%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET873+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 0.38s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : SET873+1 : TPTP v9.0.0. Released v3.2.0.
% 0.07/0.13 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n005.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Fri Jun 20 05:49:20 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.12/0.38 % SZS status Theorem
% 0.12/0.38 % SZS output start Proof
% 0.12/0.38 tff(tptp_fun_B_3_type, type, (
% 0.12/0.38 tptp_fun_B_3: $i)).
% 0.12/0.38 tff(tptp_fun_A_4_type, type, (
% 0.12/0.38 tptp_fun_A_4: $i)).
% 0.12/0.38 tff(in_type, type, (
% 0.12/0.38 in: ( $i * $i ) > $o)).
% 0.12/0.38 tff(singleton_type, type, (
% 0.12/0.38 singleton: $i > $i)).
% 0.12/0.38 tff(tptp_fun_C_0_type, type, (
% 0.12/0.38 tptp_fun_C_0: ( $i * $i ) > $i)).
% 0.12/0.38 tff(subset_type, type, (
% 0.12/0.38 subset: ( $i * $i ) > $o)).
% 0.12/0.38 tff(set_union2_type, type, (
% 0.12/0.38 set_union2: ( $i * $i ) > $i)).
% 0.12/0.38 tff(1,plain,
% 0.12/0.38 (^[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))))))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(2,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))))))) <=> ![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)))))))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[1])).
% 0.12/0.38 tff(3,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))))))) <=> ![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)))))))),
% 0.12/0.38 inference(pull_quant,[status(thm)],[])).
% 0.12/0.38 tff(4,plain,
% 0.12/0.38 (^[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)))))))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(5,plain,
% 0.12/0.38 (![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)))))))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[4])).
% 0.12/0.38 tff(6,plain,
% 0.12/0.38 (![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)))))))),
% 0.12/0.38 inference(transitivity,[status(thm)],[5, 3])).
% 0.12/0.38 tff(7,plain,
% 0.12/0.38 (![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)))))))),
% 0.12/0.38 inference(transitivity,[status(thm)],[6, 2])).
% 0.12/0.38 tff(8,plain,
% 0.12/0.38 (^[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))))))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(9,plain,
% 0.12/0.38 (![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)))))))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[8])).
% 0.12/0.38 tff(10,plain,
% 0.12/0.38 (![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)))))))),
% 0.12/0.38 inference(transitivity,[status(thm)],[9, 7])).
% 0.12/0.38 tff(11,plain,
% 0.12/0.38 (^[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)))))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(12,plain,
% 0.12/0.39 (![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)))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[11])).
% 0.12/0.39 tff(13,plain,
% 0.12/0.39 (![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)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(14,axiom,(![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d1_tarski')).
% 0.12/0.39 tff(15,plain,
% 0.12/0.39 (![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[14, 13])).
% 0.12/0.39 tff(16,plain,(
% 0.12/0.39 ![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)))))),
% 0.12/0.39 inference(skolemize,[status(sab)],[15])).
% 0.12/0.39 tff(17,plain,
% 0.12/0.39 (![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)))))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[16, 12])).
% 0.12/0.39 tff(18,plain,
% 0.12/0.39 (![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)))))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[17, 10])).
% 0.12/0.39 tff(19,plain,
% 0.12/0.39 (((~![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(B!3, singleton(A!4)) <=> (B!3 = A!4))) <=> ((~![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(B!3, singleton(A!4)) <=> (B!3 = A!4)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(20,plain,
% 0.12/0.39 ((~(~(in(B!3, singleton(A!4)) <=> (B!3 = A!4)))) <=> (in(B!3, singleton(A!4)) <=> (B!3 = A!4))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(21,plain,
% 0.12/0.39 (((~(in(B!3, singleton(A!4)) <=> (B!3 = A!4))) | $false) <=> (~(in(B!3, singleton(A!4)) <=> (B!3 = A!4)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(22,plain,
% 0.12/0.39 ((~$true) <=> $false),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(23,plain,
% 0.12/0.39 (($true | (~(in(tptp_fun_C_0(singleton(A!4), A!4), singleton(A!4)) <=> (tptp_fun_C_0(singleton(A!4), A!4) = A!4)))) <=> $true),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(24,plain,
% 0.12/0.39 ((~(in(tptp_fun_C_0(singleton(A!4), A!4), singleton(A!4)) <=> (tptp_fun_C_0(singleton(A!4), A!4) = A!4))) <=> (~(in(tptp_fun_C_0(singleton(A!4), A!4), singleton(A!4)) <=> (tptp_fun_C_0(singleton(A!4), A!4) = A!4)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(25,plain,
% 0.12/0.39 ((singleton(A!4) = singleton(A!4)) <=> $true),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(26,plain,
% 0.12/0.39 (((singleton(A!4) = singleton(A!4)) | (~(in(tptp_fun_C_0(singleton(A!4), A!4), singleton(A!4)) <=> (tptp_fun_C_0(singleton(A!4), A!4) = A!4)))) <=> ($true | (~(in(tptp_fun_C_0(singleton(A!4), A!4), singleton(A!4)) <=> (tptp_fun_C_0(singleton(A!4), A!4) = A!4))))),
% 0.12/0.39 inference(monotonicity,[status(thm)],[25, 24])).
% 0.12/0.39 tff(27,plain,
% 0.12/0.39 (((singleton(A!4) = singleton(A!4)) | (~(in(tptp_fun_C_0(singleton(A!4), A!4), singleton(A!4)) <=> (tptp_fun_C_0(singleton(A!4), A!4) = A!4)))) <=> $true),
% 0.12/0.39 inference(transitivity,[status(thm)],[26, 23])).
% 0.21/0.39 tff(28,plain,
% 0.21/0.39 ((~((singleton(A!4) = singleton(A!4)) | (~(in(tptp_fun_C_0(singleton(A!4), A!4), singleton(A!4)) <=> (tptp_fun_C_0(singleton(A!4), A!4) = A!4))))) <=> (~$true)),
% 0.21/0.39 inference(monotonicity,[status(thm)],[27])).
% 0.21/0.39 tff(29,plain,
% 0.21/0.39 ((~((singleton(A!4) = singleton(A!4)) | (~(in(tptp_fun_C_0(singleton(A!4), A!4), singleton(A!4)) <=> (tptp_fun_C_0(singleton(A!4), A!4) = A!4))))) <=> $false),
% 0.21/0.39 inference(transitivity,[status(thm)],[28, 22])).
% 0.21/0.39 tff(30,plain,
% 0.21/0.39 ((~(in(B!3, singleton(A!4)) <=> (B!3 = A!4))) <=> (~(in(B!3, singleton(A!4)) <=> (B!3 = A!4)))),
% 0.21/0.39 inference(rewrite,[status(thm)],[])).
% 0.21/0.39 tff(31,plain,
% 0.21/0.39 (($false | (in(B!3, singleton(A!4)) <=> (B!3 = A!4))) <=> (in(B!3, singleton(A!4)) <=> (B!3 = A!4))),
% 0.21/0.39 inference(rewrite,[status(thm)],[])).
% 0.21/0.39 tff(32,plain,
% 0.21/0.39 ((~(singleton(A!4) = singleton(A!4))) <=> (~$true)),
% 0.21/0.39 inference(monotonicity,[status(thm)],[25])).
% 0.21/0.39 tff(33,plain,
% 0.21/0.39 ((~(singleton(A!4) = singleton(A!4))) <=> $false),
% 0.21/0.39 inference(transitivity,[status(thm)],[32, 22])).
% 0.21/0.39 tff(34,plain,
% 0.21/0.39 (((~(singleton(A!4) = singleton(A!4))) | (in(B!3, singleton(A!4)) <=> (B!3 = A!4))) <=> ($false | (in(B!3, singleton(A!4)) <=> (B!3 = A!4)))),
% 0.21/0.39 inference(monotonicity,[status(thm)],[33])).
% 0.21/0.39 tff(35,plain,
% 0.21/0.39 (((~(singleton(A!4) = singleton(A!4))) | (in(B!3, singleton(A!4)) <=> (B!3 = A!4))) <=> (in(B!3, singleton(A!4)) <=> (B!3 = A!4))),
% 0.21/0.39 inference(transitivity,[status(thm)],[34, 31])).
% 0.21/0.39 tff(36,plain,
% 0.21/0.39 ((~((~(singleton(A!4) = singleton(A!4))) | (in(B!3, singleton(A!4)) <=> (B!3 = A!4)))) <=> (~(in(B!3, singleton(A!4)) <=> (B!3 = A!4)))),
% 0.21/0.39 inference(monotonicity,[status(thm)],[35])).
% 0.21/0.39 tff(37,plain,
% 0.21/0.39 ((~((~(singleton(A!4) = singleton(A!4))) | (in(B!3, singleton(A!4)) <=> (B!3 = A!4)))) <=> (~(in(B!3, singleton(A!4)) <=> (B!3 = A!4)))),
% 0.21/0.39 inference(transitivity,[status(thm)],[36, 30])).
% 0.21/0.39 tff(38,plain,
% 0.21/0.39 (((~((~(singleton(A!4) = singleton(A!4))) | (in(B!3, singleton(A!4)) <=> (B!3 = A!4)))) | (~((singleton(A!4) = singleton(A!4)) | (~(in(tptp_fun_C_0(singleton(A!4), A!4), singleton(A!4)) <=> (tptp_fun_C_0(singleton(A!4), A!4) = A!4)))))) <=> ((~(in(B!3, singleton(A!4)) <=> (B!3 = A!4))) | $false)),
% 0.21/0.39 inference(monotonicity,[status(thm)],[37, 29])).
% 0.21/0.39 tff(39,plain,
% 0.21/0.39 (((~((~(singleton(A!4) = singleton(A!4))) | (in(B!3, singleton(A!4)) <=> (B!3 = A!4)))) | (~((singleton(A!4) = singleton(A!4)) | (~(in(tptp_fun_C_0(singleton(A!4), A!4), singleton(A!4)) <=> (tptp_fun_C_0(singleton(A!4), A!4) = A!4)))))) <=> (~(in(B!3, singleton(A!4)) <=> (B!3 = A!4)))),
% 0.21/0.39 inference(transitivity,[status(thm)],[38, 21])).
% 0.21/0.39 tff(40,plain,
% 0.21/0.39 ((~((~((~(singleton(A!4) = singleton(A!4))) | (in(B!3, singleton(A!4)) <=> (B!3 = A!4)))) | (~((singleton(A!4) = singleton(A!4)) | (~(in(tptp_fun_C_0(singleton(A!4), A!4), singleton(A!4)) <=> (tptp_fun_C_0(singleton(A!4), A!4) = A!4))))))) <=> (~(~(in(B!3, singleton(A!4)) <=> (B!3 = A!4))))),
% 0.21/0.39 inference(monotonicity,[status(thm)],[39])).
% 0.21/0.39 tff(41,plain,
% 0.21/0.39 ((~((~((~(singleton(A!4) = singleton(A!4))) | (in(B!3, singleton(A!4)) <=> (B!3 = A!4)))) | (~((singleton(A!4) = singleton(A!4)) | (~(in(tptp_fun_C_0(singleton(A!4), A!4), singleton(A!4)) <=> (tptp_fun_C_0(singleton(A!4), A!4) = A!4))))))) <=> (in(B!3, singleton(A!4)) <=> (B!3 = A!4))),
% 0.21/0.39 inference(transitivity,[status(thm)],[40, 20])).
% 0.21/0.39 tff(42,plain,
% 0.21/0.39 (((~![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(A!4) = singleton(A!4))) | (in(B!3, singleton(A!4)) <=> (B!3 = A!4)))) | (~((singleton(A!4) = singleton(A!4)) | (~(in(tptp_fun_C_0(singleton(A!4), A!4), singleton(A!4)) <=> (tptp_fun_C_0(singleton(A!4), A!4) = A!4)))))))) <=> ((~![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(B!3, singleton(A!4)) <=> (B!3 = A!4)))),
% 0.21/0.39 inference(monotonicity,[status(thm)],[41])).
% 0.21/0.39 tff(43,plain,
% 0.21/0.39 (((~![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(A!4) = singleton(A!4))) | (in(B!3, singleton(A!4)) <=> (B!3 = A!4)))) | (~((singleton(A!4) = singleton(A!4)) | (~(in(tptp_fun_C_0(singleton(A!4), A!4), singleton(A!4)) <=> (tptp_fun_C_0(singleton(A!4), A!4) = A!4)))))))) <=> ((~![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(B!3, singleton(A!4)) <=> (B!3 = A!4)))),
% 0.21/0.39 inference(transitivity,[status(thm)],[42, 19])).
% 0.21/0.39 tff(44,plain,
% 0.21/0.39 ((~![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(A!4) = singleton(A!4))) | (in(B!3, singleton(A!4)) <=> (B!3 = A!4)))) | (~((singleton(A!4) = singleton(A!4)) | (~(in(tptp_fun_C_0(singleton(A!4), A!4), singleton(A!4)) <=> (tptp_fun_C_0(singleton(A!4), A!4) = A!4)))))))),
% 0.21/0.39 inference(quant_inst,[status(thm)],[])).
% 0.21/0.39 tff(45,plain,
% 0.21/0.39 ((~![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(B!3, singleton(A!4)) <=> (B!3 = A!4))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[44, 43])).
% 0.21/0.39 tff(46,plain,
% 0.21/0.39 (in(B!3, singleton(A!4)) <=> (B!3 = A!4)),
% 0.21/0.39 inference(unit_resolution,[status(thm)],[45, 18])).
% 0.21/0.39 tff(47,plain,
% 0.21/0.39 (^[A: $i, B: $i] : refl((set_union2(A, B) = set_union2(B, A)) <=> (set_union2(A, B) = set_union2(B, A)))),
% 0.21/0.39 inference(bind,[status(th)],[])).
% 0.21/0.39 tff(48,plain,
% 0.21/0.39 (![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A)) <=> ![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))),
% 0.21/0.39 inference(quant_intro,[status(thm)],[47])).
% 0.21/0.39 tff(49,plain,
% 0.21/0.39 (![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A)) <=> ![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))),
% 0.21/0.39 inference(rewrite,[status(thm)],[])).
% 0.21/0.39 tff(50,axiom,(![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','commutativity_k2_xboole_0')).
% 0.21/0.39 tff(51,plain,
% 0.21/0.39 (![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[50, 49])).
% 0.21/0.39 tff(52,plain,(
% 0.21/0.39 ![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))),
% 0.21/0.39 inference(skolemize,[status(sab)],[51])).
% 0.21/0.39 tff(53,plain,
% 0.21/0.39 (![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[52, 48])).
% 0.21/0.39 tff(54,plain,
% 0.21/0.39 ((~![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))) | (set_union2(singleton(A!4), singleton(B!3)) = set_union2(singleton(B!3), singleton(A!4)))),
% 0.21/0.39 inference(quant_inst,[status(thm)],[])).
% 0.21/0.39 tff(55,plain,
% 0.21/0.39 (set_union2(singleton(A!4), singleton(B!3)) = set_union2(singleton(B!3), singleton(A!4))),
% 0.21/0.39 inference(unit_resolution,[status(thm)],[54, 53])).
% 0.21/0.39 tff(56,plain,
% 0.21/0.39 ((~![A: $i, B: $i] : ((~(set_union2(singleton(A), singleton(B)) = singleton(A))) | (A = B))) <=> (~![A: $i, B: $i] : ((~(set_union2(singleton(A), singleton(B)) = singleton(A))) | (A = B)))),
% 0.21/0.39 inference(rewrite,[status(thm)],[])).
% 0.21/0.39 tff(57,plain,
% 0.21/0.39 ((~![A: $i, B: $i] : ((set_union2(singleton(A), singleton(B)) = singleton(A)) => (A = B))) <=> (~![A: $i, B: $i] : ((~(set_union2(singleton(A), singleton(B)) = singleton(A))) | (A = B)))),
% 0.21/0.39 inference(rewrite,[status(thm)],[])).
% 0.21/0.39 tff(58,axiom,(~![A: $i, B: $i] : ((set_union2(singleton(A), singleton(B)) = singleton(A)) => (A = B))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t13_zfmisc_1')).
% 0.21/0.39 tff(59,plain,
% 0.21/0.39 (~![A: $i, B: $i] : ((~(set_union2(singleton(A), singleton(B)) = singleton(A))) | (A = B))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[58, 57])).
% 0.21/0.39 tff(60,plain,
% 0.21/0.39 (~![A: $i, B: $i] : ((~(set_union2(singleton(A), singleton(B)) = singleton(A))) | (A = B))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[59, 56])).
% 0.21/0.39 tff(61,plain,
% 0.21/0.39 (~![A: $i, B: $i] : ((~(set_union2(singleton(A), singleton(B)) = singleton(A))) | (A = B))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[60, 56])).
% 0.21/0.40 tff(62,plain,
% 0.21/0.40 (~![A: $i, B: $i] : ((~(set_union2(singleton(A), singleton(B)) = singleton(A))) | (A = B))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[61, 56])).
% 0.21/0.40 tff(63,plain,(
% 0.21/0.40 ~((~(set_union2(singleton(A!4), singleton(B!3)) = singleton(A!4))) | (A!4 = B!3))),
% 0.21/0.40 inference(skolemize,[status(sab)],[62])).
% 0.21/0.40 tff(64,plain,
% 0.21/0.40 (set_union2(singleton(A!4), singleton(B!3)) = singleton(A!4)),
% 0.21/0.40 inference(or_elim,[status(thm)],[63])).
% 0.21/0.40 tff(65,plain,
% 0.21/0.40 (singleton(A!4) = set_union2(singleton(A!4), singleton(B!3))),
% 0.21/0.40 inference(symmetry,[status(thm)],[64])).
% 0.21/0.40 tff(66,plain,
% 0.21/0.40 (singleton(A!4) = set_union2(singleton(B!3), singleton(A!4))),
% 0.21/0.40 inference(transitivity,[status(thm)],[65, 55])).
% 0.21/0.40 tff(67,plain,
% 0.21/0.40 (subset(singleton(A!4), singleton(A!4)) <=> subset(set_union2(singleton(B!3), singleton(A!4)), singleton(A!4))),
% 0.21/0.40 inference(monotonicity,[status(thm)],[66])).
% 0.21/0.40 tff(68,plain,
% 0.21/0.40 (subset(set_union2(singleton(B!3), singleton(A!4)), singleton(A!4)) <=> subset(singleton(A!4), singleton(A!4))),
% 0.21/0.40 inference(symmetry,[status(thm)],[67])).
% 0.21/0.40 tff(69,plain,
% 0.21/0.40 ((~subset(set_union2(singleton(B!3), singleton(A!4)), singleton(A!4))) <=> (~subset(singleton(A!4), singleton(A!4)))),
% 0.21/0.40 inference(monotonicity,[status(thm)],[68])).
% 0.21/0.40 tff(70,assumption,(~subset(set_union2(singleton(B!3), singleton(A!4)), singleton(A!4))), introduced(assumption)).
% 0.21/0.40 tff(71,plain,
% 0.21/0.40 (~subset(singleton(A!4), singleton(A!4))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[70, 69])).
% 0.21/0.40 tff(72,plain,
% 0.21/0.40 (^[A: $i] : refl(subset(A, A) <=> subset(A, A))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(73,plain,
% 0.21/0.40 (![A: $i] : subset(A, A) <=> ![A: $i] : subset(A, A)),
% 0.21/0.40 inference(quant_intro,[status(thm)],[72])).
% 0.21/0.40 tff(74,plain,
% 0.21/0.40 (![A: $i] : subset(A, A) <=> ![A: $i] : subset(A, A)),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(75,plain,
% 0.21/0.40 (![A: $i, B: $i] : subset(A, A) <=> ![A: $i] : subset(A, A)),
% 0.21/0.40 inference(elim_unused_vars,[status(thm)],[])).
% 0.21/0.40 tff(76,axiom,(![A: $i, B: $i] : subset(A, A)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','reflexivity_r1_tarski')).
% 0.21/0.40 tff(77,plain,
% 0.21/0.40 (![A: $i] : subset(A, A)),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[76, 75])).
% 0.21/0.40 tff(78,plain,
% 0.21/0.40 (![A: $i] : subset(A, A)),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[77, 74])).
% 0.21/0.40 tff(79,plain,(
% 0.21/0.40 ![A: $i] : subset(A, A)),
% 0.21/0.40 inference(skolemize,[status(sab)],[78])).
% 0.21/0.40 tff(80,plain,
% 0.21/0.40 (![A: $i] : subset(A, A)),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[79, 73])).
% 0.21/0.40 tff(81,plain,
% 0.21/0.40 ((~![A: $i] : subset(A, A)) | subset(singleton(A!4), singleton(A!4))),
% 0.21/0.40 inference(quant_inst,[status(thm)],[])).
% 0.21/0.40 tff(82,plain,
% 0.21/0.40 (subset(singleton(A!4), singleton(A!4))),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[81, 80])).
% 0.21/0.40 tff(83,plain,
% 0.21/0.40 ($false),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[82, 71])).
% 0.21/0.40 tff(84,plain,(subset(set_union2(singleton(B!3), singleton(A!4)), singleton(A!4))), inference(lemma,lemma(discharge,[]))).
% 0.21/0.40 tff(85,plain,
% 0.21/0.40 (^[A: $i, B: $i] : refl(((~subset(set_union2(singleton(A), B), B)) | in(A, B)) <=> ((~subset(set_union2(singleton(A), B), B)) | in(A, B)))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(86,plain,
% 0.21/0.40 (![A: $i, B: $i] : ((~subset(set_union2(singleton(A), B), B)) | in(A, B)) <=> ![A: $i, B: $i] : ((~subset(set_union2(singleton(A), B), B)) | in(A, B))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[85])).
% 0.21/0.40 tff(87,plain,
% 0.21/0.40 (![A: $i, B: $i] : ((~subset(set_union2(singleton(A), B), B)) | in(A, B)) <=> ![A: $i, B: $i] : ((~subset(set_union2(singleton(A), B), B)) | in(A, B))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(88,plain,
% 0.21/0.40 (^[A: $i, B: $i] : rewrite((subset(set_union2(singleton(A), B), B) => in(A, B)) <=> ((~subset(set_union2(singleton(A), B), B)) | in(A, B)))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(89,plain,
% 0.21/0.40 (![A: $i, B: $i] : (subset(set_union2(singleton(A), B), B) => in(A, B)) <=> ![A: $i, B: $i] : ((~subset(set_union2(singleton(A), B), B)) | in(A, B))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[88])).
% 0.21/0.40 tff(90,axiom,(![A: $i, B: $i] : (subset(set_union2(singleton(A), B), B) => in(A, B))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','l21_zfmisc_1')).
% 0.21/0.40 tff(91,plain,
% 0.21/0.40 (![A: $i, B: $i] : ((~subset(set_union2(singleton(A), B), B)) | in(A, B))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[90, 89])).
% 0.21/0.40 tff(92,plain,
% 0.21/0.40 (![A: $i, B: $i] : ((~subset(set_union2(singleton(A), B), B)) | in(A, B))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[91, 87])).
% 0.21/0.40 tff(93,plain,(
% 0.21/0.40 ![A: $i, B: $i] : ((~subset(set_union2(singleton(A), B), B)) | in(A, B))),
% 0.21/0.40 inference(skolemize,[status(sab)],[92])).
% 0.21/0.40 tff(94,plain,
% 0.21/0.40 (![A: $i, B: $i] : ((~subset(set_union2(singleton(A), B), B)) | in(A, B))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[93, 86])).
% 0.21/0.40 tff(95,plain,
% 0.21/0.40 (((~![A: $i, B: $i] : ((~subset(set_union2(singleton(A), B), B)) | in(A, B))) | ((~subset(set_union2(singleton(B!3), singleton(A!4)), singleton(A!4))) | in(B!3, singleton(A!4)))) <=> ((~![A: $i, B: $i] : ((~subset(set_union2(singleton(A), B), B)) | in(A, B))) | (~subset(set_union2(singleton(B!3), singleton(A!4)), singleton(A!4))) | in(B!3, singleton(A!4)))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(96,plain,
% 0.21/0.40 ((~![A: $i, B: $i] : ((~subset(set_union2(singleton(A), B), B)) | in(A, B))) | ((~subset(set_union2(singleton(B!3), singleton(A!4)), singleton(A!4))) | in(B!3, singleton(A!4)))),
% 0.21/0.40 inference(quant_inst,[status(thm)],[])).
% 0.21/0.40 tff(97,plain,
% 0.21/0.40 ((~![A: $i, B: $i] : ((~subset(set_union2(singleton(A), B), B)) | in(A, B))) | (~subset(set_union2(singleton(B!3), singleton(A!4)), singleton(A!4))) | in(B!3, singleton(A!4))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[96, 95])).
% 0.21/0.40 tff(98,plain,
% 0.21/0.40 ((~subset(set_union2(singleton(B!3), singleton(A!4)), singleton(A!4))) | in(B!3, singleton(A!4))),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[97, 94])).
% 0.21/0.40 tff(99,plain,
% 0.21/0.40 (in(B!3, singleton(A!4))),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[98, 84])).
% 0.21/0.40 tff(100,plain,
% 0.21/0.40 ((~(in(B!3, singleton(A!4)) <=> (B!3 = A!4))) | (~in(B!3, singleton(A!4))) | (B!3 = A!4)),
% 0.21/0.40 inference(tautology,[status(thm)],[])).
% 0.21/0.40 tff(101,plain,
% 0.21/0.40 ((~(in(B!3, singleton(A!4)) <=> (B!3 = A!4))) | (B!3 = A!4)),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[100, 99])).
% 0.21/0.40 tff(102,plain,
% 0.21/0.40 (B!3 = A!4),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[101, 46])).
% 0.21/0.40 tff(103,plain,
% 0.21/0.40 (A!4 = B!3),
% 0.21/0.40 inference(symmetry,[status(thm)],[102])).
% 0.21/0.40 tff(104,plain,
% 0.21/0.40 (~(A!4 = B!3)),
% 0.21/0.40 inference(or_elim,[status(thm)],[63])).
% 0.21/0.40 tff(105,plain,
% 0.21/0.40 ($false),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[104, 103])).
% 0.21/0.40 % SZS output end Proof
% 0.21/0.40 % E exiting
%------------------------------------------------------------------------------