%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET986+1 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n006.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:26:45 AM UTC 2025
% Result : Theorem 0.12s 0.37s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : SET986+1 : TPTP v9.0.0. Released v3.2.0.
% 0.07/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n006.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Jun 20 05:54:05 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_A_7_type, type, (
% 0.12/0.37 tptp_fun_A_7: $i)).
% 0.12/0.37 tff(tptp_fun_D_1_type, type, (
% 0.12/0.37 tptp_fun_D_1: ( $i * $i * $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(singleton_type, type, (
% 0.12/0.37 singleton: $i > $i)).
% 0.12/0.37 tff(tptp_fun_B_6_type, type, (
% 0.12/0.37 tptp_fun_B_6: $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(set_union2_type, type, (
% 0.12/0.37 set_union2: ( $i * $i ) > $i)).
% 0.12/0.37 tff(1,assumption,(~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7))), introduced(assumption)).
% 0.12/0.37 tff(2,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(3,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)],[2])).
% 0.12/0.37 tff(4,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(5,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(6,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)],[5])).
% 0.12/0.37 tff(7,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)],[6, 4])).
% 0.12/0.37 tff(8,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)],[7, 3])).
% 0.12/0.37 tff(9,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(10,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)],[9])).
% 0.12/0.37 tff(11,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)],[10, 8])).
% 0.12/0.37 tff(12,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)))))))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(13,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)],[12])).
% 0.12/0.38 tff(14,plain,
% 0.12/0.38 (![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)))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(15,axiom,(![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d1_tarski')).
% 0.12/0.38 tff(16,plain,
% 0.12/0.38 (![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[15, 14])).
% 0.12/0.38 tff(17,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)))))),
% 0.12/0.38 inference(skolemize,[status(sab)],[16])).
% 0.12/0.38 tff(18,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)))))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[17, 13])).
% 0.12/0.38 tff(19,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)))))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[18, 11])).
% 0.12/0.38 tff(20,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(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7))) <=> ((~![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(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(21,plain,
% 0.12/0.38 ((~(~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)))) <=> (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(22,plain,
% 0.12/0.38 (((~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7))) | $false) <=> (~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(23,plain,
% 0.12/0.38 ((~$true) <=> $false),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(24,plain,
% 0.12/0.38 (($true | (~(in(tptp_fun_C_0(singleton(A!7), A!7), singleton(A!7)) <=> (tptp_fun_C_0(singleton(A!7), A!7) = A!7)))) <=> $true),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(25,plain,
% 0.12/0.38 ((~(in(tptp_fun_C_0(singleton(A!7), A!7), singleton(A!7)) <=> (tptp_fun_C_0(singleton(A!7), A!7) = A!7))) <=> (~(in(tptp_fun_C_0(singleton(A!7), A!7), singleton(A!7)) <=> (tptp_fun_C_0(singleton(A!7), A!7) = A!7)))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(26,plain,
% 0.12/0.38 ((singleton(A!7) = singleton(A!7)) <=> $true),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(27,plain,
% 0.12/0.38 (((singleton(A!7) = singleton(A!7)) | (~(in(tptp_fun_C_0(singleton(A!7), A!7), singleton(A!7)) <=> (tptp_fun_C_0(singleton(A!7), A!7) = A!7)))) <=> ($true | (~(in(tptp_fun_C_0(singleton(A!7), A!7), singleton(A!7)) <=> (tptp_fun_C_0(singleton(A!7), A!7) = A!7))))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[26, 25])).
% 0.12/0.38 tff(28,plain,
% 0.12/0.38 (((singleton(A!7) = singleton(A!7)) | (~(in(tptp_fun_C_0(singleton(A!7), A!7), singleton(A!7)) <=> (tptp_fun_C_0(singleton(A!7), A!7) = A!7)))) <=> $true),
% 0.12/0.38 inference(transitivity,[status(thm)],[27, 24])).
% 0.12/0.38 tff(29,plain,
% 0.12/0.38 ((~((singleton(A!7) = singleton(A!7)) | (~(in(tptp_fun_C_0(singleton(A!7), A!7), singleton(A!7)) <=> (tptp_fun_C_0(singleton(A!7), A!7) = A!7))))) <=> (~$true)),
% 0.12/0.38 inference(monotonicity,[status(thm)],[28])).
% 0.12/0.38 tff(30,plain,
% 0.12/0.38 ((~((singleton(A!7) = singleton(A!7)) | (~(in(tptp_fun_C_0(singleton(A!7), A!7), singleton(A!7)) <=> (tptp_fun_C_0(singleton(A!7), A!7) = A!7))))) <=> $false),
% 0.12/0.38 inference(transitivity,[status(thm)],[29, 23])).
% 0.12/0.38 tff(31,plain,
% 0.12/0.38 ((~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7))) <=> (~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(32,plain,
% 0.12/0.38 (($false | (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7))) <=> (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(33,plain,
% 0.12/0.38 ((~(singleton(A!7) = singleton(A!7))) <=> (~$true)),
% 0.12/0.38 inference(monotonicity,[status(thm)],[26])).
% 0.12/0.38 tff(34,plain,
% 0.12/0.38 ((~(singleton(A!7) = singleton(A!7))) <=> $false),
% 0.12/0.38 inference(transitivity,[status(thm)],[33, 23])).
% 0.12/0.38 tff(35,plain,
% 0.12/0.38 (((~(singleton(A!7) = singleton(A!7))) | (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7))) <=> ($false | (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[34])).
% 0.12/0.38 tff(36,plain,
% 0.12/0.38 (((~(singleton(A!7) = singleton(A!7))) | (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7))) <=> (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7))),
% 0.12/0.38 inference(transitivity,[status(thm)],[35, 32])).
% 0.12/0.38 tff(37,plain,
% 0.12/0.38 ((~((~(singleton(A!7) = singleton(A!7))) | (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)))) <=> (~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[36])).
% 0.12/0.38 tff(38,plain,
% 0.12/0.38 ((~((~(singleton(A!7) = singleton(A!7))) | (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)))) <=> (~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)))),
% 0.12/0.38 inference(transitivity,[status(thm)],[37, 31])).
% 0.12/0.38 tff(39,plain,
% 0.12/0.38 (((~((~(singleton(A!7) = singleton(A!7))) | (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)))) | (~((singleton(A!7) = singleton(A!7)) | (~(in(tptp_fun_C_0(singleton(A!7), A!7), singleton(A!7)) <=> (tptp_fun_C_0(singleton(A!7), A!7) = A!7)))))) <=> ((~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7))) | $false)),
% 0.12/0.38 inference(monotonicity,[status(thm)],[38, 30])).
% 0.12/0.38 tff(40,plain,
% 0.12/0.38 (((~((~(singleton(A!7) = singleton(A!7))) | (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)))) | (~((singleton(A!7) = singleton(A!7)) | (~(in(tptp_fun_C_0(singleton(A!7), A!7), singleton(A!7)) <=> (tptp_fun_C_0(singleton(A!7), A!7) = A!7)))))) <=> (~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)))),
% 0.12/0.38 inference(transitivity,[status(thm)],[39, 22])).
% 0.12/0.38 tff(41,plain,
% 0.12/0.38 ((~((~((~(singleton(A!7) = singleton(A!7))) | (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)))) | (~((singleton(A!7) = singleton(A!7)) | (~(in(tptp_fun_C_0(singleton(A!7), A!7), singleton(A!7)) <=> (tptp_fun_C_0(singleton(A!7), A!7) = A!7))))))) <=> (~(~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7))))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[40])).
% 0.12/0.38 tff(42,plain,
% 0.12/0.38 ((~((~((~(singleton(A!7) = singleton(A!7))) | (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)))) | (~((singleton(A!7) = singleton(A!7)) | (~(in(tptp_fun_C_0(singleton(A!7), A!7), singleton(A!7)) <=> (tptp_fun_C_0(singleton(A!7), A!7) = A!7))))))) <=> (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7))),
% 0.12/0.38 inference(transitivity,[status(thm)],[41, 21])).
% 0.12/0.38 tff(43,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(A!7) = singleton(A!7))) | (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)))) | (~((singleton(A!7) = singleton(A!7)) | (~(in(tptp_fun_C_0(singleton(A!7), A!7), singleton(A!7)) <=> (tptp_fun_C_0(singleton(A!7), A!7) = A!7)))))))) <=> ((~![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(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[42])).
% 0.12/0.38 tff(44,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(A!7) = singleton(A!7))) | (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)))) | (~((singleton(A!7) = singleton(A!7)) | (~(in(tptp_fun_C_0(singleton(A!7), A!7), singleton(A!7)) <=> (tptp_fun_C_0(singleton(A!7), A!7) = A!7)))))))) <=> ((~![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(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)))),
% 0.12/0.39 inference(transitivity,[status(thm)],[43, 20])).
% 0.12/0.39 tff(45,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)))))))) | (~((~((~(singleton(A!7) = singleton(A!7))) | (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)))) | (~((singleton(A!7) = singleton(A!7)) | (~(in(tptp_fun_C_0(singleton(A!7), A!7), singleton(A!7)) <=> (tptp_fun_C_0(singleton(A!7), A!7) = A!7)))))))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(46,plain,
% 0.12/0.39 ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[45, 44])).
% 0.12/0.39 tff(47,plain,
% 0.12/0.39 ($false),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[46, 19, 1])).
% 0.12/0.39 tff(48,plain,(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)), inference(lemma,lemma(discharge,[]))).
% 0.12/0.39 tff(49,assumption,(~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7))), introduced(assumption)).
% 0.12/0.39 tff(50,assumption,(((~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6)) | (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)) <=> in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7)))), introduced(assumption)).
% 0.12/0.39 tff(51,plain,
% 0.12/0.39 (^[A: $i, B: $i, C: $i] : refl((~(((~in(A, B)) | (A = C)) <=> in(A, set_difference(B, singleton(C))))) <=> (~(((~in(A, B)) | (A = C)) <=> in(A, set_difference(B, singleton(C))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(52,plain,
% 0.12/0.39 (![A: $i, B: $i, C: $i] : (~(((~in(A, B)) | (A = C)) <=> in(A, set_difference(B, singleton(C))))) <=> ![A: $i, B: $i, C: $i] : (~(((~in(A, B)) | (A = C)) <=> in(A, set_difference(B, singleton(C)))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[51])).
% 0.12/0.39 tff(53,plain,
% 0.12/0.39 (^[A: $i, B: $i, C: $i] : trans(trans(monotonicity(rewrite((in(A, B) & (~(A = C))) <=> (~((~in(A, B)) | (A = C)))), ((in(A, set_difference(B, singleton(C))) <=> (in(A, B) & (~(A = C)))) <=> (in(A, set_difference(B, singleton(C))) <=> (~((~in(A, B)) | (A = C)))))), rewrite((in(A, set_difference(B, singleton(C))) <=> (~((~in(A, B)) | (A = C)))) <=> (~(((~in(A, B)) | (A = C)) <=> in(A, set_difference(B, singleton(C)))))), ((in(A, set_difference(B, singleton(C))) <=> (in(A, B) & (~(A = C)))) <=> (~(((~in(A, B)) | (A = C)) <=> in(A, set_difference(B, singleton(C))))))), rewrite((~(((~in(A, B)) | (A = C)) <=> in(A, set_difference(B, singleton(C))))) <=> (~(((~in(A, B)) | (A = C)) <=> in(A, set_difference(B, singleton(C)))))), ((in(A, set_difference(B, singleton(C))) <=> (in(A, B) & (~(A = C)))) <=> (~(((~in(A, B)) | (A = C)) <=> in(A, set_difference(B, singleton(C)))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(54,plain,
% 0.12/0.39 (![A: $i, B: $i, C: $i] : (in(A, set_difference(B, singleton(C))) <=> (in(A, B) & (~(A = C)))) <=> ![A: $i, B: $i, C: $i] : (~(((~in(A, B)) | (A = C)) <=> in(A, set_difference(B, singleton(C)))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[53])).
% 0.12/0.39 tff(55,plain,
% 0.12/0.39 (![A: $i, B: $i, C: $i] : (in(A, set_difference(B, singleton(C))) <=> (in(A, B) & (~(A = C)))) <=> ![A: $i, B: $i, C: $i] : (in(A, set_difference(B, singleton(C))) <=> (in(A, B) & (~(A = C))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(56,axiom,(![A: $i, B: $i, C: $i] : (in(A, set_difference(B, singleton(C))) <=> (in(A, B) & (~(A = C))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t64_zfmisc_1')).
% 0.12/0.39 tff(57,plain,
% 0.12/0.39 (![A: $i, B: $i, C: $i] : (in(A, set_difference(B, singleton(C))) <=> (in(A, B) & (~(A = C))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[56, 55])).
% 0.12/0.39 tff(58,plain,(
% 0.12/0.39 ![A: $i, B: $i, C: $i] : (in(A, set_difference(B, singleton(C))) <=> (in(A, B) & (~(A = C))))),
% 0.12/0.39 inference(skolemize,[status(sab)],[57])).
% 0.12/0.39 tff(59,plain,
% 0.12/0.39 (![A: $i, B: $i, C: $i] : (~(((~in(A, B)) | (A = C)) <=> in(A, set_difference(B, singleton(C)))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[58, 54])).
% 0.12/0.39 tff(60,plain,
% 0.12/0.39 (![A: $i, B: $i, C: $i] : (~(((~in(A, B)) | (A = C)) <=> in(A, set_difference(B, singleton(C)))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[59, 52])).
% 0.12/0.39 tff(61,plain,
% 0.12/0.39 ((~![A: $i, B: $i, C: $i] : (~(((~in(A, B)) | (A = C)) <=> in(A, set_difference(B, singleton(C)))))) | (~(((~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6)) | (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)) <=> in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7)))))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(62,plain,
% 0.12/0.39 ($false),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[61, 60, 50])).
% 0.12/0.39 tff(63,plain,(~(((~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6)) | (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)) <=> in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7))))), inference(lemma,lemma(discharge,[]))).
% 0.12/0.39 tff(64,assumption,(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6)), introduced(assumption)).
% 0.12/0.39 tff(65,plain,
% 0.12/0.39 (^[A: $i, B: $i, C: $i, D: $i] : refl((~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))) <=> (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(66,plain,
% 0.12/0.39 (![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[65])).
% 0.12/0.39 tff(67,plain,
% 0.12/0.39 (![A: $i, B: $i, C: $i] : ![D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))))),
% 0.12/0.39 inference(pull_quant,[status(thm)],[])).
% 0.12/0.39 tff(68,plain,
% 0.12/0.39 (^[A: $i, B: $i, C: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(pull_quant(((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) <=> ![D: $i] : ((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))), ((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) <=> (~![D: $i] : ((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))))), pull_quant((~![D: $i] : ((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) <=> ?[D: $i] : (~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B)))))), ((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) <=> ?[D: $i] : (~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))))), (((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))) <=> (?[D: $i] : (~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))))), pull_quant((?[D: $i] : (~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))) <=> ?[D: $i] : ((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))), (((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))) <=> ?[D: $i] : ((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))))), ((~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))) <=> (~?[D: $i] : ((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))))), pull_quant((~?[D: $i] : ((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))) <=> ![D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))))), ((~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))) <=> ![D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(69,plain,
% 0.12/0.39 (![A: $i, B: $i, C: $i] : (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))) <=> ![A: $i, B: $i, C: $i] : ![D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[68])).
% 0.12/0.39 tff(70,plain,
% 0.12/0.39 (![A: $i, B: $i, C: $i] : (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))))),
% 0.12/0.39 inference(transitivity,[status(thm)],[69, 67])).
% 0.12/0.39 tff(71,plain,
% 0.12/0.39 (![A: $i, B: $i, C: $i] : (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))))),
% 0.12/0.39 inference(transitivity,[status(thm)],[70, 66])).
% 0.12/0.39 tff(72,plain,
% 0.12/0.39 (^[A: $i, B: $i, C: $i] : rewrite((~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))) <=> (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(73,plain,
% 0.12/0.39 (![A: $i, B: $i, C: $i] : (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[72])).
% 0.12/0.39 tff(74,plain,
% 0.12/0.39 (![A: $i, B: $i, C: $i] : (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))))),
% 0.12/0.39 inference(transitivity,[status(thm)],[73, 71])).
% 0.12/0.39 tff(75,plain,
% 0.12/0.39 (^[A: $i, B: $i, C: $i] : trans(monotonicity(rewrite(((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) <=> ((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))), trans(monotonicity(trans(monotonicity(rewrite((in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))) <=> (in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))), ((~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))) <=> (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))), rewrite((~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))) <=> (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))), ((~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))) <=> (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))), (((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))) <=> ((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))), rewrite(((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))) <=> ((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))), (((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))) <=> ((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))), ((((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) & ((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))) <=> (((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) & ((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))))), rewrite((((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) & ((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))) <=> (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))))), ((((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) & ((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))) <=> (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(76,plain,
% 0.12/0.39 (![A: $i, B: $i, C: $i] : (((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) & ((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B)))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[75])).
% 0.12/0.39 tff(77,plain,
% 0.12/0.39 (![A: $i, B: $i, C: $i] : ((C = set_union2(A, B)) <=> ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) <=> ![A: $i, B: $i, C: $i] : ((C = set_union2(A, B)) <=> ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(78,axiom,(![A: $i, B: $i, C: $i] : ((C = set_union2(A, B)) <=> ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d2_xboole_0')).
% 0.12/0.39 tff(79,plain,
% 0.12/0.39 (![A: $i, B: $i, C: $i] : ((C = set_union2(A, B)) <=> ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[78, 77])).
% 0.12/0.39 tff(80,plain,(
% 0.12/0.39 ![A: $i, B: $i, C: $i] : (((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) & ((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))),
% 0.12/0.39 inference(skolemize,[status(sab)],[79])).
% 0.12/0.39 tff(81,plain,
% 0.12/0.39 (![A: $i, B: $i, C: $i] : (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[80, 76])).
% 0.12/0.39 tff(82,plain,
% 0.12/0.39 (![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[81, 74])).
% 0.12/0.39 tff(83,plain,
% 0.12/0.39 ((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))))) | (~((~((~(B!6 = set_union2(set_difference(B!6, singleton(A!7)), singleton(A!7)))) | (in(A!7, B!6) <=> (in(A!7, set_difference(B!6, singleton(A!7))) | in(A!7, singleton(A!7)))))) | (~((B!6 = set_union2(set_difference(B!6, singleton(A!7)), singleton(A!7))) | (~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6) <=> (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7))) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)))))))))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(84,plain,
% 0.20/0.40 (~((~((~(B!6 = set_union2(set_difference(B!6, singleton(A!7)), singleton(A!7)))) | (in(A!7, B!6) <=> (in(A!7, set_difference(B!6, singleton(A!7))) | in(A!7, singleton(A!7)))))) | (~((B!6 = set_union2(set_difference(B!6, singleton(A!7)), singleton(A!7))) | (~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6) <=> (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7))) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7))))))))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[83, 82])).
% 0.20/0.40 tff(85,plain,
% 0.20/0.40 (((~((~(B!6 = set_union2(set_difference(B!6, singleton(A!7)), singleton(A!7)))) | (in(A!7, B!6) <=> (in(A!7, set_difference(B!6, singleton(A!7))) | in(A!7, singleton(A!7)))))) | (~((B!6 = set_union2(set_difference(B!6, singleton(A!7)), singleton(A!7))) | (~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6) <=> (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7))) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)))))))) | ((B!6 = set_union2(set_difference(B!6, singleton(A!7)), singleton(A!7))) | (~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6) <=> (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7))) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7))))))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(86,plain,
% 0.20/0.40 ((B!6 = set_union2(set_difference(B!6, singleton(A!7)), singleton(A!7))) | (~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6) <=> (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7))) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)))))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[85, 84])).
% 0.20/0.40 tff(87,plain,
% 0.20/0.40 ((B!6 = set_union2(set_difference(B!6, singleton(A!7)), singleton(A!7))) <=> (set_union2(set_difference(B!6, singleton(A!7)), singleton(A!7)) = B!6)),
% 0.20/0.40 inference(commutativity,[status(thm)],[])).
% 0.20/0.40 tff(88,plain,
% 0.20/0.40 ((set_union2(set_difference(B!6, singleton(A!7)), singleton(A!7)) = B!6) <=> (B!6 = set_union2(set_difference(B!6, singleton(A!7)), singleton(A!7)))),
% 0.20/0.40 inference(symmetry,[status(thm)],[87])).
% 0.20/0.40 tff(89,plain,
% 0.20/0.40 ((~(set_union2(set_difference(B!6, singleton(A!7)), singleton(A!7)) = B!6)) <=> (~(B!6 = set_union2(set_difference(B!6, singleton(A!7)), singleton(A!7))))),
% 0.20/0.40 inference(monotonicity,[status(thm)],[88])).
% 0.20/0.40 tff(90,plain,
% 0.20/0.40 ((~![A: $i, B: $i] : ((~in(A, B)) | (set_union2(set_difference(B, singleton(A)), singleton(A)) = B))) <=> (~![A: $i, B: $i] : ((~in(A, B)) | (set_union2(set_difference(B, singleton(A)), singleton(A)) = B)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(91,plain,
% 0.20/0.40 ((~![A: $i, B: $i] : (in(A, B) => (set_union2(set_difference(B, singleton(A)), singleton(A)) = B))) <=> (~![A: $i, B: $i] : ((~in(A, B)) | (set_union2(set_difference(B, singleton(A)), singleton(A)) = B)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(92,axiom,(~![A: $i, B: $i] : (in(A, B) => (set_union2(set_difference(B, singleton(A)), singleton(A)) = B))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t140_zfmisc_1')).
% 0.20/0.40 tff(93,plain,
% 0.20/0.40 (~![A: $i, B: $i] : ((~in(A, B)) | (set_union2(set_difference(B, singleton(A)), singleton(A)) = B))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[92, 91])).
% 0.20/0.40 tff(94,plain,
% 0.20/0.40 (~![A: $i, B: $i] : ((~in(A, B)) | (set_union2(set_difference(B, singleton(A)), singleton(A)) = B))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[93, 90])).
% 0.20/0.40 tff(95,plain,
% 0.20/0.40 (~![A: $i, B: $i] : ((~in(A, B)) | (set_union2(set_difference(B, singleton(A)), singleton(A)) = B))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[94, 90])).
% 0.20/0.40 tff(96,plain,
% 0.20/0.40 (~![A: $i, B: $i] : ((~in(A, B)) | (set_union2(set_difference(B, singleton(A)), singleton(A)) = B))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[95, 90])).
% 0.20/0.40 tff(97,plain,(
% 0.20/0.40 ~((~in(A!7, B!6)) | (set_union2(set_difference(B!6, singleton(A!7)), singleton(A!7)) = B!6))),
% 0.20/0.40 inference(skolemize,[status(sab)],[96])).
% 0.20/0.40 tff(98,plain,
% 0.20/0.40 (~(set_union2(set_difference(B!6, singleton(A!7)), singleton(A!7)) = B!6)),
% 0.20/0.40 inference(or_elim,[status(thm)],[97])).
% 0.20/0.40 tff(99,plain,
% 0.20/0.40 (~(B!6 = set_union2(set_difference(B!6, singleton(A!7)), singleton(A!7)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[98, 89])).
% 0.20/0.40 tff(100,plain,
% 0.20/0.40 ((~((B!6 = set_union2(set_difference(B!6, singleton(A!7)), singleton(A!7))) | (~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6) <=> (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7))) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7))))))) | (B!6 = set_union2(set_difference(B!6, singleton(A!7)), singleton(A!7))) | (~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6) <=> (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7))) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)))))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(101,plain,
% 0.20/0.40 (~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6) <=> (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7))) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7))))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[100, 99, 86])).
% 0.20/0.40 tff(102,plain,
% 0.20/0.40 ((in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6) <=> (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7))) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)))) | (~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6)) | (~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7))) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7))))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(103,plain,
% 0.20/0.40 ((~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6)) | (~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7))) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7))))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[102, 101])).
% 0.20/0.40 tff(104,plain,
% 0.20/0.40 (~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7))) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[103, 64])).
% 0.20/0.40 tff(105,plain,
% 0.20/0.40 ((in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7))) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7))) | (~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7))))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(106,plain,
% 0.20/0.40 (~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7)))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[105, 104])).
% 0.20/0.40 tff(107,plain,
% 0.20/0.40 ((in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)) | (~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6)) | (~(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(108,plain,
% 0.20/0.40 (~(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[107, 64, 49])).
% 0.20/0.40 tff(109,plain,
% 0.20/0.40 ((~((~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6)) | (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7))) | (~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6)) | (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(110,plain,
% 0.20/0.40 (~((~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6)) | (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[109, 108, 64])).
% 0.20/0.40 tff(111,plain,
% 0.20/0.40 ((((~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6)) | (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)) <=> in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7)))) | ((~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6)) | (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7)))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(112,plain,
% 0.20/0.40 ($false),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[111, 110, 106, 63])).
% 0.20/0.40 tff(113,plain,((~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6)) | (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.40 tff(114,plain,
% 0.20/0.40 (~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[113, 49])).
% 0.20/0.40 tff(115,plain,
% 0.20/0.40 ((in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6) | (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(116,plain,
% 0.20/0.40 (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[115, 114, 49])).
% 0.20/0.40 tff(117,plain,
% 0.20/0.40 (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6) <=> in(A!7, B!6)),
% 0.20/0.40 inference(monotonicity,[status(thm)],[116])).
% 0.20/0.40 tff(118,plain,
% 0.20/0.40 (in(A!7, B!6) <=> in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6)),
% 0.20/0.40 inference(symmetry,[status(thm)],[117])).
% 0.20/0.40 tff(119,plain,
% 0.20/0.40 (in(A!7, B!6)),
% 0.20/0.40 inference(or_elim,[status(thm)],[97])).
% 0.20/0.40 tff(120,plain,
% 0.20/0.40 (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6)),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[119, 118])).
% 0.20/0.40 tff(121,plain,
% 0.20/0.40 ($false),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[114, 120])).
% 0.20/0.40 tff(122,plain,(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.41 tff(123,assumption,(~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6)), introduced(assumption)).
% 0.20/0.41 tff(124,plain,
% 0.20/0.41 ((in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6) <=> (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7))) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)))) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6) | (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7))) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)))),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(125,plain,
% 0.20/0.41 (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6) | (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7))) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[124, 101])).
% 0.20/0.41 tff(126,plain,
% 0.20/0.41 (in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7))) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[125, 123])).
% 0.20/0.41 tff(127,plain,
% 0.20/0.41 (((~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6)) | (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6)),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(128,plain,
% 0.20/0.41 ((~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6)) | (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[127, 123])).
% 0.20/0.41 tff(129,plain,
% 0.20/0.41 ((((~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6)) | (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)) <=> in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7)))) | (~((~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6)) | (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7))) | (~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7))))),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(130,plain,
% 0.20/0.41 (~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7)))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[129, 128, 63])).
% 0.20/0.41 tff(131,plain,
% 0.20/0.41 ((~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7))) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6) | (~(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7))),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(132,plain,
% 0.20/0.41 (~(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[131, 123, 122])).
% 0.20/0.41 tff(133,plain,
% 0.20/0.41 ((~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7))) | (~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7))) | (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(134,plain,
% 0.20/0.41 (~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[133, 132, 48])).
% 0.20/0.41 tff(135,plain,
% 0.20/0.41 ((~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7))) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)))) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7))) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7))),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(136,plain,
% 0.20/0.41 ($false),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[135, 134, 130, 126])).
% 0.20/0.41 tff(137,plain,(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.41 tff(138,plain,
% 0.20/0.41 ((~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7))) | (~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), B!6)) | (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7)),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(139,plain,
% 0.20/0.41 (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[138, 137, 122])).
% 0.20/0.41 tff(140,plain,
% 0.20/0.41 (~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7))) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[103, 137])).
% 0.20/0.41 tff(141,plain,
% 0.20/0.41 ((in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), set_difference(B!6, singleton(A!7))) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7))) | (~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)))),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(142,plain,
% 0.20/0.41 (~in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[141, 140])).
% 0.20/0.41 tff(143,plain,
% 0.20/0.41 ((~(in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) <=> (tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7))) | in(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))), singleton(A!7)) | (~(tptp_fun_D_1(B!6, singleton(A!7), set_difference(B!6, singleton(A!7))) = A!7))),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(144,plain,
% 0.20/0.41 ($false),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[143, 142, 139, 48])).
% 0.20/0.41 % SZS output end Proof
% 0.20/0.41 % E exiting
%------------------------------------------------------------------------------