%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET162+4 : TPTP v9.0.0. Released v2.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n002.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 05:25:23 AM UTC 2025
% Result : Theorem 0.13s 0.38s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : SET162+4 : TPTP v9.0.0. Released v2.2.0.
% 0.07/0.13 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n002.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Jun 20 04:29:46 EDT 2025
% 0.13/0.35 % CPUTime :
% 0.13/0.38 % SZS status Theorem
% 0.13/0.38 % SZS output start Proof
% 0.13/0.38 tff(member_type, type, (
% 0.13/0.38 member: ( $i * $i ) > $o)).
% 0.13/0.38 tff(tptp_fun_A_3_type, type, (
% 0.13/0.38 tptp_fun_A_3: $i)).
% 0.13/0.38 tff(tptp_fun_X_0_type, type, (
% 0.13/0.38 tptp_fun_X_0: ( $i * $i ) > $i)).
% 0.13/0.38 tff(union_type, type, (
% 0.13/0.38 union: ( $i * $i ) > $i)).
% 0.13/0.38 tff(empty_set_type, type, (
% 0.13/0.38 empty_set: $i)).
% 0.13/0.38 tff(subset_type, type, (
% 0.13/0.38 subset: ( $i * $i ) > $o)).
% 0.13/0.38 tff(equal_set_type, type, (
% 0.13/0.38 equal_set: ( $i * $i ) > $o)).
% 0.13/0.38 tff(1,plain,
% 0.13/0.38 (^[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))))))))),
% 0.13/0.38 inference(bind,[status(th)],[])).
% 0.13/0.38 tff(2,plain,
% 0.13/0.38 (![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)))))))),
% 0.13/0.38 inference(quant_intro,[status(thm)],[1])).
% 0.13/0.38 tff(3,plain,
% 0.13/0.38 (^[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))))))))),
% 0.13/0.38 inference(bind,[status(th)],[])).
% 0.13/0.38 tff(4,plain,
% 0.13/0.38 (![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)))))))),
% 0.13/0.38 inference(quant_intro,[status(thm)],[3])).
% 0.13/0.38 tff(5,plain,
% 0.13/0.38 (![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)))))))),
% 0.13/0.38 inference(transitivity,[status(thm)],[4, 2])).
% 0.13/0.38 tff(6,plain,
% 0.13/0.38 (^[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)))))))))),
% 0.13/0.38 inference(bind,[status(th)],[])).
% 0.13/0.38 tff(7,plain,
% 0.13/0.38 (![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)))))))),
% 0.13/0.39 inference(quant_intro,[status(thm)],[6])).
% 0.13/0.39 tff(8,plain,
% 0.13/0.39 (![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)))),
% 0.13/0.39 inference(rewrite,[status(thm)],[])).
% 0.13/0.39 tff(9,plain,
% 0.13/0.39 (^[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))))),
% 0.13/0.39 inference(bind,[status(th)],[])).
% 0.13/0.39 tff(10,plain,
% 0.13/0.39 (![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)))),
% 0.13/0.39 inference(quant_intro,[status(thm)],[9])).
% 0.13/0.39 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')).
% 0.13/0.39 tff(12,plain,
% 0.13/0.39 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[11, 10])).
% 0.13/0.39 tff(13,plain,
% 0.13/0.39 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[12, 8])).
% 0.13/0.39 tff(14,plain,(
% 0.13/0.39 ![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)))))),
% 0.13/0.39 inference(skolemize,[status(sab)],[13])).
% 0.13/0.39 tff(15,plain,
% 0.13/0.39 (![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)))))))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[14, 7])).
% 0.13/0.39 tff(16,plain,
% 0.13/0.39 (![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)))))))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[15, 5])).
% 0.13/0.39 tff(17,plain,
% 0.13/0.39 ((~![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(A!3, union(A!3, empty_set))) | ![X: $i] : ((~member(X, A!3)) | member(X, union(A!3, empty_set))))) | (~(subset(A!3, union(A!3, empty_set)) | (~((~member(tptp_fun_X_0(union(A!3, empty_set), A!3), A!3)) | member(tptp_fun_X_0(union(A!3, empty_set), A!3), union(A!3, empty_set))))))))),
% 0.13/0.39 inference(quant_inst,[status(thm)],[])).
% 0.13/0.39 tff(18,plain,
% 0.13/0.39 (~((~((~subset(A!3, union(A!3, empty_set))) | ![X: $i] : ((~member(X, A!3)) | member(X, union(A!3, empty_set))))) | (~(subset(A!3, union(A!3, empty_set)) | (~((~member(tptp_fun_X_0(union(A!3, empty_set), A!3), A!3)) | member(tptp_fun_X_0(union(A!3, empty_set), A!3), union(A!3, empty_set)))))))),
% 0.13/0.39 inference(unit_resolution,[status(thm)],[17, 16])).
% 0.13/0.39 tff(19,plain,
% 0.13/0.39 (((~((~subset(A!3, union(A!3, empty_set))) | ![X: $i] : ((~member(X, A!3)) | member(X, union(A!3, empty_set))))) | (~(subset(A!3, union(A!3, empty_set)) | (~((~member(tptp_fun_X_0(union(A!3, empty_set), A!3), A!3)) | member(tptp_fun_X_0(union(A!3, empty_set), A!3), union(A!3, empty_set))))))) | (subset(A!3, union(A!3, empty_set)) | (~((~member(tptp_fun_X_0(union(A!3, empty_set), A!3), A!3)) | member(tptp_fun_X_0(union(A!3, empty_set), A!3), union(A!3, empty_set)))))),
% 0.13/0.39 inference(tautology,[status(thm)],[])).
% 0.13/0.39 tff(20,plain,
% 0.13/0.39 (subset(A!3, union(A!3, empty_set)) | (~((~member(tptp_fun_X_0(union(A!3, empty_set), A!3), A!3)) | member(tptp_fun_X_0(union(A!3, empty_set), A!3), union(A!3, empty_set))))),
% 0.13/0.39 inference(unit_resolution,[status(thm)],[19, 18])).
% 0.13/0.39 tff(21,assumption,((~((~subset(union(A!3, empty_set), A!3)) | ![X: $i] : ((~member(X, union(A!3, empty_set))) | member(X, A!3)))) | (~(subset(union(A!3, empty_set), A!3) | (~((~member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set))) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3)))))), introduced(assumption)).
% 0.13/0.39 tff(22,plain,
% 0.13/0.39 ((~![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(union(A!3, empty_set), A!3)) | ![X: $i] : ((~member(X, union(A!3, empty_set))) | member(X, A!3)))) | (~(subset(union(A!3, empty_set), A!3) | (~((~member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set))) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3)))))))),
% 0.13/0.39 inference(quant_inst,[status(thm)],[])).
% 0.13/0.39 tff(23,plain,
% 0.13/0.39 ($false),
% 0.13/0.39 inference(unit_resolution,[status(thm)],[22, 16, 21])).
% 0.13/0.39 tff(24,plain,(~((~((~subset(union(A!3, empty_set), A!3)) | ![X: $i] : ((~member(X, union(A!3, empty_set))) | member(X, A!3)))) | (~(subset(union(A!3, empty_set), A!3) | (~((~member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set))) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3))))))), inference(lemma,lemma(discharge,[]))).
% 0.13/0.39 tff(25,plain,
% 0.13/0.39 (((~((~subset(union(A!3, empty_set), A!3)) | ![X: $i] : ((~member(X, union(A!3, empty_set))) | member(X, A!3)))) | (~(subset(union(A!3, empty_set), A!3) | (~((~member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set))) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3)))))) | (subset(union(A!3, empty_set), A!3) | (~((~member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set))) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3))))),
% 0.13/0.39 inference(tautology,[status(thm)],[])).
% 0.13/0.39 tff(26,plain,
% 0.13/0.39 (subset(union(A!3, empty_set), A!3) | (~((~member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set))) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3)))),
% 0.13/0.39 inference(unit_resolution,[status(thm)],[25, 24])).
% 0.13/0.39 tff(27,assumption,(~((~member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set))) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3))), introduced(assumption)).
% 0.13/0.39 tff(28,plain,
% 0.13/0.39 (((~member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set))) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3)) | (~member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3))),
% 0.13/0.39 inference(tautology,[status(thm)],[])).
% 0.13/0.39 tff(29,plain,
% 0.13/0.39 (~member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3)),
% 0.13/0.39 inference(unit_resolution,[status(thm)],[28, 27])).
% 0.13/0.39 tff(30,plain,
% 0.13/0.39 (((~member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set))) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3)) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set))),
% 0.13/0.39 inference(tautology,[status(thm)],[])).
% 0.13/0.39 tff(31,plain,
% 0.13/0.39 (member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set))),
% 0.13/0.39 inference(unit_resolution,[status(thm)],[30, 27])).
% 0.13/0.39 tff(32,assumption,(~(member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set)) <=> (member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), empty_set)))), introduced(assumption)).
% 0.13/0.39 tff(33,plain,
% 0.13/0.39 (^[X: $i, A: $i, B: $i] : refl((member(X, union(A, B)) <=> (member(X, B) | member(X, A))) <=> (member(X, union(A, B)) <=> (member(X, B) | member(X, A))))),
% 0.13/0.39 inference(bind,[status(th)],[])).
% 0.13/0.39 tff(34,plain,
% 0.13/0.39 (![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A))) <=> ![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A)))),
% 0.13/0.39 inference(quant_intro,[status(thm)],[33])).
% 0.13/0.39 tff(35,plain,
% 0.13/0.39 (![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A))) <=> ![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A)))),
% 0.13/0.39 inference(rewrite,[status(thm)],[])).
% 0.13/0.39 tff(36,plain,
% 0.13/0.39 (^[X: $i, A: $i, B: $i] : rewrite((member(X, union(A, B)) <=> (member(X, A) | member(X, B))) <=> (member(X, union(A, B)) <=> (member(X, B) | member(X, A))))),
% 0.13/0.39 inference(bind,[status(th)],[])).
% 0.13/0.39 tff(37,plain,
% 0.13/0.39 (![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, A) | member(X, B))) <=> ![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A)))),
% 0.13/0.39 inference(quant_intro,[status(thm)],[36])).
% 0.13/0.39 tff(38,axiom,(![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, A) | member(X, B)))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+0.ax','union')).
% 0.13/0.39 tff(39,plain,
% 0.13/0.39 (![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A)))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[38, 37])).
% 0.13/0.39 tff(40,plain,
% 0.13/0.39 (![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A)))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[39, 35])).
% 0.13/0.39 tff(41,plain,(
% 0.13/0.39 ![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A)))),
% 0.13/0.39 inference(skolemize,[status(sab)],[40])).
% 0.13/0.39 tff(42,plain,
% 0.13/0.39 (![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A)))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[41, 34])).
% 0.13/0.39 tff(43,plain,
% 0.13/0.39 (((~![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A)))) | (member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set)) <=> (member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), empty_set)))) <=> ((~![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A)))) | (member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set)) <=> (member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), empty_set))))),
% 0.13/0.39 inference(rewrite,[status(thm)],[])).
% 0.13/0.39 tff(44,plain,
% 0.13/0.39 ((member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set)) <=> (member(tptp_fun_X_0(A!3, union(A!3, empty_set)), empty_set) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3))) <=> (member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set)) <=> (member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), empty_set)))),
% 0.13/0.39 inference(rewrite,[status(thm)],[])).
% 0.13/0.39 tff(45,plain,
% 0.13/0.39 (((~![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A)))) | (member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set)) <=> (member(tptp_fun_X_0(A!3, union(A!3, empty_set)), empty_set) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3)))) <=> ((~![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A)))) | (member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set)) <=> (member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), empty_set))))),
% 0.13/0.39 inference(monotonicity,[status(thm)],[44])).
% 0.13/0.39 tff(46,plain,
% 0.13/0.39 (((~![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A)))) | (member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set)) <=> (member(tptp_fun_X_0(A!3, union(A!3, empty_set)), empty_set) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3)))) <=> ((~![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A)))) | (member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set)) <=> (member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), empty_set))))),
% 0.13/0.39 inference(transitivity,[status(thm)],[45, 43])).
% 0.13/0.39 tff(47,plain,
% 0.13/0.39 ((~![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A)))) | (member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set)) <=> (member(tptp_fun_X_0(A!3, union(A!3, empty_set)), empty_set) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3)))),
% 0.13/0.39 inference(quant_inst,[status(thm)],[])).
% 0.13/0.39 tff(48,plain,
% 0.13/0.39 ((~![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A)))) | (member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set)) <=> (member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), empty_set)))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[47, 46])).
% 0.13/0.40 tff(49,plain,
% 0.13/0.40 ($false),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[48, 42, 32])).
% 0.13/0.40 tff(50,plain,(member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set)) <=> (member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), empty_set))), inference(lemma,lemma(discharge,[]))).
% 0.13/0.40 tff(51,plain,
% 0.13/0.40 ((~(member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set)) <=> (member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), empty_set)))) | (~member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set))) | (member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), empty_set))),
% 0.13/0.40 inference(tautology,[status(thm)],[])).
% 0.13/0.40 tff(52,plain,
% 0.13/0.40 ((~member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set))) | (member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), empty_set))),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[51, 50])).
% 0.13/0.40 tff(53,plain,
% 0.13/0.40 (member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), empty_set)),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[52, 31])).
% 0.13/0.40 tff(54,assumption,(member(tptp_fun_X_0(A!3, union(A!3, empty_set)), empty_set)), introduced(assumption)).
% 0.13/0.40 tff(55,plain,
% 0.13/0.40 (^[X: $i] : refl((~member(X, empty_set)) <=> (~member(X, empty_set)))),
% 0.13/0.40 inference(bind,[status(th)],[])).
% 0.13/0.40 tff(56,plain,
% 0.13/0.40 (![X: $i] : (~member(X, empty_set)) <=> ![X: $i] : (~member(X, empty_set))),
% 0.13/0.40 inference(quant_intro,[status(thm)],[55])).
% 0.13/0.40 tff(57,plain,
% 0.13/0.40 (![X: $i] : (~member(X, empty_set)) <=> ![X: $i] : (~member(X, empty_set))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(58,axiom,(![X: $i] : (~member(X, empty_set))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+0.ax','empty_set')).
% 0.13/0.40 tff(59,plain,
% 0.13/0.40 (![X: $i] : (~member(X, empty_set))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[58, 57])).
% 0.13/0.40 tff(60,plain,(
% 0.13/0.40 ![X: $i] : (~member(X, empty_set))),
% 0.13/0.40 inference(skolemize,[status(sab)],[59])).
% 0.13/0.40 tff(61,plain,
% 0.13/0.40 (![X: $i] : (~member(X, empty_set))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[60, 56])).
% 0.13/0.40 tff(62,plain,
% 0.13/0.40 ((~![X: $i] : (~member(X, empty_set))) | (~member(tptp_fun_X_0(A!3, union(A!3, empty_set)), empty_set))),
% 0.13/0.40 inference(quant_inst,[status(thm)],[])).
% 0.13/0.40 tff(63,plain,
% 0.13/0.40 ($false),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[62, 61, 54])).
% 0.13/0.40 tff(64,plain,(~member(tptp_fun_X_0(A!3, union(A!3, empty_set)), empty_set)), inference(lemma,lemma(discharge,[]))).
% 0.13/0.40 tff(65,plain,
% 0.13/0.40 ((~(member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), empty_set))) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), empty_set)),
% 0.13/0.40 inference(tautology,[status(thm)],[])).
% 0.13/0.40 tff(66,plain,
% 0.13/0.40 ((~(member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), empty_set))) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3)),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[65, 64])).
% 0.13/0.40 tff(67,plain,
% 0.13/0.40 ($false),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[66, 53, 29])).
% 0.13/0.40 tff(68,plain,((~member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set))) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3)), inference(lemma,lemma(discharge,[]))).
% 0.13/0.40 tff(69,plain,
% 0.13/0.40 ((~(subset(union(A!3, empty_set), A!3) | (~((~member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set))) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3))))) | subset(union(A!3, empty_set), A!3) | (~((~member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set))) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3)))),
% 0.13/0.40 inference(tautology,[status(thm)],[])).
% 0.13/0.40 tff(70,plain,
% 0.13/0.40 ((~(subset(union(A!3, empty_set), A!3) | (~((~member(tptp_fun_X_0(A!3, union(A!3, empty_set)), union(A!3, empty_set))) | member(tptp_fun_X_0(A!3, union(A!3, empty_set)), A!3))))) | subset(union(A!3, empty_set), A!3)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[69, 68])).
% 0.20/0.40 tff(71,plain,
% 0.20/0.40 (subset(union(A!3, empty_set), A!3)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[70, 26])).
% 0.20/0.40 tff(72,plain,
% 0.20/0.40 (^[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))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(73,plain,
% 0.20/0.40 (![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)))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[72])).
% 0.20/0.40 tff(74,plain,
% 0.20/0.40 (^[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)))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(75,plain,
% 0.20/0.40 (![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)))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[74])).
% 0.20/0.40 tff(76,plain,
% 0.20/0.40 (![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)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(77,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')).
% 0.20/0.40 tff(78,plain,
% 0.20/0.40 (![A: $i, B: $i] : (equal_set(A, B) <=> (subset(A, B) & subset(B, A)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[77, 76])).
% 0.20/0.40 tff(79,plain,(
% 0.20/0.40 ![A: $i, B: $i] : (equal_set(A, B) <=> (subset(A, B) & subset(B, A)))),
% 0.20/0.40 inference(skolemize,[status(sab)],[78])).
% 0.20/0.40 tff(80,plain,
% 0.20/0.40 (![A: $i, B: $i] : (~(((~subset(A, B)) | (~subset(B, A))) <=> equal_set(A, B)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[79, 75])).
% 0.20/0.40 tff(81,plain,
% 0.20/0.40 (![A: $i, B: $i] : (~(((~subset(A, B)) | (~subset(B, A))) <=> equal_set(A, B)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[80, 73])).
% 0.20/0.40 tff(82,plain,
% 0.20/0.40 ((~![A: $i, B: $i] : (~(((~subset(A, B)) | (~subset(B, A))) <=> equal_set(A, B)))) | (~(((~subset(union(A!3, empty_set), A!3)) | (~subset(A!3, union(A!3, empty_set)))) <=> equal_set(union(A!3, empty_set), A!3)))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(83,plain,
% 0.20/0.40 (~(((~subset(union(A!3, empty_set), A!3)) | (~subset(A!3, union(A!3, empty_set)))) <=> equal_set(union(A!3, empty_set), A!3))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[82, 81])).
% 0.20/0.40 tff(84,plain,
% 0.20/0.40 ((~![A: $i] : equal_set(union(A, empty_set), A)) <=> (~![A: $i] : equal_set(union(A, empty_set), A))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(85,axiom,(~![A: $i] : equal_set(union(A, empty_set), A)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','thI18')).
% 0.20/0.40 tff(86,plain,
% 0.20/0.40 (~![A: $i] : equal_set(union(A, empty_set), A)),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[85, 84])).
% 0.20/0.40 tff(87,plain,
% 0.20/0.40 (~![A: $i] : equal_set(union(A, empty_set), A)),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[86, 84])).
% 0.20/0.40 tff(88,plain,
% 0.20/0.40 (~![A: $i] : equal_set(union(A, empty_set), A)),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[87, 84])).
% 0.20/0.40 tff(89,plain,
% 0.20/0.40 (~![A: $i] : equal_set(union(A, empty_set), A)),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[88, 84])).
% 0.20/0.40 tff(90,plain,(
% 0.20/0.40 ~equal_set(union(A!3, empty_set), A!3)),
% 0.20/0.40 inference(skolemize,[status(sab)],[89])).
% 0.20/0.40 tff(91,plain,
% 0.20/0.40 ((((~subset(union(A!3, empty_set), A!3)) | (~subset(A!3, union(A!3, empty_set)))) <=> equal_set(union(A!3, empty_set), A!3)) | ((~subset(union(A!3, empty_set), A!3)) | (~subset(A!3, union(A!3, empty_set)))) | equal_set(union(A!3, empty_set), A!3)),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(92,plain,
% 0.20/0.40 ((((~subset(union(A!3, empty_set), A!3)) | (~subset(A!3, union(A!3, empty_set)))) <=> equal_set(union(A!3, empty_set), A!3)) | ((~subset(union(A!3, empty_set), A!3)) | (~subset(A!3, union(A!3, empty_set))))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[91, 90])).
% 0.20/0.40 tff(93,plain,
% 0.20/0.40 ((~subset(union(A!3, empty_set), A!3)) | (~subset(A!3, union(A!3, empty_set)))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[92, 83])).
% 0.20/0.40 tff(94,plain,
% 0.20/0.40 ((~((~subset(union(A!3, empty_set), A!3)) | (~subset(A!3, union(A!3, empty_set))))) | (~subset(union(A!3, empty_set), A!3)) | (~subset(A!3, union(A!3, empty_set)))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(95,plain,
% 0.20/0.40 ((~subset(union(A!3, empty_set), A!3)) | (~subset(A!3, union(A!3, empty_set)))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[94, 93])).
% 0.20/0.40 tff(96,plain,
% 0.20/0.40 (~subset(A!3, union(A!3, empty_set))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[95, 71])).
% 0.20/0.40 tff(97,plain,
% 0.20/0.40 ((~(subset(A!3, union(A!3, empty_set)) | (~((~member(tptp_fun_X_0(union(A!3, empty_set), A!3), A!3)) | member(tptp_fun_X_0(union(A!3, empty_set), A!3), union(A!3, empty_set)))))) | subset(A!3, union(A!3, empty_set)) | (~((~member(tptp_fun_X_0(union(A!3, empty_set), A!3), A!3)) | member(tptp_fun_X_0(union(A!3, empty_set), A!3), union(A!3, empty_set))))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(98,plain,
% 0.20/0.40 ((~(subset(A!3, union(A!3, empty_set)) | (~((~member(tptp_fun_X_0(union(A!3, empty_set), A!3), A!3)) | member(tptp_fun_X_0(union(A!3, empty_set), A!3), union(A!3, empty_set)))))) | (~((~member(tptp_fun_X_0(union(A!3, empty_set), A!3), A!3)) | member(tptp_fun_X_0(union(A!3, empty_set), A!3), union(A!3, empty_set))))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[97, 96])).
% 0.20/0.40 tff(99,plain,
% 0.20/0.40 (~((~member(tptp_fun_X_0(union(A!3, empty_set), A!3), A!3)) | member(tptp_fun_X_0(union(A!3, empty_set), A!3), union(A!3, empty_set)))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[98, 20])).
% 0.20/0.40 tff(100,plain,
% 0.20/0.40 (((~member(tptp_fun_X_0(union(A!3, empty_set), A!3), A!3)) | member(tptp_fun_X_0(union(A!3, empty_set), A!3), union(A!3, empty_set))) | member(tptp_fun_X_0(union(A!3, empty_set), A!3), A!3)),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(101,plain,
% 0.20/0.40 (member(tptp_fun_X_0(union(A!3, empty_set), A!3), A!3)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[100, 99])).
% 0.20/0.40 tff(102,plain,
% 0.20/0.40 ((member(tptp_fun_X_0(union(A!3, empty_set), A!3), empty_set) | member(tptp_fun_X_0(union(A!3, empty_set), A!3), A!3)) | (~member(tptp_fun_X_0(union(A!3, empty_set), A!3), A!3))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(103,plain,
% 0.20/0.40 (member(tptp_fun_X_0(union(A!3, empty_set), A!3), empty_set) | member(tptp_fun_X_0(union(A!3, empty_set), A!3), A!3)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[102, 101])).
% 0.20/0.40 tff(104,plain,
% 0.20/0.40 (((~member(tptp_fun_X_0(union(A!3, empty_set), A!3), A!3)) | member(tptp_fun_X_0(union(A!3, empty_set), A!3), union(A!3, empty_set))) | (~member(tptp_fun_X_0(union(A!3, empty_set), A!3), union(A!3, empty_set)))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(105,plain,
% 0.20/0.40 (~member(tptp_fun_X_0(union(A!3, empty_set), A!3), union(A!3, empty_set))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[104, 99])).
% 0.20/0.40 tff(106,plain,
% 0.20/0.40 ((~(member(tptp_fun_X_0(union(A!3, empty_set), A!3), union(A!3, empty_set)) <=> (member(tptp_fun_X_0(union(A!3, empty_set), A!3), empty_set) | member(tptp_fun_X_0(union(A!3, empty_set), A!3), A!3)))) | member(tptp_fun_X_0(union(A!3, empty_set), A!3), union(A!3, empty_set)) | (~(member(tptp_fun_X_0(union(A!3, empty_set), A!3), empty_set) | member(tptp_fun_X_0(union(A!3, empty_set), A!3), A!3)))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(107,plain,
% 0.20/0.40 (~(member(tptp_fun_X_0(union(A!3, empty_set), A!3), union(A!3, empty_set)) <=> (member(tptp_fun_X_0(union(A!3, empty_set), A!3), empty_set) | member(tptp_fun_X_0(union(A!3, empty_set), A!3), A!3)))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[106, 105, 103])).
% 0.20/0.40 tff(108,plain,
% 0.20/0.40 ((~![X: $i, A: $i, B: $i] : (member(X, union(A, B)) <=> (member(X, B) | member(X, A)))) | (member(tptp_fun_X_0(union(A!3, empty_set), A!3), union(A!3, empty_set)) <=> (member(tptp_fun_X_0(union(A!3, empty_set), A!3), empty_set) | member(tptp_fun_X_0(union(A!3, empty_set), A!3), A!3)))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(109,plain,
% 0.20/0.40 ($false),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[108, 42, 107])).
% 0.20/0.40 % SZS output end Proof
% 0.20/0.41 % E exiting
%------------------------------------------------------------------------------