%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET351+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 05:25:39 AM UTC 2025
% Result : Theorem 0.20s 0.37s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SET351+4 : TPTP v9.0.0. Released v2.2.0.
% 0.03/0.12 % 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 04:54:05 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.20/0.37 % SZS status Theorem
% 0.20/0.37 % SZS output start Proof
% 0.20/0.37 tff(member_type, type, (
% 0.20/0.37 member: ( $i * $i ) > $o)).
% 0.20/0.37 tff(singleton_type, type, (
% 0.20/0.37 singleton: $i > $i)).
% 0.20/0.37 tff(tptp_fun_A_3_type, type, (
% 0.20/0.37 tptp_fun_A_3: $i)).
% 0.20/0.37 tff(tptp_fun_X_0_type, type, (
% 0.20/0.37 tptp_fun_X_0: ( $i * $i ) > $i)).
% 0.20/0.37 tff(sum_type, type, (
% 0.20/0.37 sum: $i > $i)).
% 0.20/0.37 tff(tptp_fun_Y_1_type, type, (
% 0.20/0.37 tptp_fun_Y_1: ( $i * $i ) > $i)).
% 0.20/0.37 tff(subset_type, type, (
% 0.20/0.37 subset: ( $i * $i ) > $o)).
% 0.20/0.37 tff(equal_set_type, type, (
% 0.20/0.37 equal_set: ( $i * $i ) > $o)).
% 0.20/0.37 tff(1,plain,
% 0.20/0.37 (^[X: $i, A: $i] : refl((~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y))))))) <=> (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y))))))))),
% 0.20/0.37 inference(bind,[status(th)],[])).
% 0.20/0.37 tff(2,plain,
% 0.20/0.37 (![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y))))))) <=> ![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))),
% 0.20/0.37 inference(quant_intro,[status(thm)],[1])).
% 0.20/0.37 tff(3,plain,
% 0.20/0.37 (^[X: $i, A: $i] : rewrite((~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y))))))) <=> (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y))))))))),
% 0.20/0.37 inference(bind,[status(th)],[])).
% 0.20/0.37 tff(4,plain,
% 0.20/0.37 (![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y))))))) <=> ![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))),
% 0.20/0.37 inference(quant_intro,[status(thm)],[3])).
% 0.20/0.37 tff(5,plain,
% 0.20/0.37 (![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y))))))) <=> ![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))),
% 0.20/0.37 inference(transitivity,[status(thm)],[4, 2])).
% 0.20/0.37 tff(6,plain,
% 0.20/0.37 (^[X: $i, A: $i] : trans(monotonicity(rewrite(((~member(X, sum(A))) | (member(tptp_fun_Y_1(A, X), A) & member(X, tptp_fun_Y_1(A, X)))) <=> ((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))), rewrite((member(X, sum(A)) | ![Y: $i] : (~(member(Y, A) & member(X, Y)))) <=> (member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y))))), ((((~member(X, sum(A))) | (member(tptp_fun_Y_1(A, X), A) & member(X, tptp_fun_Y_1(A, X)))) & (member(X, sum(A)) | ![Y: $i] : (~(member(Y, A) & member(X, Y))))) <=> (((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X)))))) & (member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y))))))), rewrite((((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X)))))) & (member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y))))) <=> (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))), ((((~member(X, sum(A))) | (member(tptp_fun_Y_1(A, X), A) & member(X, tptp_fun_Y_1(A, X)))) & (member(X, sum(A)) | ![Y: $i] : (~(member(Y, A) & member(X, Y))))) <=> (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(7,plain,
% 0.20/0.38 (![X: $i, A: $i] : (((~member(X, sum(A))) | (member(tptp_fun_Y_1(A, X), A) & member(X, tptp_fun_Y_1(A, X)))) & (member(X, sum(A)) | ![Y: $i] : (~(member(Y, A) & member(X, Y))))) <=> ![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[6])).
% 0.20/0.38 tff(8,plain,
% 0.20/0.38 (![X: $i, A: $i] : (member(X, sum(A)) <=> ?[Y: $i] : (member(Y, A) & member(X, Y))) <=> ![X: $i, A: $i] : (member(X, sum(A)) <=> ?[Y: $i] : (member(Y, A) & member(X, Y)))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(9,axiom,(![X: $i, A: $i] : (member(X, sum(A)) <=> ?[Y: $i] : (member(Y, A) & member(X, Y)))), file('/export/starexec/sandbox/benchmark/Axioms/SET006+0.ax','sum')).
% 0.20/0.38 tff(10,plain,
% 0.20/0.38 (![X: $i, A: $i] : (member(X, sum(A)) <=> ?[Y: $i] : (member(Y, A) & member(X, Y)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[9, 8])).
% 0.20/0.38 tff(11,plain,(
% 0.20/0.38 ![X: $i, A: $i] : (((~member(X, sum(A))) | (member(tptp_fun_Y_1(A, X), A) & member(X, tptp_fun_Y_1(A, X)))) & (member(X, sum(A)) | ![Y: $i] : (~(member(Y, A) & member(X, Y)))))),
% 0.20/0.38 inference(skolemize,[status(sab)],[10])).
% 0.20/0.38 tff(12,plain,
% 0.20/0.38 (![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[11, 7])).
% 0.20/0.38 tff(13,plain,
% 0.20/0.38 (![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[12, 5])).
% 0.20/0.38 tff(14,plain,
% 0.20/0.38 (((~![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))) | (~((~(member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), Y))))) | (~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3)))) | (~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(sum(singleton(A!3)), A!3)))) | (~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(sum(singleton(A!3)), A!3)), singleton(A!3)))))))))) <=> ((~![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))) | (~((~(member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), Y))))) | (~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3)))) | (~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(sum(singleton(A!3)), A!3)))) | (~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(sum(singleton(A!3)), A!3)), singleton(A!3))))))))))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(15,plain,
% 0.20/0.38 ((~((~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3)))) | (~((~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(sum(singleton(A!3)), A!3)), singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(sum(singleton(A!3)), A!3)))))))) | (~(member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), Y))))))) <=> (~((~(member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), Y))))) | (~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3)))) | (~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(sum(singleton(A!3)), A!3)))) | (~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(sum(singleton(A!3)), A!3)), singleton(A!3)))))))))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(16,plain,
% 0.20/0.38 (((~![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))) | (~((~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3)))) | (~((~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(sum(singleton(A!3)), A!3)), singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(sum(singleton(A!3)), A!3)))))))) | (~(member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), Y)))))))) <=> ((~![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))) | (~((~(member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), Y))))) | (~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3)))) | (~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(sum(singleton(A!3)), A!3)))) | (~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(sum(singleton(A!3)), A!3)), singleton(A!3))))))))))),
% 0.20/0.38 inference(monotonicity,[status(thm)],[15])).
% 0.20/0.38 tff(17,plain,
% 0.20/0.38 (((~![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))) | (~((~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3)))) | (~((~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(sum(singleton(A!3)), A!3)), singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(sum(singleton(A!3)), A!3)))))))) | (~(member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), Y)))))))) <=> ((~![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))) | (~((~(member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), Y))))) | (~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3)))) | (~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(sum(singleton(A!3)), A!3)))) | (~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(sum(singleton(A!3)), A!3)), singleton(A!3))))))))))),
% 0.20/0.38 inference(transitivity,[status(thm)],[16, 14])).
% 0.20/0.38 tff(18,plain,
% 0.20/0.38 ((~![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))) | (~((~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3)))) | (~((~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(sum(singleton(A!3)), A!3)), singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(sum(singleton(A!3)), A!3)))))))) | (~(member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), Y)))))))),
% 0.20/0.38 inference(quant_inst,[status(thm)],[])).
% 0.20/0.38 tff(19,plain,
% 0.20/0.38 ((~![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))) | (~((~(member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), Y))))) | (~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3)))) | (~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(sum(singleton(A!3)), A!3)))) | (~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(sum(singleton(A!3)), A!3)), singleton(A!3)))))))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[18, 17])).
% 0.20/0.38 tff(20,plain,
% 0.20/0.38 (~((~(member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), Y))))) | (~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3)))) | (~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(sum(singleton(A!3)), A!3)))) | (~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(sum(singleton(A!3)), A!3)), singleton(A!3))))))))),
% 0.20/0.38 inference(unit_resolution,[status(thm)],[19, 13])).
% 0.20/0.38 tff(21,plain,
% 0.20/0.38 (((~(member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), Y))))) | (~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3)))) | (~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(sum(singleton(A!3)), A!3)))) | (~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(sum(singleton(A!3)), A!3)), singleton(A!3)))))))) | (member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), Y))))),
% 0.20/0.38 inference(tautology,[status(thm)],[])).
% 0.20/0.38 tff(22,plain,
% 0.20/0.38 (member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), Y)))),
% 0.20/0.38 inference(unit_resolution,[status(thm)],[21, 20])).
% 0.20/0.38 tff(23,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(24,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)],[23])).
% 0.20/0.38 tff(25,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(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))))))) <=> ![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)],[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))))))) <=> ![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)],[26, 24])).
% 0.20/0.38 tff(28,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(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))))) <=> ![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)],[28])).
% 0.20/0.38 tff(30,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(31,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(32,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)],[31])).
% 0.20/0.38 tff(33,axiom,(![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : (member(X, A) => member(X, B)))), file('/export/starexec/sandbox/benchmark/Axioms/SET006+0.ax','subset')).
% 0.20/0.38 tff(34,plain,
% 0.20/0.38 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[33, 32])).
% 0.20/0.38 tff(35,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)],[34, 30])).
% 0.20/0.38 tff(36,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)],[35])).
% 0.20/0.38 tff(37,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)],[36, 29])).
% 0.20/0.38 tff(38,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)],[37, 27])).
% 0.20/0.38 tff(39,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(A!3, sum(singleton(A!3)))) | ![X: $i] : ((~member(X, A!3)) | member(X, sum(singleton(A!3)))))) | (~(subset(A!3, sum(singleton(A!3))) | (~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), A!3)) | member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3)))))))))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(40,plain,
% 0.20/0.39 (~((~((~subset(A!3, sum(singleton(A!3)))) | ![X: $i] : ((~member(X, A!3)) | member(X, sum(singleton(A!3)))))) | (~(subset(A!3, sum(singleton(A!3))) | (~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), A!3)) | member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3))))))))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[39, 38])).
% 0.20/0.39 tff(41,plain,
% 0.20/0.39 (((~((~subset(A!3, sum(singleton(A!3)))) | ![X: $i] : ((~member(X, A!3)) | member(X, sum(singleton(A!3)))))) | (~(subset(A!3, sum(singleton(A!3))) | (~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), A!3)) | member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3)))))))) | (subset(A!3, sum(singleton(A!3))) | (~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), A!3)) | member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3))))))),
% 0.20/0.39 inference(tautology,[status(thm)],[])).
% 0.20/0.39 tff(42,plain,
% 0.20/0.39 (subset(A!3, sum(singleton(A!3))) | (~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), A!3)) | member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3)))))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[41, 40])).
% 0.20/0.39 tff(43,assumption,((~((~subset(sum(singleton(A!3)), A!3)) | ![X: $i] : ((~member(X, sum(singleton(A!3)))) | member(X, A!3)))) | (~(subset(sum(singleton(A!3)), A!3) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | member(tptp_fun_X_0(A!3, sum(singleton(A!3))), A!3)))))), introduced(assumption)).
% 0.20/0.39 tff(44,plain,
% 0.20/0.39 ((~![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(sum(singleton(A!3)), A!3)) | ![X: $i] : ((~member(X, sum(singleton(A!3)))) | member(X, A!3)))) | (~(subset(sum(singleton(A!3)), A!3) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | member(tptp_fun_X_0(A!3, sum(singleton(A!3))), A!3)))))))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(45,plain,
% 0.20/0.39 ($false),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[44, 38, 43])).
% 0.20/0.39 tff(46,plain,(~((~((~subset(sum(singleton(A!3)), A!3)) | ![X: $i] : ((~member(X, sum(singleton(A!3)))) | member(X, A!3)))) | (~(subset(sum(singleton(A!3)), A!3) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | member(tptp_fun_X_0(A!3, sum(singleton(A!3))), A!3))))))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.39 tff(47,plain,
% 0.20/0.39 (((~((~subset(sum(singleton(A!3)), A!3)) | ![X: $i] : ((~member(X, sum(singleton(A!3)))) | member(X, A!3)))) | (~(subset(sum(singleton(A!3)), A!3) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | member(tptp_fun_X_0(A!3, sum(singleton(A!3))), A!3)))))) | (subset(sum(singleton(A!3)), A!3) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | member(tptp_fun_X_0(A!3, sum(singleton(A!3))), A!3))))),
% 0.20/0.39 inference(tautology,[status(thm)],[])).
% 0.20/0.39 tff(48,plain,
% 0.20/0.39 (subset(sum(singleton(A!3)), A!3) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | member(tptp_fun_X_0(A!3, sum(singleton(A!3))), A!3)))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[47, 46])).
% 0.20/0.39 tff(49,plain,
% 0.20/0.39 (^[X: $i, A: $i] : refl((member(X, singleton(A)) <=> (X = A)) <=> (member(X, singleton(A)) <=> (X = A)))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(50,plain,
% 0.20/0.39 (![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A)) <=> ![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[49])).
% 0.20/0.39 tff(51,plain,
% 0.20/0.39 (![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A)) <=> ![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(52,axiom,(![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))), file('/export/starexec/sandbox/benchmark/Axioms/SET006+0.ax','singleton')).
% 0.20/0.39 tff(53,plain,
% 0.20/0.39 (![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[52, 51])).
% 0.20/0.39 tff(54,plain,(
% 0.20/0.39 ![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))),
% 0.20/0.39 inference(skolemize,[status(sab)],[53])).
% 0.20/0.39 tff(55,plain,
% 0.20/0.39 (![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[54, 50])).
% 0.20/0.39 tff(56,plain,
% 0.20/0.39 ((~![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))) | (member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3)) <=> (tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))) = A!3))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(57,plain,
% 0.20/0.39 (member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3)) <=> (tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))) = A!3)),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[56, 55])).
% 0.20/0.39 tff(58,plain,
% 0.20/0.39 (((~![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))) | (~((~(member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), Y))))) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))))) | (~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3)))))))))) <=> ((~![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))) | (~((~(member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), Y))))) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))))) | (~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3))))))))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(59,plain,
% 0.20/0.39 ((~((~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | (~((~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3))) | (~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))))))))) | (~(member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), Y))))))) <=> (~((~(member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), Y))))) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))))) | (~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3)))))))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(60,plain,
% 0.20/0.39 (((~![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))) | (~((~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | (~((~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3))) | (~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))))))))) | (~(member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), Y)))))))) <=> ((~![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))) | (~((~(member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), Y))))) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))))) | (~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3))))))))))),
% 0.20/0.39 inference(monotonicity,[status(thm)],[59])).
% 0.20/0.39 tff(61,plain,
% 0.20/0.39 (((~![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))) | (~((~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | (~((~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3))) | (~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))))))))) | (~(member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), Y)))))))) <=> ((~![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))) | (~((~(member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), Y))))) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))))) | (~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3))))))))))),
% 0.20/0.39 inference(transitivity,[status(thm)],[60, 58])).
% 0.20/0.39 tff(62,plain,
% 0.20/0.39 ((~![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))) | (~((~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | (~((~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3))) | (~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))))))))) | (~(member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), Y)))))))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(63,plain,
% 0.20/0.39 ((~![X: $i, A: $i] : (~((~((~member(X, sum(A))) | (~((~member(tptp_fun_Y_1(A, X), A)) | (~member(X, tptp_fun_Y_1(A, X))))))) | (~(member(X, sum(A)) | ![Y: $i] : ((~member(Y, A)) | (~member(X, Y)))))))) | (~((~(member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), Y))))) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))))) | (~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3)))))))))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[62, 61])).
% 0.20/0.39 tff(64,plain,
% 0.20/0.39 (~((~(member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), Y))))) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))))) | (~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3))))))))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[63, 13])).
% 0.20/0.40 tff(65,plain,
% 0.20/0.40 (((~(member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), Y))))) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))))) | (~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3)))))))) | ((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))))) | (~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3))))))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(66,plain,
% 0.20/0.40 ((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))))) | (~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3)))))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[65, 64])).
% 0.20/0.40 tff(67,assumption,(~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | member(tptp_fun_X_0(A!3, sum(singleton(A!3))), A!3))), introduced(assumption)).
% 0.20/0.40 tff(68,plain,
% 0.20/0.40 (((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | member(tptp_fun_X_0(A!3, sum(singleton(A!3))), A!3)) | member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(69,plain,
% 0.20/0.40 (member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[68, 67])).
% 0.20/0.40 tff(70,plain,
% 0.20/0.40 ((~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))))) | (~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3))))))) | (~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))))) | (~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3)))))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(71,plain,
% 0.20/0.40 ((~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))))) | (~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3))))))) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))))) | (~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3)))))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[70, 69])).
% 0.20/0.40 tff(72,plain,
% 0.20/0.40 (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))))) | (~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3))))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[71, 66])).
% 0.20/0.40 tff(73,plain,
% 0.20/0.40 (((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))))) | (~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3)))) | member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(74,plain,
% 0.20/0.40 (member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[73, 72])).
% 0.20/0.40 tff(75,plain,
% 0.20/0.40 ((~(member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3)) <=> (tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))) = A!3))) | (~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3))) | (tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))) = A!3)),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(76,plain,
% 0.20/0.40 ((~(member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3)) <=> (tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))) = A!3))) | (tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))) = A!3)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[75, 74])).
% 0.20/0.40 tff(77,plain,
% 0.20/0.40 (tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))) = A!3),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[76, 57])).
% 0.20/0.40 tff(78,plain,
% 0.20/0.40 (member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3))))) <=> member(tptp_fun_X_0(A!3, sum(singleton(A!3))), A!3)),
% 0.20/0.40 inference(monotonicity,[status(thm)],[77])).
% 0.20/0.40 tff(79,plain,
% 0.20/0.40 (member(tptp_fun_X_0(A!3, sum(singleton(A!3))), A!3) <=> member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))))),
% 0.20/0.40 inference(symmetry,[status(thm)],[78])).
% 0.20/0.40 tff(80,plain,
% 0.20/0.40 ((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), A!3)) <=> (~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3))))))),
% 0.20/0.40 inference(monotonicity,[status(thm)],[79])).
% 0.20/0.40 tff(81,plain,
% 0.20/0.40 (((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | member(tptp_fun_X_0(A!3, sum(singleton(A!3))), A!3)) | (~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), A!3))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(82,plain,
% 0.20/0.40 (~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), A!3)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[81, 67])).
% 0.20/0.40 tff(83,plain,
% 0.20/0.40 (~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[82, 80])).
% 0.20/0.40 tff(84,plain,
% 0.20/0.40 (((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))))) | (~member(tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))), singleton(A!3)))) | member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(85,plain,
% 0.20/0.40 (member(tptp_fun_X_0(A!3, sum(singleton(A!3))), tptp_fun_Y_1(singleton(A!3), tptp_fun_X_0(A!3, sum(singleton(A!3)))))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[84, 72])).
% 0.20/0.40 tff(86,plain,
% 0.20/0.40 ($false),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[85, 83])).
% 0.20/0.40 tff(87,plain,((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | member(tptp_fun_X_0(A!3, sum(singleton(A!3))), A!3)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.40 tff(88,plain,
% 0.20/0.40 ((~(subset(sum(singleton(A!3)), A!3) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | member(tptp_fun_X_0(A!3, sum(singleton(A!3))), A!3))))) | subset(sum(singleton(A!3)), A!3) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | member(tptp_fun_X_0(A!3, sum(singleton(A!3))), A!3)))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(89,plain,
% 0.20/0.40 ((~(subset(sum(singleton(A!3)), A!3) | (~((~member(tptp_fun_X_0(A!3, sum(singleton(A!3))), sum(singleton(A!3)))) | member(tptp_fun_X_0(A!3, sum(singleton(A!3))), A!3))))) | subset(sum(singleton(A!3)), A!3)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[88, 87])).
% 0.20/0.40 tff(90,plain,
% 0.20/0.40 (subset(sum(singleton(A!3)), A!3)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[89, 48])).
% 0.20/0.40 tff(91,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(92,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)],[91])).
% 0.20/0.40 tff(93,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(94,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)],[93])).
% 0.20/0.40 tff(95,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(96,axiom,(![A: $i, B: $i] : (equal_set(A, B) <=> (subset(A, B) & subset(B, A)))), file('/export/starexec/sandbox/benchmark/Axioms/SET006+0.ax','equal_set')).
% 0.20/0.40 tff(97,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)],[96, 95])).
% 0.20/0.40 tff(98,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)],[97])).
% 0.20/0.40 tff(99,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)],[98, 94])).
% 0.20/0.40 tff(100,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)],[99, 92])).
% 0.20/0.40 tff(101,plain,
% 0.20/0.40 ((~![A: $i, B: $i] : (~(((~subset(A, B)) | (~subset(B, A))) <=> equal_set(A, B)))) | (~(((~subset(sum(singleton(A!3)), A!3)) | (~subset(A!3, sum(singleton(A!3))))) <=> equal_set(sum(singleton(A!3)), A!3)))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(102,plain,
% 0.20/0.40 (~(((~subset(sum(singleton(A!3)), A!3)) | (~subset(A!3, sum(singleton(A!3))))) <=> equal_set(sum(singleton(A!3)), A!3))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[101, 100])).
% 0.20/0.40 tff(103,plain,
% 0.20/0.40 ((~![A: $i] : equal_set(sum(singleton(A)), A)) <=> (~![A: $i] : equal_set(sum(singleton(A)), A))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(104,axiom,(~![A: $i] : equal_set(sum(singleton(A)), A)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','thI39')).
% 0.20/0.40 tff(105,plain,
% 0.20/0.40 (~![A: $i] : equal_set(sum(singleton(A)), A)),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[104, 103])).
% 0.20/0.40 tff(106,plain,
% 0.20/0.40 (~![A: $i] : equal_set(sum(singleton(A)), A)),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[105, 103])).
% 0.20/0.40 tff(107,plain,
% 0.20/0.40 (~![A: $i] : equal_set(sum(singleton(A)), A)),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[106, 103])).
% 0.20/0.40 tff(108,plain,
% 0.20/0.40 (~![A: $i] : equal_set(sum(singleton(A)), A)),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[107, 103])).
% 0.20/0.40 tff(109,plain,(
% 0.20/0.40 ~equal_set(sum(singleton(A!3)), A!3)),
% 0.20/0.40 inference(skolemize,[status(sab)],[108])).
% 0.20/0.40 tff(110,plain,
% 0.20/0.40 ((((~subset(sum(singleton(A!3)), A!3)) | (~subset(A!3, sum(singleton(A!3))))) <=> equal_set(sum(singleton(A!3)), A!3)) | ((~subset(sum(singleton(A!3)), A!3)) | (~subset(A!3, sum(singleton(A!3))))) | equal_set(sum(singleton(A!3)), A!3)),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(111,plain,
% 0.20/0.40 ((((~subset(sum(singleton(A!3)), A!3)) | (~subset(A!3, sum(singleton(A!3))))) <=> equal_set(sum(singleton(A!3)), A!3)) | ((~subset(sum(singleton(A!3)), A!3)) | (~subset(A!3, sum(singleton(A!3)))))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[110, 109])).
% 0.20/0.40 tff(112,plain,
% 0.20/0.40 ((~subset(sum(singleton(A!3)), A!3)) | (~subset(A!3, sum(singleton(A!3))))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[111, 102])).
% 0.20/0.40 tff(113,plain,
% 0.20/0.40 ((~((~subset(sum(singleton(A!3)), A!3)) | (~subset(A!3, sum(singleton(A!3)))))) | (~subset(sum(singleton(A!3)), A!3)) | (~subset(A!3, sum(singleton(A!3))))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(114,plain,
% 0.20/0.40 ((~subset(sum(singleton(A!3)), A!3)) | (~subset(A!3, sum(singleton(A!3))))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[113, 112])).
% 0.20/0.40 tff(115,plain,
% 0.20/0.40 (~subset(A!3, sum(singleton(A!3)))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[114, 90])).
% 0.20/0.40 tff(116,plain,
% 0.20/0.40 ((~(subset(A!3, sum(singleton(A!3))) | (~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), A!3)) | member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3))))))) | subset(A!3, sum(singleton(A!3))) | (~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), A!3)) | member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3)))))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(117,plain,
% 0.20/0.40 ((~(subset(A!3, sum(singleton(A!3))) | (~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), A!3)) | member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3))))))) | (~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), A!3)) | member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3)))))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[116, 115])).
% 0.20/0.40 tff(118,plain,
% 0.20/0.40 (~((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), A!3)) | member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3))))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[117, 42])).
% 0.20/0.40 tff(119,plain,
% 0.20/0.40 (((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), A!3)) | member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3)))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3))))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(120,plain,
% 0.20/0.40 (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3)))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[119, 118])).
% 0.20/0.40 tff(121,plain,
% 0.20/0.40 ((~(member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), Y))))) | member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), Y)))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(122,plain,
% 0.20/0.40 ((~(member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), Y))))) | ![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), Y)))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[121, 120])).
% 0.20/0.40 tff(123,plain,
% 0.20/0.40 (![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), Y)))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[122, 22])).
% 0.20/0.40 tff(124,plain,
% 0.20/0.40 (((~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), A!3)) | member(tptp_fun_X_0(sum(singleton(A!3)), A!3), sum(singleton(A!3)))) | member(tptp_fun_X_0(sum(singleton(A!3)), A!3), A!3)),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(125,plain,
% 0.20/0.40 (member(tptp_fun_X_0(sum(singleton(A!3)), A!3), A!3)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[124, 118])).
% 0.20/0.40 tff(126,plain,
% 0.20/0.40 (((~![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), Y)))) | ((~member(A!3, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), A!3)))) <=> ((~![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), Y)))) | (~member(A!3, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), A!3)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(127,plain,
% 0.20/0.40 ((~![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), Y)))) | ((~member(A!3, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), A!3)))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(128,plain,
% 0.20/0.41 ((~![Y: $i] : ((~member(Y, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), Y)))) | (~member(A!3, singleton(A!3))) | (~member(tptp_fun_X_0(sum(singleton(A!3)), A!3), A!3))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[127, 126])).
% 0.20/0.41 tff(129,plain,
% 0.20/0.41 (~member(A!3, singleton(A!3))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[128, 125, 123])).
% 0.20/0.41 tff(130,plain,
% 0.20/0.41 (((~![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))) | member(A!3, singleton(A!3))) <=> ((~![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))) | member(A!3, singleton(A!3)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(131,plain,
% 0.20/0.41 ((member(A!3, singleton(A!3)) <=> $true) <=> member(A!3, singleton(A!3))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(132,plain,
% 0.20/0.41 ((A!3 = A!3) <=> $true),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(133,plain,
% 0.20/0.41 ((member(A!3, singleton(A!3)) <=> (A!3 = A!3)) <=> (member(A!3, singleton(A!3)) <=> $true)),
% 0.20/0.41 inference(monotonicity,[status(thm)],[132])).
% 0.20/0.41 tff(134,plain,
% 0.20/0.41 ((member(A!3, singleton(A!3)) <=> (A!3 = A!3)) <=> member(A!3, singleton(A!3))),
% 0.20/0.41 inference(transitivity,[status(thm)],[133, 131])).
% 0.20/0.41 tff(135,plain,
% 0.20/0.41 (((~![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))) | (member(A!3, singleton(A!3)) <=> (A!3 = A!3))) <=> ((~![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))) | member(A!3, singleton(A!3)))),
% 0.20/0.41 inference(monotonicity,[status(thm)],[134])).
% 0.20/0.41 tff(136,plain,
% 0.20/0.41 (((~![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))) | (member(A!3, singleton(A!3)) <=> (A!3 = A!3))) <=> ((~![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))) | member(A!3, singleton(A!3)))),
% 0.20/0.41 inference(transitivity,[status(thm)],[135, 130])).
% 0.20/0.41 tff(137,plain,
% 0.20/0.41 ((~![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))) | (member(A!3, singleton(A!3)) <=> (A!3 = A!3))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(138,plain,
% 0.20/0.41 ((~![X: $i, A: $i] : (member(X, singleton(A)) <=> (X = A))) | member(A!3, singleton(A!3))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[137, 136])).
% 0.20/0.41 tff(139,plain,
% 0.20/0.41 ($false),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[138, 55, 129])).
% 0.20/0.41 % SZS output end Proof
% 0.20/0.41 % E exiting
%------------------------------------------------------------------------------