%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET767+4 : TPTP v9.0.0. Released v2.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n001.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 05:26:18 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
% 0.03/0.13 % Problem : SET767+4 : TPTP v9.0.0. Released v2.2.0.
% 0.03/0.13 % Command : run_E %s %d THM
% 0.13/0.35 % Computer : n001.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Fri Jun 20 05:45:35 EDT 2025
% 0.13/0.35 % 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(equivalence_class_type, type, (
% 0.21/0.40 equivalence_class: ( $i * $i * $i ) > $i)).
% 0.21/0.40 tff(tptp_fun_R_21_type, type, (
% 0.21/0.40 tptp_fun_R_21: $i)).
% 0.21/0.40 tff(tptp_fun_E_22_type, type, (
% 0.21/0.40 tptp_fun_E_22: $i)).
% 0.21/0.40 tff(tptp_fun_A_20_type, type, (
% 0.21/0.40 tptp_fun_A_20: $i)).
% 0.21/0.40 tff(tptp_fun_X_0_type, type, (
% 0.21/0.40 tptp_fun_X_0: ( $i * $i ) > $i)).
% 0.21/0.40 tff(apply_type, type, (
% 0.21/0.40 apply: ( $i * $i * $i ) > $o)).
% 0.21/0.40 tff(subset_type, type, (
% 0.21/0.40 subset: ( $i * $i ) > $o)).
% 0.21/0.40 tff(equivalence_type, type, (
% 0.21/0.40 equivalence: ( $i * $i ) > $o)).
% 0.21/0.40 tff(1,plain,
% 0.21/0.40 (^[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))))))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(2,plain,
% 0.21/0.40 (![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)))))))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[1])).
% 0.21/0.40 tff(3,plain,
% 0.21/0.40 (^[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))))))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(4,plain,
% 0.21/0.40 (![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)))))))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[3])).
% 0.21/0.40 tff(5,plain,
% 0.21/0.40 (![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)))))))),
% 0.21/0.40 inference(transitivity,[status(thm)],[4, 2])).
% 0.21/0.40 tff(6,plain,
% 0.21/0.40 (^[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)))))))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(7,plain,
% 0.21/0.40 (![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)))))))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[6])).
% 0.21/0.40 tff(8,plain,
% 0.21/0.40 (![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)))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(9,plain,
% 0.21/0.40 (^[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))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(10,plain,
% 0.21/0.40 (![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)))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[9])).
% 0.21/0.40 tff(11,axiom,(![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : (member(X, A) => member(X, B)))), file('/export/starexec/sandbox/benchmark/Axioms/SET006+0.ax','subset')).
% 0.21/0.40 tff(12,plain,
% 0.21/0.40 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[11, 10])).
% 0.21/0.40 tff(13,plain,
% 0.21/0.40 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[12, 8])).
% 0.21/0.40 tff(14,plain,(
% 0.21/0.40 ![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)))))),
% 0.21/0.40 inference(skolemize,[status(sab)],[13])).
% 0.21/0.40 tff(15,plain,
% 0.21/0.40 (![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)))))))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[14, 7])).
% 0.21/0.40 tff(16,plain,
% 0.21/0.40 (![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)))))))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[15, 5])).
% 0.21/0.40 tff(17,plain,
% 0.21/0.40 ((~![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(equivalence_class(A!20, E!22, R!21), E!22)) | ![X: $i] : ((~member(X, equivalence_class(A!20, E!22, R!21))) | member(X, E!22)))) | (~(subset(equivalence_class(A!20, E!22, R!21), E!22) | (~((~member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), equivalence_class(A!20, E!22, R!21))) | member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), E!22)))))))),
% 0.21/0.40 inference(quant_inst,[status(thm)],[])).
% 0.21/0.40 tff(18,plain,
% 0.21/0.40 (~((~((~subset(equivalence_class(A!20, E!22, R!21), E!22)) | ![X: $i] : ((~member(X, equivalence_class(A!20, E!22, R!21))) | member(X, E!22)))) | (~(subset(equivalence_class(A!20, E!22, R!21), E!22) | (~((~member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), equivalence_class(A!20, E!22, R!21))) | member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), E!22))))))),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[17, 16])).
% 0.21/0.40 tff(19,plain,
% 0.21/0.40 (((~((~subset(equivalence_class(A!20, E!22, R!21), E!22)) | ![X: $i] : ((~member(X, equivalence_class(A!20, E!22, R!21))) | member(X, E!22)))) | (~(subset(equivalence_class(A!20, E!22, R!21), E!22) | (~((~member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), equivalence_class(A!20, E!22, R!21))) | member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), E!22)))))) | (subset(equivalence_class(A!20, E!22, R!21), E!22) | (~((~member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), equivalence_class(A!20, E!22, R!21))) | member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), E!22))))),
% 0.21/0.40 inference(tautology,[status(thm)],[])).
% 0.21/0.40 tff(20,plain,
% 0.21/0.40 (subset(equivalence_class(A!20, E!22, R!21), E!22) | (~((~member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), equivalence_class(A!20, E!22, R!21))) | member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), E!22)))),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[19, 18])).
% 0.21/0.40 tff(21,plain,
% 0.21/0.40 ((~![E: $i, R: $i, A: $i] : ((~equivalence(R, E)) | subset(equivalence_class(A, E, R), E))) <=> (~![E: $i, R: $i, A: $i] : ((~equivalence(R, E)) | subset(equivalence_class(A, E, R), E)))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(22,plain,
% 0.21/0.40 ((~![E: $i, R: $i, A: $i] : (equivalence(R, E) => subset(equivalence_class(A, E, R), E))) <=> (~![E: $i, R: $i, A: $i] : ((~equivalence(R, E)) | subset(equivalence_class(A, E, R), E)))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(23,axiom,(~![E: $i, R: $i, A: $i] : (equivalence(R, E) => subset(equivalence_class(A, E, R), E))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','thIII03')).
% 0.21/0.40 tff(24,plain,
% 0.21/0.40 (~![E: $i, R: $i, A: $i] : ((~equivalence(R, E)) | subset(equivalence_class(A, E, R), E))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[23, 22])).
% 0.21/0.40 tff(25,plain,
% 0.21/0.40 (~![E: $i, R: $i, A: $i] : ((~equivalence(R, E)) | subset(equivalence_class(A, E, R), E))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[24, 21])).
% 0.21/0.40 tff(26,plain,
% 0.21/0.40 (~![E: $i, R: $i, A: $i] : ((~equivalence(R, E)) | subset(equivalence_class(A, E, R), E))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[25, 21])).
% 0.21/0.40 tff(27,plain,
% 0.21/0.40 (~![E: $i, R: $i, A: $i] : ((~equivalence(R, E)) | subset(equivalence_class(A, E, R), E))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[26, 21])).
% 0.21/0.40 tff(28,plain,(
% 0.21/0.40 ~((~equivalence(R!21, E!22)) | subset(equivalence_class(A!20, E!22, R!21), E!22))),
% 0.21/0.40 inference(skolemize,[status(sab)],[27])).
% 0.21/0.40 tff(29,plain,
% 0.21/0.40 (~subset(equivalence_class(A!20, E!22, R!21), E!22)),
% 0.21/0.40 inference(or_elim,[status(thm)],[28])).
% 0.21/0.40 tff(30,plain,
% 0.21/0.40 ((~(subset(equivalence_class(A!20, E!22, R!21), E!22) | (~((~member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), equivalence_class(A!20, E!22, R!21))) | member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), E!22))))) | subset(equivalence_class(A!20, E!22, R!21), E!22) | (~((~member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), equivalence_class(A!20, E!22, R!21))) | member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), E!22)))),
% 0.21/0.40 inference(tautology,[status(thm)],[])).
% 0.21/0.40 tff(31,plain,
% 0.21/0.40 ((~(subset(equivalence_class(A!20, E!22, R!21), E!22) | (~((~member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), equivalence_class(A!20, E!22, R!21))) | member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), E!22))))) | (~((~member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), equivalence_class(A!20, E!22, R!21))) | member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), E!22)))),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[30, 29])).
% 0.21/0.40 tff(32,plain,
% 0.21/0.40 (~((~member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), equivalence_class(A!20, E!22, R!21))) | member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), E!22))),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[31, 20])).
% 0.21/0.40 tff(33,plain,
% 0.21/0.40 (((~member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), equivalence_class(A!20, E!22, R!21))) | member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), E!22)) | (~member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), E!22))),
% 0.21/0.40 inference(tautology,[status(thm)],[])).
% 0.21/0.40 tff(34,plain,
% 0.21/0.40 (~member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), E!22)),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[33, 32])).
% 0.21/0.40 tff(35,plain,
% 0.21/0.40 (((~member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), E!22)) | (~apply(R!21, A!20, tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21))))) | member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), E!22)),
% 0.21/0.40 inference(tautology,[status(thm)],[])).
% 0.21/0.40 tff(36,plain,
% 0.21/0.40 ((~member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), E!22)) | (~apply(R!21, A!20, tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21))))),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[35, 34])).
% 0.21/0.40 tff(37,plain,
% 0.21/0.40 (((~member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), equivalence_class(A!20, E!22, R!21))) | member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), E!22)) | member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), equivalence_class(A!20, E!22, R!21))),
% 0.21/0.40 inference(tautology,[status(thm)],[])).
% 0.21/0.40 tff(38,plain,
% 0.21/0.40 (member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), equivalence_class(A!20, E!22, R!21))),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[37, 32])).
% 0.21/0.40 tff(39,plain,
% 0.21/0.40 ((((~member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), E!22)) | (~apply(R!21, A!20, tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21))))) <=> member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), equivalence_class(A!20, E!22, R!21))) | (~((~member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), E!22)) | (~apply(R!21, A!20, tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)))))) | (~member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), equivalence_class(A!20, E!22, R!21)))),
% 0.21/0.40 inference(tautology,[status(thm)],[])).
% 0.21/0.40 tff(40,plain,
% 0.21/0.40 (((~member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), E!22)) | (~apply(R!21, A!20, tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21))))) <=> member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), equivalence_class(A!20, E!22, R!21))),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[39, 38, 36])).
% 0.21/0.40 tff(41,plain,
% 0.21/0.40 (^[R: $i, E: $i, A: $i, X: $i] : refl((~(((~member(X, E)) | (~apply(R, A, X))) <=> member(X, equivalence_class(A, E, R)))) <=> (~(((~member(X, E)) | (~apply(R, A, X))) <=> member(X, equivalence_class(A, E, R)))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(42,plain,
% 0.21/0.40 (![R: $i, E: $i, A: $i, X: $i] : (~(((~member(X, E)) | (~apply(R, A, X))) <=> member(X, equivalence_class(A, E, R)))) <=> ![R: $i, E: $i, A: $i, X: $i] : (~(((~member(X, E)) | (~apply(R, A, X))) <=> member(X, equivalence_class(A, E, R))))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[41])).
% 0.21/0.41 tff(43,plain,
% 0.21/0.41 (^[R: $i, E: $i, A: $i, X: $i] : trans(trans(monotonicity(rewrite((member(X, E) & apply(R, A, X)) <=> (~((~member(X, E)) | (~apply(R, A, X))))), ((member(X, equivalence_class(A, E, R)) <=> (member(X, E) & apply(R, A, X))) <=> (member(X, equivalence_class(A, E, R)) <=> (~((~member(X, E)) | (~apply(R, A, X))))))), rewrite((member(X, equivalence_class(A, E, R)) <=> (~((~member(X, E)) | (~apply(R, A, X))))) <=> (~(((~member(X, E)) | (~apply(R, A, X))) <=> member(X, equivalence_class(A, E, R))))), ((member(X, equivalence_class(A, E, R)) <=> (member(X, E) & apply(R, A, X))) <=> (~(((~member(X, E)) | (~apply(R, A, X))) <=> member(X, equivalence_class(A, E, R)))))), rewrite((~(((~member(X, E)) | (~apply(R, A, X))) <=> member(X, equivalence_class(A, E, R)))) <=> (~(((~member(X, E)) | (~apply(R, A, X))) <=> member(X, equivalence_class(A, E, R))))), ((member(X, equivalence_class(A, E, R)) <=> (member(X, E) & apply(R, A, X))) <=> (~(((~member(X, E)) | (~apply(R, A, X))) <=> member(X, equivalence_class(A, E, R))))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(44,plain,
% 0.21/0.41 (![R: $i, E: $i, A: $i, X: $i] : (member(X, equivalence_class(A, E, R)) <=> (member(X, E) & apply(R, A, X))) <=> ![R: $i, E: $i, A: $i, X: $i] : (~(((~member(X, E)) | (~apply(R, A, X))) <=> member(X, equivalence_class(A, E, R))))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[43])).
% 0.21/0.41 tff(45,plain,
% 0.21/0.41 (![R: $i, E: $i, A: $i, X: $i] : (member(X, equivalence_class(A, E, R)) <=> (member(X, E) & apply(R, A, X))) <=> ![R: $i, E: $i, A: $i, X: $i] : (member(X, equivalence_class(A, E, R)) <=> (member(X, E) & apply(R, A, X)))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(46,axiom,(![R: $i, E: $i, A: $i, X: $i] : (member(X, equivalence_class(A, E, R)) <=> (member(X, E) & apply(R, A, X)))), file('/export/starexec/sandbox/benchmark/Axioms/SET006+2.ax','equivalence_class')).
% 0.21/0.41 tff(47,plain,
% 0.21/0.41 (![R: $i, E: $i, A: $i, X: $i] : (member(X, equivalence_class(A, E, R)) <=> (member(X, E) & apply(R, A, X)))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[46, 45])).
% 0.21/0.41 tff(48,plain,(
% 0.21/0.41 ![R: $i, E: $i, A: $i, X: $i] : (member(X, equivalence_class(A, E, R)) <=> (member(X, E) & apply(R, A, X)))),
% 0.21/0.41 inference(skolemize,[status(sab)],[47])).
% 0.21/0.41 tff(49,plain,
% 0.21/0.41 (![R: $i, E: $i, A: $i, X: $i] : (~(((~member(X, E)) | (~apply(R, A, X))) <=> member(X, equivalence_class(A, E, R))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[48, 44])).
% 0.21/0.41 tff(50,plain,
% 0.21/0.41 (![R: $i, E: $i, A: $i, X: $i] : (~(((~member(X, E)) | (~apply(R, A, X))) <=> member(X, equivalence_class(A, E, R))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[49, 42])).
% 0.21/0.41 tff(51,plain,
% 0.21/0.41 ((~![R: $i, E: $i, A: $i, X: $i] : (~(((~member(X, E)) | (~apply(R, A, X))) <=> member(X, equivalence_class(A, E, R))))) | (~(((~member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), E!22)) | (~apply(R!21, A!20, tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21))))) <=> member(tptp_fun_X_0(E!22, equivalence_class(A!20, E!22, R!21)), equivalence_class(A!20, E!22, R!21))))),
% 0.21/0.41 inference(quant_inst,[status(thm)],[])).
% 0.21/0.41 tff(52,plain,
% 0.21/0.41 ($false),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[51, 50, 40])).
% 0.21/0.41 % SZS output end Proof
% 0.21/0.41 % E exiting
%------------------------------------------------------------------------------