%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET095+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 05:25:12 AM UTC 2025
% Result : Theorem 0.13s 0.38s
% Output : Proof 0.13s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.13 % Problem : SET095+4 : TPTP v9.0.0. Released v2.2.0.
% 0.10/0.13 % 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 04:18:16 EDT 2025
% 0.13/0.34 % 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_4_type, type, (
% 0.13/0.38 tptp_fun_A_4: $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(singleton_type, type, (
% 0.13/0.38 singleton: $i > $i)).
% 0.13/0.38 tff(tptp_fun_X_3_type, type, (
% 0.13/0.38 tptp_fun_X_3: $i)).
% 0.13/0.38 tff(subset_type, type, (
% 0.13/0.38 subset: ( $i * $i ) > $o)).
% 0.13/0.38 tff(1,plain,
% 0.13/0.38 (^[X: $i, A: $i] : refl((member(X, singleton(A)) <=> (X = A)) <=> (member(X, singleton(A)) <=> (X = A)))),
% 0.13/0.38 inference(bind,[status(th)],[])).
% 0.13/0.38 tff(2,plain,
% 0.13/0.38 (![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A)) <=> ![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))),
% 0.13/0.38 inference(quant_intro,[status(thm)],[1])).
% 0.13/0.38 tff(3,plain,
% 0.13/0.38 (![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A)) <=> ![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))),
% 0.13/0.38 inference(rewrite,[status(thm)],[])).
% 0.13/0.38 tff(4,axiom,(![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+0.ax','singleton')).
% 0.13/0.38 tff(5,plain,
% 0.13/0.38 (![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))),
% 0.13/0.38 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.13/0.38 tff(6,plain,(
% 0.13/0.38 ![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))),
% 0.13/0.38 inference(skolemize,[status(sab)],[5])).
% 0.13/0.38 tff(7,plain,
% 0.13/0.38 (![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))),
% 0.13/0.38 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.13/0.38 tff(8,plain,
% 0.13/0.38 ((~![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))) | (member(tptp_fun_X_0(A!4, singleton(X!3)), singleton(X!3)) <=> (tptp_fun_X_0(A!4, singleton(X!3)) = X!3))),
% 0.13/0.38 inference(quant_inst,[status(thm)],[])).
% 0.13/0.38 tff(9,plain,
% 0.13/0.38 (member(tptp_fun_X_0(A!4, singleton(X!3)), singleton(X!3)) <=> (tptp_fun_X_0(A!4, singleton(X!3)) = X!3)),
% 0.13/0.38 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.13/0.38 tff(10,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(11,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)],[10])).
% 0.13/0.38 tff(12,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(13,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)],[12])).
% 0.13/0.38 tff(14,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)],[13, 11])).
% 0.13/0.38 tff(15,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(16,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)],[15])).
% 0.13/0.38 tff(17,plain,
% 0.13/0.38 (![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)))),
% 0.13/0.38 inference(rewrite,[status(thm)],[])).
% 0.13/0.38 tff(18,plain,
% 0.13/0.38 (^[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))))),
% 0.13/0.38 inference(bind,[status(th)],[])).
% 0.13/0.38 tff(19,plain,
% 0.13/0.38 (![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)))),
% 0.13/0.38 inference(quant_intro,[status(thm)],[18])).
% 0.13/0.38 tff(20,axiom,(![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : (member(X, A) => member(X, B)))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+0.ax','subset')).
% 0.13/0.38 tff(21,plain,
% 0.13/0.38 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.13/0.38 inference(modus_ponens,[status(thm)],[20, 19])).
% 0.13/0.38 tff(22,plain,
% 0.13/0.38 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.13/0.38 inference(modus_ponens,[status(thm)],[21, 17])).
% 0.13/0.38 tff(23,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)))))),
% 0.13/0.38 inference(skolemize,[status(sab)],[22])).
% 0.13/0.38 tff(24,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)))))))),
% 0.13/0.38 inference(modus_ponens,[status(thm)],[23, 16])).
% 0.13/0.38 tff(25,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)))))))),
% 0.13/0.38 inference(modus_ponens,[status(thm)],[24, 14])).
% 0.13/0.38 tff(26,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)))))))) | (~((~((~subset(singleton(X!3), A!4)) | ![X: $i] : ((~member(X, singleton(X!3))) | member(X, A!4)))) | (~(subset(singleton(X!3), A!4) | (~((~member(tptp_fun_X_0(A!4, singleton(X!3)), singleton(X!3))) | member(tptp_fun_X_0(A!4, singleton(X!3)), A!4)))))))),
% 0.13/0.38 inference(quant_inst,[status(thm)],[])).
% 0.13/0.38 tff(27,plain,
% 0.13/0.38 (~((~((~subset(singleton(X!3), A!4)) | ![X: $i] : ((~member(X, singleton(X!3))) | member(X, A!4)))) | (~(subset(singleton(X!3), A!4) | (~((~member(tptp_fun_X_0(A!4, singleton(X!3)), singleton(X!3))) | member(tptp_fun_X_0(A!4, singleton(X!3)), A!4))))))),
% 0.13/0.38 inference(unit_resolution,[status(thm)],[26, 25])).
% 0.13/0.38 tff(28,plain,
% 0.13/0.38 (((~((~subset(singleton(X!3), A!4)) | ![X: $i] : ((~member(X, singleton(X!3))) | member(X, A!4)))) | (~(subset(singleton(X!3), A!4) | (~((~member(tptp_fun_X_0(A!4, singleton(X!3)), singleton(X!3))) | member(tptp_fun_X_0(A!4, singleton(X!3)), A!4)))))) | (subset(singleton(X!3), A!4) | (~((~member(tptp_fun_X_0(A!4, singleton(X!3)), singleton(X!3))) | member(tptp_fun_X_0(A!4, singleton(X!3)), A!4))))),
% 0.13/0.38 inference(tautology,[status(thm)],[])).
% 0.13/0.38 tff(29,plain,
% 0.13/0.38 (subset(singleton(X!3), A!4) | (~((~member(tptp_fun_X_0(A!4, singleton(X!3)), singleton(X!3))) | member(tptp_fun_X_0(A!4, singleton(X!3)), A!4)))),
% 0.13/0.38 inference(unit_resolution,[status(thm)],[28, 27])).
% 0.13/0.38 tff(30,plain,
% 0.13/0.38 ((~![A: $i, X: $i] : ((~member(X, A)) | subset(singleton(X), A))) <=> (~![A: $i, X: $i] : ((~member(X, A)) | subset(singleton(X), A)))),
% 0.13/0.38 inference(rewrite,[status(thm)],[])).
% 0.13/0.38 tff(31,plain,
% 0.13/0.38 ((~![A: $i, X: $i] : (member(X, A) => subset(singleton(X), A))) <=> (~![A: $i, X: $i] : ((~member(X, A)) | subset(singleton(X), A)))),
% 0.13/0.38 inference(rewrite,[status(thm)],[])).
% 0.13/0.38 tff(32,axiom,(~![A: $i, X: $i] : (member(X, A) => subset(singleton(X), A))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','thI44')).
% 0.13/0.38 tff(33,plain,
% 0.13/0.38 (~![A: $i, X: $i] : ((~member(X, A)) | subset(singleton(X), A))),
% 0.13/0.38 inference(modus_ponens,[status(thm)],[32, 31])).
% 0.13/0.38 tff(34,plain,
% 0.13/0.38 (~![A: $i, X: $i] : ((~member(X, A)) | subset(singleton(X), A))),
% 0.13/0.38 inference(modus_ponens,[status(thm)],[33, 30])).
% 0.13/0.38 tff(35,plain,
% 0.13/0.38 (~![A: $i, X: $i] : ((~member(X, A)) | subset(singleton(X), A))),
% 0.13/0.38 inference(modus_ponens,[status(thm)],[34, 30])).
% 0.13/0.38 tff(36,plain,
% 0.13/0.38 (~![A: $i, X: $i] : ((~member(X, A)) | subset(singleton(X), A))),
% 0.13/0.38 inference(modus_ponens,[status(thm)],[35, 30])).
% 0.13/0.38 tff(37,plain,(
% 0.13/0.38 ~((~member(X!3, A!4)) | subset(singleton(X!3), A!4))),
% 0.13/0.38 inference(skolemize,[status(sab)],[36])).
% 0.13/0.38 tff(38,plain,
% 0.13/0.38 (~subset(singleton(X!3), A!4)),
% 0.13/0.38 inference(or_elim,[status(thm)],[37])).
% 0.13/0.38 tff(39,plain,
% 0.13/0.38 ((~(subset(singleton(X!3), A!4) | (~((~member(tptp_fun_X_0(A!4, singleton(X!3)), singleton(X!3))) | member(tptp_fun_X_0(A!4, singleton(X!3)), A!4))))) | subset(singleton(X!3), A!4) | (~((~member(tptp_fun_X_0(A!4, singleton(X!3)), singleton(X!3))) | member(tptp_fun_X_0(A!4, singleton(X!3)), A!4)))),
% 0.13/0.38 inference(tautology,[status(thm)],[])).
% 0.13/0.38 tff(40,plain,
% 0.13/0.38 ((~(subset(singleton(X!3), A!4) | (~((~member(tptp_fun_X_0(A!4, singleton(X!3)), singleton(X!3))) | member(tptp_fun_X_0(A!4, singleton(X!3)), A!4))))) | (~((~member(tptp_fun_X_0(A!4, singleton(X!3)), singleton(X!3))) | member(tptp_fun_X_0(A!4, singleton(X!3)), A!4)))),
% 0.13/0.38 inference(unit_resolution,[status(thm)],[39, 38])).
% 0.13/0.38 tff(41,plain,
% 0.13/0.38 (~((~member(tptp_fun_X_0(A!4, singleton(X!3)), singleton(X!3))) | member(tptp_fun_X_0(A!4, singleton(X!3)), A!4))),
% 0.13/0.38 inference(unit_resolution,[status(thm)],[40, 29])).
% 0.13/0.38 tff(42,plain,
% 0.13/0.38 (((~member(tptp_fun_X_0(A!4, singleton(X!3)), singleton(X!3))) | member(tptp_fun_X_0(A!4, singleton(X!3)), A!4)) | member(tptp_fun_X_0(A!4, singleton(X!3)), singleton(X!3))),
% 0.13/0.38 inference(tautology,[status(thm)],[])).
% 0.13/0.38 tff(43,plain,
% 0.13/0.38 (member(tptp_fun_X_0(A!4, singleton(X!3)), singleton(X!3))),
% 0.13/0.38 inference(unit_resolution,[status(thm)],[42, 41])).
% 0.13/0.38 tff(44,plain,
% 0.13/0.38 ((~(member(tptp_fun_X_0(A!4, singleton(X!3)), singleton(X!3)) <=> (tptp_fun_X_0(A!4, singleton(X!3)) = X!3))) | (~member(tptp_fun_X_0(A!4, singleton(X!3)), singleton(X!3))) | (tptp_fun_X_0(A!4, singleton(X!3)) = X!3)),
% 0.13/0.38 inference(tautology,[status(thm)],[])).
% 0.13/0.38 tff(45,plain,
% 0.13/0.38 ((~(member(tptp_fun_X_0(A!4, singleton(X!3)), singleton(X!3)) <=> (tptp_fun_X_0(A!4, singleton(X!3)) = X!3))) | (tptp_fun_X_0(A!4, singleton(X!3)) = X!3)),
% 0.13/0.38 inference(unit_resolution,[status(thm)],[44, 43])).
% 0.13/0.39 tff(46,plain,
% 0.13/0.39 (tptp_fun_X_0(A!4, singleton(X!3)) = X!3),
% 0.13/0.39 inference(unit_resolution,[status(thm)],[45, 9])).
% 0.13/0.39 tff(47,plain,
% 0.13/0.39 (member(tptp_fun_X_0(A!4, singleton(X!3)), A!4) <=> member(X!3, A!4)),
% 0.13/0.39 inference(monotonicity,[status(thm)],[46])).
% 0.13/0.39 tff(48,plain,
% 0.13/0.39 (member(X!3, A!4) <=> member(tptp_fun_X_0(A!4, singleton(X!3)), A!4)),
% 0.13/0.39 inference(symmetry,[status(thm)],[47])).
% 0.13/0.39 tff(49,plain,
% 0.13/0.39 (member(X!3, A!4)),
% 0.13/0.39 inference(or_elim,[status(thm)],[37])).
% 0.13/0.39 tff(50,plain,
% 0.13/0.39 (member(tptp_fun_X_0(A!4, singleton(X!3)), A!4)),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[49, 48])).
% 0.13/0.39 tff(51,plain,
% 0.13/0.39 (((~member(tptp_fun_X_0(A!4, singleton(X!3)), singleton(X!3))) | member(tptp_fun_X_0(A!4, singleton(X!3)), A!4)) | (~member(tptp_fun_X_0(A!4, singleton(X!3)), A!4))),
% 0.13/0.39 inference(tautology,[status(thm)],[])).
% 0.13/0.39 tff(52,plain,
% 0.13/0.39 (~member(tptp_fun_X_0(A!4, singleton(X!3)), A!4)),
% 0.13/0.39 inference(unit_resolution,[status(thm)],[51, 41])).
% 0.13/0.39 tff(53,plain,
% 0.13/0.39 ($false),
% 0.13/0.39 inference(unit_resolution,[status(thm)],[52, 50])).
% 0.13/0.39 % SZS output end Proof
% 0.13/0.39 % E exiting
%------------------------------------------------------------------------------