair(Y!7, Y!7)), unordered_pair(Y!7, Y!7))) | (~((~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)) | (~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class))))),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[87, 86])).
% 0.19/0.47 tff(89,plain,
% 0.19/0.47 (~((~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)) | (~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)))),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[88, 23])).
% 0.19/0.47 tff(90,plain,
% 0.19/0.47 (((~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)) | (~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class))) | (apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)),
% 0.19/0.47 inference(tautology,[status(thm)],[])).
% 0.19/0.47 tff(91,plain,
% 0.19/0.47 (apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[90, 89])).
% 0.19/0.47 tff(92,plain,
% 0.19/0.47 (Y!7 = apply(XF!6, unordered_pair(Y!7, Y!7))),
% 0.19/0.47 inference(symmetry,[status(thm)],[91])).
% 0.19/0.47 tff(93,plain,
% 0.19/0.47 (member(Y!7, universal_class) <=> member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)),
% 0.19/0.47 inference(monotonicity,[status(thm)],[92])).
% 0.19/0.47 tff(94,plain,
% 0.19/0.47 (member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class) <=> member(Y!7, universal_class)),
% 0.19/0.47 inference(symmetry,[status(thm)],[93])).
% 0.19/0.48 tff(95,plain,
% 0.19/0.48 (((~(apply(XF!6, unordered_pair(Y!7, Y!7)) = Y!7)) | (~member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class))) | member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)),
% 0.19/0.48 inference(tautology,[status(thm)],[])).
% 0.19/0.48 tff(96,plain,
% 0.19/0.48 (member(apply(XF!6, unordered_pair(Y!7, Y!7)), universal_class)),
% 0.19/0.48 inference(unit_resolution,[status(thm)],[95, 89])).
% 0.19/0.48 tff(97,plain,
% 0.19/0.48 (member(Y!7, universal_class)),
% 0.19/0.48 inference(modus_ponens,[status(thm)],[96, 94])).
% 0.19/0.48 tff(98,plain,
% 0.19/0.48 (~member(Y!7, universal_class)),
% 0.19/0.48 inference(or_elim,[status(thm)],[57])).
% 0.19/0.48 tff(99,plain,
% 0.19/0.48 ($false),
% 0.19/0.48 inference(unit_resolution,[status(thm)],[98, 97])).
% 0.19/0.48 % SZS output end Proof
% 0.19/0.48 % E exiting
%------------------------------------------------------------------------------