%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET811+4 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n016.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 05:26:23 AM UTC 2025
% Result : Theorem 0.23s 0.42s
% Output : Proof 0.23s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.15 % Problem : SET811+4 : TPTP v9.0.0. Released v3.2.0.
% 0.10/0.15 % Command : run_E %s %d THM
% 0.15/0.37 % Computer : n016.cluster.edu
% 0.15/0.37 % Model : x86_64 x86_64
% 0.15/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.37 % Memory : 8042.1875MB
% 0.15/0.37 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.37 % CPULimit : 300
% 0.15/0.37 % WCLimit : 300
% 0.15/0.37 % DateTime : Fri Jun 20 05:43:47 EDT 2025
% 0.15/0.37 % CPUTime :
% 0.23/0.42 % SZS status Theorem
% 0.23/0.42 % SZS output start Proof
% 0.23/0.42 tff(apply_type, type, (
% 0.23/0.42 apply: ( $i * $i * $i ) > $o)).
% 0.23/0.42 tff(tptp_fun_X_14_type, type, (
% 0.23/0.42 tptp_fun_X_14: $i)).
% 0.23/0.42 tff(tptp_fun_X_0_type, type, (
% 0.23/0.42 tptp_fun_X_0: ( $i * $i ) > $i)).
% 0.23/0.42 tff(initial_segment_type, type, (
% 0.23/0.42 initial_segment: ( $i * $i * $i ) > $i)).
% 0.23/0.42 tff(tptp_fun_A_13_type, type, (
% 0.23/0.42 tptp_fun_A_13: $i)).
% 0.23/0.42 tff(member_predicate_type, type, (
% 0.23/0.42 member_predicate: $i)).
% 0.23/0.42 tff(member_type, type, (
% 0.23/0.42 member: ( $i * $i ) > $o)).
% 0.23/0.42 tff(subset_type, type, (
% 0.23/0.42 subset: ( $i * $i ) > $o)).
% 0.23/0.42 tff(strict_well_order_type, type, (
% 0.23/0.42 strict_well_order: ( $i * $i ) > $o)).
% 0.23/0.42 tff(set_type, type, (
% 0.23/0.42 set: $i > $o)).
% 0.23/0.42 tff(on_type, type, (
% 0.23/0.42 on: $i)).
% 0.23/0.42 tff(tptp_fun_X_3_type, type, (
% 0.23/0.42 tptp_fun_X_3: $i > $i)).
% 0.23/0.42 tff(equal_set_type, type, (
% 0.23/0.42 equal_set: ( $i * $i ) > $o)).
% 0.23/0.42 tff(1,plain,
% 0.23/0.42 (^[X: $i, Y: $i] : refl((apply(member_predicate, X, Y) <=> member(X, Y)) <=> (apply(member_predicate, X, Y) <=> member(X, Y)))),
% 0.23/0.42 inference(bind,[status(th)],[])).
% 0.23/0.42 tff(2,plain,
% 0.23/0.42 (![X: $i, Y: $i] : (apply(member_predicate, X, Y) <=> member(X, Y)) <=> ![X: $i, Y: $i] : (apply(member_predicate, X, Y) <=> member(X, Y))),
% 0.23/0.42 inference(quant_intro,[status(thm)],[1])).
% 0.23/0.42 tff(3,plain,
% 0.23/0.42 (![X: $i, Y: $i] : (apply(member_predicate, X, Y) <=> member(X, Y)) <=> ![X: $i, Y: $i] : (apply(member_predicate, X, Y) <=> member(X, Y))),
% 0.23/0.42 inference(rewrite,[status(thm)],[])).
% 0.23/0.42 tff(4,axiom,(![X: $i, Y: $i] : (apply(member_predicate, X, Y) <=> member(X, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+4.ax','rel_member')).
% 0.23/0.42 tff(5,plain,
% 0.23/0.42 (![X: $i, Y: $i] : (apply(member_predicate, X, Y) <=> member(X, Y))),
% 0.23/0.42 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.23/0.42 tff(6,plain,(
% 0.23/0.42 ![X: $i, Y: $i] : (apply(member_predicate, X, Y) <=> member(X, Y))),
% 0.23/0.42 inference(skolemize,[status(sab)],[5])).
% 0.23/0.42 tff(7,plain,
% 0.23/0.42 (![X: $i, Y: $i] : (apply(member_predicate, X, Y) <=> member(X, Y))),
% 0.23/0.42 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.23/0.42 tff(8,plain,
% 0.23/0.42 ((~![X: $i, Y: $i] : (apply(member_predicate, X, Y) <=> member(X, Y))) | (apply(member_predicate, tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14) <=> member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14))),
% 0.23/0.42 inference(quant_inst,[status(thm)],[])).
% 0.23/0.42 tff(9,plain,
% 0.23/0.42 (apply(member_predicate, tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14) <=> member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14)),
% 0.23/0.42 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.23/0.42 tff(10,plain,
% 0.23/0.42 (^[A: $i, B: $i] : refl((~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))) <=> (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))))),
% 0.23/0.42 inference(bind,[status(th)],[])).
% 0.23/0.42 tff(11,plain,
% 0.23/0.42 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))),
% 0.23/0.42 inference(quant_intro,[status(thm)],[10])).
% 0.23/0.42 tff(12,plain,
% 0.23/0.42 (^[A: $i, B: $i] : rewrite((~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))) <=> (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))))),
% 0.23/0.42 inference(bind,[status(th)],[])).
% 0.23/0.42 tff(13,plain,
% 0.23/0.42 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))),
% 0.23/0.42 inference(quant_intro,[status(thm)],[12])).
% 0.23/0.42 tff(14,plain,
% 0.23/0.42 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))),
% 0.23/0.42 inference(transitivity,[status(thm)],[13, 11])).
% 0.23/0.42 tff(15,plain,
% 0.23/0.42 (^[A: $i, B: $i] : trans(monotonicity(rewrite(((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) <=> ((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))), rewrite((subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))) <=> (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))), ((((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) & (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))) <=> (((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) & (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))), rewrite((((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) & (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))) <=> (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))), ((((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) & (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))) <=> (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))))),
% 0.23/0.42 inference(bind,[status(th)],[])).
% 0.23/0.42 tff(16,plain,
% 0.23/0.42 (![A: $i, B: $i] : (((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) & (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))),
% 0.23/0.42 inference(quant_intro,[status(thm)],[15])).
% 0.23/0.42 tff(17,plain,
% 0.23/0.42 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B))) <=> ![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.23/0.42 inference(rewrite,[status(thm)],[])).
% 0.23/0.42 tff(18,plain,
% 0.23/0.42 (^[A: $i, B: $i] : rewrite((subset(A, B) <=> ![X: $i] : (member(X, A) => member(X, B))) <=> (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B))))),
% 0.23/0.42 inference(bind,[status(th)],[])).
% 0.23/0.42 tff(19,plain,
% 0.23/0.42 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : (member(X, A) => member(X, B))) <=> ![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.23/0.42 inference(quant_intro,[status(thm)],[18])).
% 0.23/0.42 tff(20,axiom,(![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : (member(X, A) => member(X, B)))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+0.ax','subset')).
% 0.23/0.42 tff(21,plain,
% 0.23/0.42 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.23/0.42 inference(modus_ponens,[status(thm)],[20, 19])).
% 0.23/0.42 tff(22,plain,
% 0.23/0.42 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.23/0.42 inference(modus_ponens,[status(thm)],[21, 17])).
% 0.23/0.42 tff(23,plain,(
% 0.23/0.42 ![A: $i, B: $i] : (((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) & (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))),
% 0.23/0.42 inference(skolemize,[status(sab)],[22])).
% 0.23/0.42 tff(24,plain,
% 0.23/0.42 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))),
% 0.23/0.42 inference(modus_ponens,[status(thm)],[23, 16])).
% 0.23/0.42 tff(25,plain,
% 0.23/0.42 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))),
% 0.23/0.42 inference(modus_ponens,[status(thm)],[24, 14])).
% 0.23/0.42 tff(26,plain,
% 0.23/0.42 ((~![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))) | (~((~((~subset(initial_segment(X!14, member_predicate, A!13), X!14)) | ![X: $i] : ((~member(X, initial_segment(X!14, member_predicate, A!13))) | member(X, X!14)))) | (~(subset(initial_segment(X!14, member_predicate, A!13), X!14) | (~((~member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), initial_segment(X!14, member_predicate, A!13))) | member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14)))))))),
% 0.23/0.42 inference(quant_inst,[status(thm)],[])).
% 0.23/0.42 tff(27,plain,
% 0.23/0.42 (~((~((~subset(initial_segment(X!14, member_predicate, A!13), X!14)) | ![X: $i] : ((~member(X, initial_segment(X!14, member_predicate, A!13))) | member(X, X!14)))) | (~(subset(initial_segment(X!14, member_predicate, A!13), X!14) | (~((~member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), initial_segment(X!14, member_predicate, A!13))) | member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14))))))),
% 0.23/0.42 inference(unit_resolution,[status(thm)],[26, 25])).
% 0.23/0.42 tff(28,plain,
% 0.23/0.42 (((~((~subset(initial_segment(X!14, member_predicate, A!13), X!14)) | ![X: $i] : ((~member(X, initial_segment(X!14, member_predicate, A!13))) | member(X, X!14)))) | (~(subset(initial_segment(X!14, member_predicate, A!13), X!14) | (~((~member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), initial_segment(X!14, member_predicate, A!13))) | member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14)))))) | (subset(initial_segment(X!14, member_predicate, A!13), X!14) | (~((~member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), initial_segment(X!14, member_predicate, A!13))) | member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14))))),
% 0.23/0.42 inference(tautology,[status(thm)],[])).
% 0.23/0.42 tff(29,plain,
% 0.23/0.42 (subset(initial_segment(X!14, member_predicate, A!13), X!14) | (~((~member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), initial_segment(X!14, member_predicate, A!13))) | member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14)))),
% 0.23/0.42 inference(unit_resolution,[status(thm)],[28, 27])).
% 0.23/0.42 tff(30,assumption,((~((~subset(X!14, initial_segment(X!14, member_predicate, A!13))) | ![X: $i] : ((~member(X, X!14)) | member(X, initial_segment(X!14, member_predicate, A!13))))) | (~(subset(X!14, initial_segment(X!14, member_predicate, A!13)) | (~((~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14)) | member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), initial_segment(X!14, member_predicate, A!13))))))), introduced(assumption)).
% 0.23/0.42 tff(31,plain,
% 0.23/0.42 ((~![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))) | (~((~((~subset(X!14, initial_segment(X!14, member_predicate, A!13))) | ![X: $i] : ((~member(X, X!14)) | member(X, initial_segment(X!14, member_predicate, A!13))))) | (~(subset(X!14, initial_segment(X!14, member_predicate, A!13)) | (~((~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14)) | member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), initial_segment(X!14, member_predicate, A!13))))))))),
% 0.23/0.42 inference(quant_inst,[status(thm)],[])).
% 0.23/0.42 tff(32,plain,
% 0.23/0.42 ($false),
% 0.23/0.42 inference(unit_resolution,[status(thm)],[31, 25, 30])).
% 0.23/0.42 tff(33,plain,(~((~((~subset(X!14, initial_segment(X!14, member_predicate, A!13))) | ![X: $i] : ((~member(X, X!14)) | member(X, initial_segment(X!14, member_predicate, A!13))))) | (~(subset(X!14, initial_segment(X!14, member_predicate, A!13)) | (~((~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14)) | member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), initial_segment(X!14, member_predicate, A!13)))))))), inference(lemma,lemma(discharge,[]))).
% 0.23/0.43 tff(34,plain,
% 0.23/0.43 (((~((~subset(X!14, initial_segment(X!14, member_predicate, A!13))) | ![X: $i] : ((~member(X, X!14)) | member(X, initial_segment(X!14, member_predicate, A!13))))) | (~(subset(X!14, initial_segment(X!14, member_predicate, A!13)) | (~((~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14)) | member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), initial_segment(X!14, member_predicate, A!13))))))) | (subset(X!14, initial_segment(X!14, member_predicate, A!13)) | (~((~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14)) | member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), initial_segment(X!14, member_predicate, A!13)))))),
% 0.23/0.43 inference(tautology,[status(thm)],[])).
% 0.23/0.43 tff(35,plain,
% 0.23/0.43 (subset(X!14, initial_segment(X!14, member_predicate, A!13)) | (~((~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14)) | member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), initial_segment(X!14, member_predicate, A!13))))),
% 0.23/0.43 inference(unit_resolution,[status(thm)],[34, 33])).
% 0.23/0.43 tff(36,plain,
% 0.23/0.43 ((~![X: $i, Y: $i] : (apply(member_predicate, X, Y) <=> member(X, Y))) | (apply(member_predicate, tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14) <=> member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14))),
% 0.23/0.43 inference(quant_inst,[status(thm)],[])).
% 0.23/0.43 tff(37,plain,
% 0.23/0.43 (apply(member_predicate, tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14) <=> member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14)),
% 0.23/0.43 inference(unit_resolution,[status(thm)],[36, 7])).
% 0.23/0.43 tff(38,plain,
% 0.23/0.43 (^[X: $i, R: $i, A: $i, Y: $i] : refl((~(((~member(Y, A)) | (~apply(R, Y, X))) <=> member(Y, initial_segment(X, R, A)))) <=> (~(((~member(Y, A)) | (~apply(R, Y, X))) <=> member(Y, initial_segment(X, R, A)))))),
% 0.23/0.43 inference(bind,[status(th)],[])).
% 0.23/0.43 tff(39,plain,
% 0.23/0.43 (![X: $i, R: $i, A: $i, Y: $i] : (~(((~member(Y, A)) | (~apply(R, Y, X))) <=> member(Y, initial_segment(X, R, A)))) <=> ![X: $i, R: $i, A: $i, Y: $i] : (~(((~member(Y, A)) | (~apply(R, Y, X))) <=> member(Y, initial_segment(X, R, A))))),
% 0.23/0.43 inference(quant_intro,[status(thm)],[38])).
% 0.23/0.43 tff(40,plain,
% 0.23/0.43 (^[X: $i, R: $i, A: $i, Y: $i] : trans(trans(monotonicity(rewrite((member(Y, A) & apply(R, Y, X)) <=> (~((~member(Y, A)) | (~apply(R, Y, X))))), ((member(Y, initial_segment(X, R, A)) <=> (member(Y, A) & apply(R, Y, X))) <=> (member(Y, initial_segment(X, R, A)) <=> (~((~member(Y, A)) | (~apply(R, Y, X))))))), rewrite((member(Y, initial_segment(X, R, A)) <=> (~((~member(Y, A)) | (~apply(R, Y, X))))) <=> (~(((~member(Y, A)) | (~apply(R, Y, X))) <=> member(Y, initial_segment(X, R, A))))), ((member(Y, initial_segment(X, R, A)) <=> (member(Y, A) & apply(R, Y, X))) <=> (~(((~member(Y, A)) | (~apply(R, Y, X))) <=> member(Y, initial_segment(X, R, A)))))), rewrite((~(((~member(Y, A)) | (~apply(R, Y, X))) <=> member(Y, initial_segment(X, R, A)))) <=> (~(((~member(Y, A)) | (~apply(R, Y, X))) <=> member(Y, initial_segment(X, R, A))))), ((member(Y, initial_segment(X, R, A)) <=> (member(Y, A) & apply(R, Y, X))) <=> (~(((~member(Y, A)) | (~apply(R, Y, X))) <=> member(Y, initial_segment(X, R, A))))))),
% 0.23/0.43 inference(bind,[status(th)],[])).
% 0.23/0.43 tff(41,plain,
% 0.23/0.43 (![X: $i, R: $i, A: $i, Y: $i] : (member(Y, initial_segment(X, R, A)) <=> (member(Y, A) & apply(R, Y, X))) <=> ![X: $i, R: $i, A: $i, Y: $i] : (~(((~member(Y, A)) | (~apply(R, Y, X))) <=> member(Y, initial_segment(X, R, A))))),
% 0.23/0.43 inference(quant_intro,[status(thm)],[40])).
% 0.23/0.43 tff(42,plain,
% 0.23/0.43 (![X: $i, R: $i, A: $i, Y: $i] : (member(Y, initial_segment(X, R, A)) <=> (member(Y, A) & apply(R, Y, X))) <=> ![X: $i, R: $i, A: $i, Y: $i] : (member(Y, initial_segment(X, R, A)) <=> (member(Y, A) & apply(R, Y, X)))),
% 0.23/0.43 inference(rewrite,[status(thm)],[])).
% 0.23/0.43 tff(43,axiom,(![X: $i, R: $i, A: $i, Y: $i] : (member(Y, initial_segment(X, R, A)) <=> (member(Y, A) & apply(R, Y, X)))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+4.ax','initial_segment')).
% 0.23/0.43 tff(44,plain,
% 0.23/0.43 (![X: $i, R: $i, A: $i, Y: $i] : (member(Y, initial_segment(X, R, A)) <=> (member(Y, A) & apply(R, Y, X)))),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[43, 42])).
% 0.23/0.43 tff(45,plain,(
% 0.23/0.43 ![X: $i, R: $i, A: $i, Y: $i] : (member(Y, initial_segment(X, R, A)) <=> (member(Y, A) & apply(R, Y, X)))),
% 0.23/0.43 inference(skolemize,[status(sab)],[44])).
% 0.23/0.43 tff(46,plain,
% 0.23/0.43 (![X: $i, R: $i, A: $i, Y: $i] : (~(((~member(Y, A)) | (~apply(R, Y, X))) <=> member(Y, initial_segment(X, R, A))))),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[45, 41])).
% 0.23/0.43 tff(47,plain,
% 0.23/0.43 (![X: $i, R: $i, A: $i, Y: $i] : (~(((~member(Y, A)) | (~apply(R, Y, X))) <=> member(Y, initial_segment(X, R, A))))),
% 0.23/0.43 inference(modus_ponens,[status(thm)],[46, 39])).
% 0.23/0.43 tff(48,plain,
% 0.23/0.43 ((~![X: $i, R: $i, A: $i, Y: $i] : (~(((~member(Y, A)) | (~apply(R, Y, X))) <=> member(Y, initial_segment(X, R, A))))) | (~(((~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), A!13)) | (~apply(member_predicate, tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14))) <=> member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), initial_segment(X!14, member_predicate, A!13))))),
% 0.23/0.43 inference(quant_inst,[status(thm)],[])).
% 0.23/0.43 tff(49,plain,
% 0.23/0.43 (~(((~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), A!13)) | (~apply(member_predicate, tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14))) <=> member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), initial_segment(X!14, member_predicate, A!13)))),
% 0.23/0.43 inference(unit_resolution,[status(thm)],[48, 47])).
% 0.23/0.43 tff(50,assumption,(~((~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14)) | member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), initial_segment(X!14, member_predicate, A!13)))), introduced(assumption)).
% 0.23/0.43 tff(51,plain,
% 0.23/0.43 (((~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14)) | member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), initial_segment(X!14, member_predicate, A!13))) | (~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), initial_segment(X!14, member_predicate, A!13)))),
% 0.23/0.43 inference(tautology,[status(thm)],[])).
% 0.23/0.43 tff(52,plain,
% 0.23/0.43 (~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), initial_segment(X!14, member_predicate, A!13))),
% 0.23/0.43 inference(unit_resolution,[status(thm)],[51, 50])).
% 0.23/0.43 tff(53,plain,
% 0.23/0.43 ((((~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), A!13)) | (~apply(member_predicate, tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14))) <=> member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), initial_segment(X!14, member_predicate, A!13))) | ((~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), A!13)) | (~apply(member_predicate, tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14))) | member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), initial_segment(X!14, member_predicate, A!13))),
% 0.23/0.43 inference(tautology,[status(thm)],[])).
% 0.23/0.43 tff(54,plain,
% 0.23/0.43 ((((~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), A!13)) | (~apply(member_predicate, tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14))) <=> member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), initial_segment(X!14, member_predicate, A!13))) | ((~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), A!13)) | (~apply(member_predicate, tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14)))),
% 0.23/0.43 inference(unit_resolution,[status(thm)],[53, 52])).
% 0.23/0.43 tff(55,plain,
% 0.23/0.43 ((~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), A!13)) | (~apply(member_predicate, tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14))),
% 0.23/0.43 inference(unit_resolution,[status(thm)],[54, 49])).
% 0.23/0.43 tff(56,plain,
% 0.23/0.43 (((~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14)) | member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), initial_segment(X!14, member_predicate, A!13))) | member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14)),
% 0.23/0.43 inference(tautology,[status(thm)],[])).
% 0.23/0.43 tff(57,plain,
% 0.23/0.43 (member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14)),
% 0.23/0.43 inference(unit_resolution,[status(thm)],[56, 50])).
% 0.23/0.43 tff(58,plain,
% 0.23/0.43 ((~![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))) | (~((~((~subset(X!14, A!13)) | ![X: $i] : ((~member(X, X!14)) | member(X, A!13)))) | (~(subset(X!14, A!13) | (~((~member(tptp_fun_X_0(A!13, X!14), X!14)) | member(tptp_fun_X_0(A!13, X!14), A!13)))))))),
% 0.23/0.43 inference(quant_inst,[status(thm)],[])).
% 0.23/0.43 tff(59,plain,
% 0.23/0.43 (~((~((~subset(X!14, A!13)) | ![X: $i] : ((~member(X, X!14)) | member(X, A!13)))) | (~(subset(X!14, A!13) | (~((~member(tptp_fun_X_0(A!13, X!14), X!14)) | member(tptp_fun_X_0(A!13, X!14), A!13))))))),
% 0.23/0.43 inference(unit_resolution,[status(thm)],[58, 25])).
% 0.23/0.43 tff(60,plain,
% 0.23/0.43 (((~((~subset(X!14, A!13)) | ![X: $i] : ((~member(X, X!14)) | member(X, A!13)))) | (~(subset(X!14, A!13) | (~((~member(tptp_fun_X_0(A!13, X!14), X!14)) | member(tptp_fun_X_0(A!13, X!14), A!13)))))) | ((~subset(X!14, A!13)) | ![X: $i] : ((~member(X, X!14)) | member(X, A!13)))),
% 0.23/0.43 inference(tautology,[status(thm)],[])).
% 0.23/0.43 tff(61,plain,
% 0.23/0.43 ((~subset(X!14, A!13)) | ![X: $i] : ((~member(X, X!14)) | member(X, A!13))),
% 0.23/0.43 inference(unit_resolution,[status(thm)],[60, 59])).
% 0.23/0.43 tff(62,plain,
% 0.23/0.43 (^[A: $i] : rewrite((~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))) <=> (~((~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) | (~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))))))),
% 0.23/0.43 inference(bind,[status(th)],[])).
% 0.23/0.43 tff(63,plain,
% 0.23/0.43 (![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))) <=> ![A: $i] : (~((~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) | (~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))))))),
% 0.23/0.43 inference(quant_intro,[status(thm)],[62])).
% 0.23/0.43 tff(64,plain,
% 0.23/0.43 (^[A: $i] : refl((~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))) <=> (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))))),
% 0.23/0.43 inference(bind,[status(th)],[])).
% 0.23/0.43 tff(65,plain,
% 0.23/0.43 (![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))) <=> ![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))))),
% 0.23/0.44 inference(quant_intro,[status(thm)],[64])).
% 0.23/0.44 tff(66,plain,
% 0.23/0.44 (^[A: $i] : rewrite((~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))) <=> (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))))),
% 0.23/0.44 inference(bind,[status(th)],[])).
% 0.23/0.44 tff(67,plain,
% 0.23/0.44 (![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))) <=> ![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))))),
% 0.23/0.44 inference(quant_intro,[status(thm)],[66])).
% 0.23/0.44 tff(68,plain,
% 0.23/0.44 (![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))) <=> ![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))))),
% 0.23/0.44 inference(transitivity,[status(thm)],[67, 65])).
% 0.23/0.44 tff(69,plain,
% 0.23/0.44 (^[A: $i] : trans(monotonicity(rewrite(((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) <=> ((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))), rewrite((member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))) <=> (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))), ((((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) <=> (((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))) & (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))), rewrite((((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))) & (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) <=> (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))))), ((((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) <=> (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))))))),
% 0.23/0.44 inference(bind,[status(th)],[])).
% 0.23/0.44 tff(70,plain,
% 0.23/0.44 (![A: $i] : (((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) <=> ![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))))),
% 0.23/0.44 inference(quant_intro,[status(thm)],[69])).
% 0.23/0.44 tff(71,plain,
% 0.23/0.44 (^[A: $i] : rewrite((((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | ((~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))) <=> (((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))),
% 0.23/0.44 inference(bind,[status(th)],[])).
% 0.23/0.44 tff(72,plain,
% 0.23/0.44 (![A: $i] : (((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | ((~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))) <=> ![A: $i] : (((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))),
% 0.23/0.44 inference(quant_intro,[status(thm)],[71])).
% 0.23/0.44 tff(73,plain,
% 0.23/0.44 (![A: $i] : (member(A, on) <=> (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) <=> ![A: $i] : (member(A, on) <=> (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A))))),
% 0.23/0.44 inference(rewrite,[status(thm)],[])).
% 0.23/0.44 tff(74,plain,
% 0.23/0.44 (^[A: $i] : rewrite((member(A, on) <=> ((set(A) & strict_well_order(member_predicate, A)) & ![X: $i] : (member(X, A) => subset(X, A)))) <=> (member(A, on) <=> (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))))),
% 0.23/0.44 inference(bind,[status(th)],[])).
% 0.23/0.44 tff(75,plain,
% 0.23/0.44 (![A: $i] : (member(A, on) <=> ((set(A) & strict_well_order(member_predicate, A)) & ![X: $i] : (member(X, A) => subset(X, A)))) <=> ![A: $i] : (member(A, on) <=> (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A))))),
% 0.23/0.44 inference(quant_intro,[status(thm)],[74])).
% 0.23/0.44 tff(76,axiom,(![A: $i] : (member(A, on) <=> ((set(A) & strict_well_order(member_predicate, A)) & ![X: $i] : (member(X, A) => subset(X, A))))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+4.ax','ordinal_number')).
% 0.23/0.44 tff(77,plain,
% 0.23/0.44 (![A: $i] : (member(A, on) <=> (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A))))),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[76, 75])).
% 0.23/0.44 tff(78,plain,
% 0.23/0.44 (![A: $i] : (member(A, on) <=> (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A))))),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[77, 73])).
% 0.23/0.44 tff(79,plain,(
% 0.23/0.44 ![A: $i] : (((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | ((~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))),
% 0.23/0.44 inference(skolemize,[status(sab)],[78])).
% 0.23/0.44 tff(80,plain,
% 0.23/0.44 (![A: $i] : (((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[79, 72])).
% 0.23/0.44 tff(81,plain,
% 0.23/0.44 (![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))))),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[80, 70])).
% 0.23/0.44 tff(82,plain,
% 0.23/0.44 (![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))))),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[81, 68])).
% 0.23/0.44 tff(83,plain,
% 0.23/0.44 (![A: $i] : (~((~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) | (~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))))))),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[82, 63])).
% 0.23/0.44 tff(84,plain,
% 0.23/0.44 (((~![A: $i] : (~((~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) | (~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))))))) | (~((~(member(A!13, on) | (~strict_well_order(member_predicate, A!13)) | (~set(A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))))))))) <=> ((~![A: $i] : (~((~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) | (~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))))))) | (~((~(member(A!13, on) | (~strict_well_order(member_predicate, A!13)) | (~set(A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))))))))))),
% 0.23/0.44 inference(rewrite,[status(thm)],[])).
% 0.23/0.44 tff(85,plain,
% 0.23/0.44 ((~((~(member(A!13, on) | (~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))))))))) <=> (~((~(member(A!13, on) | (~strict_well_order(member_predicate, A!13)) | (~set(A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))))))))),
% 0.23/0.44 inference(rewrite,[status(thm)],[])).
% 0.23/0.44 tff(86,plain,
% 0.23/0.44 (((~![A: $i] : (~((~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) | (~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))))))) | (~((~(member(A!13, on) | (~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))))))))) <=> ((~![A: $i] : (~((~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) | (~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))))))) | (~((~(member(A!13, on) | (~strict_well_order(member_predicate, A!13)) | (~set(A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))))))))))),
% 0.23/0.44 inference(monotonicity,[status(thm)],[85])).
% 0.23/0.44 tff(87,plain,
% 0.23/0.44 (((~![A: $i] : (~((~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) | (~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))))))) | (~((~(member(A!13, on) | (~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))))))))) <=> ((~![A: $i] : (~((~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) | (~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))))))) | (~((~(member(A!13, on) | (~strict_well_order(member_predicate, A!13)) | (~set(A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))))))))))),
% 0.23/0.44 inference(transitivity,[status(thm)],[86, 84])).
% 0.23/0.44 tff(88,plain,
% 0.23/0.44 ((~![A: $i] : (~((~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) | (~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))))))) | (~((~(member(A!13, on) | (~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))))))))),
% 0.23/0.44 inference(quant_inst,[status(thm)],[])).
% 0.23/0.44 tff(89,plain,
% 0.23/0.44 ((~![A: $i] : (~((~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) | (~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))))))) | (~((~(member(A!13, on) | (~strict_well_order(member_predicate, A!13)) | (~set(A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))))))))),
% 0.23/0.44 inference(modus_ponens,[status(thm)],[88, 87])).
% 0.23/0.44 tff(90,plain,
% 0.23/0.44 (~((~(member(A!13, on) | (~strict_well_order(member_predicate, A!13)) | (~set(A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))))))))),
% 0.23/0.44 inference(unit_resolution,[status(thm)],[89, 83])).
% 0.23/0.44 tff(91,plain,
% 0.23/0.44 (((~(member(A!13, on) | (~strict_well_order(member_predicate, A!13)) | (~set(A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))))))) | ((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))))))),
% 0.23/0.45 inference(tautology,[status(thm)],[])).
% 0.23/0.45 tff(92,plain,
% 0.23/0.45 ((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))))),
% 0.23/0.45 inference(unit_resolution,[status(thm)],[91, 90])).
% 0.23/0.45 tff(93,plain,
% 0.23/0.45 ((~![A: $i] : ((~member(A, on)) | ![X: $i] : ((~member(X, A)) | equal_set(X, initial_segment(X, member_predicate, A))))) <=> (~![A: $i] : ((~member(A, on)) | ![X: $i] : ((~member(X, A)) | equal_set(X, initial_segment(X, member_predicate, A)))))),
% 0.23/0.45 inference(rewrite,[status(thm)],[])).
% 0.23/0.45 tff(94,plain,
% 0.23/0.45 ((~![A: $i] : (member(A, on) => ![X: $i] : (member(X, A) => equal_set(X, initial_segment(X, member_predicate, A))))) <=> (~![A: $i] : ((~member(A, on)) | ![X: $i] : ((~member(X, A)) | equal_set(X, initial_segment(X, member_predicate, A)))))),
% 0.23/0.45 inference(rewrite,[status(thm)],[])).
% 0.23/0.45 tff(95,axiom,(~![A: $i] : (member(A, on) => ![X: $i] : (member(X, A) => equal_set(X, initial_segment(X, member_predicate, A))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','thV5')).
% 0.23/0.45 tff(96,plain,
% 0.23/0.45 (~![A: $i] : ((~member(A, on)) | ![X: $i] : ((~member(X, A)) | equal_set(X, initial_segment(X, member_predicate, A))))),
% 0.23/0.45 inference(modus_ponens,[status(thm)],[95, 94])).
% 0.23/0.45 tff(97,plain,
% 0.23/0.45 (~![A: $i] : ((~member(A, on)) | ![X: $i] : ((~member(X, A)) | equal_set(X, initial_segment(X, member_predicate, A))))),
% 0.23/0.45 inference(modus_ponens,[status(thm)],[96, 93])).
% 0.23/0.45 tff(98,plain,
% 0.23/0.45 (~![A: $i] : ((~member(A, on)) | ![X: $i] : ((~member(X, A)) | equal_set(X, initial_segment(X, member_predicate, A))))),
% 0.23/0.45 inference(modus_ponens,[status(thm)],[97, 93])).
% 0.23/0.45 tff(99,plain,
% 0.23/0.45 (~![A: $i] : ((~member(A, on)) | ![X: $i] : ((~member(X, A)) | equal_set(X, initial_segment(X, member_predicate, A))))),
% 0.23/0.45 inference(modus_ponens,[status(thm)],[98, 93])).
% 0.23/0.45 tff(100,plain,
% 0.23/0.45 (member(A!13, on)),
% 0.23/0.45 inference(and_elim,[status(thm)],[99])).
% 0.23/0.45 tff(101,plain,
% 0.23/0.45 ((~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))))))) | (~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))))),
% 0.23/0.45 inference(tautology,[status(thm)],[])).
% 0.23/0.45 tff(102,plain,
% 0.23/0.45 ((~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))))))) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))))),
% 0.23/0.45 inference(unit_resolution,[status(thm)],[101, 100])).
% 0.23/0.45 tff(103,plain,
% 0.23/0.45 (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))))),
% 0.23/0.45 inference(unit_resolution,[status(thm)],[102, 92])).
% 0.23/0.45 tff(104,plain,
% 0.23/0.45 (((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))) | ![X: $i] : ((~member(X, A!13)) | subset(X, A!13))),
% 0.23/0.45 inference(tautology,[status(thm)],[])).
% 0.23/0.45 tff(105,plain,
% 0.23/0.45 (![X: $i] : ((~member(X, A!13)) | subset(X, A!13))),
% 0.23/0.45 inference(unit_resolution,[status(thm)],[104, 103])).
% 0.23/0.45 tff(106,plain,
% 0.23/0.45 (~((~member(X!14, A!13)) | equal_set(X!14, initial_segment(X!14, member_predicate, A!13)))),
% 0.23/0.45 inference(and_elim,[status(thm)],[99])).
% 0.23/0.45 tff(107,plain,
% 0.23/0.45 (member(X!14, A!13)),
% 0.23/0.45 inference(or_elim,[status(thm)],[106])).
% 0.23/0.45 tff(108,plain,
% 0.23/0.45 (((~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))) | ((~member(X!14, A!13)) | subset(X!14, A!13))) <=> ((~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))) | (~member(X!14, A!13)) | subset(X!14, A!13))),
% 0.23/0.45 inference(rewrite,[status(thm)],[])).
% 0.23/0.45 tff(109,plain,
% 0.23/0.45 ((~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))) | ((~member(X!14, A!13)) | subset(X!14, A!13))),
% 0.23/0.45 inference(quant_inst,[status(thm)],[])).
% 0.23/0.45 tff(110,plain,
% 0.23/0.45 ((~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))) | (~member(X!14, A!13)) | subset(X!14, A!13)),
% 0.23/0.45 inference(modus_ponens,[status(thm)],[109, 108])).
% 0.23/0.45 tff(111,plain,
% 0.23/0.45 (subset(X!14, A!13)),
% 0.23/0.45 inference(unit_resolution,[status(thm)],[110, 107, 105])).
% 0.23/0.45 tff(112,plain,
% 0.23/0.45 ((~((~subset(X!14, A!13)) | ![X: $i] : ((~member(X, X!14)) | member(X, A!13)))) | (~subset(X!14, A!13)) | ![X: $i] : ((~member(X, X!14)) | member(X, A!13))),
% 0.23/0.45 inference(tautology,[status(thm)],[])).
% 0.23/0.45 tff(113,plain,
% 0.23/0.45 ((~((~subset(X!14, A!13)) | ![X: $i] : ((~member(X, X!14)) | member(X, A!13)))) | ![X: $i] : ((~member(X, X!14)) | member(X, A!13))),
% 0.23/0.45 inference(unit_resolution,[status(thm)],[112, 111])).
% 0.23/0.45 tff(114,plain,
% 0.23/0.45 (![X: $i] : ((~member(X, X!14)) | member(X, A!13))),
% 0.23/0.45 inference(unit_resolution,[status(thm)],[113, 61])).
% 0.23/0.45 tff(115,plain,
% 0.23/0.45 (((~![X: $i] : ((~member(X, X!14)) | member(X, A!13))) | ((~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14)) | member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), A!13))) <=> ((~![X: $i] : ((~member(X, X!14)) | member(X, A!13))) | (~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14)) | member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), A!13))),
% 0.23/0.45 inference(rewrite,[status(thm)],[])).
% 0.23/0.45 tff(116,plain,
% 0.23/0.45 ((~![X: $i] : ((~member(X, X!14)) | member(X, A!13))) | ((~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14)) | member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), A!13))),
% 0.23/0.45 inference(quant_inst,[status(thm)],[])).
% 0.23/0.45 tff(117,plain,
% 0.23/0.45 ((~![X: $i] : ((~member(X, X!14)) | member(X, A!13))) | (~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14)) | member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), A!13)),
% 0.23/0.45 inference(modus_ponens,[status(thm)],[116, 115])).
% 0.23/0.45 tff(118,plain,
% 0.23/0.45 (member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), A!13)),
% 0.23/0.45 inference(unit_resolution,[status(thm)],[117, 114, 57])).
% 0.23/0.45 tff(119,plain,
% 0.23/0.45 ((~((~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), A!13)) | (~apply(member_predicate, tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14)))) | (~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), A!13)) | (~apply(member_predicate, tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14))),
% 0.23/0.45 inference(tautology,[status(thm)],[])).
% 0.23/0.45 tff(120,plain,
% 0.23/0.45 (~apply(member_predicate, tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14)),
% 0.23/0.45 inference(unit_resolution,[status(thm)],[119, 118, 55])).
% 0.23/0.45 tff(121,plain,
% 0.23/0.45 ((~(apply(member_predicate, tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14) <=> member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14))) | apply(member_predicate, tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14) | (~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14))),
% 0.23/0.45 inference(tautology,[status(thm)],[])).
% 0.23/0.45 tff(122,plain,
% 0.23/0.45 ((~(apply(member_predicate, tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14) <=> member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14))) | apply(member_predicate, tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14)),
% 0.23/0.45 inference(unit_resolution,[status(thm)],[121, 57])).
% 0.23/0.45 tff(123,plain,
% 0.23/0.45 ($false),
% 0.23/0.45 inference(unit_resolution,[status(thm)],[122, 120, 37])).
% 0.23/0.45 tff(124,plain,((~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14)) | member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), initial_segment(X!14, member_predicate, A!13))), inference(lemma,lemma(discharge,[]))).
% 0.23/0.45 tff(125,plain,
% 0.23/0.45 ((~(subset(X!14, initial_segment(X!14, member_predicate, A!13)) | (~((~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14)) | member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), initial_segment(X!14, member_predicate, A!13)))))) | subset(X!14, initial_segment(X!14, member_predicate, A!13)) | (~((~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14)) | member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), initial_segment(X!14, member_predicate, A!13))))),
% 0.23/0.45 inference(tautology,[status(thm)],[])).
% 0.23/0.45 tff(126,plain,
% 0.23/0.45 ((~(subset(X!14, initial_segment(X!14, member_predicate, A!13)) | (~((~member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), X!14)) | member(tptp_fun_X_0(initial_segment(X!14, member_predicate, A!13), X!14), initial_segment(X!14, member_predicate, A!13)))))) | subset(X!14, initial_segment(X!14, member_predicate, A!13))),
% 0.23/0.45 inference(unit_resolution,[status(thm)],[125, 124])).
% 0.23/0.45 tff(127,plain,
% 0.23/0.45 (subset(X!14, initial_segment(X!14, member_predicate, A!13))),
% 0.23/0.45 inference(unit_resolution,[status(thm)],[126, 35])).
% 0.23/0.45 tff(128,plain,
% 0.23/0.45 (^[A: $i, B: $i] : refl((~(((~subset(A, B)) | (~subset(B, A))) <=> equal_set(A, B))) <=> (~(((~subset(A, B)) | (~subset(B, A))) <=> equal_set(A, B))))),
% 0.23/0.45 inference(bind,[status(th)],[])).
% 0.23/0.45 tff(129,plain,
% 0.23/0.45 (![A: $i, B: $i] : (~(((~subset(A, B)) | (~subset(B, A))) <=> equal_set(A, B))) <=> ![A: $i, B: $i] : (~(((~subset(A, B)) | (~subset(B, A))) <=> equal_set(A, B)))),
% 0.23/0.45 inference(quant_intro,[status(thm)],[128])).
% 0.23/0.45 tff(130,plain,
% 0.23/0.45 (^[A: $i, B: $i] : trans(trans(monotonicity(rewrite((subset(A, B) & subset(B, A)) <=> (~((~subset(A, B)) | (~subset(B, A))))), ((equal_set(A, B) <=> (subset(A, B) & subset(B, A))) <=> (equal_set(A, B) <=> (~((~subset(A, B)) | (~subset(B, A))))))), rewrite((equal_set(A, B) <=> (~((~subset(A, B)) | (~subset(B, A))))) <=> (~(((~subset(A, B)) | (~subset(B, A))) <=> equal_set(A, B)))), ((equal_set(A, B) <=> (subset(A, B) & subset(B, A))) <=> (~(((~subset(A, B)) | (~subset(B, A))) <=> equal_set(A, B))))), rewrite((~(((~subset(A, B)) | (~subset(B, A))) <=> equal_set(A, B))) <=> (~(((~subset(A, B)) | (~subset(B, A))) <=> equal_set(A, B)))), ((equal_set(A, B) <=> (subset(A, B) & subset(B, A))) <=> (~(((~subset(A, B)) | (~subset(B, A))) <=> equal_set(A, B)))))),
% 0.23/0.45 inference(bind,[status(th)],[])).
% 0.23/0.45 tff(131,plain,
% 0.23/0.45 (![A: $i, B: $i] : (equal_set(A, B) <=> (subset(A, B) & subset(B, A))) <=> ![A: $i, B: $i] : (~(((~subset(A, B)) | (~subset(B, A))) <=> equal_set(A, B)))),
% 0.23/0.45 inference(quant_intro,[status(thm)],[130])).
% 0.23/0.45 tff(132,plain,
% 0.23/0.45 (![A: $i, B: $i] : (equal_set(A, B) <=> (subset(A, B) & subset(B, A))) <=> ![A: $i, B: $i] : (equal_set(A, B) <=> (subset(A, B) & subset(B, A)))),
% 0.23/0.45 inference(rewrite,[status(thm)],[])).
% 0.23/0.45 tff(133,axiom,(![A: $i, B: $i] : (equal_set(A, B) <=> (subset(A, B) & subset(B, A)))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+0.ax','equal_set')).
% 0.23/0.45 tff(134,plain,
% 0.23/0.45 (![A: $i, B: $i] : (equal_set(A, B) <=> (subset(A, B) & subset(B, A)))),
% 0.23/0.45 inference(modus_ponens,[status(thm)],[133, 132])).
% 0.23/0.45 tff(135,plain,(
% 0.23/0.45 ![A: $i, B: $i] : (equal_set(A, B) <=> (subset(A, B) & subset(B, A)))),
% 0.23/0.45 inference(skolemize,[status(sab)],[134])).
% 0.23/0.45 tff(136,plain,
% 0.23/0.45 (![A: $i, B: $i] : (~(((~subset(A, B)) | (~subset(B, A))) <=> equal_set(A, B)))),
% 0.23/0.45 inference(modus_ponens,[status(thm)],[135, 131])).
% 0.23/0.45 tff(137,plain,
% 0.23/0.45 (![A: $i, B: $i] : (~(((~subset(A, B)) | (~subset(B, A))) <=> equal_set(A, B)))),
% 0.23/0.45 inference(modus_ponens,[status(thm)],[136, 129])).
% 0.23/0.45 tff(138,plain,
% 0.23/0.45 ((~![A: $i, B: $i] : (~(((~subset(A, B)) | (~subset(B, A))) <=> equal_set(A, B)))) | (~(((~subset(X!14, initial_segment(X!14, member_predicate, A!13))) | (~subset(initial_segment(X!14, member_predicate, A!13), X!14))) <=> equal_set(X!14, initial_segment(X!14, member_predicate, A!13))))),
% 0.23/0.45 inference(quant_inst,[status(thm)],[])).
% 0.23/0.45 tff(139,plain,
% 0.23/0.45 (~(((~subset(X!14, initial_segment(X!14, member_predicate, A!13))) | (~subset(initial_segment(X!14, member_predicate, A!13), X!14))) <=> equal_set(X!14, initial_segment(X!14, member_predicate, A!13)))),
% 0.23/0.46 inference(unit_resolution,[status(thm)],[138, 137])).
% 0.23/0.46 tff(140,plain,
% 0.23/0.46 (~equal_set(X!14, initial_segment(X!14, member_predicate, A!13))),
% 0.23/0.46 inference(or_elim,[status(thm)],[106])).
% 0.23/0.46 tff(141,plain,
% 0.23/0.46 ((((~subset(X!14, initial_segment(X!14, member_predicate, A!13))) | (~subset(initial_segment(X!14, member_predicate, A!13), X!14))) <=> equal_set(X!14, initial_segment(X!14, member_predicate, A!13))) | ((~subset(X!14, initial_segment(X!14, member_predicate, A!13))) | (~subset(initial_segment(X!14, member_predicate, A!13), X!14))) | equal_set(X!14, initial_segment(X!14, member_predicate, A!13))),
% 0.23/0.46 inference(tautology,[status(thm)],[])).
% 0.23/0.46 tff(142,plain,
% 0.23/0.46 ((((~subset(X!14, initial_segment(X!14, member_predicate, A!13))) | (~subset(initial_segment(X!14, member_predicate, A!13), X!14))) <=> equal_set(X!14, initial_segment(X!14, member_predicate, A!13))) | ((~subset(X!14, initial_segment(X!14, member_predicate, A!13))) | (~subset(initial_segment(X!14, member_predicate, A!13), X!14)))),
% 0.23/0.46 inference(unit_resolution,[status(thm)],[141, 140])).
% 0.23/0.46 tff(143,plain,
% 0.23/0.46 ((~subset(X!14, initial_segment(X!14, member_predicate, A!13))) | (~subset(initial_segment(X!14, member_predicate, A!13), X!14))),
% 0.23/0.46 inference(unit_resolution,[status(thm)],[142, 139])).
% 0.23/0.46 tff(144,plain,
% 0.23/0.46 ((~((~subset(X!14, initial_segment(X!14, member_predicate, A!13))) | (~subset(initial_segment(X!14, member_predicate, A!13), X!14)))) | (~subset(X!14, initial_segment(X!14, member_predicate, A!13))) | (~subset(initial_segment(X!14, member_predicate, A!13), X!14))),
% 0.23/0.46 inference(tautology,[status(thm)],[])).
% 0.23/0.46 tff(145,plain,
% 0.23/0.46 ((~subset(X!14, initial_segment(X!14, member_predicate, A!13))) | (~subset(initial_segment(X!14, member_predicate, A!13), X!14))),
% 0.23/0.46 inference(unit_resolution,[status(thm)],[144, 143])).
% 0.23/0.46 tff(146,plain,
% 0.23/0.46 (~subset(initial_segment(X!14, member_predicate, A!13), X!14)),
% 0.23/0.46 inference(unit_resolution,[status(thm)],[145, 127])).
% 0.23/0.46 tff(147,plain,
% 0.23/0.46 ((~(subset(initial_segment(X!14, member_predicate, A!13), X!14) | (~((~member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), initial_segment(X!14, member_predicate, A!13))) | member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14))))) | subset(initial_segment(X!14, member_predicate, A!13), X!14) | (~((~member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), initial_segment(X!14, member_predicate, A!13))) | member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14)))),
% 0.23/0.46 inference(tautology,[status(thm)],[])).
% 0.23/0.46 tff(148,plain,
% 0.23/0.46 ((~(subset(initial_segment(X!14, member_predicate, A!13), X!14) | (~((~member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), initial_segment(X!14, member_predicate, A!13))) | member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14))))) | (~((~member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), initial_segment(X!14, member_predicate, A!13))) | member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14)))),
% 0.23/0.46 inference(unit_resolution,[status(thm)],[147, 146])).
% 0.23/0.46 tff(149,plain,
% 0.23/0.46 (~((~member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), initial_segment(X!14, member_predicate, A!13))) | member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14))),
% 0.23/0.46 inference(unit_resolution,[status(thm)],[148, 29])).
% 0.23/0.46 tff(150,plain,
% 0.23/0.46 (((~member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), initial_segment(X!14, member_predicate, A!13))) | member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14)) | (~member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14))),
% 0.23/0.46 inference(tautology,[status(thm)],[])).
% 0.23/0.46 tff(151,plain,
% 0.23/0.46 (~member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14)),
% 0.23/0.46 inference(unit_resolution,[status(thm)],[150, 149])).
% 0.23/0.46 tff(152,plain,
% 0.23/0.46 ((~(apply(member_predicate, tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14) <=> member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14))) | (~apply(member_predicate, tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14)) | member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14)),
% 0.23/0.46 inference(tautology,[status(thm)],[])).
% 0.23/0.46 tff(153,plain,
% 0.23/0.46 ((~(apply(member_predicate, tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14) <=> member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14))) | (~apply(member_predicate, tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14))),
% 0.23/0.46 inference(unit_resolution,[status(thm)],[152, 151])).
% 0.23/0.46 tff(154,plain,
% 0.23/0.46 (~apply(member_predicate, tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14)),
% 0.23/0.46 inference(unit_resolution,[status(thm)],[153, 9])).
% 0.23/0.46 tff(155,plain,
% 0.23/0.46 ((~![X: $i, R: $i, A: $i, Y: $i] : (~(((~member(Y, A)) | (~apply(R, Y, X))) <=> member(Y, initial_segment(X, R, A))))) | (~(((~member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), A!13)) | (~apply(member_predicate, tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14))) <=> member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), initial_segment(X!14, member_predicate, A!13))))),
% 0.23/0.46 inference(quant_inst,[status(thm)],[])).
% 0.23/0.46 tff(156,plain,
% 0.23/0.46 (~(((~member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), A!13)) | (~apply(member_predicate, tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14))) <=> member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), initial_segment(X!14, member_predicate, A!13)))),
% 0.23/0.46 inference(unit_resolution,[status(thm)],[155, 47])).
% 0.23/0.46 tff(157,plain,
% 0.23/0.46 (((~member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), initial_segment(X!14, member_predicate, A!13))) | member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14)) | member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), initial_segment(X!14, member_predicate, A!13))),
% 0.23/0.46 inference(tautology,[status(thm)],[])).
% 0.23/0.46 tff(158,plain,
% 0.23/0.46 (member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), initial_segment(X!14, member_predicate, A!13))),
% 0.23/0.46 inference(unit_resolution,[status(thm)],[157, 149])).
% 0.23/0.46 tff(159,plain,
% 0.23/0.46 ((((~member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), A!13)) | (~apply(member_predicate, tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14))) <=> member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), initial_segment(X!14, member_predicate, A!13))) | (~((~member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), A!13)) | (~apply(member_predicate, tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14)))) | (~member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), initial_segment(X!14, member_predicate, A!13)))),
% 0.23/0.46 inference(tautology,[status(thm)],[])).
% 0.23/0.46 tff(160,plain,
% 0.23/0.46 ((((~member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), A!13)) | (~apply(member_predicate, tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14))) <=> member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), initial_segment(X!14, member_predicate, A!13))) | (~((~member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), A!13)) | (~apply(member_predicate, tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14))))),
% 0.23/0.46 inference(unit_resolution,[status(thm)],[159, 158])).
% 0.23/0.46 tff(161,plain,
% 0.23/0.46 (~((~member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), A!13)) | (~apply(member_predicate, tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14)))),
% 0.23/0.46 inference(unit_resolution,[status(thm)],[160, 156])).
% 0.23/0.46 tff(162,plain,
% 0.23/0.46 (((~member(tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), A!13)) | (~apply(member_predicate, tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14))) | apply(member_predicate, tptp_fun_X_0(X!14, initial_segment(X!14, member_predicate, A!13)), X!14)),
% 0.23/0.46 inference(tautology,[status(thm)],[])).
% 0.23/0.46 tff(163,plain,
% 0.23/0.46 ($false),
% 0.23/0.46 inference(unit_resolution,[status(thm)],[162, 161, 154])).
% 0.23/0.46 % SZS output end Proof
% 0.23/0.46 % E exiting
%------------------------------------------------------------------------------