%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET602+4 : TPTP v9.0.0. Released v2.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n006.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 05:25:58 AM UTC 2025
% Result : Theorem 0.12s 0.37s
% Output : Proof 0.12s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : SET602+4 : TPTP v9.0.0. Released v2.2.0.
% 0.07/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n006.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Jun 20 05:30:20 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.12/0.37 % SZS status Theorem
% 0.12/0.37 % SZS output start Proof
% 0.12/0.37 tff(member_type, type, (
% 0.12/0.37 member: ( $i * $i ) > $o)).
% 0.12/0.37 tff(empty_set_type, type, (
% 0.12/0.37 empty_set: $i)).
% 0.12/0.37 tff(tptp_fun_X_0_type, type, (
% 0.12/0.37 tptp_fun_X_0: ( $i * $i ) > $i)).
% 0.12/0.37 tff(difference_type, type, (
% 0.12/0.37 difference: ( $i * $i ) > $i)).
% 0.12/0.37 tff(tptp_fun_E_3_type, type, (
% 0.12/0.37 tptp_fun_E_3: $i)).
% 0.12/0.37 tff(subset_type, type, (
% 0.12/0.37 subset: ( $i * $i ) > $o)).
% 0.12/0.37 tff(equal_set_type, type, (
% 0.12/0.37 equal_set: ( $i * $i ) > $o)).
% 0.12/0.37 tff(1,plain,
% 0.12/0.37 (^[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))))))))),
% 0.12/0.37 inference(bind,[status(th)],[])).
% 0.12/0.37 tff(2,plain,
% 0.12/0.37 (![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)))))))),
% 0.12/0.37 inference(quant_intro,[status(thm)],[1])).
% 0.12/0.37 tff(3,plain,
% 0.12/0.37 (^[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))))))))),
% 0.12/0.37 inference(bind,[status(th)],[])).
% 0.12/0.37 tff(4,plain,
% 0.12/0.37 (![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)))))))),
% 0.12/0.37 inference(quant_intro,[status(thm)],[3])).
% 0.12/0.37 tff(5,plain,
% 0.12/0.37 (![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)))))))),
% 0.12/0.37 inference(transitivity,[status(thm)],[4, 2])).
% 0.12/0.37 tff(6,plain,
% 0.12/0.37 (^[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)))))))))),
% 0.12/0.37 inference(bind,[status(th)],[])).
% 0.12/0.37 tff(7,plain,
% 0.12/0.37 (![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)))))))),
% 0.12/0.37 inference(quant_intro,[status(thm)],[6])).
% 0.12/0.37 tff(8,plain,
% 0.12/0.37 (![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)))),
% 0.12/0.37 inference(rewrite,[status(thm)],[])).
% 0.12/0.37 tff(9,plain,
% 0.12/0.37 (^[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))))),
% 0.12/0.37 inference(bind,[status(th)],[])).
% 0.12/0.37 tff(10,plain,
% 0.12/0.37 (![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)))),
% 0.12/0.37 inference(quant_intro,[status(thm)],[9])).
% 0.12/0.37 tff(11,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')).
% 0.12/0.37 tff(12,plain,
% 0.12/0.37 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[11, 10])).
% 0.12/0.37 tff(13,plain,
% 0.12/0.37 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[12, 8])).
% 0.12/0.37 tff(14,plain,(
% 0.12/0.37 ![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)))))),
% 0.12/0.37 inference(skolemize,[status(sab)],[13])).
% 0.12/0.37 tff(15,plain,
% 0.12/0.37 (![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)))))))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[14, 7])).
% 0.12/0.37 tff(16,plain,
% 0.12/0.37 (![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)))))))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[15, 5])).
% 0.12/0.37 tff(17,plain,
% 0.12/0.37 ((~![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(empty_set, difference(E!3, E!3))) | ![X: $i] : ((~member(X, empty_set)) | member(X, difference(E!3, E!3))))) | (~(subset(empty_set, difference(E!3, E!3)) | (~((~member(tptp_fun_X_0(difference(E!3, E!3), empty_set), empty_set)) | member(tptp_fun_X_0(difference(E!3, E!3), empty_set), difference(E!3, E!3))))))))),
% 0.12/0.37 inference(quant_inst,[status(thm)],[])).
% 0.12/0.37 tff(18,plain,
% 0.12/0.37 (~((~((~subset(empty_set, difference(E!3, E!3))) | ![X: $i] : ((~member(X, empty_set)) | member(X, difference(E!3, E!3))))) | (~(subset(empty_set, difference(E!3, E!3)) | (~((~member(tptp_fun_X_0(difference(E!3, E!3), empty_set), empty_set)) | member(tptp_fun_X_0(difference(E!3, E!3), empty_set), difference(E!3, E!3)))))))),
% 0.12/0.37 inference(unit_resolution,[status(thm)],[17, 16])).
% 0.12/0.37 tff(19,plain,
% 0.12/0.37 (((~((~subset(empty_set, difference(E!3, E!3))) | ![X: $i] : ((~member(X, empty_set)) | member(X, difference(E!3, E!3))))) | (~(subset(empty_set, difference(E!3, E!3)) | (~((~member(tptp_fun_X_0(difference(E!3, E!3), empty_set), empty_set)) | member(tptp_fun_X_0(difference(E!3, E!3), empty_set), difference(E!3, E!3))))))) | (subset(empty_set, difference(E!3, E!3)) | (~((~member(tptp_fun_X_0(difference(E!3, E!3), empty_set), empty_set)) | member(tptp_fun_X_0(difference(E!3, E!3), empty_set), difference(E!3, E!3)))))),
% 0.12/0.37 inference(tautology,[status(thm)],[])).
% 0.12/0.37 tff(20,plain,
% 0.12/0.37 (subset(empty_set, difference(E!3, E!3)) | (~((~member(tptp_fun_X_0(difference(E!3, E!3), empty_set), empty_set)) | member(tptp_fun_X_0(difference(E!3, E!3), empty_set), difference(E!3, E!3))))),
% 0.12/0.37 inference(unit_resolution,[status(thm)],[19, 18])).
% 0.12/0.37 tff(21,assumption,((~((~subset(difference(E!3, E!3), empty_set)) | ![X: $i] : ((~member(X, difference(E!3, E!3))) | member(X, empty_set)))) | (~(subset(difference(E!3, E!3), empty_set) | (~((~member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), difference(E!3, E!3))) | member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), empty_set)))))), introduced(assumption)).
% 0.12/0.37 tff(22,plain,
% 0.12/0.37 ((~![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(difference(E!3, E!3), empty_set)) | ![X: $i] : ((~member(X, difference(E!3, E!3))) | member(X, empty_set)))) | (~(subset(difference(E!3, E!3), empty_set) | (~((~member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), difference(E!3, E!3))) | member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), empty_set)))))))),
% 0.12/0.37 inference(quant_inst,[status(thm)],[])).
% 0.12/0.37 tff(23,plain,
% 0.12/0.37 ($false),
% 0.12/0.37 inference(unit_resolution,[status(thm)],[22, 16, 21])).
% 0.12/0.37 tff(24,plain,(~((~((~subset(difference(E!3, E!3), empty_set)) | ![X: $i] : ((~member(X, difference(E!3, E!3))) | member(X, empty_set)))) | (~(subset(difference(E!3, E!3), empty_set) | (~((~member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), difference(E!3, E!3))) | member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), empty_set))))))), inference(lemma,lemma(discharge,[]))).
% 0.12/0.37 tff(25,plain,
% 0.12/0.37 (((~((~subset(difference(E!3, E!3), empty_set)) | ![X: $i] : ((~member(X, difference(E!3, E!3))) | member(X, empty_set)))) | (~(subset(difference(E!3, E!3), empty_set) | (~((~member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), difference(E!3, E!3))) | member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), empty_set)))))) | (subset(difference(E!3, E!3), empty_set) | (~((~member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), difference(E!3, E!3))) | member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), empty_set))))),
% 0.12/0.37 inference(tautology,[status(thm)],[])).
% 0.12/0.37 tff(26,plain,
% 0.12/0.37 (subset(difference(E!3, E!3), empty_set) | (~((~member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), difference(E!3, E!3))) | member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), empty_set)))),
% 0.12/0.37 inference(unit_resolution,[status(thm)],[25, 24])).
% 0.12/0.37 tff(27,assumption,(member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), difference(E!3, E!3))), introduced(assumption)).
% 0.12/0.37 tff(28,plain,
% 0.12/0.37 (^[B: $i, A: $i, E: $i] : refl((~(((~member(B, E)) | member(B, A)) <=> member(B, difference(E, A)))) <=> (~(((~member(B, E)) | member(B, A)) <=> member(B, difference(E, A)))))),
% 0.12/0.37 inference(bind,[status(th)],[])).
% 0.12/0.37 tff(29,plain,
% 0.12/0.37 (![B: $i, A: $i, E: $i] : (~(((~member(B, E)) | member(B, A)) <=> member(B, difference(E, A)))) <=> ![B: $i, A: $i, E: $i] : (~(((~member(B, E)) | member(B, A)) <=> member(B, difference(E, A))))),
% 0.12/0.37 inference(quant_intro,[status(thm)],[28])).
% 0.12/0.37 tff(30,plain,
% 0.12/0.37 (^[B: $i, A: $i, E: $i] : trans(trans(monotonicity(rewrite((member(B, E) & (~member(B, A))) <=> (~((~member(B, E)) | member(B, A)))), ((member(B, difference(E, A)) <=> (member(B, E) & (~member(B, A)))) <=> (member(B, difference(E, A)) <=> (~((~member(B, E)) | member(B, A)))))), rewrite((member(B, difference(E, A)) <=> (~((~member(B, E)) | member(B, A)))) <=> (~(((~member(B, E)) | member(B, A)) <=> member(B, difference(E, A))))), ((member(B, difference(E, A)) <=> (member(B, E) & (~member(B, A)))) <=> (~(((~member(B, E)) | member(B, A)) <=> member(B, difference(E, A)))))), rewrite((~(((~member(B, E)) | member(B, A)) <=> member(B, difference(E, A)))) <=> (~(((~member(B, E)) | member(B, A)) <=> member(B, difference(E, A))))), ((member(B, difference(E, A)) <=> (member(B, E) & (~member(B, A)))) <=> (~(((~member(B, E)) | member(B, A)) <=> member(B, difference(E, A))))))),
% 0.12/0.37 inference(bind,[status(th)],[])).
% 0.12/0.37 tff(31,plain,
% 0.12/0.37 (![B: $i, A: $i, E: $i] : (member(B, difference(E, A)) <=> (member(B, E) & (~member(B, A)))) <=> ![B: $i, A: $i, E: $i] : (~(((~member(B, E)) | member(B, A)) <=> member(B, difference(E, A))))),
% 0.12/0.37 inference(quant_intro,[status(thm)],[30])).
% 0.12/0.37 tff(32,plain,
% 0.12/0.37 (![B: $i, A: $i, E: $i] : (member(B, difference(E, A)) <=> (member(B, E) & (~member(B, A)))) <=> ![B: $i, A: $i, E: $i] : (member(B, difference(E, A)) <=> (member(B, E) & (~member(B, A))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(33,axiom,(![B: $i, A: $i, E: $i] : (member(B, difference(E, A)) <=> (member(B, E) & (~member(B, A))))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+0.ax','difference')).
% 0.12/0.38 tff(34,plain,
% 0.12/0.38 (![B: $i, A: $i, E: $i] : (member(B, difference(E, A)) <=> (member(B, E) & (~member(B, A))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[33, 32])).
% 0.12/0.38 tff(35,plain,(
% 0.12/0.38 ![B: $i, A: $i, E: $i] : (member(B, difference(E, A)) <=> (member(B, E) & (~member(B, A))))),
% 0.12/0.38 inference(skolemize,[status(sab)],[34])).
% 0.12/0.38 tff(36,plain,
% 0.12/0.38 (![B: $i, A: $i, E: $i] : (~(((~member(B, E)) | member(B, A)) <=> member(B, difference(E, A))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[35, 31])).
% 0.12/0.38 tff(37,plain,
% 0.12/0.38 (![B: $i, A: $i, E: $i] : (~(((~member(B, E)) | member(B, A)) <=> member(B, difference(E, A))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[36, 29])).
% 0.12/0.38 tff(38,plain,
% 0.12/0.38 (((~![B: $i, A: $i, E: $i] : (~(((~member(B, E)) | member(B, A)) <=> member(B, difference(E, A))))) | (~member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), difference(E!3, E!3)))) <=> ((~![B: $i, A: $i, E: $i] : (~(((~member(B, E)) | member(B, A)) <=> member(B, difference(E, A))))) | (~member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), difference(E!3, E!3))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(39,plain,
% 0.12/0.38 ((~(((~member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), E!3)) | member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), E!3)) <=> member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), difference(E!3, E!3)))) <=> (~member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), difference(E!3, E!3)))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(40,plain,
% 0.12/0.38 (((~![B: $i, A: $i, E: $i] : (~(((~member(B, E)) | member(B, A)) <=> member(B, difference(E, A))))) | (~(((~member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), E!3)) | member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), E!3)) <=> member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), difference(E!3, E!3))))) <=> ((~![B: $i, A: $i, E: $i] : (~(((~member(B, E)) | member(B, A)) <=> member(B, difference(E, A))))) | (~member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), difference(E!3, E!3))))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[39])).
% 0.12/0.38 tff(41,plain,
% 0.12/0.38 (((~![B: $i, A: $i, E: $i] : (~(((~member(B, E)) | member(B, A)) <=> member(B, difference(E, A))))) | (~(((~member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), E!3)) | member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), E!3)) <=> member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), difference(E!3, E!3))))) <=> ((~![B: $i, A: $i, E: $i] : (~(((~member(B, E)) | member(B, A)) <=> member(B, difference(E, A))))) | (~member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), difference(E!3, E!3))))),
% 0.12/0.38 inference(transitivity,[status(thm)],[40, 38])).
% 0.12/0.38 tff(42,plain,
% 0.12/0.38 ((~![B: $i, A: $i, E: $i] : (~(((~member(B, E)) | member(B, A)) <=> member(B, difference(E, A))))) | (~(((~member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), E!3)) | member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), E!3)) <=> member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), difference(E!3, E!3))))),
% 0.12/0.38 inference(quant_inst,[status(thm)],[])).
% 0.12/0.38 tff(43,plain,
% 0.12/0.38 ((~![B: $i, A: $i, E: $i] : (~(((~member(B, E)) | member(B, A)) <=> member(B, difference(E, A))))) | (~member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), difference(E!3, E!3)))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[42, 41])).
% 0.12/0.38 tff(44,plain,
% 0.12/0.38 ($false),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[43, 37, 27])).
% 0.12/0.38 tff(45,plain,(~member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), difference(E!3, E!3))), inference(lemma,lemma(discharge,[]))).
% 0.12/0.38 tff(46,plain,
% 0.12/0.38 (((~member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), difference(E!3, E!3))) | member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), empty_set)) | member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), difference(E!3, E!3))),
% 0.12/0.38 inference(tautology,[status(thm)],[])).
% 0.12/0.38 tff(47,plain,
% 0.12/0.38 ((~member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), difference(E!3, E!3))) | member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), empty_set)),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[46, 45])).
% 0.12/0.38 tff(48,plain,
% 0.12/0.38 ((~(subset(difference(E!3, E!3), empty_set) | (~((~member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), difference(E!3, E!3))) | member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), empty_set))))) | subset(difference(E!3, E!3), empty_set) | (~((~member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), difference(E!3, E!3))) | member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), empty_set)))),
% 0.12/0.38 inference(tautology,[status(thm)],[])).
% 0.12/0.38 tff(49,plain,
% 0.12/0.38 ((~(subset(difference(E!3, E!3), empty_set) | (~((~member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), difference(E!3, E!3))) | member(tptp_fun_X_0(empty_set, difference(E!3, E!3)), empty_set))))) | subset(difference(E!3, E!3), empty_set)),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[48, 47])).
% 0.12/0.38 tff(50,plain,
% 0.12/0.38 (subset(difference(E!3, E!3), empty_set)),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[49, 26])).
% 0.12/0.38 tff(51,plain,
% 0.12/0.38 (^[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))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(52,plain,
% 0.12/0.38 (![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)))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[51])).
% 0.12/0.38 tff(53,plain,
% 0.12/0.38 (^[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)))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(54,plain,
% 0.12/0.38 (![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)))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[53])).
% 0.12/0.38 tff(55,plain,
% 0.12/0.38 (![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)))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(56,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')).
% 0.12/0.38 tff(57,plain,
% 0.12/0.38 (![A: $i, B: $i] : (equal_set(A, B) <=> (subset(A, B) & subset(B, A)))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[56, 55])).
% 0.12/0.38 tff(58,plain,(
% 0.12/0.38 ![A: $i, B: $i] : (equal_set(A, B) <=> (subset(A, B) & subset(B, A)))),
% 0.12/0.38 inference(skolemize,[status(sab)],[57])).
% 0.12/0.38 tff(59,plain,
% 0.12/0.38 (![A: $i, B: $i] : (~(((~subset(A, B)) | (~subset(B, A))) <=> equal_set(A, B)))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[58, 54])).
% 0.12/0.38 tff(60,plain,
% 0.12/0.38 (![A: $i, B: $i] : (~(((~subset(A, B)) | (~subset(B, A))) <=> equal_set(A, B)))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[59, 52])).
% 0.12/0.38 tff(61,plain,
% 0.12/0.38 ((~![A: $i, B: $i] : (~(((~subset(A, B)) | (~subset(B, A))) <=> equal_set(A, B)))) | (~(((~subset(difference(E!3, E!3), empty_set)) | (~subset(empty_set, difference(E!3, E!3)))) <=> equal_set(difference(E!3, E!3), empty_set)))),
% 0.12/0.38 inference(quant_inst,[status(thm)],[])).
% 0.12/0.38 tff(62,plain,
% 0.12/0.38 (~(((~subset(difference(E!3, E!3), empty_set)) | (~subset(empty_set, difference(E!3, E!3)))) <=> equal_set(difference(E!3, E!3), empty_set))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[61, 60])).
% 0.12/0.38 tff(63,plain,
% 0.12/0.38 ((~![E: $i] : equal_set(difference(E, E), empty_set)) <=> (~![E: $i] : equal_set(difference(E, E), empty_set))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(64,axiom,(~![E: $i] : equal_set(difference(E, E), empty_set)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','thI29')).
% 0.12/0.38 tff(65,plain,
% 0.12/0.38 (~![E: $i] : equal_set(difference(E, E), empty_set)),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[64, 63])).
% 0.12/0.38 tff(66,plain,
% 0.12/0.38 (~![E: $i] : equal_set(difference(E, E), empty_set)),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[65, 63])).
% 0.12/0.38 tff(67,plain,
% 0.12/0.38 (~![E: $i] : equal_set(difference(E, E), empty_set)),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[66, 63])).
% 0.12/0.38 tff(68,plain,
% 0.12/0.38 (~![E: $i] : equal_set(difference(E, E), empty_set)),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[67, 63])).
% 0.12/0.38 tff(69,plain,(
% 0.12/0.38 ~equal_set(difference(E!3, E!3), empty_set)),
% 0.12/0.38 inference(skolemize,[status(sab)],[68])).
% 0.12/0.38 tff(70,plain,
% 0.12/0.38 ((((~subset(difference(E!3, E!3), empty_set)) | (~subset(empty_set, difference(E!3, E!3)))) <=> equal_set(difference(E!3, E!3), empty_set)) | ((~subset(difference(E!3, E!3), empty_set)) | (~subset(empty_set, difference(E!3, E!3)))) | equal_set(difference(E!3, E!3), empty_set)),
% 0.12/0.38 inference(tautology,[status(thm)],[])).
% 0.12/0.38 tff(71,plain,
% 0.12/0.38 ((((~subset(difference(E!3, E!3), empty_set)) | (~subset(empty_set, difference(E!3, E!3)))) <=> equal_set(difference(E!3, E!3), empty_set)) | ((~subset(difference(E!3, E!3), empty_set)) | (~subset(empty_set, difference(E!3, E!3))))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[70, 69])).
% 0.12/0.38 tff(72,plain,
% 0.12/0.38 ((~subset(difference(E!3, E!3), empty_set)) | (~subset(empty_set, difference(E!3, E!3)))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[71, 62])).
% 0.12/0.38 tff(73,plain,
% 0.12/0.38 ((~((~subset(difference(E!3, E!3), empty_set)) | (~subset(empty_set, difference(E!3, E!3))))) | (~subset(difference(E!3, E!3), empty_set)) | (~subset(empty_set, difference(E!3, E!3)))),
% 0.12/0.38 inference(tautology,[status(thm)],[])).
% 0.12/0.38 tff(74,plain,
% 0.12/0.38 ((~subset(difference(E!3, E!3), empty_set)) | (~subset(empty_set, difference(E!3, E!3)))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[73, 72])).
% 0.12/0.38 tff(75,plain,
% 0.12/0.38 (~subset(empty_set, difference(E!3, E!3))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[74, 50])).
% 0.12/0.38 tff(76,plain,
% 0.12/0.38 ((~(subset(empty_set, difference(E!3, E!3)) | (~((~member(tptp_fun_X_0(difference(E!3, E!3), empty_set), empty_set)) | member(tptp_fun_X_0(difference(E!3, E!3), empty_set), difference(E!3, E!3)))))) | subset(empty_set, difference(E!3, E!3)) | (~((~member(tptp_fun_X_0(difference(E!3, E!3), empty_set), empty_set)) | member(tptp_fun_X_0(difference(E!3, E!3), empty_set), difference(E!3, E!3))))),
% 0.12/0.38 inference(tautology,[status(thm)],[])).
% 0.12/0.38 tff(77,plain,
% 0.12/0.38 ((~(subset(empty_set, difference(E!3, E!3)) | (~((~member(tptp_fun_X_0(difference(E!3, E!3), empty_set), empty_set)) | member(tptp_fun_X_0(difference(E!3, E!3), empty_set), difference(E!3, E!3)))))) | (~((~member(tptp_fun_X_0(difference(E!3, E!3), empty_set), empty_set)) | member(tptp_fun_X_0(difference(E!3, E!3), empty_set), difference(E!3, E!3))))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[76, 75])).
% 0.12/0.38 tff(78,plain,
% 0.12/0.38 (~((~member(tptp_fun_X_0(difference(E!3, E!3), empty_set), empty_set)) | member(tptp_fun_X_0(difference(E!3, E!3), empty_set), difference(E!3, E!3)))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[77, 20])).
% 0.12/0.38 tff(79,plain,
% 0.12/0.38 (((~member(tptp_fun_X_0(difference(E!3, E!3), empty_set), empty_set)) | member(tptp_fun_X_0(difference(E!3, E!3), empty_set), difference(E!3, E!3))) | member(tptp_fun_X_0(difference(E!3, E!3), empty_set), empty_set)),
% 0.12/0.38 inference(tautology,[status(thm)],[])).
% 0.12/0.38 tff(80,plain,
% 0.12/0.38 (member(tptp_fun_X_0(difference(E!3, E!3), empty_set), empty_set)),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[79, 78])).
% 0.12/0.38 tff(81,plain,
% 0.12/0.38 (^[X: $i] : refl((~member(X, empty_set)) <=> (~member(X, empty_set)))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(82,plain,
% 0.12/0.38 (![X: $i] : (~member(X, empty_set)) <=> ![X: $i] : (~member(X, empty_set))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[81])).
% 0.12/0.38 tff(83,plain,
% 0.12/0.38 (![X: $i] : (~member(X, empty_set)) <=> ![X: $i] : (~member(X, empty_set))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(84,axiom,(![X: $i] : (~member(X, empty_set))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+0.ax','empty_set')).
% 0.12/0.38 tff(85,plain,
% 0.12/0.38 (![X: $i] : (~member(X, empty_set))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[84, 83])).
% 0.12/0.38 tff(86,plain,(
% 0.12/0.38 ![X: $i] : (~member(X, empty_set))),
% 0.12/0.38 inference(skolemize,[status(sab)],[85])).
% 0.12/0.38 tff(87,plain,
% 0.12/0.38 (![X: $i] : (~member(X, empty_set))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[86, 82])).
% 0.12/0.38 tff(88,plain,
% 0.12/0.38 ((~![X: $i] : (~member(X, empty_set))) | (~member(tptp_fun_X_0(difference(E!3, E!3), empty_set), empty_set))),
% 0.12/0.38 inference(quant_inst,[status(thm)],[])).
% 0.12/0.38 tff(89,plain,
% 0.12/0.38 ($false),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[88, 87, 80])).
% 0.12/0.38 % SZS output end Proof
% 0.12/0.38 % E exiting
%------------------------------------------------------------------------------