%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET082+1 : TPTP v9.0.0. Bugfixed v5.4.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n012.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 05:25:09 AM UTC 2025
% Result : Theorem 0.21s 0.40s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : SET082+1 : TPTP v9.0.0. Bugfixed v5.4.0.
% 0.07/0.13 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n012.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Jun 20 04:12:40 EDT 2025
% 0.13/0.34 % CPUTime :
% 0.21/0.40 % SZS status Theorem
% 0.21/0.40 % SZS output start Proof
% 0.21/0.40 tff(member_type, type, (
% 0.21/0.40 member: ( $i * $i ) > $o)).
% 0.21/0.40 tff(universal_class_type, type, (
% 0.21/0.40 universal_class: $i)).
% 0.21/0.40 tff(tptp_fun_X_7_type, type, (
% 0.21/0.40 tptp_fun_X_7: $i)).
% 0.21/0.40 tff(apply_type, type, (
% 0.21/0.40 apply: ( $i * $i ) > $i)).
% 0.21/0.40 tff(unordered_pair_type, type, (
% 0.21/0.40 unordered_pair: ( $i * $i ) > $i)).
% 0.21/0.40 tff(tptp_fun_XF_6_type, type, (
% 0.21/0.40 tptp_fun_XF_6: $i)).
% 0.21/0.40 tff(null_class_type, type, (
% 0.21/0.40 null_class: $i)).
% 0.21/0.40 tff(singleton_type, type, (
% 0.21/0.40 singleton: $i > $i)).
% 0.21/0.40 tff(function_type, type, (
% 0.21/0.40 function: $i > $o)).
% 0.21/0.40 tff(1,plain,
% 0.21/0.40 (^[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)))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(2,plain,
% 0.21/0.40 (![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))))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[1])).
% 0.21/0.40 tff(3,plain,
% 0.21/0.40 (^[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))))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(4,plain,
% 0.21/0.40 (![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))))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[3])).
% 0.21/0.40 tff(5,plain,
% 0.21/0.40 (![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))))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(6,plain,
% 0.21/0.40 (^[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)))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(7,plain,
% 0.21/0.40 (![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))))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[6])).
% 0.21/0.40 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')).
% 0.21/0.40 tff(9,plain,
% 0.21/0.40 (![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X))))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[8, 7])).
% 0.21/0.40 tff(10,plain,
% 0.21/0.40 (![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X))))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[9, 5])).
% 0.21/0.40 tff(11,plain,(
% 0.21/0.40 ![U: $i, X: $i, Y: $i] : (member(U, unordered_pair(X, Y)) <=> (member(U, universal_class) & ((U = Y) | (U = X))))),
% 0.21/0.40 inference(skolemize,[status(sab)],[10])).
% 0.21/0.40 tff(12,plain,
% 0.21/0.40 (![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[11, 4])).
% 0.21/0.40 tff(13,plain,
% 0.21/0.40 (![U: $i, X: $i, Y: $i] : (~(((~member(U, universal_class)) | (~((U = Y) | (U = X)))) <=> member(U, unordered_pair(X, Y))))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[12, 2])).
% 0.21/0.40 tff(14,plain,
% 0.21/0.40 (((~![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(X!7, X!7)), universal_class)) | (~(apply(XF!6, unordered_pair(X!7, X!7)) = X!7))) <=> member(apply(XF!6, unordered_pair(X!7, X!7)), unordered_pair(X!7, X!7))))) <=> ((~![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(X!7, X!7)), universal_class)) | (~(apply(XF!6, unordered_pair(X!7, X!7)) = X!7))) <=> member(apply(XF!6, unordered_pair(X!7, X!7)), unordered_pair(X!7, X!7)))))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(15,plain,
% 0.21/0.40 ((~(((~member(apply(XF!6, unordered_pair(X!7, X!7)), universal_class)) | (~(apply(XF!6, unordered_pair(X!7, X!7)) = X!7))) <=> member(apply(XF!6, unordered_pair(X!7, X!7)), unordered_pair(X!7, X!7)))) <=> (~(((~member(apply(XF!6, unordered_pair(X!7, X!7)), universal_class)) | (~(apply(XF!6, unordered_pair(X!7, X!7)) = X!7))) <=> member(apply(XF!6, unordered_pair(X!7, X!7)), unordered_pair(X!7, X!7))))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(16,plain,
% 0.21/0.40 ((((~member(apply(XF!6, unordered_pair(X!7, X!7)), universal_class)) | (~((apply(XF!6, unordered_pair(X!7, X!7)) = X!7) | (apply(XF!6, unordered_pair(X!7, X!7)) = X!7)))) <=> member(apply(XF!6, unordered_pair(X!7, X!7)), unordered_pair(X!7, X!7))) <=> (((~member(apply(XF!6, unordered_pair(X!7, X!7)), universal_class)) | (~(apply(XF!6, unordered_pair(X!7, X!7)) = X!7))) <=> member(apply(XF!6, unordered_pair(X!7, X!7)), unordered_pair(X!7, X!7)))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(17,plain,
% 0.21/0.40 ((~(((~member(apply(XF!6, unordered_pair(X!7, X!7)), universal_class)) | (~((apply(XF!6, unordered_pair(X!7, X!7)) = X!7) | (apply(XF!6, unordered_pair(X!7, X!7)) = X!7)))) <=> member(apply(XF!6, unordered_pair(X!7, X!7)), unordered_pair(X!7, X!7)))) <=> (~(((~member(apply(XF!6, unordered_pair(X!7, X!7)), universal_class)) | (~(apply(XF!6, unordered_pair(X!7, X!7)) = X!7))) <=> member(apply(XF!6, unordered_pair(X!7, X!7)), unordered_pair(X!7, X!7))))),
% 0.21/0.40 inference(monotonicity,[status(thm)],[16])).
% 0.21/0.40 tff(18,plain,
% 0.21/0.40 ((~(((~member(apply(XF!6, unordered_pair(X!7, X!7)), universal_class)) | (~((apply(XF!6, unordered_pair(X!7, X!7)) = X!7) | (apply(XF!6, unordered_pair(X!7, X!7)) = X!7)))) <=> member(apply(XF!6, unordered_pair(X!7, X!7)), unordered_pair(X!7, X!7)))) <=> (~(((~member(apply(XF!6, unordered_pair(X!7, X!7)), universal_class)) | (~(apply(XF!6, unordered_pair(X!7, X!7)) = X!7))) <=> member(apply(XF!6, unordered_pair(X!7, X!7)), unordered_pair(X!7, X!7))))),
% 0.21/0.40 inference(transitivity,[status(thm)],[17, 15])).
% 0.21/0.40 tff(19,plain,
% 0.21/0.40 (((~![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(X!7, X!7)), universal_class)) | (~((apply(XF!6, unordered_pair(X!7, X!7)) = X!7) | (apply(XF!6, unordered_pair(X!7, X!7)) = X!7)))) <=> member(apply(XF!6, unordered_pair(X!7, X!7)), unordered_pair(X!7, X!7))))) <=> ((~![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(X!7, X!7)), universal_class)) | (~(apply(XF!6, unordered_pair(X!7, X!7)) = X!7))) <=> member(apply(XF!6, unordered_pair(X!7, X!7)), unordered_pair(X!7, X!7)))))),
% 0.21/0.40 inference(monotonicity,[status(thm)],[18])).
% 0.21/0.40 tff(20,plain,
% 0.21/0.40 (((~![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(X!7, X!7)), universal_class)) | (~((apply(XF!6, unordered_pair(X!7, X!7)) = X!7) | (apply(XF!6, unordered_pair(X!7, X!7)) = X!7)))) <=> member(apply(XF!6, unordered_pair(X!7, X!7)), unordered_pair(X!7, X!7))))) <=> ((~![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(X!7, X!7)), universal_class)) | (~(apply(XF!6, unordered_pair(X!7, X!7)) = X!7))) <=> member(apply(XF!6, unordered_pair(X!7, X!7)), unordered_pair(X!7, X!7)))))),
% 0.21/0.40 inference(transitivity,[status(thm)],[19, 14])).
% 0.21/0.40 tff(21,plain,
% 0.21/0.40 ((~![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(X!7, X!7)), universal_class)) | (~((apply(XF!6, unordered_pair(X!7, X!7)) = X!7) | (apply(XF!6, unordered_pair(X!7, X!7)) = X!7)))) <=> member(apply(XF!6, unordered_pair(X!7, X!7)), unordered_pair(X!7, X!7))))),
% 0.21/0.40 inference(quant_inst,[status(thm)],[])).
% 0.21/0.40 tff(22,plain,
% 0.21/0.40 ((~![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(X!7, X!7)), universal_class)) | (~(apply(XF!6, unordered_pair(X!7, X!7)) = X!7))) <=> member(apply(XF!6, unordered_pair(X!7, X!7)), unordered_pair(X!7, X!7))))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[21, 20])).
% 0.21/0.40 tff(23,plain,
% 0.21/0.40 (~(((~member(apply(XF!6, unordered_pair(X!7, X!7)), universal_class)) | (~(apply(XF!6, unordered_pair(X!7, X!7)) = X!7))) <=> member(apply(XF!6, unordered_pair(X!7, X!7)), unordered_pair(X!7, X!7)))),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[22, 13])).
% 0.21/0.40 tff(24,plain,
% 0.21/0.40 (^[X: $i] : refl((singleton(X) = unordered_pair(X, X)) <=> (singleton(X) = unordered_pair(X, X)))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(25,plain,
% 0.21/0.40 (![X: $i] : (singleton(X) = unordered_pair(X, X)) <=> ![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[24])).
% 0.21/0.40 tff(26,plain,
% 0.21/0.40 (![X: $i] : (singleton(X) = unordered_pair(X, X)) <=> ![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(27,axiom,(![X: $i] : (singleton(X) = unordered_pair(X, X))), file('/export/starexec/sandbox/benchmark/Axioms/SET005+0.ax','singleton_set_defn')).
% 0.21/0.40 tff(28,plain,
% 0.21/0.40 (![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[27, 26])).
% 0.21/0.40 tff(29,plain,(
% 0.21/0.40 ![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.21/0.40 inference(skolemize,[status(sab)],[28])).
% 0.21/0.40 tff(30,plain,
% 0.21/0.40 (![X: $i] : (singleton(X) = unordered_pair(X, X))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[29, 25])).
% 0.21/0.40 tff(31,plain,
% 0.21/0.40 ((~![X: $i] : (singleton(X) = unordered_pair(X, X))) | (singleton(X!7) = unordered_pair(X!7, X!7))),
% 0.21/0.40 inference(quant_inst,[status(thm)],[])).
% 0.21/0.40 tff(32,plain,
% 0.21/0.40 (singleton(X!7) = unordered_pair(X!7, X!7)),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[31, 30])).
% 0.21/0.40 tff(33,plain,
% 0.21/0.40 (unordered_pair(X!7, X!7) = singleton(X!7)),
% 0.21/0.40 inference(symmetry,[status(thm)],[32])).
% 0.21/0.40 tff(34,plain,
% 0.21/0.40 ((unordered_pair(X!7, X!7) = null_class) <=> (singleton(X!7) = null_class)),
% 0.21/0.40 inference(monotonicity,[status(thm)],[33])).
% 0.21/0.40 tff(35,plain,
% 0.21/0.40 ((singleton(X!7) = null_class) <=> (unordered_pair(X!7, X!7) = null_class)),
% 0.21/0.40 inference(symmetry,[status(thm)],[34])).
% 0.21/0.40 tff(36,plain,
% 0.21/0.40 ((~(singleton(X!7) = null_class)) <=> (~(unordered_pair(X!7, X!7) = null_class))),
% 0.21/0.40 inference(monotonicity,[status(thm)],[35])).
% 0.21/0.40 tff(37,plain,
% 0.21/0.40 ((~![X: $i] : (member(X, universal_class) | (singleton(X) = null_class))) <=> (~![X: $i] : (member(X, universal_class) | (singleton(X) = null_class)))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(38,plain,
% 0.21/0.40 ((~![X: $i] : ((~member(X, universal_class)) => (singleton(X) = null_class))) <=> (~![X: $i] : (member(X, universal_class) | (singleton(X) = null_class)))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(39,axiom,(~![X: $i] : ((~member(X, universal_class)) => (singleton(X) = null_class))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','singleton_is_null_class')).
% 0.21/0.40 tff(40,plain,
% 0.21/0.40 (~![X: $i] : (member(X, universal_class) | (singleton(X) = null_class))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[39, 38])).
% 0.21/0.40 tff(41,plain,
% 0.21/0.40 (~![X: $i] : (member(X, universal_class) | (singleton(X) = null_class))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[40, 37])).
% 0.21/0.40 tff(42,plain,
% 0.21/0.40 (~![X: $i] : (member(X, universal_class) | (singleton(X) = null_class))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[41, 37])).
% 0.21/0.40 tff(43,plain,
% 0.21/0.40 (~![X: $i] : (member(X, universal_class) | (singleton(X) = null_class))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[42, 37])).
% 0.21/0.40 tff(44,plain,(
% 0.21/0.40 ~(member(X!7, universal_class) | (singleton(X!7) = null_class))),
% 0.21/0.40 inference(skolemize,[status(sab)],[43])).
% 0.21/0.40 tff(45,plain,
% 0.21/0.40 (~(singleton(X!7) = null_class)),
% 0.21/0.40 inference(or_elim,[status(thm)],[44])).
% 0.21/0.40 tff(46,plain,
% 0.21/0.40 (~(unordered_pair(X!7, X!7) = null_class)),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[45, 36])).
% 0.21/0.40 tff(47,plain,
% 0.21/0.40 (^[X: $i, Y: $i] : refl(member(unordered_pair(X, Y), universal_class) <=> member(unordered_pair(X, Y), universal_class))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(48,plain,
% 0.21/0.40 (![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class) <=> ![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.21/0.40 inference(quant_intro,[status(thm)],[47])).
% 0.21/0.40 tff(49,plain,
% 0.21/0.40 (![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class) <=> ![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(50,axiom,(![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)), file('/export/starexec/sandbox/benchmark/Axioms/SET005+0.ax','unordered_pair')).
% 0.21/0.40 tff(51,plain,
% 0.21/0.40 (![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[50, 49])).
% 0.21/0.40 tff(52,plain,(
% 0.21/0.40 ![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.21/0.40 inference(skolemize,[status(sab)],[51])).
% 0.21/0.40 tff(53,plain,
% 0.21/0.40 (![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[52, 48])).
% 0.21/0.40 tff(54,plain,
% 0.21/0.40 ((~![X: $i, Y: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(X!7, X!7), universal_class)),
% 0.21/0.40 inference(quant_inst,[status(thm)],[])).
% 0.21/0.40 tff(55,plain,
% 0.21/0.40 (member(unordered_pair(X!7, X!7), universal_class)),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[54, 53])).
% 0.21/0.40 tff(56,plain,
% 0.21/0.40 (^[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)))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(57,plain,
% 0.21/0.40 (![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))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[56])).
% 0.21/0.40 tff(58,plain,
% 0.21/0.40 ((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)))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(59,plain,
% 0.21/0.40 (?[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))))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(60,plain,
% 0.21/0.40 (^[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)))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(61,plain,
% 0.21/0.40 (?[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))))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[60])).
% 0.21/0.41 tff(62,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')).
% 0.21/0.41 tff(63,plain,
% 0.21/0.41 (?[XF: $i] : (function(XF) & ![Y: $i] : (member(apply(XF, Y), Y) | (Y = null_class) | (~member(Y, universal_class))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[62, 61])).
% 0.21/0.41 tff(64,plain,
% 0.21/0.41 (?[XF: $i] : (function(XF) & ![Y: $i] : (member(apply(XF, Y), Y) | (Y = null_class) | (~member(Y, universal_class))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[63, 59])).
% 0.21/0.41 tff(65,plain,
% 0.21/0.41 (function(XF!6) & ![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[64, 58])).
% 0.21/0.41 tff(66,plain,
% 0.21/0.41 (![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))),
% 0.21/0.41 inference(and_elim,[status(thm)],[65])).
% 0.21/0.41 tff(67,plain,
% 0.21/0.41 (![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[66, 57])).
% 0.21/0.41 tff(68,plain,
% 0.21/0.41 (((~![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))) | ((unordered_pair(X!7, X!7) = null_class) | (~member(unordered_pair(X!7, X!7), universal_class)) | member(apply(XF!6, unordered_pair(X!7, X!7)), unordered_pair(X!7, X!7)))) <=> ((~![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))) | (unordered_pair(X!7, X!7) = null_class) | (~member(unordered_pair(X!7, X!7), universal_class)) | member(apply(XF!6, unordered_pair(X!7, X!7)), unordered_pair(X!7, X!7)))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(69,plain,
% 0.21/0.41 ((~![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))) | ((unordered_pair(X!7, X!7) = null_class) | (~member(unordered_pair(X!7, X!7), universal_class)) | member(apply(XF!6, unordered_pair(X!7, X!7)), unordered_pair(X!7, X!7)))),
% 0.21/0.41 inference(quant_inst,[status(thm)],[])).
% 0.21/0.41 tff(70,plain,
% 0.21/0.41 ((~![Y: $i] : ((Y = null_class) | (~member(Y, universal_class)) | member(apply(XF!6, Y), Y))) | (unordered_pair(X!7, X!7) = null_class) | (~member(unordered_pair(X!7, X!7), universal_class)) | member(apply(XF!6, unordered_pair(X!7, X!7)), unordered_pair(X!7, X!7))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[69, 68])).
% 0.21/0.41 tff(71,plain,
% 0.21/0.41 ((unordered_pair(X!7, X!7) = null_class) | member(apply(XF!6, unordered_pair(X!7, X!7)), unordered_pair(X!7, X!7))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[70, 67, 55])).
% 0.21/0.41 tff(72,plain,
% 0.21/0.41 (member(apply(XF!6, unordered_pair(X!7, X!7)), unordered_pair(X!7, X!7))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[71, 46])).
% 0.21/0.41 tff(73,plain,
% 0.21/0.41 ((((~member(apply(XF!6, unordered_pair(X!7, X!7)), universal_class)) | (~(apply(XF!6, unordered_pair(X!7, X!7)) = X!7))) <=> member(apply(XF!6, unordered_pair(X!7, X!7)), unordered_pair(X!7, X!7))) | (~((~member(apply(XF!6, unordered_pair(X!7, X!7)), universal_class)) | (~(apply(XF!6, unordered_pair(X!7, X!7)) = X!7)))) | (~member(apply(XF!6, unordered_pair(X!7, X!7)), unordered_pair(X!7, X!7)))),
% 0.21/0.41 inference(tautology,[status(thm)],[])).
% 0.21/0.41 tff(74,plain,
% 0.21/0.41 ((((~member(apply(XF!6, unordered_pair(X!7, X!7)), universal_class)) | (~(apply(XF!6, unordered_pair(X!7, X!7)) = X!7))) <=> member(apply(XF!6, unordered_pair(X!7, X!7)), unordered_pair(X!7, X!7))) | (~((~member(apply(XF!6, unordered_pair(X!7, X!7)), universal_class)) | (~(apply(XF!6, unordered_pair(X!7, X!7)) = X!7))))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[73, 72])).
% 0.21/0.41 tff(75,plain,
% 0.21/0.41 (~((~member(apply(XF!6, unordered_pair(X!7, X!7)), universal_class)) | (~(apply(XF!6, unordered_pair(X!7, X!7)) = X!7)))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[74, 23])).
% 0.21/0.41 tff(76,plain,
% 0.21/0.41 (((~member(apply(XF!6, unordered_pair(X!7, X!7)), universal_class)) | (~(apply(XF!6, unordered_pair(X!7, X!7)) = X!7))) | (apply(XF!6, unordered_pair(X!7, X!7)) = X!7)),
% 0.21/0.41 inference(tautology,[status(thm)],[])).
% 0.21/0.41 tff(77,plain,
% 0.21/0.41 (apply(XF!6, unordered_pair(X!7, X!7)) = X!7),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[76, 75])).
% 0.21/0.41 tff(78,plain,
% 0.21/0.41 (X!7 = apply(XF!6, unordered_pair(X!7, X!7))),
% 0.21/0.41 inference(symmetry,[status(thm)],[77])).
% 0.21/0.41 tff(79,plain,
% 0.21/0.41 (member(X!7, universal_class) <=> member(apply(XF!6, unordered_pair(X!7, X!7)), universal_class)),
% 0.21/0.41 inference(monotonicity,[status(thm)],[78])).
% 0.21/0.41 tff(80,plain,
% 0.21/0.41 (member(apply(XF!6, unordered_pair(X!7, X!7)), universal_class) <=> member(X!7, universal_class)),
% 0.21/0.41 inference(symmetry,[status(thm)],[79])).
% 0.21/0.41 tff(81,plain,
% 0.21/0.41 (((~member(apply(XF!6, unordered_pair(X!7, X!7)), universal_class)) | (~(apply(XF!6, unordered_pair(X!7, X!7)) = X!7))) | member(apply(XF!6, unordered_pair(X!7, X!7)), universal_class)),
% 0.21/0.41 inference(tautology,[status(thm)],[])).
% 0.21/0.41 tff(82,plain,
% 0.21/0.41 (member(apply(XF!6, unordered_pair(X!7, X!7)), universal_class)),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[81, 75])).
% 0.21/0.41 tff(83,plain,
% 0.21/0.41 (member(X!7, universal_class)),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[82, 80])).
% 0.21/0.41 tff(84,plain,
% 0.21/0.41 (~member(X!7, universal_class)),
% 0.21/0.41 inference(or_elim,[status(thm)],[44])).
% 0.21/0.41 tff(85,plain,
% 0.21/0.41 ($false),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[84, 83])).
% 0.21/0.41 % SZS output end Proof
% 0.21/0.41 % E exiting
%------------------------------------------------------------------------------