%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : NUM386+1 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n005.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:20:52 AM UTC 2025
% Result : Theorem 0.18s 0.36s
% Output : Proof 0.18s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.11 % Problem : NUM386+1 : TPTP v9.0.0. Released v3.2.0.
% 0.11/0.11 % Command : run_E %s %d THM
% 0.11/0.32 % Computer : n005.cluster.edu
% 0.11/0.32 % Model : x86_64 x86_64
% 0.11/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.32 % Memory : 8042.1875MB
% 0.11/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.32 % CPULimit : 300
% 0.11/0.32 % WCLimit : 300
% 0.11/0.32 % DateTime : Fri Jun 20 01:01:50 EDT 2025
% 0.11/0.32 % CPUTime :
% 0.18/0.36 % SZS status Theorem
% 0.18/0.36 % SZS output start Proof
% 0.18/0.36 tff(tptp_fun_B_13_type, type, (
% 0.18/0.36 tptp_fun_B_13: $i)).
% 0.18/0.36 tff(tptp_fun_A_14_type, type, (
% 0.18/0.36 tptp_fun_A_14: $i)).
% 0.18/0.36 tff(in_type, type, (
% 0.18/0.36 in: ( $i * $i ) > $o)).
% 0.18/0.36 tff(singleton_type, type, (
% 0.18/0.36 singleton: $i > $i)).
% 0.18/0.36 tff(tptp_fun_C_0_type, type, (
% 0.18/0.36 tptp_fun_C_0: ( $i * $i ) > $i)).
% 0.18/0.36 tff(set_union2_type, type, (
% 0.18/0.36 set_union2: ( $i * $i ) > $i)).
% 0.18/0.36 tff(succ_type, type, (
% 0.18/0.36 succ: $i > $i)).
% 0.18/0.36 tff(tptp_fun_D_1_type, type, (
% 0.18/0.36 tptp_fun_D_1: ( $i * $i * $i ) > $i)).
% 0.18/0.36 tff(1,assumption,(~(in(A!14, singleton(B!13)) <=> (A!14 = B!13))), introduced(assumption)).
% 0.18/0.36 tff(2,plain,
% 0.18/0.36 (^[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))))))))),
% 0.18/0.36 inference(bind,[status(th)],[])).
% 0.18/0.36 tff(3,plain,
% 0.18/0.36 (![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)))))))),
% 0.18/0.36 inference(quant_intro,[status(thm)],[2])).
% 0.18/0.36 tff(4,plain,
% 0.18/0.36 (![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)))))))),
% 0.18/0.36 inference(pull_quant,[status(thm)],[])).
% 0.18/0.36 tff(5,plain,
% 0.18/0.36 (^[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)))))))))),
% 0.18/0.37 inference(bind,[status(th)],[])).
% 0.18/0.37 tff(6,plain,
% 0.18/0.37 (![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)))))))),
% 0.18/0.37 inference(quant_intro,[status(thm)],[5])).
% 0.18/0.37 tff(7,plain,
% 0.18/0.37 (![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)))))))),
% 0.18/0.37 inference(transitivity,[status(thm)],[6, 4])).
% 0.18/0.37 tff(8,plain,
% 0.18/0.37 (![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)))))))),
% 0.18/0.37 inference(transitivity,[status(thm)],[7, 3])).
% 0.18/0.37 tff(9,plain,
% 0.18/0.37 (^[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))))))))),
% 0.18/0.37 inference(bind,[status(th)],[])).
% 0.18/0.37 tff(10,plain,
% 0.18/0.37 (![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)))))))),
% 0.18/0.37 inference(quant_intro,[status(thm)],[9])).
% 0.18/0.37 tff(11,plain,
% 0.18/0.37 (![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)))))))),
% 0.18/0.37 inference(transitivity,[status(thm)],[10, 8])).
% 0.18/0.37 tff(12,plain,
% 0.18/0.37 (^[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)))))))))),
% 0.18/0.37 inference(bind,[status(th)],[])).
% 0.18/0.37 tff(13,plain,
% 0.18/0.37 (![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)))))))),
% 0.18/0.37 inference(quant_intro,[status(thm)],[12])).
% 0.18/0.37 tff(14,plain,
% 0.18/0.37 (![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A))) <=> ![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))),
% 0.18/0.37 inference(rewrite,[status(thm)],[])).
% 0.18/0.37 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')).
% 0.18/0.37 tff(16,plain,
% 0.18/0.37 (![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))),
% 0.18/0.37 inference(modus_ponens,[status(thm)],[15, 14])).
% 0.18/0.37 tff(17,plain,(
% 0.18/0.37 ![A: $i, B: $i] : (((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))),
% 0.18/0.37 inference(skolemize,[status(sab)],[16])).
% 0.18/0.37 tff(18,plain,
% 0.18/0.37 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.18/0.37 inference(modus_ponens,[status(thm)],[17, 13])).
% 0.18/0.37 tff(19,plain,
% 0.18/0.37 (![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.18/0.37 inference(modus_ponens,[status(thm)],[18, 11])).
% 0.18/0.37 tff(20,plain,
% 0.18/0.37 (((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13))) <=> ((~![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!14, singleton(B!13)) <=> (A!14 = B!13)))),
% 0.18/0.37 inference(rewrite,[status(thm)],[])).
% 0.18/0.37 tff(21,plain,
% 0.18/0.37 ((~(~(in(A!14, singleton(B!13)) <=> (A!14 = B!13)))) <=> (in(A!14, singleton(B!13)) <=> (A!14 = B!13))),
% 0.18/0.37 inference(rewrite,[status(thm)],[])).
% 0.18/0.37 tff(22,plain,
% 0.18/0.37 (((~(in(A!14, singleton(B!13)) <=> (A!14 = B!13))) | $false) <=> (~(in(A!14, singleton(B!13)) <=> (A!14 = B!13)))),
% 0.18/0.37 inference(rewrite,[status(thm)],[])).
% 0.18/0.37 tff(23,plain,
% 0.18/0.37 ((~$true) <=> $false),
% 0.18/0.37 inference(rewrite,[status(thm)],[])).
% 0.18/0.37 tff(24,plain,
% 0.18/0.37 (($true | (~(in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13)) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13)))) <=> $true),
% 0.18/0.37 inference(rewrite,[status(thm)],[])).
% 0.18/0.37 tff(25,plain,
% 0.18/0.37 ((~(in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13)) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13))) <=> (~(in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13)) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13)))),
% 0.18/0.37 inference(rewrite,[status(thm)],[])).
% 0.18/0.37 tff(26,plain,
% 0.18/0.37 ((singleton(B!13) = singleton(B!13)) <=> $true),
% 0.18/0.37 inference(rewrite,[status(thm)],[])).
% 0.18/0.37 tff(27,plain,
% 0.18/0.37 (((singleton(B!13) = singleton(B!13)) | (~(in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13)) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13)))) <=> ($true | (~(in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13)) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13))))),
% 0.18/0.37 inference(monotonicity,[status(thm)],[26, 25])).
% 0.18/0.37 tff(28,plain,
% 0.18/0.37 (((singleton(B!13) = singleton(B!13)) | (~(in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13)) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13)))) <=> $true),
% 0.18/0.37 inference(transitivity,[status(thm)],[27, 24])).
% 0.18/0.37 tff(29,plain,
% 0.18/0.37 ((~((singleton(B!13) = singleton(B!13)) | (~(in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13)) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13))))) <=> (~$true)),
% 0.18/0.37 inference(monotonicity,[status(thm)],[28])).
% 0.18/0.37 tff(30,plain,
% 0.18/0.37 ((~((singleton(B!13) = singleton(B!13)) | (~(in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13)) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13))))) <=> $false),
% 0.18/0.37 inference(transitivity,[status(thm)],[29, 23])).
% 0.18/0.37 tff(31,plain,
% 0.18/0.37 ((~(in(A!14, singleton(B!13)) <=> (A!14 = B!13))) <=> (~(in(A!14, singleton(B!13)) <=> (A!14 = B!13)))),
% 0.18/0.37 inference(rewrite,[status(thm)],[])).
% 0.18/0.37 tff(32,plain,
% 0.18/0.37 (($false | (in(A!14, singleton(B!13)) <=> (A!14 = B!13))) <=> (in(A!14, singleton(B!13)) <=> (A!14 = B!13))),
% 0.18/0.37 inference(rewrite,[status(thm)],[])).
% 0.18/0.37 tff(33,plain,
% 0.18/0.37 ((~(singleton(B!13) = singleton(B!13))) <=> (~$true)),
% 0.18/0.37 inference(monotonicity,[status(thm)],[26])).
% 0.18/0.37 tff(34,plain,
% 0.18/0.37 ((~(singleton(B!13) = singleton(B!13))) <=> $false),
% 0.18/0.37 inference(transitivity,[status(thm)],[33, 23])).
% 0.18/0.37 tff(35,plain,
% 0.18/0.37 (((~(singleton(B!13) = singleton(B!13))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13))) <=> ($false | (in(A!14, singleton(B!13)) <=> (A!14 = B!13)))),
% 0.18/0.37 inference(monotonicity,[status(thm)],[34])).
% 0.18/0.37 tff(36,plain,
% 0.18/0.37 (((~(singleton(B!13) = singleton(B!13))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13))) <=> (in(A!14, singleton(B!13)) <=> (A!14 = B!13))),
% 0.18/0.37 inference(transitivity,[status(thm)],[35, 32])).
% 0.18/0.37 tff(37,plain,
% 0.18/0.37 ((~((~(singleton(B!13) = singleton(B!13))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13)))) <=> (~(in(A!14, singleton(B!13)) <=> (A!14 = B!13)))),
% 0.18/0.37 inference(monotonicity,[status(thm)],[36])).
% 0.18/0.37 tff(38,plain,
% 0.18/0.37 ((~((~(singleton(B!13) = singleton(B!13))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13)))) <=> (~(in(A!14, singleton(B!13)) <=> (A!14 = B!13)))),
% 0.18/0.37 inference(transitivity,[status(thm)],[37, 31])).
% 0.18/0.37 tff(39,plain,
% 0.18/0.37 (((~((~(singleton(B!13) = singleton(B!13))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13)))) | (~((singleton(B!13) = singleton(B!13)) | (~(in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13)) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13)))))) <=> ((~(in(A!14, singleton(B!13)) <=> (A!14 = B!13))) | $false)),
% 0.18/0.37 inference(monotonicity,[status(thm)],[38, 30])).
% 0.18/0.37 tff(40,plain,
% 0.18/0.37 (((~((~(singleton(B!13) = singleton(B!13))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13)))) | (~((singleton(B!13) = singleton(B!13)) | (~(in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13)) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13)))))) <=> (~(in(A!14, singleton(B!13)) <=> (A!14 = B!13)))),
% 0.18/0.37 inference(transitivity,[status(thm)],[39, 22])).
% 0.18/0.37 tff(41,plain,
% 0.18/0.37 ((~((~((~(singleton(B!13) = singleton(B!13))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13)))) | (~((singleton(B!13) = singleton(B!13)) | (~(in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13)) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13))))))) <=> (~(~(in(A!14, singleton(B!13)) <=> (A!14 = B!13))))),
% 0.18/0.37 inference(monotonicity,[status(thm)],[40])).
% 0.18/0.37 tff(42,plain,
% 0.18/0.37 ((~((~((~(singleton(B!13) = singleton(B!13))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13)))) | (~((singleton(B!13) = singleton(B!13)) | (~(in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13)) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13))))))) <=> (in(A!14, singleton(B!13)) <=> (A!14 = B!13))),
% 0.18/0.37 inference(transitivity,[status(thm)],[41, 21])).
% 0.18/0.37 tff(43,plain,
% 0.18/0.37 (((~![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)))))))) | (~((~((~(singleton(B!13) = singleton(B!13))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13)))) | (~((singleton(B!13) = singleton(B!13)) | (~(in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13)) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13)))))))) <=> ((~![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!14, singleton(B!13)) <=> (A!14 = B!13)))),
% 0.18/0.37 inference(monotonicity,[status(thm)],[42])).
% 0.18/0.37 tff(44,plain,
% 0.18/0.37 (((~![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)))))))) | (~((~((~(singleton(B!13) = singleton(B!13))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13)))) | (~((singleton(B!13) = singleton(B!13)) | (~(in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13)) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13)))))))) <=> ((~![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!14, singleton(B!13)) <=> (A!14 = B!13)))),
% 0.18/0.37 inference(transitivity,[status(thm)],[43, 20])).
% 0.18/0.38 tff(45,plain,
% 0.18/0.38 ((~![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!13) = singleton(B!13))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13)))) | (~((singleton(B!13) = singleton(B!13)) | (~(in(tptp_fun_C_0(singleton(B!13), B!13), singleton(B!13)) <=> (tptp_fun_C_0(singleton(B!13), B!13) = B!13)))))))),
% 0.18/0.38 inference(quant_inst,[status(thm)],[])).
% 0.18/0.38 tff(46,plain,
% 0.18/0.38 ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(A!14, singleton(B!13)) <=> (A!14 = B!13))),
% 0.18/0.38 inference(modus_ponens,[status(thm)],[45, 44])).
% 0.18/0.38 tff(47,plain,
% 0.18/0.38 ($false),
% 0.18/0.38 inference(unit_resolution,[status(thm)],[46, 19, 1])).
% 0.18/0.38 tff(48,plain,(in(A!14, singleton(B!13)) <=> (A!14 = B!13)), inference(lemma,lemma(discharge,[]))).
% 0.18/0.38 tff(49,plain,
% 0.18/0.38 (^[A: $i] : refl((succ(A) = set_union2(A, singleton(A))) <=> (succ(A) = set_union2(A, singleton(A))))),
% 0.18/0.38 inference(bind,[status(th)],[])).
% 0.18/0.38 tff(50,plain,
% 0.18/0.38 (![A: $i] : (succ(A) = set_union2(A, singleton(A))) <=> ![A: $i] : (succ(A) = set_union2(A, singleton(A)))),
% 0.18/0.38 inference(quant_intro,[status(thm)],[49])).
% 0.18/0.38 tff(51,plain,
% 0.18/0.38 (![A: $i] : (succ(A) = set_union2(A, singleton(A))) <=> ![A: $i] : (succ(A) = set_union2(A, singleton(A)))),
% 0.18/0.38 inference(rewrite,[status(thm)],[])).
% 0.18/0.38 tff(52,axiom,(![A: $i] : (succ(A) = set_union2(A, singleton(A)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d1_ordinal1')).
% 0.18/0.38 tff(53,plain,
% 0.18/0.38 (![A: $i] : (succ(A) = set_union2(A, singleton(A)))),
% 0.18/0.38 inference(modus_ponens,[status(thm)],[52, 51])).
% 0.18/0.38 tff(54,plain,(
% 0.18/0.38 ![A: $i] : (succ(A) = set_union2(A, singleton(A)))),
% 0.18/0.38 inference(skolemize,[status(sab)],[53])).
% 0.18/0.38 tff(55,plain,
% 0.18/0.38 (![A: $i] : (succ(A) = set_union2(A, singleton(A)))),
% 0.18/0.38 inference(modus_ponens,[status(thm)],[54, 50])).
% 0.18/0.38 tff(56,plain,
% 0.18/0.38 ((~![A: $i] : (succ(A) = set_union2(A, singleton(A)))) | (succ(B!13) = set_union2(B!13, singleton(B!13)))),
% 0.18/0.38 inference(quant_inst,[status(thm)],[])).
% 0.18/0.38 tff(57,plain,
% 0.18/0.38 (succ(B!13) = set_union2(B!13, singleton(B!13))),
% 0.18/0.38 inference(unit_resolution,[status(thm)],[56, 55])).
% 0.18/0.38 tff(58,plain,
% 0.18/0.38 (set_union2(B!13, singleton(B!13)) = succ(B!13)),
% 0.18/0.38 inference(symmetry,[status(thm)],[57])).
% 0.18/0.38 tff(59,plain,
% 0.18/0.38 (in(A!14, set_union2(B!13, singleton(B!13))) <=> in(A!14, succ(B!13))),
% 0.18/0.38 inference(monotonicity,[status(thm)],[58])).
% 0.18/0.38 tff(60,plain,
% 0.18/0.38 (in(A!14, succ(B!13)) <=> in(A!14, set_union2(B!13, singleton(B!13)))),
% 0.18/0.38 inference(symmetry,[status(thm)],[59])).
% 0.18/0.38 tff(61,plain,
% 0.18/0.38 ((~in(A!14, succ(B!13))) <=> (~in(A!14, set_union2(B!13, singleton(B!13))))),
% 0.18/0.38 inference(monotonicity,[status(thm)],[60])).
% 0.18/0.38 tff(62,assumption,(~(in(A!14, B!13) | (A!14 = B!13))), introduced(assumption)).
% 0.18/0.38 tff(63,plain,
% 0.18/0.38 ((~(in(A!14, succ(B!13)) <=> (in(A!14, B!13) | (A!14 = B!13)))) <=> (~(in(A!14, succ(B!13)) <=> (in(A!14, B!13) | (A!14 = B!13))))),
% 0.18/0.38 inference(rewrite,[status(thm)],[])).
% 0.18/0.38 tff(64,plain,
% 0.18/0.38 ((in(A!14, succ(B!13)) <=> (in(A!14, B!13) | (A!14 = B!13))) <=> (in(A!14, succ(B!13)) <=> (in(A!14, B!13) | (A!14 = B!13)))),
% 0.18/0.38 inference(rewrite,[status(thm)],[])).
% 0.18/0.38 tff(65,plain,
% 0.18/0.38 ((~(in(A!14, succ(B!13)) <=> (in(A!14, B!13) | (A!14 = B!13)))) <=> (~(in(A!14, succ(B!13)) <=> (in(A!14, B!13) | (A!14 = B!13))))),
% 0.18/0.38 inference(monotonicity,[status(thm)],[64])).
% 0.18/0.38 tff(66,plain,
% 0.18/0.38 ((~(in(A!14, succ(B!13)) <=> (in(A!14, B!13) | (A!14 = B!13)))) <=> (~(in(A!14, succ(B!13)) <=> (in(A!14, B!13) | (A!14 = B!13))))),
% 0.18/0.38 inference(transitivity,[status(thm)],[65, 63])).
% 0.18/0.38 tff(67,plain,
% 0.18/0.38 ((~![A: $i, B: $i] : (in(A, succ(B)) <=> (in(A, B) | (A = B)))) <=> (~![A: $i, B: $i] : (in(A, succ(B)) <=> (in(A, B) | (A = B))))),
% 0.18/0.38 inference(rewrite,[status(thm)],[])).
% 0.18/0.38 tff(68,axiom,(~![A: $i, B: $i] : (in(A, succ(B)) <=> (in(A, B) | (A = B)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t13_ordinal1')).
% 0.18/0.38 tff(69,plain,
% 0.18/0.38 (~![A: $i, B: $i] : (in(A, succ(B)) <=> (in(A, B) | (A = B)))),
% 0.18/0.38 inference(modus_ponens,[status(thm)],[68, 67])).
% 0.18/0.38 tff(70,plain,
% 0.18/0.38 (~![A: $i, B: $i] : (in(A, succ(B)) <=> (in(A, B) | (A = B)))),
% 0.18/0.38 inference(modus_ponens,[status(thm)],[69, 67])).
% 0.18/0.38 tff(71,plain,
% 0.18/0.38 (~![A: $i, B: $i] : (in(A, succ(B)) <=> (in(A, B) | (A = B)))),
% 0.18/0.38 inference(modus_ponens,[status(thm)],[70, 67])).
% 0.18/0.38 tff(72,plain,
% 0.18/0.38 (~![A: $i, B: $i] : (in(A, succ(B)) <=> (in(A, B) | (A = B)))),
% 0.18/0.38 inference(modus_ponens,[status(thm)],[71, 67])).
% 0.18/0.38 tff(73,plain,(
% 0.18/0.38 ~(in(A!14, succ(B!13)) <=> (in(A!14, B!13) | (A!14 = B!13)))),
% 0.18/0.38 inference(skolemize,[status(sab)],[72])).
% 0.18/0.38 tff(74,plain,
% 0.18/0.38 (~(in(A!14, succ(B!13)) <=> (in(A!14, B!13) | (A!14 = B!13)))),
% 0.18/0.38 inference(modus_ponens,[status(thm)],[73, 66])).
% 0.18/0.38 tff(75,plain,
% 0.18/0.38 ((in(A!14, succ(B!13)) <=> (in(A!14, B!13) | (A!14 = B!13))) | in(A!14, succ(B!13)) | (in(A!14, B!13) | (A!14 = B!13))),
% 0.18/0.38 inference(tautology,[status(thm)],[])).
% 0.18/0.38 tff(76,plain,
% 0.18/0.38 (in(A!14, succ(B!13)) | (in(A!14, B!13) | (A!14 = B!13))),
% 0.18/0.38 inference(unit_resolution,[status(thm)],[75, 74])).
% 0.18/0.38 tff(77,plain,
% 0.18/0.38 (in(A!14, succ(B!13))),
% 0.18/0.38 inference(unit_resolution,[status(thm)],[76, 62])).
% 0.18/0.38 tff(78,plain,
% 0.18/0.38 (in(A!14, set_union2(B!13, singleton(B!13)))),
% 0.18/0.38 inference(modus_ponens,[status(thm)],[77, 60])).
% 0.18/0.38 tff(79,plain,
% 0.18/0.38 ((in(A!14, B!13) | (A!14 = B!13)) | (~in(A!14, B!13))),
% 0.18/0.38 inference(tautology,[status(thm)],[])).
% 0.18/0.38 tff(80,plain,
% 0.18/0.38 (~in(A!14, B!13)),
% 0.18/0.38 inference(unit_resolution,[status(thm)],[79, 62])).
% 0.18/0.38 tff(81,plain,
% 0.18/0.38 ((in(A!14, B!13) | (A!14 = B!13)) | (~(A!14 = B!13))),
% 0.18/0.38 inference(tautology,[status(thm)],[])).
% 0.18/0.38 tff(82,plain,
% 0.18/0.38 (~(A!14 = B!13)),
% 0.18/0.38 inference(unit_resolution,[status(thm)],[81, 62])).
% 0.18/0.38 tff(83,plain,
% 0.18/0.38 ((~(in(A!14, singleton(B!13)) <=> (A!14 = B!13))) | (~in(A!14, singleton(B!13))) | (A!14 = B!13)),
% 0.18/0.38 inference(tautology,[status(thm)],[])).
% 0.18/0.38 tff(84,plain,
% 0.18/0.38 (~in(A!14, singleton(B!13))),
% 0.18/0.38 inference(unit_resolution,[status(thm)],[83, 82, 48])).
% 0.18/0.38 tff(85,plain,
% 0.18/0.38 ((~(in(A!14, B!13) | in(A!14, singleton(B!13)))) | in(A!14, B!13) | in(A!14, singleton(B!13))),
% 0.18/0.38 inference(tautology,[status(thm)],[])).
% 0.18/0.38 tff(86,plain,
% 0.18/0.38 (~(in(A!14, B!13) | in(A!14, singleton(B!13)))),
% 0.18/0.38 inference(unit_resolution,[status(thm)],[85, 84, 80])).
% 0.18/0.38 tff(87,plain,
% 0.18/0.38 (^[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)))))))))),
% 0.18/0.38 inference(bind,[status(th)],[])).
% 0.18/0.38 tff(88,plain,
% 0.18/0.38 (![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))))))))),
% 0.18/0.38 inference(quant_intro,[status(thm)],[87])).
% 0.18/0.38 tff(89,plain,
% 0.18/0.38 (![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))))))))),
% 0.18/0.38 inference(pull_quant,[status(thm)],[])).
% 0.18/0.38 tff(90,plain,
% 0.18/0.38 (^[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))))))))))),
% 0.18/0.38 inference(bind,[status(th)],[])).
% 0.18/0.38 tff(91,plain,
% 0.18/0.38 (![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))))))))),
% 0.18/0.38 inference(quant_intro,[status(thm)],[90])).
% 0.18/0.38 tff(92,plain,
% 0.18/0.38 (![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))))))))),
% 0.18/0.38 inference(transitivity,[status(thm)],[91, 89])).
% 0.18/0.38 tff(93,plain,
% 0.18/0.38 (![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))))))))),
% 0.18/0.38 inference(transitivity,[status(thm)],[92, 88])).
% 0.18/0.38 tff(94,plain,
% 0.18/0.38 (^[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)))))))))),
% 0.18/0.38 inference(bind,[status(th)],[])).
% 0.18/0.38 tff(95,plain,
% 0.18/0.38 (![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))))))))),
% 0.18/0.38 inference(quant_intro,[status(thm)],[94])).
% 0.18/0.38 tff(96,plain,
% 0.18/0.38 (![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))))))))),
% 0.18/0.38 inference(transitivity,[status(thm)],[95, 93])).
% 0.18/0.38 tff(97,plain,
% 0.18/0.38 (^[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))))))))))),
% 0.18/0.38 inference(bind,[status(th)],[])).
% 0.18/0.38 tff(98,plain,
% 0.18/0.38 (![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))))))))),
% 0.18/0.38 inference(quant_intro,[status(thm)],[97])).
% 0.18/0.38 tff(99,plain,
% 0.18/0.38 (![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))))),
% 0.18/0.38 inference(rewrite,[status(thm)],[])).
% 0.18/0.38 tff(100,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')).
% 0.18/0.38 tff(101,plain,
% 0.18/0.38 (![A: $i, B: $i, C: $i] : ((C = set_union2(A, B)) <=> ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))),
% 0.18/0.38 inference(modus_ponens,[status(thm)],[100, 99])).
% 0.18/0.38 tff(102,plain,(
% 0.18/0.38 ![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))))))),
% 0.18/0.39 inference(skolemize,[status(sab)],[101])).
% 0.18/0.39 tff(103,plain,
% 0.18/0.39 (![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))))))))),
% 0.18/0.39 inference(modus_ponens,[status(thm)],[102, 98])).
% 0.18/0.39 tff(104,plain,
% 0.18/0.39 (![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))))))))),
% 0.18/0.39 inference(modus_ponens,[status(thm)],[103, 96])).
% 0.18/0.39 tff(105,plain,
% 0.18/0.39 (((~![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))))))))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))) <=> ((~![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))))))))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))),
% 0.18/0.39 inference(rewrite,[status(thm)],[])).
% 0.18/0.39 tff(106,plain,
% 0.18/0.39 ((~(~(in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))) <=> (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))),
% 0.18/0.39 inference(rewrite,[status(thm)],[])).
% 0.18/0.39 tff(107,plain,
% 0.18/0.39 (((~(in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))) | $false) <=> (~(in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))),
% 0.18/0.39 inference(rewrite,[status(thm)],[])).
% 0.18/0.39 tff(108,plain,
% 0.18/0.39 (($true | (~(in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13)))))) <=> $true),
% 0.18/0.39 inference(rewrite,[status(thm)],[])).
% 0.18/0.39 tff(109,plain,
% 0.18/0.39 ((~(in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13))))) <=> (~(in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13)))))),
% 0.18/0.39 inference(rewrite,[status(thm)],[])).
% 0.18/0.39 tff(110,plain,
% 0.18/0.39 ((set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13))) <=> $true),
% 0.18/0.39 inference(rewrite,[status(thm)],[])).
% 0.18/0.39 tff(111,plain,
% 0.18/0.39 (((set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13))) | (~(in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13)))))) <=> ($true | (~(in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13))))))),
% 0.18/0.39 inference(monotonicity,[status(thm)],[110, 109])).
% 0.18/0.39 tff(112,plain,
% 0.18/0.39 (((set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13))) | (~(in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13)))))) <=> $true),
% 0.18/0.39 inference(transitivity,[status(thm)],[111, 108])).
% 0.18/0.39 tff(113,plain,
% 0.18/0.39 ((~((set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13))) | (~(in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13))))))) <=> (~$true)),
% 0.18/0.39 inference(monotonicity,[status(thm)],[112])).
% 0.18/0.39 tff(114,plain,
% 0.18/0.39 ((~((set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13))) | (~(in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13))))))) <=> $false),
% 0.18/0.39 inference(transitivity,[status(thm)],[113, 23])).
% 0.18/0.39 tff(115,plain,
% 0.18/0.39 ((~(in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))) <=> (~(in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))),
% 0.18/0.39 inference(rewrite,[status(thm)],[])).
% 0.18/0.39 tff(116,plain,
% 0.18/0.39 (($false | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))) <=> (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))),
% 0.18/0.39 inference(rewrite,[status(thm)],[])).
% 0.18/0.39 tff(117,plain,
% 0.18/0.39 ((~(set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13)))) <=> (~$true)),
% 0.18/0.39 inference(monotonicity,[status(thm)],[110])).
% 0.18/0.39 tff(118,plain,
% 0.18/0.39 ((~(set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13)))) <=> $false),
% 0.18/0.39 inference(transitivity,[status(thm)],[117, 23])).
% 0.18/0.39 tff(119,plain,
% 0.18/0.39 (((~(set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13)))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))) <=> ($false | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))),
% 0.18/0.39 inference(monotonicity,[status(thm)],[118])).
% 0.18/0.39 tff(120,plain,
% 0.18/0.39 (((~(set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13)))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))) <=> (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))),
% 0.18/0.39 inference(transitivity,[status(thm)],[119, 116])).
% 0.18/0.39 tff(121,plain,
% 0.18/0.39 ((~((~(set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13)))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))) <=> (~(in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))),
% 0.18/0.39 inference(monotonicity,[status(thm)],[120])).
% 0.18/0.39 tff(122,plain,
% 0.18/0.39 ((~((~(set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13)))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))) <=> (~(in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))),
% 0.18/0.39 inference(transitivity,[status(thm)],[121, 115])).
% 0.18/0.39 tff(123,plain,
% 0.18/0.39 (((~((~(set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13)))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))) | (~((set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13))) | (~(in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13)))))))) <=> ((~(in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))) | $false)),
% 0.18/0.39 inference(monotonicity,[status(thm)],[122, 114])).
% 0.18/0.39 tff(124,plain,
% 0.18/0.39 (((~((~(set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13)))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))) | (~((set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13))) | (~(in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13)))))))) <=> (~(in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))),
% 0.18/0.39 inference(transitivity,[status(thm)],[123, 107])).
% 0.18/0.39 tff(125,plain,
% 0.18/0.39 ((~((~((~(set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13)))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))) | (~((set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13))) | (~(in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13))))))))) <=> (~(~(in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))))),
% 0.18/0.39 inference(monotonicity,[status(thm)],[124])).
% 0.18/0.39 tff(126,plain,
% 0.18/0.39 ((~((~((~(set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13)))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))) | (~((set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13))) | (~(in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13))))))))) <=> (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))),
% 0.18/0.39 inference(transitivity,[status(thm)],[125, 106])).
% 0.18/0.39 tff(127,plain,
% 0.18/0.39 (((~![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))))))))) | (~((~((~(set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13)))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))) | (~((set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13))) | (~(in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13)))))))))) <=> ((~![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))))))))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))),
% 0.18/0.39 inference(monotonicity,[status(thm)],[126])).
% 0.18/0.39 tff(128,plain,
% 0.18/0.39 (((~![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))))))))) | (~((~((~(set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13)))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))) | (~((set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13))) | (~(in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13)))))))))) <=> ((~![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))))))))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))),
% 0.18/0.40 inference(transitivity,[status(thm)],[127, 105])).
% 0.18/0.40 tff(129,plain,
% 0.18/0.40 ((~![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))))))))) | (~((~((~(set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13)))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))))) | (~((set_union2(B!13, singleton(B!13)) = set_union2(B!13, singleton(B!13))) | (~(in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), set_union2(B!13, singleton(B!13))) <=> (in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), B!13) | in(tptp_fun_D_1(set_union2(B!13, singleton(B!13)), singleton(B!13), B!13), singleton(B!13)))))))))),
% 0.18/0.40 inference(quant_inst,[status(thm)],[])).
% 0.18/0.40 tff(130,plain,
% 0.18/0.40 ((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) | in(tptp_fun_D_1(C, B, A), B))))))))) | (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))),
% 0.18/0.40 inference(modus_ponens,[status(thm)],[129, 128])).
% 0.18/0.40 tff(131,plain,
% 0.18/0.40 (in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13)))),
% 0.18/0.40 inference(unit_resolution,[status(thm)],[130, 104])).
% 0.18/0.40 tff(132,plain,
% 0.18/0.40 ((~(in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))) | (~in(A!14, set_union2(B!13, singleton(B!13)))) | (in(A!14, B!13) | in(A!14, singleton(B!13)))),
% 0.18/0.40 inference(tautology,[status(thm)],[])).
% 0.18/0.40 tff(133,plain,
% 0.18/0.40 ((~in(A!14, set_union2(B!13, singleton(B!13)))) | (in(A!14, B!13) | in(A!14, singleton(B!13)))),
% 0.18/0.40 inference(unit_resolution,[status(thm)],[132, 131])).
% 0.18/0.40 tff(134,plain,
% 0.18/0.40 (~in(A!14, set_union2(B!13, singleton(B!13)))),
% 0.18/0.40 inference(unit_resolution,[status(thm)],[133, 86])).
% 0.18/0.40 tff(135,plain,
% 0.18/0.40 ($false),
% 0.18/0.40 inference(unit_resolution,[status(thm)],[134, 78])).
% 0.18/0.40 tff(136,plain,(in(A!14, B!13) | (A!14 = B!13)), inference(lemma,lemma(discharge,[]))).
% 0.18/0.40 tff(137,plain,
% 0.18/0.40 ((in(A!14, succ(B!13)) <=> (in(A!14, B!13) | (A!14 = B!13))) | (~in(A!14, succ(B!13))) | (~(in(A!14, B!13) | (A!14 = B!13)))),
% 0.18/0.40 inference(tautology,[status(thm)],[])).
% 0.18/0.40 tff(138,plain,
% 0.18/0.40 ((~in(A!14, succ(B!13))) | (~(in(A!14, B!13) | (A!14 = B!13)))),
% 0.18/0.40 inference(unit_resolution,[status(thm)],[137, 74])).
% 0.18/0.40 tff(139,plain,
% 0.18/0.40 (~in(A!14, succ(B!13))),
% 0.18/0.40 inference(unit_resolution,[status(thm)],[138, 136])).
% 0.18/0.40 tff(140,plain,
% 0.18/0.40 (~in(A!14, set_union2(B!13, singleton(B!13)))),
% 0.18/0.40 inference(modus_ponens,[status(thm)],[139, 61])).
% 0.18/0.40 tff(141,plain,
% 0.18/0.40 ((~(in(A!14, set_union2(B!13, singleton(B!13))) <=> (in(A!14, B!13) | in(A!14, singleton(B!13))))) | in(A!14, set_union2(B!13, singleton(B!13))) | (~(in(A!14, B!13) | in(A!14, singleton(B!13))))),
% 0.18/0.40 inference(tautology,[status(thm)],[])).
% 0.18/0.40 tff(142,plain,
% 0.18/0.40 (in(A!14, set_union2(B!13, singleton(B!13))) | (~(in(A!14, B!13) | in(A!14, singleton(B!13))))),
% 0.18/0.40 inference(unit_resolution,[status(thm)],[141, 131])).
% 0.18/0.40 tff(143,plain,
% 0.18/0.40 (~(in(A!14, B!13) | in(A!14, singleton(B!13)))),
% 0.18/0.40 inference(unit_resolution,[status(thm)],[142, 140])).
% 0.18/0.40 tff(144,plain,
% 0.18/0.40 ((in(A!14, B!13) | in(A!14, singleton(B!13))) | (~in(A!14, singleton(B!13)))),
% 0.18/0.40 inference(tautology,[status(thm)],[])).
% 0.18/0.40 tff(145,plain,
% 0.18/0.40 (~in(A!14, singleton(B!13))),
% 0.18/0.40 inference(unit_resolution,[status(thm)],[144, 143])).
% 0.18/0.40 tff(146,plain,
% 0.18/0.40 ((in(A!14, B!13) | in(A!14, singleton(B!13))) | (~in(A!14, B!13))),
% 0.18/0.40 inference(tautology,[status(thm)],[])).
% 0.18/0.40 tff(147,plain,
% 0.18/0.40 (~in(A!14, B!13)),
% 0.18/0.40 inference(unit_resolution,[status(thm)],[146, 143])).
% 0.18/0.40 tff(148,plain,
% 0.18/0.40 ((~(in(A!14, B!13) | (A!14 = B!13))) | in(A!14, B!13) | (A!14 = B!13)),
% 0.18/0.40 inference(tautology,[status(thm)],[])).
% 0.18/0.40 tff(149,plain,
% 0.18/0.40 (A!14 = B!13),
% 0.18/0.40 inference(unit_resolution,[status(thm)],[148, 147, 136])).
% 0.18/0.40 tff(150,plain,
% 0.18/0.40 ((~(in(A!14, singleton(B!13)) <=> (A!14 = B!13))) | in(A!14, singleton(B!13)) | (~(A!14 = B!13))),
% 0.18/0.40 inference(tautology,[status(thm)],[])).
% 0.18/0.40 tff(151,plain,
% 0.18/0.40 ($false),
% 0.18/0.40 inference(unit_resolution,[status(thm)],[150, 149, 145, 48])).
% 0.18/0.40 % SZS output end Proof
% 0.18/0.40 % E exiting
%------------------------------------------------------------------------------