%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SEU145+1 : TPTP v9.0.0. Released v3.3.0.
% 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 05:27:01 AM UTC 2025
% Result : Theorem 0.10s 0.36s
% Output : Proof 0.18s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.11 % Problem : SEU145+1 : TPTP v9.0.0. Released v3.3.0.
% 0.10/0.11 % Command : run_E %s %d THM
% 0.10/0.32 % Computer : n006.cluster.edu
% 0.10/0.32 % Model : x86_64 x86_64
% 0.10/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.32 % Memory : 8042.1875MB
% 0.10/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.10/0.32 % CPULimit : 300
% 0.10/0.32 % WCLimit : 300
% 0.10/0.32 % DateTime : Fri Jun 20 06:07:05 EDT 2025
% 0.10/0.32 % CPUTime :
% 0.10/0.36 % SZS status Theorem
% 0.10/0.36 % SZS output start Proof
% 0.10/0.36 tff(in_type, type, (
% 0.10/0.36 in: ( $i * $i ) > $o)).
% 0.10/0.36 tff(tptp_fun_A_5_type, type, (
% 0.10/0.36 tptp_fun_A_5: $i)).
% 0.10/0.36 tff(tptp_fun_C_1_type, type, (
% 0.10/0.36 tptp_fun_C_1: ( $i * $i ) > $i)).
% 0.10/0.36 tff(set_difference_type, type, (
% 0.10/0.36 set_difference: ( $i * $i ) > $i)).
% 0.10/0.36 tff(singleton_type, type, (
% 0.10/0.36 singleton: $i > $i)).
% 0.10/0.36 tff(tptp_fun_C_3_type, type, (
% 0.10/0.36 tptp_fun_C_3: $i)).
% 0.10/0.36 tff(tptp_fun_B_4_type, type, (
% 0.10/0.36 tptp_fun_B_4: $i)).
% 0.10/0.36 tff(tptp_fun_C_0_type, type, (
% 0.10/0.36 tptp_fun_C_0: ( $i * $i ) > $i)).
% 0.10/0.36 tff(tptp_fun_D_2_type, type, (
% 0.10/0.36 tptp_fun_D_2: ( $i * $i * $i ) > $i)).
% 0.10/0.36 tff(subset_type, type, (
% 0.10/0.36 subset: ( $i * $i ) > $o)).
% 0.10/0.36 tff(1,plain,
% 0.10/0.36 (^[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))))))))),
% 0.10/0.36 inference(bind,[status(th)],[])).
% 0.10/0.36 tff(2,plain,
% 0.10/0.36 (![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)))))))),
% 0.10/0.36 inference(quant_intro,[status(thm)],[1])).
% 0.10/0.36 tff(3,plain,
% 0.10/0.36 (![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)))))))),
% 0.10/0.36 inference(pull_quant,[status(thm)],[])).
% 0.10/0.36 tff(4,plain,
% 0.10/0.36 (^[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)))))))))),
% 0.10/0.37 inference(bind,[status(th)],[])).
% 0.10/0.37 tff(5,plain,
% 0.10/0.37 (![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)))))))),
% 0.10/0.37 inference(quant_intro,[status(thm)],[4])).
% 0.10/0.37 tff(6,plain,
% 0.10/0.37 (![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)))))))),
% 0.10/0.37 inference(transitivity,[status(thm)],[5, 3])).
% 0.10/0.37 tff(7,plain,
% 0.10/0.37 (![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)))))))),
% 0.10/0.37 inference(transitivity,[status(thm)],[6, 2])).
% 0.10/0.37 tff(8,plain,
% 0.10/0.37 (^[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))))))))),
% 0.10/0.37 inference(bind,[status(th)],[])).
% 0.10/0.37 tff(9,plain,
% 0.10/0.37 (![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)))))))),
% 0.10/0.37 inference(quant_intro,[status(thm)],[8])).
% 0.10/0.37 tff(10,plain,
% 0.10/0.37 (![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)))))))),
% 0.10/0.37 inference(transitivity,[status(thm)],[9, 7])).
% 0.10/0.37 tff(11,plain,
% 0.10/0.37 (^[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)))))))))),
% 0.10/0.37 inference(bind,[status(th)],[])).
% 0.10/0.37 tff(12,plain,
% 0.10/0.37 (![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)))))))),
% 0.10/0.37 inference(quant_intro,[status(thm)],[11])).
% 0.10/0.37 tff(13,plain,
% 0.10/0.37 (![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)))),
% 0.10/0.37 inference(rewrite,[status(thm)],[])).
% 0.10/0.37 tff(14,axiom,(![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','d1_tarski')).
% 0.10/0.37 tff(15,plain,
% 0.10/0.37 (![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))),
% 0.10/0.37 inference(modus_ponens,[status(thm)],[14, 13])).
% 0.10/0.37 tff(16,plain,(
% 0.10/0.37 ![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)))))),
% 0.10/0.37 inference(skolemize,[status(sab)],[15])).
% 0.10/0.37 tff(17,plain,
% 0.10/0.37 (![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)))))))),
% 0.10/0.37 inference(modus_ponens,[status(thm)],[16, 12])).
% 0.10/0.37 tff(18,plain,
% 0.10/0.37 (![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)))))))),
% 0.10/0.37 inference(modus_ponens,[status(thm)],[17, 10])).
% 0.10/0.37 tff(19,plain,
% 0.10/0.37 (((~![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(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3))) <=> ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3)))),
% 0.10/0.37 inference(rewrite,[status(thm)],[])).
% 0.10/0.37 tff(20,plain,
% 0.10/0.37 ((~(~(in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3)))) <=> (in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3))),
% 0.10/0.37 inference(rewrite,[status(thm)],[])).
% 0.10/0.37 tff(21,plain,
% 0.10/0.37 (((~(in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3))) | $false) <=> (~(in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3)))),
% 0.10/0.37 inference(rewrite,[status(thm)],[])).
% 0.10/0.37 tff(22,plain,
% 0.10/0.37 ((~$true) <=> $false),
% 0.10/0.37 inference(rewrite,[status(thm)],[])).
% 0.10/0.37 tff(23,plain,
% 0.10/0.37 (($true | (~(in(tptp_fun_C_0(singleton(C!3), C!3), singleton(C!3)) <=> (tptp_fun_C_0(singleton(C!3), C!3) = C!3)))) <=> $true),
% 0.10/0.37 inference(rewrite,[status(thm)],[])).
% 0.10/0.37 tff(24,plain,
% 0.10/0.37 ((~(in(tptp_fun_C_0(singleton(C!3), C!3), singleton(C!3)) <=> (tptp_fun_C_0(singleton(C!3), C!3) = C!3))) <=> (~(in(tptp_fun_C_0(singleton(C!3), C!3), singleton(C!3)) <=> (tptp_fun_C_0(singleton(C!3), C!3) = C!3)))),
% 0.10/0.37 inference(rewrite,[status(thm)],[])).
% 0.10/0.37 tff(25,plain,
% 0.10/0.37 ((singleton(C!3) = singleton(C!3)) <=> $true),
% 0.10/0.37 inference(rewrite,[status(thm)],[])).
% 0.10/0.37 tff(26,plain,
% 0.10/0.37 (((singleton(C!3) = singleton(C!3)) | (~(in(tptp_fun_C_0(singleton(C!3), C!3), singleton(C!3)) <=> (tptp_fun_C_0(singleton(C!3), C!3) = C!3)))) <=> ($true | (~(in(tptp_fun_C_0(singleton(C!3), C!3), singleton(C!3)) <=> (tptp_fun_C_0(singleton(C!3), C!3) = C!3))))),
% 0.10/0.37 inference(monotonicity,[status(thm)],[25, 24])).
% 0.10/0.37 tff(27,plain,
% 0.10/0.37 (((singleton(C!3) = singleton(C!3)) | (~(in(tptp_fun_C_0(singleton(C!3), C!3), singleton(C!3)) <=> (tptp_fun_C_0(singleton(C!3), C!3) = C!3)))) <=> $true),
% 0.10/0.37 inference(transitivity,[status(thm)],[26, 23])).
% 0.10/0.37 tff(28,plain,
% 0.10/0.37 ((~((singleton(C!3) = singleton(C!3)) | (~(in(tptp_fun_C_0(singleton(C!3), C!3), singleton(C!3)) <=> (tptp_fun_C_0(singleton(C!3), C!3) = C!3))))) <=> (~$true)),
% 0.10/0.37 inference(monotonicity,[status(thm)],[27])).
% 0.10/0.37 tff(29,plain,
% 0.10/0.37 ((~((singleton(C!3) = singleton(C!3)) | (~(in(tptp_fun_C_0(singleton(C!3), C!3), singleton(C!3)) <=> (tptp_fun_C_0(singleton(C!3), C!3) = C!3))))) <=> $false),
% 0.10/0.37 inference(transitivity,[status(thm)],[28, 22])).
% 0.10/0.37 tff(30,plain,
% 0.10/0.37 ((~(in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3))) <=> (~(in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3)))),
% 0.10/0.37 inference(rewrite,[status(thm)],[])).
% 0.10/0.37 tff(31,plain,
% 0.10/0.37 (($false | (in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3))) <=> (in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3))),
% 0.10/0.37 inference(rewrite,[status(thm)],[])).
% 0.10/0.37 tff(32,plain,
% 0.10/0.37 ((~(singleton(C!3) = singleton(C!3))) <=> (~$true)),
% 0.10/0.37 inference(monotonicity,[status(thm)],[25])).
% 0.10/0.37 tff(33,plain,
% 0.10/0.37 ((~(singleton(C!3) = singleton(C!3))) <=> $false),
% 0.10/0.37 inference(transitivity,[status(thm)],[32, 22])).
% 0.10/0.37 tff(34,plain,
% 0.10/0.37 (((~(singleton(C!3) = singleton(C!3))) | (in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3))) <=> ($false | (in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3)))),
% 0.10/0.37 inference(monotonicity,[status(thm)],[33])).
% 0.10/0.37 tff(35,plain,
% 0.10/0.37 (((~(singleton(C!3) = singleton(C!3))) | (in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3))) <=> (in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3))),
% 0.10/0.37 inference(transitivity,[status(thm)],[34, 31])).
% 0.10/0.37 tff(36,plain,
% 0.10/0.37 ((~((~(singleton(C!3) = singleton(C!3))) | (in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3)))) <=> (~(in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3)))),
% 0.10/0.37 inference(monotonicity,[status(thm)],[35])).
% 0.10/0.37 tff(37,plain,
% 0.10/0.37 ((~((~(singleton(C!3) = singleton(C!3))) | (in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3)))) <=> (~(in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3)))),
% 0.10/0.37 inference(transitivity,[status(thm)],[36, 30])).
% 0.10/0.37 tff(38,plain,
% 0.10/0.37 (((~((~(singleton(C!3) = singleton(C!3))) | (in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3)))) | (~((singleton(C!3) = singleton(C!3)) | (~(in(tptp_fun_C_0(singleton(C!3), C!3), singleton(C!3)) <=> (tptp_fun_C_0(singleton(C!3), C!3) = C!3)))))) <=> ((~(in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3))) | $false)),
% 0.10/0.37 inference(monotonicity,[status(thm)],[37, 29])).
% 0.10/0.37 tff(39,plain,
% 0.10/0.37 (((~((~(singleton(C!3) = singleton(C!3))) | (in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3)))) | (~((singleton(C!3) = singleton(C!3)) | (~(in(tptp_fun_C_0(singleton(C!3), C!3), singleton(C!3)) <=> (tptp_fun_C_0(singleton(C!3), C!3) = C!3)))))) <=> (~(in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3)))),
% 0.10/0.37 inference(transitivity,[status(thm)],[38, 21])).
% 0.10/0.37 tff(40,plain,
% 0.10/0.37 ((~((~((~(singleton(C!3) = singleton(C!3))) | (in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3)))) | (~((singleton(C!3) = singleton(C!3)) | (~(in(tptp_fun_C_0(singleton(C!3), C!3), singleton(C!3)) <=> (tptp_fun_C_0(singleton(C!3), C!3) = C!3))))))) <=> (~(~(in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3))))),
% 0.10/0.37 inference(monotonicity,[status(thm)],[39])).
% 0.10/0.37 tff(41,plain,
% 0.10/0.37 ((~((~((~(singleton(C!3) = singleton(C!3))) | (in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3)))) | (~((singleton(C!3) = singleton(C!3)) | (~(in(tptp_fun_C_0(singleton(C!3), C!3), singleton(C!3)) <=> (tptp_fun_C_0(singleton(C!3), C!3) = C!3))))))) <=> (in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3))),
% 0.10/0.37 inference(transitivity,[status(thm)],[40, 20])).
% 0.10/0.37 tff(42,plain,
% 0.10/0.37 (((~![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(C!3) = singleton(C!3))) | (in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3)))) | (~((singleton(C!3) = singleton(C!3)) | (~(in(tptp_fun_C_0(singleton(C!3), C!3), singleton(C!3)) <=> (tptp_fun_C_0(singleton(C!3), C!3) = C!3)))))))) <=> ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3)))),
% 0.10/0.37 inference(monotonicity,[status(thm)],[41])).
% 0.10/0.37 tff(43,plain,
% 0.10/0.37 (((~![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(C!3) = singleton(C!3))) | (in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3)))) | (~((singleton(C!3) = singleton(C!3)) | (~(in(tptp_fun_C_0(singleton(C!3), C!3), singleton(C!3)) <=> (tptp_fun_C_0(singleton(C!3), C!3) = C!3)))))))) <=> ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3)))),
% 0.10/0.37 inference(transitivity,[status(thm)],[42, 19])).
% 0.10/0.37 tff(44,plain,
% 0.10/0.37 ((~![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(C!3) = singleton(C!3))) | (in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3)))) | (~((singleton(C!3) = singleton(C!3)) | (~(in(tptp_fun_C_0(singleton(C!3), C!3), singleton(C!3)) <=> (tptp_fun_C_0(singleton(C!3), C!3) = C!3)))))))),
% 0.10/0.38 inference(quant_inst,[status(thm)],[])).
% 0.10/0.38 tff(45,plain,
% 0.10/0.38 ((~![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_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3))),
% 0.10/0.38 inference(modus_ponens,[status(thm)],[44, 43])).
% 0.10/0.38 tff(46,plain,
% 0.10/0.38 (in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3)),
% 0.10/0.38 inference(unit_resolution,[status(thm)],[45, 18])).
% 0.10/0.38 tff(47,plain,
% 0.10/0.38 (^[A: $i, B: $i, C: $i, D: $i] : refl((~((~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C)))))) <=> (~((~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C)))))))),
% 0.10/0.38 inference(bind,[status(th)],[])).
% 0.10/0.38 tff(48,plain,
% 0.10/0.38 (![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C)))))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))))),
% 0.10/0.38 inference(quant_intro,[status(thm)],[47])).
% 0.10/0.38 tff(49,plain,
% 0.10/0.38 (![A: $i, B: $i, C: $i] : ![D: $i] : (~((~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C)))))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))))),
% 0.10/0.38 inference(pull_quant,[status(thm)],[])).
% 0.10/0.38 tff(50,plain,
% 0.10/0.38 (^[A: $i, B: $i, C: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(pull_quant(((~(C = set_difference(A, B))) | ![D: $i] : (~(((~in(D, A)) | in(D, B)) <=> in(D, C)))) <=> ![D: $i] : ((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))), ((~((~(C = set_difference(A, B))) | ![D: $i] : (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) <=> (~![D: $i] : ((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))))), pull_quant((~![D: $i] : ((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) <=> ?[D: $i] : (~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C)))))), ((~((~(C = set_difference(A, B))) | ![D: $i] : (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) <=> ?[D: $i] : (~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))))), (((~((~(C = set_difference(A, B))) | ![D: $i] : (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))) <=> (?[D: $i] : (~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))))), pull_quant((?[D: $i] : (~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))) <=> ?[D: $i] : ((~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C)))))), (((~((~(C = set_difference(A, B))) | ![D: $i] : (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))) <=> ?[D: $i] : ((~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))))), ((~((~((~(C = set_difference(A, B))) | ![D: $i] : (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C)))))) <=> (~?[D: $i] : ((~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C)))))))), pull_quant((~?[D: $i] : ((~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C)))))) <=> ![D: $i] : (~((~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))))), ((~((~((~(C = set_difference(A, B))) | ![D: $i] : (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C)))))) <=> ![D: $i] : (~((~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))))))),
% 0.10/0.38 inference(bind,[status(th)],[])).
% 0.10/0.38 tff(51,plain,
% 0.10/0.38 (![A: $i, B: $i, C: $i] : (~((~((~(C = set_difference(A, B))) | ![D: $i] : (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C)))))) <=> ![A: $i, B: $i, C: $i] : ![D: $i] : (~((~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))))),
% 0.10/0.38 inference(quant_intro,[status(thm)],[50])).
% 0.10/0.38 tff(52,plain,
% 0.10/0.38 (![A: $i, B: $i, C: $i] : (~((~((~(C = set_difference(A, B))) | ![D: $i] : (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C)))))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))))),
% 0.10/0.38 inference(transitivity,[status(thm)],[51, 49])).
% 0.10/0.38 tff(53,plain,
% 0.10/0.38 (![A: $i, B: $i, C: $i] : (~((~((~(C = set_difference(A, B))) | ![D: $i] : (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C)))))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))))),
% 0.10/0.38 inference(transitivity,[status(thm)],[52, 48])).
% 0.10/0.38 tff(54,plain,
% 0.10/0.38 (^[A: $i, B: $i, C: $i] : rewrite((~((~((~(C = set_difference(A, B))) | ![D: $i] : (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C)))))) <=> (~((~((~(C = set_difference(A, B))) | ![D: $i] : (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C)))))))),
% 0.10/0.38 inference(bind,[status(th)],[])).
% 0.10/0.38 tff(55,plain,
% 0.10/0.38 (![A: $i, B: $i, C: $i] : (~((~((~(C = set_difference(A, B))) | ![D: $i] : (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C)))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(C = set_difference(A, B))) | ![D: $i] : (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))))),
% 0.10/0.38 inference(quant_intro,[status(thm)],[54])).
% 0.10/0.38 tff(56,plain,
% 0.10/0.38 (![A: $i, B: $i, C: $i] : (~((~((~(C = set_difference(A, B))) | ![D: $i] : (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C)))))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))))),
% 0.10/0.38 inference(transitivity,[status(thm)],[55, 53])).
% 0.10/0.38 tff(57,plain,
% 0.10/0.38 (^[A: $i, B: $i, C: $i] : trans(monotonicity(rewrite(((~(C = set_difference(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) & (~in(D, B))))) <=> ((~(C = set_difference(A, B))) | ![D: $i] : (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))), rewrite(((C = set_difference(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) & (~in(tptp_fun_D_2(C, B, A), B)))))) <=> ((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C)))), ((((~(C = set_difference(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) & (~in(D, B))))) & ((C = set_difference(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) & (~in(tptp_fun_D_2(C, B, A), B))))))) <=> (((~(C = set_difference(A, B))) | ![D: $i] : (~(((~in(D, A)) | in(D, B)) <=> in(D, C)))) & ((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C)))))), rewrite((((~(C = set_difference(A, B))) | ![D: $i] : (~(((~in(D, A)) | in(D, B)) <=> in(D, C)))) & ((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C)))) <=> (~((~((~(C = set_difference(A, B))) | ![D: $i] : (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))))), ((((~(C = set_difference(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) & (~in(D, B))))) & ((C = set_difference(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) & (~in(tptp_fun_D_2(C, B, A), B))))))) <=> (~((~((~(C = set_difference(A, B))) | ![D: $i] : (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))))))),
% 0.10/0.38 inference(bind,[status(th)],[])).
% 0.10/0.38 tff(58,plain,
% 0.10/0.38 (![A: $i, B: $i, C: $i] : (((~(C = set_difference(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) & (~in(D, B))))) & ((C = set_difference(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) & (~in(tptp_fun_D_2(C, B, A), B))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(C = set_difference(A, B))) | ![D: $i] : (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))))),
% 0.10/0.38 inference(quant_intro,[status(thm)],[57])).
% 0.10/0.38 tff(59,plain,
% 0.10/0.38 (![A: $i, B: $i, C: $i] : ((C = set_difference(A, B)) <=> ![D: $i] : (in(D, C) <=> (in(D, A) & (~in(D, B))))) <=> ![A: $i, B: $i, C: $i] : ((C = set_difference(A, B)) <=> ![D: $i] : (in(D, C) <=> (in(D, A) & (~in(D, B)))))),
% 0.10/0.38 inference(rewrite,[status(thm)],[])).
% 0.10/0.38 tff(60,axiom,(![A: $i, B: $i, C: $i] : ((C = set_difference(A, B)) <=> ![D: $i] : (in(D, C) <=> (in(D, A) & (~in(D, B)))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','d4_xboole_0')).
% 0.10/0.38 tff(61,plain,
% 0.10/0.38 (![A: $i, B: $i, C: $i] : ((C = set_difference(A, B)) <=> ![D: $i] : (in(D, C) <=> (in(D, A) & (~in(D, B)))))),
% 0.10/0.38 inference(modus_ponens,[status(thm)],[60, 59])).
% 0.10/0.38 tff(62,plain,(
% 0.10/0.38 ![A: $i, B: $i, C: $i] : (((~(C = set_difference(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) & (~in(D, B))))) & ((C = set_difference(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) & (~in(tptp_fun_D_2(C, B, A), B)))))))),
% 0.10/0.38 inference(skolemize,[status(sab)],[61])).
% 0.10/0.38 tff(63,plain,
% 0.10/0.38 (![A: $i, B: $i, C: $i] : (~((~((~(C = set_difference(A, B))) | ![D: $i] : (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))))),
% 0.10/0.38 inference(modus_ponens,[status(thm)],[62, 58])).
% 0.10/0.38 tff(64,plain,
% 0.10/0.38 (![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))))),
% 0.10/0.38 inference(modus_ponens,[status(thm)],[63, 56])).
% 0.10/0.38 tff(65,plain,
% 0.10/0.38 (((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))))) | (~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3)))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))))) | (~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3))))))),
% 0.10/0.38 inference(rewrite,[status(thm)],[])).
% 0.10/0.38 tff(66,plain,
% 0.10/0.38 ((~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3))))) <=> (~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3)))))),
% 0.10/0.38 inference(rewrite,[status(thm)],[])).
% 0.10/0.38 tff(67,plain,
% 0.10/0.38 (((((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3)))) | $false) <=> (((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3))))),
% 0.10/0.38 inference(rewrite,[status(thm)],[])).
% 0.10/0.38 tff(68,plain,
% 0.10/0.38 (($true | (((~in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), B!4)) | in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), singleton(C!3))) <=> in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), set_difference(B!4, singleton(C!3))))) <=> $true),
% 0.10/0.39 inference(rewrite,[status(thm)],[])).
% 0.10/0.39 tff(69,plain,
% 0.10/0.39 ((set_difference(B!4, singleton(C!3)) = set_difference(B!4, singleton(C!3))) <=> $true),
% 0.10/0.39 inference(rewrite,[status(thm)],[])).
% 0.10/0.39 tff(70,plain,
% 0.10/0.39 (((set_difference(B!4, singleton(C!3)) = set_difference(B!4, singleton(C!3))) | (((~in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), B!4)) | in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), singleton(C!3))) <=> in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), set_difference(B!4, singleton(C!3))))) <=> ($true | (((~in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), B!4)) | in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), singleton(C!3))) <=> in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), set_difference(B!4, singleton(C!3)))))),
% 0.10/0.39 inference(monotonicity,[status(thm)],[69])).
% 0.10/0.39 tff(71,plain,
% 0.10/0.39 (((set_difference(B!4, singleton(C!3)) = set_difference(B!4, singleton(C!3))) | (((~in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), B!4)) | in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), singleton(C!3))) <=> in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), set_difference(B!4, singleton(C!3))))) <=> $true),
% 0.10/0.39 inference(transitivity,[status(thm)],[70, 68])).
% 0.10/0.39 tff(72,plain,
% 0.10/0.39 ((~((set_difference(B!4, singleton(C!3)) = set_difference(B!4, singleton(C!3))) | (((~in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), B!4)) | in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), singleton(C!3))) <=> in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), set_difference(B!4, singleton(C!3)))))) <=> (~$true)),
% 0.10/0.39 inference(monotonicity,[status(thm)],[71])).
% 0.10/0.39 tff(73,plain,
% 0.10/0.39 ((~((set_difference(B!4, singleton(C!3)) = set_difference(B!4, singleton(C!3))) | (((~in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), B!4)) | in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), singleton(C!3))) <=> in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), set_difference(B!4, singleton(C!3)))))) <=> $false),
% 0.10/0.39 inference(transitivity,[status(thm)],[72, 22])).
% 0.10/0.39 tff(74,plain,
% 0.10/0.39 ((~(~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3)))))) <=> (((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3))))),
% 0.10/0.39 inference(rewrite,[status(thm)],[])).
% 0.10/0.39 tff(75,plain,
% 0.10/0.39 (($false | (~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3)))))) <=> (~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3)))))),
% 0.10/0.39 inference(rewrite,[status(thm)],[])).
% 0.10/0.39 tff(76,plain,
% 0.10/0.39 ((~(set_difference(B!4, singleton(C!3)) = set_difference(B!4, singleton(C!3)))) <=> (~$true)),
% 0.10/0.39 inference(monotonicity,[status(thm)],[69])).
% 0.10/0.39 tff(77,plain,
% 0.10/0.39 ((~(set_difference(B!4, singleton(C!3)) = set_difference(B!4, singleton(C!3)))) <=> $false),
% 0.10/0.39 inference(transitivity,[status(thm)],[76, 22])).
% 0.10/0.39 tff(78,plain,
% 0.10/0.39 (((~(set_difference(B!4, singleton(C!3)) = set_difference(B!4, singleton(C!3)))) | (~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3)))))) <=> ($false | (~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3))))))),
% 0.10/0.39 inference(monotonicity,[status(thm)],[77, 66])).
% 0.10/0.39 tff(79,plain,
% 0.10/0.39 (((~(set_difference(B!4, singleton(C!3)) = set_difference(B!4, singleton(C!3)))) | (~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3)))))) <=> (~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3)))))),
% 0.10/0.39 inference(transitivity,[status(thm)],[78, 75])).
% 0.10/0.39 tff(80,plain,
% 0.10/0.39 ((~((~(set_difference(B!4, singleton(C!3)) = set_difference(B!4, singleton(C!3)))) | (~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3))))))) <=> (~(~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3))))))),
% 0.10/0.39 inference(monotonicity,[status(thm)],[79])).
% 0.10/0.39 tff(81,plain,
% 0.10/0.39 ((~((~(set_difference(B!4, singleton(C!3)) = set_difference(B!4, singleton(C!3)))) | (~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3))))))) <=> (((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3))))),
% 0.10/0.39 inference(transitivity,[status(thm)],[80, 74])).
% 0.10/0.39 tff(82,plain,
% 0.10/0.39 (((~((~(set_difference(B!4, singleton(C!3)) = set_difference(B!4, singleton(C!3)))) | (~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3))))))) | (~((set_difference(B!4, singleton(C!3)) = set_difference(B!4, singleton(C!3))) | (((~in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), B!4)) | in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), singleton(C!3))) <=> in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), set_difference(B!4, singleton(C!3))))))) <=> ((((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3)))) | $false)),
% 0.10/0.39 inference(monotonicity,[status(thm)],[81, 73])).
% 0.10/0.39 tff(83,plain,
% 0.10/0.39 (((~((~(set_difference(B!4, singleton(C!3)) = set_difference(B!4, singleton(C!3)))) | (~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3))))))) | (~((set_difference(B!4, singleton(C!3)) = set_difference(B!4, singleton(C!3))) | (((~in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), B!4)) | in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), singleton(C!3))) <=> in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), set_difference(B!4, singleton(C!3))))))) <=> (((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3))))),
% 0.10/0.39 inference(transitivity,[status(thm)],[82, 67])).
% 0.10/0.39 tff(84,plain,
% 0.10/0.39 ((~((~((~(set_difference(B!4, singleton(C!3)) = set_difference(B!4, singleton(C!3)))) | (~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3))))))) | (~((set_difference(B!4, singleton(C!3)) = set_difference(B!4, singleton(C!3))) | (((~in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), B!4)) | in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), singleton(C!3))) <=> in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), set_difference(B!4, singleton(C!3)))))))) <=> (~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3)))))),
% 0.10/0.39 inference(monotonicity,[status(thm)],[83])).
% 0.10/0.39 tff(85,plain,
% 0.10/0.39 ((~((~((~(set_difference(B!4, singleton(C!3)) = set_difference(B!4, singleton(C!3)))) | (~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3))))))) | (~((set_difference(B!4, singleton(C!3)) = set_difference(B!4, singleton(C!3))) | (((~in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), B!4)) | in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), singleton(C!3))) <=> in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), set_difference(B!4, singleton(C!3)))))))) <=> (~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3)))))),
% 0.10/0.39 inference(transitivity,[status(thm)],[84, 66])).
% 0.10/0.39 tff(86,plain,
% 0.10/0.39 (((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))))) | (~((~((~(set_difference(B!4, singleton(C!3)) = set_difference(B!4, singleton(C!3)))) | (~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3))))))) | (~((set_difference(B!4, singleton(C!3)) = set_difference(B!4, singleton(C!3))) | (((~in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), B!4)) | in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), singleton(C!3))) <=> in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), set_difference(B!4, singleton(C!3))))))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))))) | (~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3))))))),
% 0.10/0.40 inference(monotonicity,[status(thm)],[85])).
% 0.10/0.40 tff(87,plain,
% 0.10/0.40 (((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))))) | (~((~((~(set_difference(B!4, singleton(C!3)) = set_difference(B!4, singleton(C!3)))) | (~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3))))))) | (~((set_difference(B!4, singleton(C!3)) = set_difference(B!4, singleton(C!3))) | (((~in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), B!4)) | in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), singleton(C!3))) <=> in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), set_difference(B!4, singleton(C!3))))))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))))) | (~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3))))))),
% 0.10/0.40 inference(transitivity,[status(thm)],[86, 65])).
% 0.10/0.40 tff(88,plain,
% 0.10/0.40 ((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))))) | (~((~((~(set_difference(B!4, singleton(C!3)) = set_difference(B!4, singleton(C!3)))) | (~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3))))))) | (~((set_difference(B!4, singleton(C!3)) = set_difference(B!4, singleton(C!3))) | (((~in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), B!4)) | in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), singleton(C!3))) <=> in(tptp_fun_D_2(set_difference(B!4, singleton(C!3)), singleton(C!3), B!4), set_difference(B!4, singleton(C!3))))))))),
% 0.10/0.40 inference(quant_inst,[status(thm)],[])).
% 0.10/0.40 tff(89,plain,
% 0.10/0.40 ((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_2(C, B, A), A)) | in(tptp_fun_D_2(C, B, A), B)) <=> in(tptp_fun_D_2(C, B, A), C))))))) | (~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3)))))),
% 0.10/0.40 inference(modus_ponens,[status(thm)],[88, 87])).
% 0.10/0.40 tff(90,plain,
% 0.10/0.40 (~(((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3))))),
% 0.10/0.40 inference(unit_resolution,[status(thm)],[89, 64])).
% 0.10/0.40 tff(91,plain,
% 0.10/0.40 (^[A: $i, B: $i] : refl((~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_1(B, A), A)) | in(tptp_fun_C_1(B, A), B))))))) <=> (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_1(B, A), A)) | in(tptp_fun_C_1(B, A), B))))))))),
% 0.10/0.40 inference(bind,[status(th)],[])).
% 0.10/0.40 tff(92,plain,
% 0.10/0.40 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_1(B, A), A)) | in(tptp_fun_C_1(B, A), B))))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_1(B, A), A)) | in(tptp_fun_C_1(B, A), B)))))))),
% 0.10/0.40 inference(quant_intro,[status(thm)],[91])).
% 0.10/0.40 tff(93,plain,
% 0.10/0.40 (^[A: $i, B: $i] : rewrite((~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_1(B, A), A)) | in(tptp_fun_C_1(B, A), B))))))) <=> (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_1(B, A), A)) | in(tptp_fun_C_1(B, A), B))))))))),
% 0.10/0.40 inference(bind,[status(th)],[])).
% 0.10/0.40 tff(94,plain,
% 0.10/0.40 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_1(B, A), A)) | in(tptp_fun_C_1(B, A), B))))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_1(B, A), A)) | in(tptp_fun_C_1(B, A), B)))))))),
% 0.10/0.40 inference(quant_intro,[status(thm)],[93])).
% 0.10/0.40 tff(95,plain,
% 0.10/0.40 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_1(B, A), A)) | in(tptp_fun_C_1(B, A), B))))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_1(B, A), A)) | in(tptp_fun_C_1(B, A), B)))))))),
% 0.10/0.40 inference(transitivity,[status(thm)],[94, 92])).
% 0.10/0.40 tff(96,plain,
% 0.10/0.40 (^[A: $i, B: $i] : trans(monotonicity(rewrite(((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) <=> ((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))), rewrite((subset(A, B) | (~((~in(tptp_fun_C_1(B, A), A)) | in(tptp_fun_C_1(B, A), B)))) <=> (subset(A, B) | (~((~in(tptp_fun_C_1(B, A), A)) | in(tptp_fun_C_1(B, A), B))))), ((((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) & (subset(A, B) | (~((~in(tptp_fun_C_1(B, A), A)) | in(tptp_fun_C_1(B, A), B))))) <=> (((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) & (subset(A, B) | (~((~in(tptp_fun_C_1(B, A), A)) | in(tptp_fun_C_1(B, A), B))))))), rewrite((((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) & (subset(A, B) | (~((~in(tptp_fun_C_1(B, A), A)) | in(tptp_fun_C_1(B, A), B))))) <=> (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_1(B, A), A)) | in(tptp_fun_C_1(B, A), B)))))))), ((((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) & (subset(A, B) | (~((~in(tptp_fun_C_1(B, A), A)) | in(tptp_fun_C_1(B, A), B))))) <=> (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_1(B, A), A)) | in(tptp_fun_C_1(B, A), B)))))))))),
% 0.10/0.40 inference(bind,[status(th)],[])).
% 0.10/0.40 tff(97,plain,
% 0.10/0.40 (![A: $i, B: $i] : (((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) & (subset(A, B) | (~((~in(tptp_fun_C_1(B, A), A)) | in(tptp_fun_C_1(B, A), B))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_1(B, A), A)) | in(tptp_fun_C_1(B, A), B)))))))),
% 0.10/0.40 inference(quant_intro,[status(thm)],[96])).
% 0.10/0.40 tff(98,plain,
% 0.10/0.40 (![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : ((~in(C, A)) | in(C, B))) <=> ![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : ((~in(C, A)) | in(C, B)))),
% 0.10/0.40 inference(rewrite,[status(thm)],[])).
% 0.10/0.40 tff(99,plain,
% 0.10/0.40 (^[A: $i, B: $i] : rewrite((subset(A, B) <=> ![C: $i] : (in(C, A) => in(C, B))) <=> (subset(A, B) <=> ![C: $i] : ((~in(C, A)) | in(C, B))))),
% 0.10/0.40 inference(bind,[status(th)],[])).
% 0.10/0.40 tff(100,plain,
% 0.10/0.40 (![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : (in(C, A) => in(C, B))) <=> ![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : ((~in(C, A)) | in(C, B)))),
% 0.10/0.40 inference(quant_intro,[status(thm)],[99])).
% 0.10/0.40 tff(101,axiom,(![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : (in(C, A) => in(C, B)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','d3_tarski')).
% 0.10/0.40 tff(102,plain,
% 0.10/0.40 (![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : ((~in(C, A)) | in(C, B)))),
% 0.10/0.40 inference(modus_ponens,[status(thm)],[101, 100])).
% 0.10/0.40 tff(103,plain,
% 0.10/0.40 (![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : ((~in(C, A)) | in(C, B)))),
% 0.10/0.40 inference(modus_ponens,[status(thm)],[102, 98])).
% 0.10/0.40 tff(104,plain,(
% 0.10/0.40 ![A: $i, B: $i] : (((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) & (subset(A, B) | (~((~in(tptp_fun_C_1(B, A), A)) | in(tptp_fun_C_1(B, A), B)))))),
% 0.10/0.40 inference(skolemize,[status(sab)],[103])).
% 0.10/0.40 tff(105,plain,
% 0.10/0.40 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_1(B, A), A)) | in(tptp_fun_C_1(B, A), B)))))))),
% 0.10/0.40 inference(modus_ponens,[status(thm)],[104, 97])).
% 0.10/0.40 tff(106,plain,
% 0.10/0.40 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_1(B, A), A)) | in(tptp_fun_C_1(B, A), B)))))))),
% 0.10/0.40 inference(modus_ponens,[status(thm)],[105, 95])).
% 0.10/0.40 tff(107,plain,
% 0.10/0.40 ((~![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_1(B, A), A)) | in(tptp_fun_C_1(B, A), B)))))))) | (~((~((~subset(A!5, set_difference(B!4, singleton(C!3)))) | ![C: $i] : ((~in(C, A!5)) | in(C, set_difference(B!4, singleton(C!3)))))) | (~(subset(A!5, set_difference(B!4, singleton(C!3))) | (~((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), A!5)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3)))))))))),
% 0.10/0.40 inference(quant_inst,[status(thm)],[])).
% 0.10/0.40 tff(108,plain,
% 0.10/0.40 (~((~((~subset(A!5, set_difference(B!4, singleton(C!3)))) | ![C: $i] : ((~in(C, A!5)) | in(C, set_difference(B!4, singleton(C!3)))))) | (~(subset(A!5, set_difference(B!4, singleton(C!3))) | (~((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), A!5)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3))))))))),
% 0.10/0.40 inference(unit_resolution,[status(thm)],[107, 106])).
% 0.10/0.40 tff(109,plain,
% 0.10/0.40 (((~((~subset(A!5, set_difference(B!4, singleton(C!3)))) | ![C: $i] : ((~in(C, A!5)) | in(C, set_difference(B!4, singleton(C!3)))))) | (~(subset(A!5, set_difference(B!4, singleton(C!3))) | (~((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), A!5)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3)))))))) | (subset(A!5, set_difference(B!4, singleton(C!3))) | (~((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), A!5)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3))))))),
% 0.10/0.40 inference(tautology,[status(thm)],[])).
% 0.10/0.40 tff(110,plain,
% 0.10/0.40 (subset(A!5, set_difference(B!4, singleton(C!3))) | (~((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), A!5)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3)))))),
% 0.10/0.40 inference(unit_resolution,[status(thm)],[109, 108])).
% 0.10/0.40 tff(111,plain,
% 0.10/0.40 ((~(in(C!3, A!5) | subset(A!5, set_difference(B!4, singleton(C!3))) | (~subset(A!5, B!4)))) <=> (~((~subset(A!5, B!4)) | subset(A!5, set_difference(B!4, singleton(C!3))) | in(C!3, A!5)))),
% 0.10/0.40 inference(rewrite,[status(thm)],[])).
% 0.10/0.40 tff(112,plain,
% 0.10/0.40 ((~![A: $i, B: $i, C: $i] : (in(C, A) | subset(A, set_difference(B, singleton(C))) | (~subset(A, B)))) <=> (~![A: $i, B: $i, C: $i] : (in(C, A) | subset(A, set_difference(B, singleton(C))) | (~subset(A, B))))),
% 0.10/0.40 inference(rewrite,[status(thm)],[])).
% 0.10/0.40 tff(113,plain,
% 0.10/0.40 ((~![A: $i, B: $i, C: $i] : (subset(A, B) => (in(C, A) | subset(A, set_difference(B, singleton(C)))))) <=> (~![A: $i, B: $i, C: $i] : (in(C, A) | subset(A, set_difference(B, singleton(C))) | (~subset(A, B))))),
% 0.10/0.40 inference(rewrite,[status(thm)],[])).
% 0.10/0.40 tff(114,axiom,(~![A: $i, B: $i, C: $i] : (subset(A, B) => (in(C, A) | subset(A, set_difference(B, singleton(C)))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','l3_zfmisc_1')).
% 0.10/0.40 tff(115,plain,
% 0.10/0.40 (~![A: $i, B: $i, C: $i] : (in(C, A) | subset(A, set_difference(B, singleton(C))) | (~subset(A, B)))),
% 0.18/0.40 inference(modus_ponens,[status(thm)],[114, 113])).
% 0.18/0.40 tff(116,plain,
% 0.18/0.40 (~![A: $i, B: $i, C: $i] : (in(C, A) | subset(A, set_difference(B, singleton(C))) | (~subset(A, B)))),
% 0.18/0.40 inference(modus_ponens,[status(thm)],[115, 112])).
% 0.18/0.40 tff(117,plain,
% 0.18/0.40 (~![A: $i, B: $i, C: $i] : (in(C, A) | subset(A, set_difference(B, singleton(C))) | (~subset(A, B)))),
% 0.18/0.40 inference(modus_ponens,[status(thm)],[116, 112])).
% 0.18/0.40 tff(118,plain,
% 0.18/0.40 (~![A: $i, B: $i, C: $i] : (in(C, A) | subset(A, set_difference(B, singleton(C))) | (~subset(A, B)))),
% 0.18/0.40 inference(modus_ponens,[status(thm)],[117, 112])).
% 0.18/0.40 tff(119,plain,(
% 0.18/0.40 ~(in(C!3, A!5) | subset(A!5, set_difference(B!4, singleton(C!3))) | (~subset(A!5, B!4)))),
% 0.18/0.40 inference(skolemize,[status(sab)],[118])).
% 0.18/0.40 tff(120,plain,
% 0.18/0.40 (~((~subset(A!5, B!4)) | subset(A!5, set_difference(B!4, singleton(C!3))) | in(C!3, A!5))),
% 0.18/0.40 inference(modus_ponens,[status(thm)],[119, 111])).
% 0.18/0.40 tff(121,plain,
% 0.18/0.40 (~subset(A!5, set_difference(B!4, singleton(C!3)))),
% 0.18/0.40 inference(or_elim,[status(thm)],[120])).
% 0.18/0.40 tff(122,plain,
% 0.18/0.40 ((~(subset(A!5, set_difference(B!4, singleton(C!3))) | (~((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), A!5)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3))))))) | subset(A!5, set_difference(B!4, singleton(C!3))) | (~((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), A!5)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3)))))),
% 0.18/0.40 inference(tautology,[status(thm)],[])).
% 0.18/0.40 tff(123,plain,
% 0.18/0.40 ((~(subset(A!5, set_difference(B!4, singleton(C!3))) | (~((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), A!5)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3))))))) | (~((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), A!5)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3)))))),
% 0.18/0.40 inference(unit_resolution,[status(thm)],[122, 121])).
% 0.18/0.40 tff(124,plain,
% 0.18/0.40 (~((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), A!5)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3))))),
% 0.18/0.40 inference(unit_resolution,[status(thm)],[123, 110])).
% 0.18/0.40 tff(125,plain,
% 0.18/0.40 (((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), A!5)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3)))) | (~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3))))),
% 0.18/0.40 inference(tautology,[status(thm)],[])).
% 0.18/0.40 tff(126,plain,
% 0.18/0.40 (~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3)))),
% 0.18/0.40 inference(unit_resolution,[status(thm)],[125, 124])).
% 0.18/0.40 tff(127,plain,
% 0.18/0.40 ((((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3)))) | ((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3)))),
% 0.18/0.40 inference(tautology,[status(thm)],[])).
% 0.18/0.40 tff(128,plain,
% 0.18/0.40 ((((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3)))) | ((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)))),
% 0.18/0.40 inference(unit_resolution,[status(thm)],[127, 126])).
% 0.18/0.40 tff(129,plain,
% 0.18/0.40 ((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))),
% 0.18/0.40 inference(unit_resolution,[status(thm)],[128, 90])).
% 0.18/0.41 tff(130,plain,
% 0.18/0.41 (((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), A!5)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), set_difference(B!4, singleton(C!3)))) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), A!5)),
% 0.18/0.41 inference(tautology,[status(thm)],[])).
% 0.18/0.41 tff(131,plain,
% 0.18/0.41 (in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), A!5)),
% 0.18/0.41 inference(unit_resolution,[status(thm)],[130, 124])).
% 0.18/0.41 tff(132,plain,
% 0.18/0.41 ((~![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_1(B, A), A)) | in(tptp_fun_C_1(B, A), B)))))))) | (~((~((~subset(A!5, B!4)) | ![C: $i] : ((~in(C, A!5)) | in(C, B!4)))) | (~(subset(A!5, B!4) | (~((~in(tptp_fun_C_1(B!4, A!5), A!5)) | in(tptp_fun_C_1(B!4, A!5), B!4)))))))),
% 0.18/0.41 inference(quant_inst,[status(thm)],[])).
% 0.18/0.41 tff(133,plain,
% 0.18/0.41 (~((~((~subset(A!5, B!4)) | ![C: $i] : ((~in(C, A!5)) | in(C, B!4)))) | (~(subset(A!5, B!4) | (~((~in(tptp_fun_C_1(B!4, A!5), A!5)) | in(tptp_fun_C_1(B!4, A!5), B!4))))))),
% 0.18/0.41 inference(unit_resolution,[status(thm)],[132, 106])).
% 0.18/0.41 tff(134,plain,
% 0.18/0.41 (((~((~subset(A!5, B!4)) | ![C: $i] : ((~in(C, A!5)) | in(C, B!4)))) | (~(subset(A!5, B!4) | (~((~in(tptp_fun_C_1(B!4, A!5), A!5)) | in(tptp_fun_C_1(B!4, A!5), B!4)))))) | ((~subset(A!5, B!4)) | ![C: $i] : ((~in(C, A!5)) | in(C, B!4)))),
% 0.18/0.41 inference(tautology,[status(thm)],[])).
% 0.18/0.41 tff(135,plain,
% 0.18/0.41 ((~subset(A!5, B!4)) | ![C: $i] : ((~in(C, A!5)) | in(C, B!4))),
% 0.18/0.41 inference(unit_resolution,[status(thm)],[134, 133])).
% 0.18/0.41 tff(136,plain,
% 0.18/0.41 (subset(A!5, B!4)),
% 0.18/0.41 inference(or_elim,[status(thm)],[120])).
% 0.18/0.41 tff(137,plain,
% 0.18/0.41 ((~((~subset(A!5, B!4)) | ![C: $i] : ((~in(C, A!5)) | in(C, B!4)))) | (~subset(A!5, B!4)) | ![C: $i] : ((~in(C, A!5)) | in(C, B!4))),
% 0.18/0.41 inference(tautology,[status(thm)],[])).
% 0.18/0.41 tff(138,plain,
% 0.18/0.41 ((~((~subset(A!5, B!4)) | ![C: $i] : ((~in(C, A!5)) | in(C, B!4)))) | ![C: $i] : ((~in(C, A!5)) | in(C, B!4))),
% 0.18/0.41 inference(unit_resolution,[status(thm)],[137, 136])).
% 0.18/0.41 tff(139,plain,
% 0.18/0.41 (![C: $i] : ((~in(C, A!5)) | in(C, B!4))),
% 0.18/0.41 inference(unit_resolution,[status(thm)],[138, 135])).
% 0.18/0.41 tff(140,plain,
% 0.18/0.41 (((~![C: $i] : ((~in(C, A!5)) | in(C, B!4))) | ((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), A!5)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4))) <=> ((~![C: $i] : ((~in(C, A!5)) | in(C, B!4))) | (~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), A!5)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4))),
% 0.18/0.41 inference(rewrite,[status(thm)],[])).
% 0.18/0.41 tff(141,plain,
% 0.18/0.41 ((~![C: $i] : ((~in(C, A!5)) | in(C, B!4))) | ((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), A!5)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4))),
% 0.18/0.41 inference(quant_inst,[status(thm)],[])).
% 0.18/0.41 tff(142,plain,
% 0.18/0.41 ((~![C: $i] : ((~in(C, A!5)) | in(C, B!4))) | (~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), A!5)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)),
% 0.18/0.41 inference(modus_ponens,[status(thm)],[141, 140])).
% 0.18/0.41 tff(143,plain,
% 0.18/0.41 (in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)),
% 0.18/0.41 inference(unit_resolution,[status(thm)],[142, 139, 131])).
% 0.18/0.41 tff(144,plain,
% 0.18/0.41 ((~((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)))) | (~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))),
% 0.18/0.41 inference(tautology,[status(thm)],[])).
% 0.18/0.41 tff(145,plain,
% 0.18/0.41 ((~((~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), B!4)) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)))) | in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))),
% 0.18/0.41 inference(unit_resolution,[status(thm)],[144, 143])).
% 0.18/0.41 tff(146,plain,
% 0.18/0.41 (in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))),
% 0.18/0.41 inference(unit_resolution,[status(thm)],[145, 129])).
% 0.18/0.41 tff(147,plain,
% 0.18/0.41 ((~(in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3))) | (~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3))) | (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3)),
% 0.18/0.41 inference(tautology,[status(thm)],[])).
% 0.18/0.41 tff(148,plain,
% 0.18/0.41 ((~(in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), singleton(C!3)) <=> (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3))) | (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3)),
% 0.18/0.41 inference(unit_resolution,[status(thm)],[147, 146])).
% 0.18/0.41 tff(149,plain,
% 0.18/0.41 (tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5) = C!3),
% 0.18/0.41 inference(unit_resolution,[status(thm)],[148, 46])).
% 0.18/0.41 tff(150,plain,
% 0.18/0.41 (in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), A!5) <=> in(C!3, A!5)),
% 0.18/0.41 inference(monotonicity,[status(thm)],[149])).
% 0.18/0.41 tff(151,plain,
% 0.18/0.41 (in(C!3, A!5) <=> in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), A!5)),
% 0.18/0.41 inference(symmetry,[status(thm)],[150])).
% 0.18/0.41 tff(152,plain,
% 0.18/0.41 ((~in(C!3, A!5)) <=> (~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), A!5))),
% 0.18/0.41 inference(monotonicity,[status(thm)],[151])).
% 0.18/0.41 tff(153,plain,
% 0.18/0.41 (~in(C!3, A!5)),
% 0.18/0.41 inference(or_elim,[status(thm)],[120])).
% 0.18/0.41 tff(154,plain,
% 0.18/0.41 (~in(tptp_fun_C_1(set_difference(B!4, singleton(C!3)), A!5), A!5)),
% 0.18/0.41 inference(modus_ponens,[status(thm)],[153, 152])).
% 0.18/0.41 tff(155,plain,
% 0.18/0.41 ($false),
% 0.18/0.41 inference(unit_resolution,[status(thm)],[131, 154])).
% 0.18/0.41 % SZS output end Proof
% 0.18/0.41 % E exiting
%------------------------------------------------------------------------------