%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET879+1 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n005.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 05:26:35 AM UTC 2025
% Result : Theorem 0.12s 0.37s
% Output : Proof 0.12s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : SET879+1 : TPTP v9.0.0. Released v3.2.0.
% 0.07/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n005.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Jun 20 05:49:35 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.12/0.37 % SZS status Theorem
% 0.12/0.37 % SZS output start Proof
% 0.12/0.37 tff(tptp_fun_B_3_type, type, (
% 0.12/0.37 tptp_fun_B_3: $i)).
% 0.12/0.37 tff(tptp_fun_A_4_type, type, (
% 0.12/0.37 tptp_fun_A_4: $i)).
% 0.12/0.37 tff(singleton_type, type, (
% 0.12/0.37 singleton: $i > $i)).
% 0.12/0.37 tff(set_difference_type, type, (
% 0.12/0.37 set_difference: ( $i * $i ) > $i)).
% 0.12/0.37 tff(in_type, type, (
% 0.12/0.37 in: ( $i * $i ) > $o)).
% 0.12/0.37 tff(tptp_fun_C_0_type, type, (
% 0.12/0.37 tptp_fun_C_0: ( $i * $i ) > $i)).
% 0.12/0.37 tff(1,assumption,(~(set_difference(singleton(A!4), singleton(B!3)) = singleton(A!4))), introduced(assumption)).
% 0.12/0.37 tff(2,plain,
% 0.12/0.37 ((~![A: $i, B: $i] : (~((A = B) <=> (set_difference(singleton(A), singleton(B)) = singleton(A))))) <=> (~![A: $i, B: $i] : (~((A = B) <=> (set_difference(singleton(A), singleton(B)) = singleton(A)))))),
% 0.12/0.37 inference(rewrite,[status(thm)],[])).
% 0.12/0.37 tff(3,plain,
% 0.12/0.37 ((~![A: $i, B: $i] : ((set_difference(singleton(A), singleton(B)) = singleton(A)) <=> (~(A = B)))) <=> (~![A: $i, B: $i] : (~((A = B) <=> (set_difference(singleton(A), singleton(B)) = singleton(A)))))),
% 0.12/0.37 inference(rewrite,[status(thm)],[])).
% 0.12/0.37 tff(4,axiom,(~![A: $i, B: $i] : ((set_difference(singleton(A), singleton(B)) = singleton(A)) <=> (~(A = B)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t20_zfmisc_1')).
% 0.12/0.37 tff(5,plain,
% 0.12/0.37 (~![A: $i, B: $i] : (~((A = B) <=> (set_difference(singleton(A), singleton(B)) = singleton(A))))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.12/0.37 tff(6,plain,
% 0.12/0.37 (~![A: $i, B: $i] : (~((A = B) <=> (set_difference(singleton(A), singleton(B)) = singleton(A))))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[5, 2])).
% 0.12/0.37 tff(7,plain,
% 0.12/0.37 (~![A: $i, B: $i] : (~((A = B) <=> (set_difference(singleton(A), singleton(B)) = singleton(A))))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.12/0.37 tff(8,plain,
% 0.12/0.37 (~![A: $i, B: $i] : (~((A = B) <=> (set_difference(singleton(A), singleton(B)) = singleton(A))))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[7, 2])).
% 0.12/0.37 tff(9,plain,(
% 0.12/0.37 (A!4 = B!3) <=> (set_difference(singleton(A!4), singleton(B!3)) = singleton(A!4))),
% 0.12/0.37 inference(skolemize,[status(sab)],[8])).
% 0.12/0.37 tff(10,plain,
% 0.12/0.37 ((~(A!4 = B!3)) | (set_difference(singleton(A!4), singleton(B!3)) = singleton(A!4)) | (~((A!4 = B!3) <=> (set_difference(singleton(A!4), singleton(B!3)) = singleton(A!4))))),
% 0.12/0.37 inference(tautology,[status(thm)],[])).
% 0.12/0.37 tff(11,plain,
% 0.12/0.37 ((~(A!4 = B!3)) | (set_difference(singleton(A!4), singleton(B!3)) = singleton(A!4))),
% 0.12/0.37 inference(unit_resolution,[status(thm)],[10, 9])).
% 0.12/0.37 tff(12,plain,
% 0.12/0.37 (~(A!4 = B!3)),
% 0.12/0.37 inference(unit_resolution,[status(thm)],[11, 1])).
% 0.12/0.37 tff(13,plain,
% 0.12/0.37 (^[A: $i, B: $i] : refl((~(in(A, B) <=> (set_difference(singleton(A), B) = singleton(A)))) <=> (~(in(A, B) <=> (set_difference(singleton(A), B) = singleton(A)))))),
% 0.12/0.37 inference(bind,[status(th)],[])).
% 0.12/0.37 tff(14,plain,
% 0.12/0.37 (![A: $i, B: $i] : (~(in(A, B) <=> (set_difference(singleton(A), B) = singleton(A)))) <=> ![A: $i, B: $i] : (~(in(A, B) <=> (set_difference(singleton(A), B) = singleton(A))))),
% 0.12/0.37 inference(quant_intro,[status(thm)],[13])).
% 0.12/0.37 tff(15,plain,
% 0.12/0.37 (![A: $i, B: $i] : (~(in(A, B) <=> (set_difference(singleton(A), B) = singleton(A)))) <=> ![A: $i, B: $i] : (~(in(A, B) <=> (set_difference(singleton(A), B) = singleton(A))))),
% 0.12/0.37 inference(rewrite,[status(thm)],[])).
% 0.12/0.37 tff(16,plain,
% 0.12/0.37 (^[A: $i, B: $i] : trans(rewrite(((set_difference(singleton(A), B) = singleton(A)) <=> (~in(A, B))) <=> (~(in(A, B) <=> (set_difference(singleton(A), B) = singleton(A))))), rewrite((~(in(A, B) <=> (set_difference(singleton(A), B) = singleton(A)))) <=> (~(in(A, B) <=> (set_difference(singleton(A), B) = singleton(A))))), (((set_difference(singleton(A), B) = singleton(A)) <=> (~in(A, B))) <=> (~(in(A, B) <=> (set_difference(singleton(A), B) = singleton(A))))))),
% 0.12/0.37 inference(bind,[status(th)],[])).
% 0.12/0.37 tff(17,plain,
% 0.12/0.37 (![A: $i, B: $i] : ((set_difference(singleton(A), B) = singleton(A)) <=> (~in(A, B))) <=> ![A: $i, B: $i] : (~(in(A, B) <=> (set_difference(singleton(A), B) = singleton(A))))),
% 0.12/0.37 inference(quant_intro,[status(thm)],[16])).
% 0.12/0.37 tff(18,axiom,(![A: $i, B: $i] : ((set_difference(singleton(A), B) = singleton(A)) <=> (~in(A, B)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','l34_zfmisc_1')).
% 0.12/0.37 tff(19,plain,
% 0.12/0.37 (![A: $i, B: $i] : (~(in(A, B) <=> (set_difference(singleton(A), B) = singleton(A))))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[18, 17])).
% 0.12/0.37 tff(20,plain,
% 0.12/0.37 (![A: $i, B: $i] : (~(in(A, B) <=> (set_difference(singleton(A), B) = singleton(A))))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[19, 15])).
% 0.12/0.37 tff(21,plain,(
% 0.12/0.37 ![A: $i, B: $i] : (~(in(A, B) <=> (set_difference(singleton(A), B) = singleton(A))))),
% 0.12/0.37 inference(skolemize,[status(sab)],[20])).
% 0.12/0.37 tff(22,plain,
% 0.12/0.37 (![A: $i, B: $i] : (~(in(A, B) <=> (set_difference(singleton(A), B) = singleton(A))))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[21, 14])).
% 0.12/0.37 tff(23,plain,
% 0.12/0.37 ((~![A: $i, B: $i] : (~(in(A, B) <=> (set_difference(singleton(A), B) = singleton(A))))) | (~(in(A!4, singleton(B!3)) <=> (set_difference(singleton(A!4), singleton(B!3)) = singleton(A!4))))),
% 0.12/0.37 inference(quant_inst,[status(thm)],[])).
% 0.12/0.37 tff(24,plain,
% 0.12/0.37 (~(in(A!4, singleton(B!3)) <=> (set_difference(singleton(A!4), singleton(B!3)) = singleton(A!4)))),
% 0.12/0.37 inference(unit_resolution,[status(thm)],[23, 22])).
% 0.12/0.37 tff(25,plain,
% 0.12/0.37 ((in(A!4, singleton(B!3)) <=> (set_difference(singleton(A!4), singleton(B!3)) = singleton(A!4))) | in(A!4, singleton(B!3)) | (set_difference(singleton(A!4), singleton(B!3)) = singleton(A!4))),
% 0.12/0.37 inference(tautology,[status(thm)],[])).
% 0.12/0.37 tff(26,plain,
% 0.12/0.37 (in(A!4, singleton(B!3)) | (set_difference(singleton(A!4), singleton(B!3)) = singleton(A!4))),
% 0.12/0.37 inference(unit_resolution,[status(thm)],[25, 24])).
% 0.12/0.37 tff(27,plain,
% 0.12/0.37 (in(A!4, singleton(B!3))),
% 0.12/0.37 inference(unit_resolution,[status(thm)],[26, 1])).
% 0.12/0.37 tff(28,plain,
% 0.12/0.37 (^[A: $i, B: $i, C: $i] : refl((~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))))),
% 0.12/0.37 inference(bind,[status(th)],[])).
% 0.12/0.37 tff(29,plain,
% 0.12/0.37 (![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.37 inference(quant_intro,[status(thm)],[28])).
% 0.12/0.37 tff(30,plain,
% 0.12/0.37 (![A: $i, B: $i] : ![C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.37 inference(pull_quant,[status(thm)],[])).
% 0.12/0.37 tff(31,plain,
% 0.12/0.37 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(pull_quant(((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) <=> ![C: $i] : ((~(B = singleton(A))) | (in(C, B) <=> (C = A)))), ((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) <=> (~![C: $i] : ((~(B = singleton(A))) | (in(C, B) <=> (C = A)))))), pull_quant((~![C: $i] : ((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) <=> ?[C: $i] : (~((~(B = singleton(A))) | (in(C, B) <=> (C = A))))), ((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) <=> ?[C: $i] : (~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))))), (((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))) <=> (?[C: $i] : (~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))), pull_quant((?[C: $i] : (~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))) <=> ?[C: $i] : ((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))), (((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))) <=> ?[C: $i] : ((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))), ((~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> (~?[C: $i] : ((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))))), pull_quant((~?[C: $i] : ((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))), ((~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))))),
% 0.12/0.37 inference(bind,[status(th)],[])).
% 0.12/0.37 tff(32,plain,
% 0.12/0.37 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i] : ![C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.37 inference(quant_intro,[status(thm)],[31])).
% 0.12/0.37 tff(33,plain,
% 0.12/0.37 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.37 inference(transitivity,[status(thm)],[32, 30])).
% 0.12/0.37 tff(34,plain,
% 0.12/0.37 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.37 inference(transitivity,[status(thm)],[33, 29])).
% 0.12/0.37 tff(35,plain,
% 0.12/0.37 (^[A: $i, B: $i] : rewrite((~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))))),
% 0.12/0.37 inference(bind,[status(th)],[])).
% 0.12/0.37 tff(36,plain,
% 0.12/0.37 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.37 inference(quant_intro,[status(thm)],[35])).
% 0.12/0.37 tff(37,plain,
% 0.12/0.37 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.37 inference(transitivity,[status(thm)],[36, 34])).
% 0.12/0.37 tff(38,plain,
% 0.12/0.37 (^[A: $i, B: $i] : trans(monotonicity(rewrite(((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) <=> ((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))), trans(monotonicity(rewrite((~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))) <=> (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))), (((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))) <=> ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))), rewrite(((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))) <=> ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))), (((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))) <=> ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))), ((((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))) <=> (((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))), rewrite((((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))) <=> (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))), ((((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))) <=> (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))))),
% 0.12/0.37 inference(bind,[status(th)],[])).
% 0.12/0.37 tff(39,plain,
% 0.12/0.37 (![A: $i, B: $i] : (((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))) <=> ![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.37 inference(quant_intro,[status(thm)],[38])).
% 0.12/0.37 tff(40,plain,
% 0.12/0.37 (![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A))) <=> ![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))),
% 0.12/0.37 inference(rewrite,[status(thm)],[])).
% 0.12/0.37 tff(41,axiom,(![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d1_tarski')).
% 0.12/0.37 tff(42,plain,
% 0.12/0.37 (![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[41, 40])).
% 0.12/0.37 tff(43,plain,(
% 0.12/0.37 ![A: $i, B: $i] : (((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))),
% 0.12/0.37 inference(skolemize,[status(sab)],[42])).
% 0.12/0.37 tff(44,plain,
% 0.12/0.37 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[43, 39])).
% 0.12/0.37 tff(45,plain,
% 0.12/0.37 (![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[44, 37])).
% 0.12/0.37 tff(46,plain,
% 0.12/0.37 (((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(A!4, singleton(B!3)) <=> (A!4 = B!3))) <=> ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(A!4, singleton(B!3)) <=> (A!4 = B!3)))),
% 0.12/0.37 inference(rewrite,[status(thm)],[])).
% 0.12/0.37 tff(47,plain,
% 0.12/0.37 ((~(~(in(A!4, singleton(B!3)) <=> (A!4 = B!3)))) <=> (in(A!4, singleton(B!3)) <=> (A!4 = B!3))),
% 0.12/0.37 inference(rewrite,[status(thm)],[])).
% 0.12/0.37 tff(48,plain,
% 0.12/0.37 (((~(in(A!4, singleton(B!3)) <=> (A!4 = B!3))) | $false) <=> (~(in(A!4, singleton(B!3)) <=> (A!4 = B!3)))),
% 0.12/0.37 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(49,plain,
% 0.12/0.38 ((~$true) <=> $false),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(50,plain,
% 0.12/0.38 (($true | (~(in(tptp_fun_C_0(singleton(B!3), B!3), singleton(B!3)) <=> (tptp_fun_C_0(singleton(B!3), B!3) = B!3)))) <=> $true),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(51,plain,
% 0.12/0.38 ((~(in(tptp_fun_C_0(singleton(B!3), B!3), singleton(B!3)) <=> (tptp_fun_C_0(singleton(B!3), B!3) = B!3))) <=> (~(in(tptp_fun_C_0(singleton(B!3), B!3), singleton(B!3)) <=> (tptp_fun_C_0(singleton(B!3), B!3) = B!3)))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(52,plain,
% 0.12/0.38 ((singleton(B!3) = singleton(B!3)) <=> $true),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(53,plain,
% 0.12/0.38 (((singleton(B!3) = singleton(B!3)) | (~(in(tptp_fun_C_0(singleton(B!3), B!3), singleton(B!3)) <=> (tptp_fun_C_0(singleton(B!3), B!3) = B!3)))) <=> ($true | (~(in(tptp_fun_C_0(singleton(B!3), B!3), singleton(B!3)) <=> (tptp_fun_C_0(singleton(B!3), B!3) = B!3))))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[52, 51])).
% 0.12/0.38 tff(54,plain,
% 0.12/0.38 (((singleton(B!3) = singleton(B!3)) | (~(in(tptp_fun_C_0(singleton(B!3), B!3), singleton(B!3)) <=> (tptp_fun_C_0(singleton(B!3), B!3) = B!3)))) <=> $true),
% 0.12/0.38 inference(transitivity,[status(thm)],[53, 50])).
% 0.12/0.38 tff(55,plain,
% 0.12/0.38 ((~((singleton(B!3) = singleton(B!3)) | (~(in(tptp_fun_C_0(singleton(B!3), B!3), singleton(B!3)) <=> (tptp_fun_C_0(singleton(B!3), B!3) = B!3))))) <=> (~$true)),
% 0.12/0.38 inference(monotonicity,[status(thm)],[54])).
% 0.12/0.38 tff(56,plain,
% 0.12/0.38 ((~((singleton(B!3) = singleton(B!3)) | (~(in(tptp_fun_C_0(singleton(B!3), B!3), singleton(B!3)) <=> (tptp_fun_C_0(singleton(B!3), B!3) = B!3))))) <=> $false),
% 0.12/0.38 inference(transitivity,[status(thm)],[55, 49])).
% 0.12/0.38 tff(57,plain,
% 0.12/0.38 ((~(in(A!4, singleton(B!3)) <=> (A!4 = B!3))) <=> (~(in(A!4, singleton(B!3)) <=> (A!4 = B!3)))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(58,plain,
% 0.12/0.38 (($false | (in(A!4, singleton(B!3)) <=> (A!4 = B!3))) <=> (in(A!4, singleton(B!3)) <=> (A!4 = B!3))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(59,plain,
% 0.12/0.38 ((~(singleton(B!3) = singleton(B!3))) <=> (~$true)),
% 0.12/0.38 inference(monotonicity,[status(thm)],[52])).
% 0.12/0.38 tff(60,plain,
% 0.12/0.38 ((~(singleton(B!3) = singleton(B!3))) <=> $false),
% 0.12/0.38 inference(transitivity,[status(thm)],[59, 49])).
% 0.12/0.38 tff(61,plain,
% 0.12/0.38 (((~(singleton(B!3) = singleton(B!3))) | (in(A!4, singleton(B!3)) <=> (A!4 = B!3))) <=> ($false | (in(A!4, singleton(B!3)) <=> (A!4 = B!3)))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[60])).
% 0.12/0.38 tff(62,plain,
% 0.12/0.38 (((~(singleton(B!3) = singleton(B!3))) | (in(A!4, singleton(B!3)) <=> (A!4 = B!3))) <=> (in(A!4, singleton(B!3)) <=> (A!4 = B!3))),
% 0.12/0.38 inference(transitivity,[status(thm)],[61, 58])).
% 0.12/0.38 tff(63,plain,
% 0.12/0.38 ((~((~(singleton(B!3) = singleton(B!3))) | (in(A!4, singleton(B!3)) <=> (A!4 = B!3)))) <=> (~(in(A!4, singleton(B!3)) <=> (A!4 = B!3)))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[62])).
% 0.12/0.38 tff(64,plain,
% 0.12/0.38 ((~((~(singleton(B!3) = singleton(B!3))) | (in(A!4, singleton(B!3)) <=> (A!4 = B!3)))) <=> (~(in(A!4, singleton(B!3)) <=> (A!4 = B!3)))),
% 0.12/0.38 inference(transitivity,[status(thm)],[63, 57])).
% 0.12/0.38 tff(65,plain,
% 0.12/0.38 (((~((~(singleton(B!3) = singleton(B!3))) | (in(A!4, singleton(B!3)) <=> (A!4 = B!3)))) | (~((singleton(B!3) = singleton(B!3)) | (~(in(tptp_fun_C_0(singleton(B!3), B!3), singleton(B!3)) <=> (tptp_fun_C_0(singleton(B!3), B!3) = B!3)))))) <=> ((~(in(A!4, singleton(B!3)) <=> (A!4 = B!3))) | $false)),
% 0.12/0.38 inference(monotonicity,[status(thm)],[64, 56])).
% 0.12/0.38 tff(66,plain,
% 0.12/0.38 (((~((~(singleton(B!3) = singleton(B!3))) | (in(A!4, singleton(B!3)) <=> (A!4 = B!3)))) | (~((singleton(B!3) = singleton(B!3)) | (~(in(tptp_fun_C_0(singleton(B!3), B!3), singleton(B!3)) <=> (tptp_fun_C_0(singleton(B!3), B!3) = B!3)))))) <=> (~(in(A!4, singleton(B!3)) <=> (A!4 = B!3)))),
% 0.12/0.38 inference(transitivity,[status(thm)],[65, 48])).
% 0.12/0.38 tff(67,plain,
% 0.12/0.38 ((~((~((~(singleton(B!3) = singleton(B!3))) | (in(A!4, singleton(B!3)) <=> (A!4 = B!3)))) | (~((singleton(B!3) = singleton(B!3)) | (~(in(tptp_fun_C_0(singleton(B!3), B!3), singleton(B!3)) <=> (tptp_fun_C_0(singleton(B!3), B!3) = B!3))))))) <=> (~(~(in(A!4, singleton(B!3)) <=> (A!4 = B!3))))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[66])).
% 0.12/0.38 tff(68,plain,
% 0.12/0.38 ((~((~((~(singleton(B!3) = singleton(B!3))) | (in(A!4, singleton(B!3)) <=> (A!4 = B!3)))) | (~((singleton(B!3) = singleton(B!3)) | (~(in(tptp_fun_C_0(singleton(B!3), B!3), singleton(B!3)) <=> (tptp_fun_C_0(singleton(B!3), B!3) = B!3))))))) <=> (in(A!4, singleton(B!3)) <=> (A!4 = B!3))),
% 0.12/0.38 inference(transitivity,[status(thm)],[67, 47])).
% 0.12/0.38 tff(69,plain,
% 0.12/0.38 (((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (~((~((~(singleton(B!3) = singleton(B!3))) | (in(A!4, singleton(B!3)) <=> (A!4 = B!3)))) | (~((singleton(B!3) = singleton(B!3)) | (~(in(tptp_fun_C_0(singleton(B!3), B!3), singleton(B!3)) <=> (tptp_fun_C_0(singleton(B!3), B!3) = B!3)))))))) <=> ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(A!4, singleton(B!3)) <=> (A!4 = B!3)))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[68])).
% 0.12/0.38 tff(70,plain,
% 0.12/0.38 (((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (~((~((~(singleton(B!3) = singleton(B!3))) | (in(A!4, singleton(B!3)) <=> (A!4 = B!3)))) | (~((singleton(B!3) = singleton(B!3)) | (~(in(tptp_fun_C_0(singleton(B!3), B!3), singleton(B!3)) <=> (tptp_fun_C_0(singleton(B!3), B!3) = B!3)))))))) <=> ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(A!4, singleton(B!3)) <=> (A!4 = B!3)))),
% 0.12/0.38 inference(transitivity,[status(thm)],[69, 46])).
% 0.12/0.38 tff(71,plain,
% 0.12/0.38 ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (~((~((~(singleton(B!3) = singleton(B!3))) | (in(A!4, singleton(B!3)) <=> (A!4 = B!3)))) | (~((singleton(B!3) = singleton(B!3)) | (~(in(tptp_fun_C_0(singleton(B!3), B!3), singleton(B!3)) <=> (tptp_fun_C_0(singleton(B!3), B!3) = B!3)))))))),
% 0.12/0.38 inference(quant_inst,[status(thm)],[])).
% 0.12/0.38 tff(72,plain,
% 0.12/0.38 ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(A!4, singleton(B!3)) <=> (A!4 = B!3))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[71, 70])).
% 0.12/0.38 tff(73,plain,
% 0.12/0.38 (in(A!4, singleton(B!3)) <=> (A!4 = B!3)),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[72, 45])).
% 0.12/0.38 tff(74,plain,
% 0.12/0.38 ((~(in(A!4, singleton(B!3)) <=> (A!4 = B!3))) | (~in(A!4, singleton(B!3))) | (A!4 = B!3)),
% 0.12/0.38 inference(tautology,[status(thm)],[])).
% 0.12/0.38 tff(75,plain,
% 0.12/0.38 ((~in(A!4, singleton(B!3))) | (A!4 = B!3)),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[74, 73])).
% 0.12/0.38 tff(76,plain,
% 0.12/0.38 ($false),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[75, 27, 12])).
% 0.12/0.38 tff(77,plain,(set_difference(singleton(A!4), singleton(B!3)) = singleton(A!4)), inference(lemma,lemma(discharge,[]))).
% 0.12/0.38 tff(78,plain,
% 0.12/0.38 ((A!4 = B!3) | (~(set_difference(singleton(A!4), singleton(B!3)) = singleton(A!4))) | (~((A!4 = B!3) <=> (set_difference(singleton(A!4), singleton(B!3)) = singleton(A!4))))),
% 0.12/0.38 inference(tautology,[status(thm)],[])).
% 0.12/0.38 tff(79,plain,
% 0.12/0.38 ((A!4 = B!3) | (~(set_difference(singleton(A!4), singleton(B!3)) = singleton(A!4)))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[78, 9])).
% 0.12/0.38 tff(80,plain,
% 0.12/0.38 (A!4 = B!3),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[79, 77])).
% 0.12/0.38 tff(81,plain,
% 0.12/0.38 ((in(A!4, singleton(B!3)) <=> (set_difference(singleton(A!4), singleton(B!3)) = singleton(A!4))) | (~in(A!4, singleton(B!3))) | (~(set_difference(singleton(A!4), singleton(B!3)) = singleton(A!4)))),
% 0.12/0.38 inference(tautology,[status(thm)],[])).
% 0.12/0.38 tff(82,plain,
% 0.12/0.38 ((~in(A!4, singleton(B!3))) | (~(set_difference(singleton(A!4), singleton(B!3)) = singleton(A!4)))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[81, 24])).
% 0.12/0.38 tff(83,plain,
% 0.12/0.38 (~in(A!4, singleton(B!3))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[82, 77])).
% 0.12/0.38 tff(84,plain,
% 0.12/0.38 ((~(in(A!4, singleton(B!3)) <=> (A!4 = B!3))) | in(A!4, singleton(B!3)) | (~(A!4 = B!3))),
% 0.12/0.38 inference(tautology,[status(thm)],[])).
% 0.12/0.38 tff(85,plain,
% 0.12/0.38 (in(A!4, singleton(B!3)) | (~(A!4 = B!3))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[84, 73])).
% 0.12/0.38 tff(86,plain,
% 0.12/0.38 ($false),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[85, 83, 80])).
% 0.12/0.38 % SZS output end Proof
% 0.12/0.38 % E exiting
%------------------------------------------------------------------------------