%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET603+4 : TPTP v9.0.0. Released v2.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n013.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 0.20s 0.38s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SET603+4 : TPTP v9.0.0. Released v2.2.0.
% 0.03/0.12 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n013.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Fri Jun 20 05:30:53 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.20/0.38 % SZS status Theorem
% 0.20/0.38 % SZS output start Proof
% 0.20/0.38 tff(member_type, type, (
% 0.20/0.38 member: ( $i * $i ) > $o)).
% 0.20/0.38 tff(empty_set_type, type, (
% 0.20/0.38 empty_set: $i)).
% 0.20/0.38 tff(tptp_fun_X_0_type, type, (
% 0.20/0.38 tptp_fun_X_0: ( $i * $i ) > $i)).
% 0.20/0.38 tff(tptp_fun_E_3_type, type, (
% 0.20/0.38 tptp_fun_E_3: $i)).
% 0.20/0.38 tff(difference_type, type, (
% 0.20/0.38 difference: ( $i * $i ) > $i)).
% 0.20/0.38 tff(subset_type, type, (
% 0.20/0.38 subset: ( $i * $i ) > $o)).
% 0.20/0.38 tff(equal_set_type, type, (
% 0.20/0.38 equal_set: ( $i * $i ) > $o)).
% 0.20/0.38 tff(1,plain,
% 0.20/0.38 (^[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)))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(2,plain,
% 0.20/0.38 (![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))))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[1])).
% 0.20/0.38 tff(3,plain,
% 0.20/0.38 (^[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))))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(4,plain,
% 0.20/0.38 (![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))))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[3])).
% 0.20/0.38 tff(5,plain,
% 0.20/0.38 (![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))))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(6,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')).
% 0.20/0.38 tff(7,plain,
% 0.20/0.38 (![B: $i, A: $i, E: $i] : (member(B, difference(E, A)) <=> (member(B, E) & (~member(B, A))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[6, 5])).
% 0.20/0.38 tff(8,plain,(
% 0.20/0.38 ![B: $i, A: $i, E: $i] : (member(B, difference(E, A)) <=> (member(B, E) & (~member(B, A))))),
% 0.20/0.38 inference(skolemize,[status(sab)],[7])).
% 0.20/0.38 tff(9,plain,
% 0.20/0.38 (![B: $i, A: $i, E: $i] : (~(((~member(B, E)) | member(B, A)) <=> member(B, difference(E, A))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[8, 4])).
% 0.20/0.38 tff(10,plain,
% 0.20/0.38 (![B: $i, A: $i, E: $i] : (~(((~member(B, E)) | member(B, A)) <=> member(B, difference(E, A))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[9, 2])).
% 0.20/0.38 tff(11,plain,
% 0.20/0.38 ((~![B: $i, A: $i, E: $i] : (~(((~member(B, E)) | member(B, A)) <=> member(B, difference(E, A))))) | (~(((~member(tptp_fun_X_0(difference(E!3, empty_set), E!3), E!3)) | member(tptp_fun_X_0(difference(E!3, empty_set), E!3), empty_set)) <=> member(tptp_fun_X_0(difference(E!3, empty_set), E!3), difference(E!3, empty_set))))),
% 0.20/0.38 inference(quant_inst,[status(thm)],[])).
% 0.20/0.38 tff(12,plain,
% 0.20/0.38 (~(((~member(tptp_fun_X_0(difference(E!3, empty_set), E!3), E!3)) | member(tptp_fun_X_0(difference(E!3, empty_set), E!3), empty_set)) <=> member(tptp_fun_X_0(difference(E!3, empty_set), E!3), difference(E!3, empty_set)))),
% 0.20/0.38 inference(unit_resolution,[status(thm)],[11, 10])).
% 0.20/0.38 tff(13,plain,
% 0.20/0.38 (^[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))))))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(14,plain,
% 0.20/0.38 (![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)))))))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[13])).
% 0.20/0.38 tff(15,plain,
% 0.20/0.38 (^[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))))))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(16,plain,
% 0.20/0.38 (![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)))))))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[15])).
% 0.20/0.38 tff(17,plain,
% 0.20/0.38 (![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)))))))),
% 0.20/0.38 inference(transitivity,[status(thm)],[16, 14])).
% 0.20/0.38 tff(18,plain,
% 0.20/0.38 (^[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)))))))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(19,plain,
% 0.20/0.38 (![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)))))))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[18])).
% 0.20/0.38 tff(20,plain,
% 0.20/0.38 (![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)))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(21,plain,
% 0.20/0.38 (^[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))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(22,plain,
% 0.20/0.38 (![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)))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[21])).
% 0.20/0.38 tff(23,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')).
% 0.20/0.38 tff(24,plain,
% 0.20/0.38 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[23, 22])).
% 0.20/0.38 tff(25,plain,
% 0.20/0.38 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[24, 20])).
% 0.20/0.38 tff(26,plain,(
% 0.20/0.38 ![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)))))),
% 0.20/0.38 inference(skolemize,[status(sab)],[25])).
% 0.20/0.38 tff(27,plain,
% 0.20/0.38 (![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)))))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[26, 19])).
% 0.20/0.38 tff(28,plain,
% 0.20/0.38 (![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)))))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[27, 17])).
% 0.20/0.38 tff(29,plain,
% 0.20/0.38 ((~![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(E!3, difference(E!3, empty_set))) | ![X: $i] : ((~member(X, E!3)) | member(X, difference(E!3, empty_set))))) | (~(subset(E!3, difference(E!3, empty_set)) | (~((~member(tptp_fun_X_0(difference(E!3, empty_set), E!3), E!3)) | member(tptp_fun_X_0(difference(E!3, empty_set), E!3), difference(E!3, empty_set))))))))),
% 0.20/0.38 inference(quant_inst,[status(thm)],[])).
% 0.20/0.38 tff(30,plain,
% 0.20/0.38 (~((~((~subset(E!3, difference(E!3, empty_set))) | ![X: $i] : ((~member(X, E!3)) | member(X, difference(E!3, empty_set))))) | (~(subset(E!3, difference(E!3, empty_set)) | (~((~member(tptp_fun_X_0(difference(E!3, empty_set), E!3), E!3)) | member(tptp_fun_X_0(difference(E!3, empty_set), E!3), difference(E!3, empty_set)))))))),
% 0.20/0.38 inference(unit_resolution,[status(thm)],[29, 28])).
% 0.20/0.38 tff(31,plain,
% 0.20/0.38 (((~((~subset(E!3, difference(E!3, empty_set))) | ![X: $i] : ((~member(X, E!3)) | member(X, difference(E!3, empty_set))))) | (~(subset(E!3, difference(E!3, empty_set)) | (~((~member(tptp_fun_X_0(difference(E!3, empty_set), E!3), E!3)) | member(tptp_fun_X_0(difference(E!3, empty_set), E!3), difference(E!3, empty_set))))))) | (subset(E!3, difference(E!3, empty_set)) | (~((~member(tptp_fun_X_0(difference(E!3, empty_set), E!3), E!3)) | member(tptp_fun_X_0(difference(E!3, empty_set), E!3), difference(E!3, empty_set)))))),
% 0.20/0.38 inference(tautology,[status(thm)],[])).
% 0.20/0.38 tff(32,plain,
% 0.20/0.38 (subset(E!3, difference(E!3, empty_set)) | (~((~member(tptp_fun_X_0(difference(E!3, empty_set), E!3), E!3)) | member(tptp_fun_X_0(difference(E!3, empty_set), E!3), difference(E!3, empty_set))))),
% 0.20/0.38 inference(unit_resolution,[status(thm)],[31, 30])).
% 0.20/0.38 tff(33,assumption,((~((~subset(difference(E!3, empty_set), E!3)) | ![X: $i] : ((~member(X, difference(E!3, empty_set))) | member(X, E!3)))) | (~(subset(difference(E!3, empty_set), E!3) | (~((~member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), difference(E!3, empty_set))) | member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), E!3)))))), introduced(assumption)).
% 0.20/0.38 tff(34,plain,
% 0.20/0.38 ((~![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, empty_set), E!3)) | ![X: $i] : ((~member(X, difference(E!3, empty_set))) | member(X, E!3)))) | (~(subset(difference(E!3, empty_set), E!3) | (~((~member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), difference(E!3, empty_set))) | member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), E!3)))))))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(35,plain,
% 0.20/0.39 ($false),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[34, 28, 33])).
% 0.20/0.39 tff(36,plain,(~((~((~subset(difference(E!3, empty_set), E!3)) | ![X: $i] : ((~member(X, difference(E!3, empty_set))) | member(X, E!3)))) | (~(subset(difference(E!3, empty_set), E!3) | (~((~member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), difference(E!3, empty_set))) | member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), E!3))))))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.39 tff(37,plain,
% 0.20/0.39 (((~((~subset(difference(E!3, empty_set), E!3)) | ![X: $i] : ((~member(X, difference(E!3, empty_set))) | member(X, E!3)))) | (~(subset(difference(E!3, empty_set), E!3) | (~((~member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), difference(E!3, empty_set))) | member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), E!3)))))) | (subset(difference(E!3, empty_set), E!3) | (~((~member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), difference(E!3, empty_set))) | member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), E!3))))),
% 0.20/0.39 inference(tautology,[status(thm)],[])).
% 0.20/0.39 tff(38,plain,
% 0.20/0.39 (subset(difference(E!3, empty_set), E!3) | (~((~member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), difference(E!3, empty_set))) | member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), E!3)))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[37, 36])).
% 0.20/0.39 tff(39,assumption,(~((~member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), difference(E!3, empty_set))) | member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), E!3))), introduced(assumption)).
% 0.20/0.39 tff(40,plain,
% 0.20/0.39 (((~member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), difference(E!3, empty_set))) | member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), E!3)) | (~member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), E!3))),
% 0.20/0.39 inference(tautology,[status(thm)],[])).
% 0.20/0.39 tff(41,plain,
% 0.20/0.39 (~member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), E!3)),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[40, 39])).
% 0.20/0.39 tff(42,plain,
% 0.20/0.39 (((~member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), difference(E!3, empty_set))) | member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), E!3)) | member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), difference(E!3, empty_set))),
% 0.20/0.39 inference(tautology,[status(thm)],[])).
% 0.20/0.39 tff(43,plain,
% 0.20/0.39 (member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), difference(E!3, empty_set))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[42, 39])).
% 0.20/0.39 tff(44,assumption,(((~member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), E!3)) | member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), empty_set)) <=> member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), difference(E!3, empty_set))), introduced(assumption)).
% 0.20/0.39 tff(45,plain,
% 0.20/0.39 ((~![B: $i, A: $i, E: $i] : (~(((~member(B, E)) | member(B, A)) <=> member(B, difference(E, A))))) | (~(((~member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), E!3)) | member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), empty_set)) <=> member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), difference(E!3, empty_set))))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(46,plain,
% 0.20/0.39 ($false),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[45, 10, 44])).
% 0.20/0.39 tff(47,plain,(~(((~member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), E!3)) | member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), empty_set)) <=> member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), difference(E!3, empty_set)))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.39 tff(48,plain,
% 0.20/0.39 ((((~member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), E!3)) | member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), empty_set)) <=> member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), difference(E!3, empty_set))) | (~((~member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), E!3)) | member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), empty_set))) | (~member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), difference(E!3, empty_set)))),
% 0.20/0.39 inference(tautology,[status(thm)],[])).
% 0.20/0.39 tff(49,plain,
% 0.20/0.39 ((~((~member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), E!3)) | member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), empty_set))) | (~member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), difference(E!3, empty_set)))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[48, 47])).
% 0.20/0.39 tff(50,plain,
% 0.20/0.39 (~((~member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), E!3)) | member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), empty_set))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[49, 43])).
% 0.20/0.39 tff(51,plain,
% 0.20/0.39 (((~member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), E!3)) | member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), empty_set)) | member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), E!3)),
% 0.20/0.39 inference(tautology,[status(thm)],[])).
% 0.20/0.39 tff(52,plain,
% 0.20/0.39 ($false),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[51, 50, 41])).
% 0.20/0.39 tff(53,plain,((~member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), difference(E!3, empty_set))) | member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), E!3)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.39 tff(54,plain,
% 0.20/0.39 ((~(subset(difference(E!3, empty_set), E!3) | (~((~member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), difference(E!3, empty_set))) | member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), E!3))))) | subset(difference(E!3, empty_set), E!3) | (~((~member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), difference(E!3, empty_set))) | member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), E!3)))),
% 0.20/0.39 inference(tautology,[status(thm)],[])).
% 0.20/0.39 tff(55,plain,
% 0.20/0.39 ((~(subset(difference(E!3, empty_set), E!3) | (~((~member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), difference(E!3, empty_set))) | member(tptp_fun_X_0(E!3, difference(E!3, empty_set)), E!3))))) | subset(difference(E!3, empty_set), E!3)),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[54, 53])).
% 0.20/0.39 tff(56,plain,
% 0.20/0.39 (subset(difference(E!3, empty_set), E!3)),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[55, 38])).
% 0.20/0.39 tff(57,plain,
% 0.20/0.39 (^[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))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(58,plain,
% 0.20/0.39 (![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)))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[57])).
% 0.20/0.39 tff(59,plain,
% 0.20/0.39 (^[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)))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(60,plain,
% 0.20/0.39 (![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)))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[59])).
% 0.20/0.39 tff(61,plain,
% 0.20/0.39 (![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)))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(62,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')).
% 0.20/0.39 tff(63,plain,
% 0.20/0.39 (![A: $i, B: $i] : (equal_set(A, B) <=> (subset(A, B) & subset(B, A)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[62, 61])).
% 0.20/0.39 tff(64,plain,(
% 0.20/0.39 ![A: $i, B: $i] : (equal_set(A, B) <=> (subset(A, B) & subset(B, A)))),
% 0.20/0.39 inference(skolemize,[status(sab)],[63])).
% 0.20/0.39 tff(65,plain,
% 0.20/0.39 (![A: $i, B: $i] : (~(((~subset(A, B)) | (~subset(B, A))) <=> equal_set(A, B)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[64, 60])).
% 0.20/0.39 tff(66,plain,
% 0.20/0.39 (![A: $i, B: $i] : (~(((~subset(A, B)) | (~subset(B, A))) <=> equal_set(A, B)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[65, 58])).
% 0.20/0.39 tff(67,plain,
% 0.20/0.39 ((~![A: $i, B: $i] : (~(((~subset(A, B)) | (~subset(B, A))) <=> equal_set(A, B)))) | (~(((~subset(difference(E!3, empty_set), E!3)) | (~subset(E!3, difference(E!3, empty_set)))) <=> equal_set(difference(E!3, empty_set), E!3)))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(68,plain,
% 0.20/0.39 (~(((~subset(difference(E!3, empty_set), E!3)) | (~subset(E!3, difference(E!3, empty_set)))) <=> equal_set(difference(E!3, empty_set), E!3))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[67, 66])).
% 0.20/0.39 tff(69,plain,
% 0.20/0.39 ((~![E: $i] : equal_set(difference(E, empty_set), E)) <=> (~![E: $i] : equal_set(difference(E, empty_set), E))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(70,axiom,(~![E: $i] : equal_set(difference(E, empty_set), E)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','thI30')).
% 0.20/0.39 tff(71,plain,
% 0.20/0.39 (~![E: $i] : equal_set(difference(E, empty_set), E)),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[70, 69])).
% 0.20/0.39 tff(72,plain,
% 0.20/0.39 (~![E: $i] : equal_set(difference(E, empty_set), E)),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[71, 69])).
% 0.20/0.39 tff(73,plain,
% 0.20/0.39 (~![E: $i] : equal_set(difference(E, empty_set), E)),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[72, 69])).
% 0.20/0.39 tff(74,plain,
% 0.20/0.39 (~![E: $i] : equal_set(difference(E, empty_set), E)),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[73, 69])).
% 0.20/0.39 tff(75,plain,(
% 0.20/0.39 ~equal_set(difference(E!3, empty_set), E!3)),
% 0.20/0.39 inference(skolemize,[status(sab)],[74])).
% 0.20/0.39 tff(76,plain,
% 0.20/0.39 ((((~subset(difference(E!3, empty_set), E!3)) | (~subset(E!3, difference(E!3, empty_set)))) <=> equal_set(difference(E!3, empty_set), E!3)) | ((~subset(difference(E!3, empty_set), E!3)) | (~subset(E!3, difference(E!3, empty_set)))) | equal_set(difference(E!3, empty_set), E!3)),
% 0.20/0.39 inference(tautology,[status(thm)],[])).
% 0.20/0.39 tff(77,plain,
% 0.20/0.39 ((((~subset(difference(E!3, empty_set), E!3)) | (~subset(E!3, difference(E!3, empty_set)))) <=> equal_set(difference(E!3, empty_set), E!3)) | ((~subset(difference(E!3, empty_set), E!3)) | (~subset(E!3, difference(E!3, empty_set))))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[76, 75])).
% 0.20/0.39 tff(78,plain,
% 0.20/0.39 ((~subset(difference(E!3, empty_set), E!3)) | (~subset(E!3, difference(E!3, empty_set)))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[77, 68])).
% 0.20/0.39 tff(79,plain,
% 0.20/0.39 ((~((~subset(difference(E!3, empty_set), E!3)) | (~subset(E!3, difference(E!3, empty_set))))) | (~subset(difference(E!3, empty_set), E!3)) | (~subset(E!3, difference(E!3, empty_set)))),
% 0.20/0.39 inference(tautology,[status(thm)],[])).
% 0.20/0.39 tff(80,plain,
% 0.20/0.39 ((~subset(difference(E!3, empty_set), E!3)) | (~subset(E!3, difference(E!3, empty_set)))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[79, 78])).
% 0.20/0.39 tff(81,plain,
% 0.20/0.39 (~subset(E!3, difference(E!3, empty_set))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[80, 56])).
% 0.20/0.39 tff(82,plain,
% 0.20/0.39 ((~(subset(E!3, difference(E!3, empty_set)) | (~((~member(tptp_fun_X_0(difference(E!3, empty_set), E!3), E!3)) | member(tptp_fun_X_0(difference(E!3, empty_set), E!3), difference(E!3, empty_set)))))) | subset(E!3, difference(E!3, empty_set)) | (~((~member(tptp_fun_X_0(difference(E!3, empty_set), E!3), E!3)) | member(tptp_fun_X_0(difference(E!3, empty_set), E!3), difference(E!3, empty_set))))),
% 0.20/0.39 inference(tautology,[status(thm)],[])).
% 0.20/0.39 tff(83,plain,
% 0.20/0.39 ((~(subset(E!3, difference(E!3, empty_set)) | (~((~member(tptp_fun_X_0(difference(E!3, empty_set), E!3), E!3)) | member(tptp_fun_X_0(difference(E!3, empty_set), E!3), difference(E!3, empty_set)))))) | (~((~member(tptp_fun_X_0(difference(E!3, empty_set), E!3), E!3)) | member(tptp_fun_X_0(difference(E!3, empty_set), E!3), difference(E!3, empty_set))))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[82, 81])).
% 0.20/0.39 tff(84,plain,
% 0.20/0.39 (~((~member(tptp_fun_X_0(difference(E!3, empty_set), E!3), E!3)) | member(tptp_fun_X_0(difference(E!3, empty_set), E!3), difference(E!3, empty_set)))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[83, 32])).
% 0.20/0.40 tff(85,plain,
% 0.20/0.40 (((~member(tptp_fun_X_0(difference(E!3, empty_set), E!3), E!3)) | member(tptp_fun_X_0(difference(E!3, empty_set), E!3), difference(E!3, empty_set))) | (~member(tptp_fun_X_0(difference(E!3, empty_set), E!3), difference(E!3, empty_set)))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(86,plain,
% 0.20/0.40 (~member(tptp_fun_X_0(difference(E!3, empty_set), E!3), difference(E!3, empty_set))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[85, 84])).
% 0.20/0.40 tff(87,plain,
% 0.20/0.40 ((((~member(tptp_fun_X_0(difference(E!3, empty_set), E!3), E!3)) | member(tptp_fun_X_0(difference(E!3, empty_set), E!3), empty_set)) <=> member(tptp_fun_X_0(difference(E!3, empty_set), E!3), difference(E!3, empty_set))) | ((~member(tptp_fun_X_0(difference(E!3, empty_set), E!3), E!3)) | member(tptp_fun_X_0(difference(E!3, empty_set), E!3), empty_set)) | member(tptp_fun_X_0(difference(E!3, empty_set), E!3), difference(E!3, empty_set))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(88,plain,
% 0.20/0.40 ((((~member(tptp_fun_X_0(difference(E!3, empty_set), E!3), E!3)) | member(tptp_fun_X_0(difference(E!3, empty_set), E!3), empty_set)) <=> member(tptp_fun_X_0(difference(E!3, empty_set), E!3), difference(E!3, empty_set))) | ((~member(tptp_fun_X_0(difference(E!3, empty_set), E!3), E!3)) | member(tptp_fun_X_0(difference(E!3, empty_set), E!3), empty_set))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[87, 86])).
% 0.20/0.40 tff(89,plain,
% 0.20/0.40 ((~member(tptp_fun_X_0(difference(E!3, empty_set), E!3), E!3)) | member(tptp_fun_X_0(difference(E!3, empty_set), E!3), empty_set)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[88, 12])).
% 0.20/0.40 tff(90,plain,
% 0.20/0.40 (((~member(tptp_fun_X_0(difference(E!3, empty_set), E!3), E!3)) | member(tptp_fun_X_0(difference(E!3, empty_set), E!3), difference(E!3, empty_set))) | member(tptp_fun_X_0(difference(E!3, empty_set), E!3), E!3)),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(91,plain,
% 0.20/0.40 (member(tptp_fun_X_0(difference(E!3, empty_set), E!3), E!3)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[90, 84])).
% 0.20/0.40 tff(92,plain,
% 0.20/0.40 ((~((~member(tptp_fun_X_0(difference(E!3, empty_set), E!3), E!3)) | member(tptp_fun_X_0(difference(E!3, empty_set), E!3), empty_set))) | (~member(tptp_fun_X_0(difference(E!3, empty_set), E!3), E!3)) | member(tptp_fun_X_0(difference(E!3, empty_set), E!3), empty_set)),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(93,plain,
% 0.20/0.40 ((~((~member(tptp_fun_X_0(difference(E!3, empty_set), E!3), E!3)) | member(tptp_fun_X_0(difference(E!3, empty_set), E!3), empty_set))) | member(tptp_fun_X_0(difference(E!3, empty_set), E!3), empty_set)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[92, 91])).
% 0.20/0.40 tff(94,plain,
% 0.20/0.40 (member(tptp_fun_X_0(difference(E!3, empty_set), E!3), empty_set)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[93, 89])).
% 0.20/0.40 tff(95,plain,
% 0.20/0.40 (^[X: $i] : refl((~member(X, empty_set)) <=> (~member(X, empty_set)))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(96,plain,
% 0.20/0.40 (![X: $i] : (~member(X, empty_set)) <=> ![X: $i] : (~member(X, empty_set))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[95])).
% 0.20/0.40 tff(97,plain,
% 0.20/0.40 (![X: $i] : (~member(X, empty_set)) <=> ![X: $i] : (~member(X, empty_set))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(98,axiom,(![X: $i] : (~member(X, empty_set))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+0.ax','empty_set')).
% 0.20/0.40 tff(99,plain,
% 0.20/0.40 (![X: $i] : (~member(X, empty_set))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[98, 97])).
% 0.20/0.40 tff(100,plain,(
% 0.20/0.40 ![X: $i] : (~member(X, empty_set))),
% 0.20/0.40 inference(skolemize,[status(sab)],[99])).
% 0.20/0.40 tff(101,plain,
% 0.20/0.40 (![X: $i] : (~member(X, empty_set))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[100, 96])).
% 0.20/0.40 tff(102,plain,
% 0.20/0.40 ((~![X: $i] : (~member(X, empty_set))) | (~member(tptp_fun_X_0(difference(E!3, empty_set), E!3), empty_set))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(103,plain,
% 0.20/0.40 ($false),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[102, 101, 94])).
% 0.20/0.40 % SZS output end Proof
% 0.20/0.40 % E exiting
%------------------------------------------------------------------------------