%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET882+1 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n009.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 05:26:36 AM UTC 2025
% Result : Theorem 0.20s 0.38s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : SET882+1 : TPTP v9.0.0. Released v3.2.0.
% 0.04/0.12 % Command : run_E %s %d THM
% 0.14/0.34 % Computer : n009.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Fri Jun 20 05:49:48 EDT 2025
% 0.14/0.34 % CPUTime :
% 0.20/0.38 % SZS status Theorem
% 0.20/0.38 % SZS output start Proof
% 0.20/0.38 tff(tptp_fun_B_3_type, type, (
% 0.20/0.38 tptp_fun_B_3: $i)).
% 0.20/0.38 tff(tptp_fun_A_4_type, type, (
% 0.20/0.38 tptp_fun_A_4: $i)).
% 0.20/0.38 tff(in_type, type, (
% 0.20/0.38 in: ( $i * $i ) > $o)).
% 0.20/0.38 tff(singleton_type, type, (
% 0.20/0.38 singleton: $i > $i)).
% 0.20/0.38 tff(tptp_fun_C_0_type, type, (
% 0.20/0.38 tptp_fun_C_0: ( $i * $i ) > $i)).
% 0.20/0.38 tff(set_difference_type, type, (
% 0.20/0.38 set_difference: ( $i * $i ) > $i)).
% 0.20/0.38 tff(unordered_pair_type, type, (
% 0.20/0.38 unordered_pair: ( $i * $i ) > $i)).
% 0.20/0.38 tff(1,assumption,(~in(B!3, singleton(B!3))), introduced(assumption)).
% 0.20/0.38 tff(2,plain,
% 0.20/0.38 (^[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))))))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(3,plain,
% 0.20/0.38 (![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)))))))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[2])).
% 0.20/0.38 tff(4,plain,
% 0.20/0.38 (![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)))))))),
% 0.20/0.38 inference(pull_quant,[status(thm)],[])).
% 0.20/0.38 tff(5,plain,
% 0.20/0.38 (^[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)))))))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(6,plain,
% 0.20/0.38 (![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)))))))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[5])).
% 0.20/0.38 tff(7,plain,
% 0.20/0.38 (![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)))))))),
% 0.20/0.38 inference(transitivity,[status(thm)],[6, 4])).
% 0.20/0.38 tff(8,plain,
% 0.20/0.38 (![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)))))))),
% 0.20/0.38 inference(transitivity,[status(thm)],[7, 3])).
% 0.20/0.38 tff(9,plain,
% 0.20/0.38 (^[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))))))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(10,plain,
% 0.20/0.38 (![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)))))))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[9])).
% 0.20/0.38 tff(11,plain,
% 0.20/0.38 (![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)))))))),
% 0.20/0.38 inference(transitivity,[status(thm)],[10, 8])).
% 0.20/0.38 tff(12,plain,
% 0.20/0.38 (^[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)))))))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(13,plain,
% 0.20/0.38 (![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)))))))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[12])).
% 0.20/0.38 tff(14,plain,
% 0.20/0.38 (![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)))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(15,axiom,(![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','d1_tarski')).
% 0.20/0.38 tff(16,plain,
% 0.20/0.38 (![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[15, 14])).
% 0.20/0.38 tff(17,plain,(
% 0.20/0.38 ![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)))))),
% 0.20/0.38 inference(skolemize,[status(sab)],[16])).
% 0.20/0.38 tff(18,plain,
% 0.20/0.38 (![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)))))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[17, 13])).
% 0.20/0.38 tff(19,plain,
% 0.20/0.38 (![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)))))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[18, 11])).
% 0.20/0.38 tff(20,plain,
% 0.20/0.38 (((~![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(B!3, singleton(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(B!3, singleton(B!3)))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(21,plain,
% 0.20/0.38 ((~(~in(B!3, singleton(B!3)))) <=> in(B!3, singleton(B!3))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(22,plain,
% 0.20/0.38 (((~in(B!3, singleton(B!3))) | $false) <=> (~in(B!3, singleton(B!3)))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(23,plain,
% 0.20/0.38 ((~$true) <=> $false),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(24,plain,
% 0.20/0.38 (($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),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(25,plain,
% 0.20/0.38 ((~(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)))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(26,plain,
% 0.20/0.38 ((singleton(B!3) = singleton(B!3)) <=> $true),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(27,plain,
% 0.20/0.38 (((singleton(B!3) = singleton(B!3)) | (~(in(tptp_fun_C_0(singleton(B!3), B!3), singleton(B!3)) <=> (tptp_fun_C_0(singleton(B!3), B!3) = B!3)))) <=> ($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))))),
% 0.20/0.38 inference(monotonicity,[status(thm)],[26, 25])).
% 0.20/0.38 tff(28,plain,
% 0.20/0.38 (((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)))) <=> $true),
% 0.20/0.38 inference(transitivity,[status(thm)],[27, 24])).
% 0.20/0.38 tff(29,plain,
% 0.20/0.38 ((~((singleton(B!3) = singleton(B!3)) | (~(in(tptp_fun_C_0(singleton(B!3), B!3), singleton(B!3)) <=> (tptp_fun_C_0(singleton(B!3), B!3) = B!3))))) <=> (~$true)),
% 0.20/0.38 inference(monotonicity,[status(thm)],[28])).
% 0.20/0.38 tff(30,plain,
% 0.20/0.38 ((~((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),
% 0.20/0.38 inference(transitivity,[status(thm)],[29, 23])).
% 0.20/0.38 tff(31,plain,
% 0.20/0.38 ((~((~(singleton(B!3) = singleton(B!3))) | (in(B!3, singleton(B!3)) <=> (B!3 = B!3)))) <=> (~in(B!3, singleton(B!3)))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(32,plain,
% 0.20/0.38 (((~((~(singleton(B!3) = singleton(B!3))) | (in(B!3, singleton(B!3)) <=> (B!3 = 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(B!3, singleton(B!3))) | $false)),
% 0.20/0.38 inference(monotonicity,[status(thm)],[31, 30])).
% 0.20/0.38 tff(33,plain,
% 0.20/0.38 (((~((~(singleton(B!3) = singleton(B!3))) | (in(B!3, singleton(B!3)) <=> (B!3 = 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(B!3, singleton(B!3)))),
% 0.20/0.38 inference(transitivity,[status(thm)],[32, 22])).
% 0.20/0.38 tff(34,plain,
% 0.20/0.38 ((~((~((~(singleton(B!3) = singleton(B!3))) | (in(B!3, singleton(B!3)) <=> (B!3 = 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(B!3, singleton(B!3))))),
% 0.20/0.38 inference(monotonicity,[status(thm)],[33])).
% 0.20/0.38 tff(35,plain,
% 0.20/0.38 ((~((~((~(singleton(B!3) = singleton(B!3))) | (in(B!3, singleton(B!3)) <=> (B!3 = 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(B!3, singleton(B!3))),
% 0.20/0.38 inference(transitivity,[status(thm)],[34, 21])).
% 0.20/0.38 tff(36,plain,
% 0.20/0.38 (((~![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(B!3, singleton(B!3)) <=> (B!3 = 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(B!3, singleton(B!3)))),
% 0.20/0.38 inference(monotonicity,[status(thm)],[35])).
% 0.20/0.38 tff(37,plain,
% 0.20/0.38 (((~![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(B!3, singleton(B!3)) <=> (B!3 = 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(B!3, singleton(B!3)))),
% 0.20/0.38 inference(transitivity,[status(thm)],[36, 20])).
% 0.20/0.38 tff(38,plain,
% 0.20/0.38 ((~![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(B!3, singleton(B!3)) <=> (B!3 = 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)))))))),
% 0.20/0.38 inference(quant_inst,[status(thm)],[])).
% 0.20/0.38 tff(39,plain,
% 0.20/0.38 ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | in(B!3, singleton(B!3))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[38, 37])).
% 0.20/0.39 tff(40,plain,
% 0.20/0.39 ($false),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[39, 19, 1])).
% 0.20/0.39 tff(41,plain,(in(B!3, singleton(B!3))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.39 tff(42,plain,
% 0.20/0.39 (((A!4 = B!3) | in(B!3, singleton(B!3))) | (~in(B!3, singleton(B!3)))),
% 0.20/0.39 inference(tautology,[status(thm)],[])).
% 0.20/0.39 tff(43,plain,
% 0.20/0.39 ((A!4 = B!3) | in(B!3, singleton(B!3))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[42, 41])).
% 0.20/0.39 tff(44,plain,
% 0.20/0.39 (^[A: $i, B: $i, C: $i] : refl((~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A)))) <=> (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A)))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(45,plain,
% 0.20/0.39 (![A: $i, B: $i, C: $i] : (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A)))) <=> ![A: $i, B: $i, C: $i] : (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A))))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[44])).
% 0.20/0.39 tff(46,plain,
% 0.20/0.39 (^[A: $i, B: $i, C: $i] : trans(trans(monotonicity(trans(monotonicity(rewrite((in(B, C) | (A = B)) <=> (in(B, C) | (A = B))), (((~in(A, C)) & (in(B, C) | (A = B))) <=> ((~in(A, C)) & (in(B, C) | (A = B))))), rewrite(((~in(A, C)) & (in(B, C) | (A = B))) <=> (~(in(A, C) | (~(in(B, C) | (A = B)))))), (((~in(A, C)) & (in(B, C) | (A = B))) <=> (~(in(A, C) | (~(in(B, C) | (A = B))))))), (((set_difference(unordered_pair(A, B), C) = singleton(A)) <=> ((~in(A, C)) & (in(B, C) | (A = B)))) <=> ((set_difference(unordered_pair(A, B), C) = singleton(A)) <=> (~(in(A, C) | (~(in(B, C) | (A = B)))))))), rewrite(((set_difference(unordered_pair(A, B), C) = singleton(A)) <=> (~(in(A, C) | (~(in(B, C) | (A = B)))))) <=> (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A))))), (((set_difference(unordered_pair(A, B), C) = singleton(A)) <=> ((~in(A, C)) & (in(B, C) | (A = B)))) <=> (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A)))))), rewrite((~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A)))) <=> (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A))))), (((set_difference(unordered_pair(A, B), C) = singleton(A)) <=> ((~in(A, C)) & (in(B, C) | (A = B)))) <=> (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A))))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(47,plain,
% 0.20/0.39 (![A: $i, B: $i, C: $i] : ((set_difference(unordered_pair(A, B), C) = singleton(A)) <=> ((~in(A, C)) & (in(B, C) | (A = B)))) <=> ![A: $i, B: $i, C: $i] : (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A))))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[46])).
% 0.20/0.39 tff(48,plain,
% 0.20/0.39 (![A: $i, B: $i, C: $i] : ((set_difference(unordered_pair(A, B), C) = singleton(A)) <=> ((~in(A, C)) & (in(B, C) | (A = B)))) <=> ![A: $i, B: $i, C: $i] : ((set_difference(unordered_pair(A, B), C) = singleton(A)) <=> ((~in(A, C)) & (in(B, C) | (A = B))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(49,axiom,(![A: $i, B: $i, C: $i] : ((set_difference(unordered_pair(A, B), C) = singleton(A)) <=> ((~in(A, C)) & (in(B, C) | (A = B))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','l39_zfmisc_1')).
% 0.20/0.39 tff(50,plain,
% 0.20/0.39 (![A: $i, B: $i, C: $i] : ((set_difference(unordered_pair(A, B), C) = singleton(A)) <=> ((~in(A, C)) & (in(B, C) | (A = B))))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[49, 48])).
% 0.20/0.39 tff(51,plain,(
% 0.20/0.39 ![A: $i, B: $i, C: $i] : ((set_difference(unordered_pair(A, B), C) = singleton(A)) <=> ((~in(A, C)) & (in(B, C) | (A = B))))),
% 0.20/0.39 inference(skolemize,[status(sab)],[50])).
% 0.20/0.39 tff(52,plain,
% 0.20/0.39 (![A: $i, B: $i, C: $i] : (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A))))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[51, 47])).
% 0.20/0.39 tff(53,plain,
% 0.20/0.39 (![A: $i, B: $i, C: $i] : (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A))))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[52, 45])).
% 0.20/0.39 tff(54,plain,
% 0.20/0.39 (((~![A: $i, B: $i, C: $i] : (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A))))) | (~((in(A!4, singleton(B!3)) | (~((A!4 = B!3) | in(B!3, singleton(B!3))))) <=> (set_difference(unordered_pair(A!4, B!3), singleton(B!3)) = singleton(A!4))))) <=> ((~![A: $i, B: $i, C: $i] : (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A))))) | (~((in(A!4, singleton(B!3)) | (~((A!4 = B!3) | in(B!3, singleton(B!3))))) <=> (set_difference(unordered_pair(A!4, B!3), singleton(B!3)) = singleton(A!4)))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(55,plain,
% 0.20/0.39 ((~((in(A!4, singleton(B!3)) | (~((A!4 = B!3) | in(B!3, singleton(B!3))))) <=> (set_difference(unordered_pair(A!4, B!3), singleton(B!3)) = singleton(A!4)))) <=> (~((in(A!4, singleton(B!3)) | (~((A!4 = B!3) | in(B!3, singleton(B!3))))) <=> (set_difference(unordered_pair(A!4, B!3), singleton(B!3)) = singleton(A!4))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(56,plain,
% 0.20/0.39 (((in(A!4, singleton(B!3)) | (~(in(B!3, singleton(B!3)) | (A!4 = B!3)))) <=> (set_difference(unordered_pair(A!4, B!3), singleton(B!3)) = singleton(A!4))) <=> ((in(A!4, singleton(B!3)) | (~((A!4 = B!3) | in(B!3, singleton(B!3))))) <=> (set_difference(unordered_pair(A!4, B!3), singleton(B!3)) = singleton(A!4)))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(57,plain,
% 0.20/0.39 ((~((in(A!4, singleton(B!3)) | (~(in(B!3, singleton(B!3)) | (A!4 = B!3)))) <=> (set_difference(unordered_pair(A!4, B!3), singleton(B!3)) = singleton(A!4)))) <=> (~((in(A!4, singleton(B!3)) | (~((A!4 = B!3) | in(B!3, singleton(B!3))))) <=> (set_difference(unordered_pair(A!4, B!3), singleton(B!3)) = singleton(A!4))))),
% 0.20/0.39 inference(monotonicity,[status(thm)],[56])).
% 0.20/0.39 tff(58,plain,
% 0.20/0.39 ((~((in(A!4, singleton(B!3)) | (~(in(B!3, singleton(B!3)) | (A!4 = B!3)))) <=> (set_difference(unordered_pair(A!4, B!3), singleton(B!3)) = singleton(A!4)))) <=> (~((in(A!4, singleton(B!3)) | (~((A!4 = B!3) | in(B!3, singleton(B!3))))) <=> (set_difference(unordered_pair(A!4, B!3), singleton(B!3)) = singleton(A!4))))),
% 0.20/0.39 inference(transitivity,[status(thm)],[57, 55])).
% 0.20/0.39 tff(59,plain,
% 0.20/0.39 (((~![A: $i, B: $i, C: $i] : (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A))))) | (~((in(A!4, singleton(B!3)) | (~(in(B!3, singleton(B!3)) | (A!4 = B!3)))) <=> (set_difference(unordered_pair(A!4, B!3), singleton(B!3)) = singleton(A!4))))) <=> ((~![A: $i, B: $i, C: $i] : (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A))))) | (~((in(A!4, singleton(B!3)) | (~((A!4 = B!3) | in(B!3, singleton(B!3))))) <=> (set_difference(unordered_pair(A!4, B!3), singleton(B!3)) = singleton(A!4)))))),
% 0.20/0.39 inference(monotonicity,[status(thm)],[58])).
% 0.20/0.39 tff(60,plain,
% 0.20/0.39 (((~![A: $i, B: $i, C: $i] : (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A))))) | (~((in(A!4, singleton(B!3)) | (~(in(B!3, singleton(B!3)) | (A!4 = B!3)))) <=> (set_difference(unordered_pair(A!4, B!3), singleton(B!3)) = singleton(A!4))))) <=> ((~![A: $i, B: $i, C: $i] : (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A))))) | (~((in(A!4, singleton(B!3)) | (~((A!4 = B!3) | in(B!3, singleton(B!3))))) <=> (set_difference(unordered_pair(A!4, B!3), singleton(B!3)) = singleton(A!4)))))),
% 0.20/0.39 inference(transitivity,[status(thm)],[59, 54])).
% 0.20/0.39 tff(61,plain,
% 0.20/0.39 ((~![A: $i, B: $i, C: $i] : (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A))))) | (~((in(A!4, singleton(B!3)) | (~(in(B!3, singleton(B!3)) | (A!4 = B!3)))) <=> (set_difference(unordered_pair(A!4, B!3), singleton(B!3)) = singleton(A!4))))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(62,plain,
% 0.20/0.39 ((~![A: $i, B: $i, C: $i] : (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A))))) | (~((in(A!4, singleton(B!3)) | (~((A!4 = B!3) | in(B!3, singleton(B!3))))) <=> (set_difference(unordered_pair(A!4, B!3), singleton(B!3)) = singleton(A!4))))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[61, 60])).
% 0.20/0.39 tff(63,plain,
% 0.20/0.39 (~((in(A!4, singleton(B!3)) | (~((A!4 = B!3) | in(B!3, singleton(B!3))))) <=> (set_difference(unordered_pair(A!4, B!3), singleton(B!3)) = singleton(A!4)))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[62, 53])).
% 0.20/0.39 tff(64,plain,
% 0.20/0.39 ((~![A: $i, B: $i] : ((A = B) | (set_difference(unordered_pair(A, B), singleton(B)) = singleton(A)))) <=> (~![A: $i, B: $i] : ((A = B) | (set_difference(unordered_pair(A, B), singleton(B)) = singleton(A))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(65,plain,
% 0.20/0.39 ((~![A: $i, B: $i] : ((~(A = B)) => (set_difference(unordered_pair(A, B), singleton(B)) = singleton(A)))) <=> (~![A: $i, B: $i] : ((A = B) | (set_difference(unordered_pair(A, B), singleton(B)) = singleton(A))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(66,axiom,(~![A: $i, B: $i] : ((~(A = B)) => (set_difference(unordered_pair(A, B), singleton(B)) = singleton(A)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t23_zfmisc_1')).
% 0.20/0.39 tff(67,plain,
% 0.20/0.39 (~![A: $i, B: $i] : ((A = B) | (set_difference(unordered_pair(A, B), singleton(B)) = singleton(A)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[66, 65])).
% 0.20/0.39 tff(68,plain,
% 0.20/0.39 (~![A: $i, B: $i] : ((A = B) | (set_difference(unordered_pair(A, B), singleton(B)) = singleton(A)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[67, 64])).
% 0.20/0.39 tff(69,plain,
% 0.20/0.39 (~![A: $i, B: $i] : ((A = B) | (set_difference(unordered_pair(A, B), singleton(B)) = singleton(A)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[68, 64])).
% 0.20/0.39 tff(70,plain,
% 0.20/0.39 (~![A: $i, B: $i] : ((A = B) | (set_difference(unordered_pair(A, B), singleton(B)) = singleton(A)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[69, 64])).
% 0.20/0.39 tff(71,plain,(
% 0.20/0.39 ~((A!4 = B!3) | (set_difference(unordered_pair(A!4, B!3), singleton(B!3)) = singleton(A!4)))),
% 0.20/0.39 inference(skolemize,[status(sab)],[70])).
% 0.20/0.39 tff(72,plain,
% 0.20/0.39 (~(set_difference(unordered_pair(A!4, B!3), singleton(B!3)) = singleton(A!4))),
% 0.20/0.39 inference(or_elim,[status(thm)],[71])).
% 0.20/0.39 tff(73,plain,
% 0.20/0.39 (((in(A!4, singleton(B!3)) | (~((A!4 = B!3) | in(B!3, singleton(B!3))))) <=> (set_difference(unordered_pair(A!4, B!3), singleton(B!3)) = singleton(A!4))) | (in(A!4, singleton(B!3)) | (~((A!4 = B!3) | in(B!3, singleton(B!3))))) | (set_difference(unordered_pair(A!4, B!3), singleton(B!3)) = singleton(A!4))),
% 0.20/0.39 inference(tautology,[status(thm)],[])).
% 0.20/0.39 tff(74,plain,
% 0.20/0.39 (((in(A!4, singleton(B!3)) | (~((A!4 = B!3) | in(B!3, singleton(B!3))))) <=> (set_difference(unordered_pair(A!4, B!3), singleton(B!3)) = singleton(A!4))) | (in(A!4, singleton(B!3)) | (~((A!4 = B!3) | in(B!3, singleton(B!3)))))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[73, 72])).
% 0.20/0.39 tff(75,plain,
% 0.20/0.39 (in(A!4, singleton(B!3)) | (~((A!4 = B!3) | in(B!3, singleton(B!3))))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[74, 63])).
% 0.20/0.39 tff(76,plain,
% 0.20/0.39 ((~(in(A!4, singleton(B!3)) | (~((A!4 = B!3) | in(B!3, singleton(B!3)))))) | in(A!4, singleton(B!3)) | (~((A!4 = B!3) | in(B!3, singleton(B!3))))),
% 0.20/0.39 inference(tautology,[status(thm)],[])).
% 0.20/0.39 tff(77,plain,
% 0.20/0.39 (in(A!4, singleton(B!3)) | (~((A!4 = B!3) | in(B!3, singleton(B!3))))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[76, 75])).
% 0.20/0.39 tff(78,plain,
% 0.20/0.39 (in(A!4, singleton(B!3))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[77, 43])).
% 0.20/0.39 tff(79,plain,
% 0.20/0.39 (~(A!4 = B!3)),
% 0.20/0.39 inference(or_elim,[status(thm)],[71])).
% 0.20/0.39 tff(80,plain,
% 0.20/0.39 ((~(in(A!4, singleton(B!3)) <=> (A!4 = B!3))) | (~in(A!4, singleton(B!3))) | (A!4 = B!3)),
% 0.20/0.39 inference(tautology,[status(thm)],[])).
% 0.20/0.39 tff(81,plain,
% 0.20/0.39 (~(in(A!4, singleton(B!3)) <=> (A!4 = B!3))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[80, 79, 78])).
% 0.20/0.39 tff(82,plain,
% 0.20/0.39 (((~![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(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)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(83,plain,
% 0.20/0.40 ((~(~(in(A!4, singleton(B!3)) <=> (A!4 = B!3)))) <=> (in(A!4, singleton(B!3)) <=> (A!4 = B!3))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(84,plain,
% 0.20/0.40 (((~(in(A!4, singleton(B!3)) <=> (A!4 = B!3))) | $false) <=> (~(in(A!4, singleton(B!3)) <=> (A!4 = B!3)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(85,plain,
% 0.20/0.40 ((~(in(A!4, singleton(B!3)) <=> (A!4 = B!3))) <=> (~(in(A!4, singleton(B!3)) <=> (A!4 = B!3)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(86,plain,
% 0.20/0.40 (($false | (in(A!4, singleton(B!3)) <=> (A!4 = B!3))) <=> (in(A!4, singleton(B!3)) <=> (A!4 = B!3))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(87,plain,
% 0.20/0.40 ((~(singleton(B!3) = singleton(B!3))) <=> (~$true)),
% 0.20/0.40 inference(monotonicity,[status(thm)],[26])).
% 0.20/0.40 tff(88,plain,
% 0.20/0.40 ((~(singleton(B!3) = singleton(B!3))) <=> $false),
% 0.20/0.40 inference(transitivity,[status(thm)],[87, 23])).
% 0.20/0.40 tff(89,plain,
% 0.20/0.40 (((~(singleton(B!3) = singleton(B!3))) | (in(A!4, singleton(B!3)) <=> (A!4 = B!3))) <=> ($false | (in(A!4, singleton(B!3)) <=> (A!4 = B!3)))),
% 0.20/0.40 inference(monotonicity,[status(thm)],[88])).
% 0.20/0.40 tff(90,plain,
% 0.20/0.40 (((~(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))),
% 0.20/0.40 inference(transitivity,[status(thm)],[89, 86])).
% 0.20/0.40 tff(91,plain,
% 0.20/0.40 ((~((~(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)))),
% 0.20/0.40 inference(monotonicity,[status(thm)],[90])).
% 0.20/0.40 tff(92,plain,
% 0.20/0.40 ((~((~(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)))),
% 0.20/0.40 inference(transitivity,[status(thm)],[91, 85])).
% 0.20/0.40 tff(93,plain,
% 0.20/0.40 (((~((~(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)),
% 0.20/0.40 inference(monotonicity,[status(thm)],[92, 30])).
% 0.20/0.40 tff(94,plain,
% 0.20/0.40 (((~((~(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)))),
% 0.20/0.40 inference(transitivity,[status(thm)],[93, 84])).
% 0.20/0.40 tff(95,plain,
% 0.20/0.40 ((~((~((~(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))))),
% 0.20/0.40 inference(monotonicity,[status(thm)],[94])).
% 0.20/0.40 tff(96,plain,
% 0.20/0.40 ((~((~((~(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))),
% 0.20/0.40 inference(transitivity,[status(thm)],[95, 83])).
% 0.20/0.40 tff(97,plain,
% 0.20/0.40 (((~![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)))),
% 0.20/0.40 inference(monotonicity,[status(thm)],[96])).
% 0.20/0.40 tff(98,plain,
% 0.20/0.40 (((~![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)))),
% 0.20/0.40 inference(transitivity,[status(thm)],[97, 82])).
% 0.20/0.40 tff(99,plain,
% 0.20/0.40 ((~![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)))))))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(100,plain,
% 0.20/0.40 ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(A!4, singleton(B!3)) <=> (A!4 = B!3))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[99, 98])).
% 0.20/0.40 tff(101,plain,
% 0.20/0.40 ($false),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[100, 19, 81])).
% 0.20/0.40 % SZS output end Proof
% 0.20/0.40 % E exiting
%------------------------------------------------------------------------------