%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET975+1 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n013.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:26:44 AM UTC 2025
% Result : Theorem 0.12s 0.37s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : SET975+1 : TPTP v9.0.0. Released v3.2.0.
% 0.07/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n013.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Jun 20 05:53:53 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.12/0.37 % SZS status Theorem
% 0.12/0.37 % SZS output start Proof
% 0.12/0.37 tff(tptp_fun_C_4_type, type, (
% 0.12/0.37 tptp_fun_C_4: $i)).
% 0.12/0.37 tff(tptp_fun_A_6_type, type, (
% 0.12/0.37 tptp_fun_A_6: $i)).
% 0.12/0.37 tff(in_type, type, (
% 0.12/0.37 in: ( $i * $i ) > $o)).
% 0.12/0.37 tff(singleton_type, type, (
% 0.12/0.37 singleton: $i > $i)).
% 0.12/0.37 tff(tptp_fun_D_3_type, type, (
% 0.12/0.37 tptp_fun_D_3: $i)).
% 0.12/0.37 tff(tptp_fun_B_5_type, type, (
% 0.12/0.37 tptp_fun_B_5: $i)).
% 0.12/0.37 tff(cartesian_product2_type, type, (
% 0.12/0.37 cartesian_product2: ( $i * $i ) > $i)).
% 0.12/0.37 tff(ordered_pair_type, type, (
% 0.12/0.37 ordered_pair: ( $i * $i ) > $i)).
% 0.12/0.37 tff(tptp_fun_C_0_type, type, (
% 0.12/0.37 tptp_fun_C_0: ( $i * $i ) > $i)).
% 0.12/0.37 tff(1,assumption,(~in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3))), introduced(assumption)).
% 0.12/0.37 tff(2,plain,
% 0.12/0.37 ((~(~(((~(A!6 = C!4)) | (~in(B!5, D!3))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3))))) <=> (((~(A!6 = C!4)) | (~in(B!5, D!3))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)))),
% 0.12/0.37 inference(rewrite,[status(thm)],[])).
% 0.12/0.37 tff(3,plain,
% 0.12/0.37 ((~(((~(A!6 = C!4)) | (~in(B!5, D!3))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)))) <=> (~(((~(A!6 = C!4)) | (~in(B!5, D!3))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3))))),
% 0.12/0.37 inference(rewrite,[status(thm)],[])).
% 0.12/0.37 tff(4,plain,
% 0.12/0.37 ((in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)) <=> (~((~(A!6 = C!4)) | (~in(B!5, D!3))))) <=> (~(((~(A!6 = C!4)) | (~in(B!5, D!3))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3))))),
% 0.12/0.37 inference(rewrite,[status(thm)],[])).
% 0.12/0.37 tff(5,plain,
% 0.12/0.37 (((A!6 = C!4) & in(B!5, D!3)) <=> (~((~(A!6 = C!4)) | (~in(B!5, D!3))))),
% 0.12/0.37 inference(rewrite,[status(thm)],[])).
% 0.12/0.37 tff(6,plain,
% 0.12/0.37 ((in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)) <=> ((A!6 = C!4) & in(B!5, D!3))) <=> (in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)) <=> (~((~(A!6 = C!4)) | (~in(B!5, D!3)))))),
% 0.12/0.37 inference(monotonicity,[status(thm)],[5])).
% 0.12/0.37 tff(7,plain,
% 0.12/0.37 ((in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)) <=> ((A!6 = C!4) & in(B!5, D!3))) <=> (~(((~(A!6 = C!4)) | (~in(B!5, D!3))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3))))),
% 0.12/0.38 inference(transitivity,[status(thm)],[6, 4])).
% 0.12/0.38 tff(8,plain,
% 0.12/0.38 ((in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)) <=> ((A!6 = C!4) & in(B!5, D!3))) <=> (~(((~(A!6 = C!4)) | (~in(B!5, D!3))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3))))),
% 0.12/0.38 inference(transitivity,[status(thm)],[7, 3])).
% 0.12/0.38 tff(9,plain,
% 0.12/0.38 ((~(in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)) <=> ((A!6 = C!4) & in(B!5, D!3)))) <=> (~(~(((~(A!6 = C!4)) | (~in(B!5, D!3))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)))))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[8])).
% 0.12/0.38 tff(10,plain,
% 0.12/0.38 ((~(in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)) <=> ((A!6 = C!4) & in(B!5, D!3)))) <=> (((~(A!6 = C!4)) | (~in(B!5, D!3))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)))),
% 0.12/0.38 inference(transitivity,[status(thm)],[9, 2])).
% 0.12/0.38 tff(11,plain,
% 0.12/0.38 ((~(in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)) <=> ((A!6 = C!4) & in(B!5, D!3)))) <=> (~(in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)) <=> ((A!6 = C!4) & in(B!5, D!3))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(12,plain,
% 0.12/0.38 ((~![A: $i, B: $i, C: $i, D: $i] : (in(ordered_pair(A, B), cartesian_product2(singleton(C), D)) <=> ((A = C) & in(B, D)))) <=> (~![A: $i, B: $i, C: $i, D: $i] : (in(ordered_pair(A, B), cartesian_product2(singleton(C), D)) <=> ((A = C) & in(B, D))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(13,axiom,(~![A: $i, B: $i, C: $i, D: $i] : (in(ordered_pair(A, B), cartesian_product2(singleton(C), D)) <=> ((A = C) & in(B, D)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t128_zfmisc_1')).
% 0.12/0.38 tff(14,plain,
% 0.12/0.38 (~![A: $i, B: $i, C: $i, D: $i] : (in(ordered_pair(A, B), cartesian_product2(singleton(C), D)) <=> ((A = C) & in(B, D)))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[13, 12])).
% 0.12/0.38 tff(15,plain,
% 0.12/0.38 (~![A: $i, B: $i, C: $i, D: $i] : (in(ordered_pair(A, B), cartesian_product2(singleton(C), D)) <=> ((A = C) & in(B, D)))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[14, 12])).
% 0.12/0.38 tff(16,plain,
% 0.12/0.38 (~![A: $i, B: $i, C: $i, D: $i] : (in(ordered_pair(A, B), cartesian_product2(singleton(C), D)) <=> ((A = C) & in(B, D)))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[15, 12])).
% 0.12/0.38 tff(17,plain,
% 0.12/0.38 (~![A: $i, B: $i, C: $i, D: $i] : (in(ordered_pair(A, B), cartesian_product2(singleton(C), D)) <=> ((A = C) & in(B, D)))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[16, 12])).
% 0.12/0.38 tff(18,plain,(
% 0.12/0.38 ~(in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)) <=> ((A!6 = C!4) & in(B!5, D!3)))),
% 0.12/0.38 inference(skolemize,[status(sab)],[17])).
% 0.12/0.38 tff(19,plain,
% 0.12/0.38 (~(in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)) <=> ((A!6 = C!4) & in(B!5, D!3)))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[18, 11])).
% 0.12/0.38 tff(20,plain,
% 0.12/0.38 (((~(A!6 = C!4)) | (~in(B!5, D!3))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[19, 10])).
% 0.12/0.38 tff(21,plain,
% 0.12/0.38 ((~((~(A!6 = C!4)) | (~in(B!5, D!3)))) | in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)) | (~(((~(A!6 = C!4)) | (~in(B!5, D!3))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3))))),
% 0.12/0.38 inference(tautology,[status(thm)],[])).
% 0.12/0.38 tff(22,plain,
% 0.12/0.38 ((~((~(A!6 = C!4)) | (~in(B!5, D!3)))) | in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[21, 20])).
% 0.12/0.38 tff(23,plain,
% 0.12/0.38 (~((~(A!6 = C!4)) | (~in(B!5, D!3)))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[22, 1])).
% 0.12/0.38 tff(24,plain,
% 0.12/0.38 (((~(A!6 = C!4)) | (~in(B!5, D!3))) | (A!6 = C!4)),
% 0.12/0.38 inference(tautology,[status(thm)],[])).
% 0.12/0.38 tff(25,plain,
% 0.12/0.38 (A!6 = C!4),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[24, 23])).
% 0.12/0.38 tff(26,plain,
% 0.12/0.38 (C!4 = A!6),
% 0.12/0.38 inference(symmetry,[status(thm)],[25])).
% 0.12/0.38 tff(27,plain,
% 0.12/0.38 (in(C!4, singleton(C!4)) <=> in(A!6, singleton(C!4))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[26])).
% 0.12/0.38 tff(28,plain,
% 0.12/0.38 (in(A!6, singleton(C!4)) <=> in(C!4, singleton(C!4))),
% 0.12/0.38 inference(symmetry,[status(thm)],[27])).
% 0.12/0.38 tff(29,plain,
% 0.12/0.38 ((~in(A!6, singleton(C!4))) <=> (~in(C!4, singleton(C!4)))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[28])).
% 0.12/0.38 tff(30,plain,
% 0.12/0.38 (^[A: $i, B: $i, C: $i, D: $i] : refl((~(((~in(A, C)) | (~in(B, D))) <=> in(ordered_pair(A, B), cartesian_product2(C, D)))) <=> (~(((~in(A, C)) | (~in(B, D))) <=> in(ordered_pair(A, B), cartesian_product2(C, D)))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(31,plain,
% 0.12/0.38 (![A: $i, B: $i, C: $i, D: $i] : (~(((~in(A, C)) | (~in(B, D))) <=> in(ordered_pair(A, B), cartesian_product2(C, D)))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~(((~in(A, C)) | (~in(B, D))) <=> in(ordered_pair(A, B), cartesian_product2(C, D))))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[30])).
% 0.12/0.38 tff(32,plain,
% 0.12/0.38 (^[A: $i, B: $i, C: $i, D: $i] : trans(trans(monotonicity(rewrite((in(A, C) & in(B, D)) <=> (~((~in(A, C)) | (~in(B, D))))), ((in(ordered_pair(A, B), cartesian_product2(C, D)) <=> (in(A, C) & in(B, D))) <=> (in(ordered_pair(A, B), cartesian_product2(C, D)) <=> (~((~in(A, C)) | (~in(B, D))))))), rewrite((in(ordered_pair(A, B), cartesian_product2(C, D)) <=> (~((~in(A, C)) | (~in(B, D))))) <=> (~(((~in(A, C)) | (~in(B, D))) <=> in(ordered_pair(A, B), cartesian_product2(C, D))))), ((in(ordered_pair(A, B), cartesian_product2(C, D)) <=> (in(A, C) & in(B, D))) <=> (~(((~in(A, C)) | (~in(B, D))) <=> in(ordered_pair(A, B), cartesian_product2(C, D)))))), rewrite((~(((~in(A, C)) | (~in(B, D))) <=> in(ordered_pair(A, B), cartesian_product2(C, D)))) <=> (~(((~in(A, C)) | (~in(B, D))) <=> in(ordered_pair(A, B), cartesian_product2(C, D))))), ((in(ordered_pair(A, B), cartesian_product2(C, D)) <=> (in(A, C) & in(B, D))) <=> (~(((~in(A, C)) | (~in(B, D))) <=> in(ordered_pair(A, B), cartesian_product2(C, D))))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(33,plain,
% 0.12/0.38 (![A: $i, B: $i, C: $i, D: $i] : (in(ordered_pair(A, B), cartesian_product2(C, D)) <=> (in(A, C) & in(B, D))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~(((~in(A, C)) | (~in(B, D))) <=> in(ordered_pair(A, B), cartesian_product2(C, D))))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[32])).
% 0.12/0.38 tff(34,plain,
% 0.12/0.38 (![A: $i, B: $i, C: $i, D: $i] : (in(ordered_pair(A, B), cartesian_product2(C, D)) <=> (in(A, C) & in(B, D))) <=> ![A: $i, B: $i, C: $i, D: $i] : (in(ordered_pair(A, B), cartesian_product2(C, D)) <=> (in(A, C) & in(B, D)))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(35,axiom,(![A: $i, B: $i, C: $i, D: $i] : (in(ordered_pair(A, B), cartesian_product2(C, D)) <=> (in(A, C) & in(B, D)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','l55_zfmisc_1')).
% 0.12/0.38 tff(36,plain,
% 0.12/0.38 (![A: $i, B: $i, C: $i, D: $i] : (in(ordered_pair(A, B), cartesian_product2(C, D)) <=> (in(A, C) & in(B, D)))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.12/0.38 tff(37,plain,(
% 0.12/0.38 ![A: $i, B: $i, C: $i, D: $i] : (in(ordered_pair(A, B), cartesian_product2(C, D)) <=> (in(A, C) & in(B, D)))),
% 0.12/0.38 inference(skolemize,[status(sab)],[36])).
% 0.12/0.38 tff(38,plain,
% 0.12/0.38 (![A: $i, B: $i, C: $i, D: $i] : (~(((~in(A, C)) | (~in(B, D))) <=> in(ordered_pair(A, B), cartesian_product2(C, D))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[37, 33])).
% 0.12/0.38 tff(39,plain,
% 0.12/0.38 (![A: $i, B: $i, C: $i, D: $i] : (~(((~in(A, C)) | (~in(B, D))) <=> in(ordered_pair(A, B), cartesian_product2(C, D))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[38, 31])).
% 0.12/0.38 tff(40,plain,
% 0.12/0.38 (((~![A: $i, B: $i, C: $i, D: $i] : (~(((~in(A, C)) | (~in(B, D))) <=> in(ordered_pair(A, B), cartesian_product2(C, D))))) | (~(((~in(B!5, D!3)) | (~in(A!6, singleton(C!4)))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : (~(((~in(A, C)) | (~in(B, D))) <=> in(ordered_pair(A, B), cartesian_product2(C, D))))) | (~(((~in(B!5, D!3)) | (~in(A!6, singleton(C!4)))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(41,plain,
% 0.12/0.38 ((~(((~in(B!5, D!3)) | (~in(A!6, singleton(C!4)))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)))) <=> (~(((~in(B!5, D!3)) | (~in(A!6, singleton(C!4)))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(42,plain,
% 0.12/0.38 ((((~in(A!6, singleton(C!4))) | (~in(B!5, D!3))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3))) <=> (((~in(B!5, D!3)) | (~in(A!6, singleton(C!4)))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(43,plain,
% 0.12/0.38 ((~(((~in(A!6, singleton(C!4))) | (~in(B!5, D!3))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)))) <=> (~(((~in(B!5, D!3)) | (~in(A!6, singleton(C!4)))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3))))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[42])).
% 0.12/0.38 tff(44,plain,
% 0.12/0.38 ((~(((~in(A!6, singleton(C!4))) | (~in(B!5, D!3))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)))) <=> (~(((~in(B!5, D!3)) | (~in(A!6, singleton(C!4)))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3))))),
% 0.12/0.38 inference(transitivity,[status(thm)],[43, 41])).
% 0.12/0.38 tff(45,plain,
% 0.12/0.38 (((~![A: $i, B: $i, C: $i, D: $i] : (~(((~in(A, C)) | (~in(B, D))) <=> in(ordered_pair(A, B), cartesian_product2(C, D))))) | (~(((~in(A!6, singleton(C!4))) | (~in(B!5, D!3))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : (~(((~in(A, C)) | (~in(B, D))) <=> in(ordered_pair(A, B), cartesian_product2(C, D))))) | (~(((~in(B!5, D!3)) | (~in(A!6, singleton(C!4)))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)))))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[44])).
% 0.12/0.38 tff(46,plain,
% 0.12/0.38 (((~![A: $i, B: $i, C: $i, D: $i] : (~(((~in(A, C)) | (~in(B, D))) <=> in(ordered_pair(A, B), cartesian_product2(C, D))))) | (~(((~in(A!6, singleton(C!4))) | (~in(B!5, D!3))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : (~(((~in(A, C)) | (~in(B, D))) <=> in(ordered_pair(A, B), cartesian_product2(C, D))))) | (~(((~in(B!5, D!3)) | (~in(A!6, singleton(C!4)))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)))))),
% 0.12/0.38 inference(transitivity,[status(thm)],[45, 40])).
% 0.12/0.38 tff(47,plain,
% 0.12/0.38 ((~![A: $i, B: $i, C: $i, D: $i] : (~(((~in(A, C)) | (~in(B, D))) <=> in(ordered_pair(A, B), cartesian_product2(C, D))))) | (~(((~in(A!6, singleton(C!4))) | (~in(B!5, D!3))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3))))),
% 0.12/0.38 inference(quant_inst,[status(thm)],[])).
% 0.12/0.38 tff(48,plain,
% 0.12/0.38 ((~![A: $i, B: $i, C: $i, D: $i] : (~(((~in(A, C)) | (~in(B, D))) <=> in(ordered_pair(A, B), cartesian_product2(C, D))))) | (~(((~in(B!5, D!3)) | (~in(A!6, singleton(C!4)))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[47, 46])).
% 0.12/0.38 tff(49,plain,
% 0.12/0.38 (~(((~in(B!5, D!3)) | (~in(A!6, singleton(C!4)))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[48, 39])).
% 0.12/0.38 tff(50,plain,
% 0.12/0.38 ((((~in(B!5, D!3)) | (~in(A!6, singleton(C!4)))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3))) | ((~in(B!5, D!3)) | (~in(A!6, singleton(C!4)))) | in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3))),
% 0.12/0.38 inference(tautology,[status(thm)],[])).
% 0.12/0.38 tff(51,plain,
% 0.12/0.38 (((~in(B!5, D!3)) | (~in(A!6, singleton(C!4)))) | in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[50, 49])).
% 0.12/0.38 tff(52,plain,
% 0.12/0.38 ((~in(B!5, D!3)) | (~in(A!6, singleton(C!4)))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[51, 1])).
% 0.12/0.38 tff(53,plain,
% 0.12/0.38 (((~(A!6 = C!4)) | (~in(B!5, D!3))) | in(B!5, D!3)),
% 0.12/0.38 inference(tautology,[status(thm)],[])).
% 0.12/0.38 tff(54,plain,
% 0.12/0.38 (in(B!5, D!3)),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[53, 23])).
% 0.12/0.38 tff(55,plain,
% 0.12/0.38 ((~((~in(B!5, D!3)) | (~in(A!6, singleton(C!4))))) | (~in(B!5, D!3)) | (~in(A!6, singleton(C!4)))),
% 0.12/0.38 inference(tautology,[status(thm)],[])).
% 0.12/0.38 tff(56,plain,
% 0.12/0.38 (~in(A!6, singleton(C!4))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[55, 54, 52])).
% 0.12/0.38 tff(57,plain,
% 0.12/0.38 (~in(C!4, singleton(C!4))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[56, 29])).
% 0.12/0.38 tff(58,plain,
% 0.12/0.38 (^[A: $i, B: $i, C: $i] : refl((~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(59,plain,
% 0.12/0.38 (![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[58])).
% 0.12/0.38 tff(60,plain,
% 0.12/0.38 (![A: $i, B: $i] : ![C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.38 inference(pull_quant,[status(thm)],[])).
% 0.12/0.38 tff(61,plain,
% 0.12/0.38 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(pull_quant(((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) <=> ![C: $i] : ((~(B = singleton(A))) | (in(C, B) <=> (C = A)))), ((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) <=> (~![C: $i] : ((~(B = singleton(A))) | (in(C, B) <=> (C = A)))))), pull_quant((~![C: $i] : ((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) <=> ?[C: $i] : (~((~(B = singleton(A))) | (in(C, B) <=> (C = A))))), ((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) <=> ?[C: $i] : (~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))))), (((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))) <=> (?[C: $i] : (~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))), pull_quant((?[C: $i] : (~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))) <=> ?[C: $i] : ((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))), (((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))) <=> ?[C: $i] : ((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))), ((~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> (~?[C: $i] : ((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))))), pull_quant((~?[C: $i] : ((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))), ((~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(62,plain,
% 0.19/0.39 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i] : ![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)))))))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[61])).
% 0.19/0.39 tff(63,plain,
% 0.19/0.39 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i, 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)))))))),
% 0.19/0.39 inference(transitivity,[status(thm)],[62, 60])).
% 0.19/0.39 tff(64,plain,
% 0.19/0.39 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i, 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)))))))),
% 0.19/0.39 inference(transitivity,[status(thm)],[63, 59])).
% 0.19/0.39 tff(65,plain,
% 0.19/0.39 (^[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))))))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(66,plain,
% 0.19/0.39 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[65])).
% 0.19/0.39 tff(67,plain,
% 0.19/0.39 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i, 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)))))))),
% 0.19/0.39 inference(transitivity,[status(thm)],[66, 64])).
% 0.19/0.39 tff(68,plain,
% 0.19/0.39 (^[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)))))))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(69,plain,
% 0.19/0.39 (![A: $i, B: $i] : (((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))) <=> ![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[68])).
% 0.19/0.39 tff(70,plain,
% 0.19/0.39 (![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A))) <=> ![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(71,axiom,(![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','d1_tarski')).
% 0.19/0.39 tff(72,plain,
% 0.19/0.39 (![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[71, 70])).
% 0.19/0.39 tff(73,plain,(
% 0.19/0.39 ![A: $i, B: $i] : (((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))),
% 0.19/0.39 inference(skolemize,[status(sab)],[72])).
% 0.19/0.39 tff(74,plain,
% 0.19/0.39 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[73, 69])).
% 0.19/0.39 tff(75,plain,
% 0.19/0.39 (![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[74, 67])).
% 0.19/0.39 tff(76,plain,
% 0.19/0.39 (((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | in(C!4, singleton(C!4))) <=> ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | in(C!4, singleton(C!4)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(77,plain,
% 0.19/0.39 ((~(~in(C!4, singleton(C!4)))) <=> in(C!4, singleton(C!4))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(78,plain,
% 0.19/0.39 (((~in(C!4, singleton(C!4))) | $false) <=> (~in(C!4, singleton(C!4)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(79,plain,
% 0.19/0.39 ((~$true) <=> $false),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(80,plain,
% 0.19/0.39 (($true | (~(in(tptp_fun_C_0(singleton(C!4), C!4), singleton(C!4)) <=> (tptp_fun_C_0(singleton(C!4), C!4) = C!4)))) <=> $true),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(81,plain,
% 0.19/0.39 ((~(in(tptp_fun_C_0(singleton(C!4), C!4), singleton(C!4)) <=> (tptp_fun_C_0(singleton(C!4), C!4) = C!4))) <=> (~(in(tptp_fun_C_0(singleton(C!4), C!4), singleton(C!4)) <=> (tptp_fun_C_0(singleton(C!4), C!4) = C!4)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(82,plain,
% 0.19/0.39 ((singleton(C!4) = singleton(C!4)) <=> $true),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(83,plain,
% 0.19/0.39 (((singleton(C!4) = singleton(C!4)) | (~(in(tptp_fun_C_0(singleton(C!4), C!4), singleton(C!4)) <=> (tptp_fun_C_0(singleton(C!4), C!4) = C!4)))) <=> ($true | (~(in(tptp_fun_C_0(singleton(C!4), C!4), singleton(C!4)) <=> (tptp_fun_C_0(singleton(C!4), C!4) = C!4))))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[82, 81])).
% 0.19/0.39 tff(84,plain,
% 0.19/0.39 (((singleton(C!4) = singleton(C!4)) | (~(in(tptp_fun_C_0(singleton(C!4), C!4), singleton(C!4)) <=> (tptp_fun_C_0(singleton(C!4), C!4) = C!4)))) <=> $true),
% 0.19/0.39 inference(transitivity,[status(thm)],[83, 80])).
% 0.19/0.39 tff(85,plain,
% 0.19/0.39 ((~((singleton(C!4) = singleton(C!4)) | (~(in(tptp_fun_C_0(singleton(C!4), C!4), singleton(C!4)) <=> (tptp_fun_C_0(singleton(C!4), C!4) = C!4))))) <=> (~$true)),
% 0.19/0.39 inference(monotonicity,[status(thm)],[84])).
% 0.19/0.39 tff(86,plain,
% 0.19/0.39 ((~((singleton(C!4) = singleton(C!4)) | (~(in(tptp_fun_C_0(singleton(C!4), C!4), singleton(C!4)) <=> (tptp_fun_C_0(singleton(C!4), C!4) = C!4))))) <=> $false),
% 0.19/0.39 inference(transitivity,[status(thm)],[85, 79])).
% 0.19/0.39 tff(87,plain,
% 0.19/0.39 ((~((~(singleton(C!4) = singleton(C!4))) | (in(C!4, singleton(C!4)) <=> (C!4 = C!4)))) <=> (~in(C!4, singleton(C!4)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(88,plain,
% 0.19/0.39 (((~((~(singleton(C!4) = singleton(C!4))) | (in(C!4, singleton(C!4)) <=> (C!4 = C!4)))) | (~((singleton(C!4) = singleton(C!4)) | (~(in(tptp_fun_C_0(singleton(C!4), C!4), singleton(C!4)) <=> (tptp_fun_C_0(singleton(C!4), C!4) = C!4)))))) <=> ((~in(C!4, singleton(C!4))) | $false)),
% 0.19/0.39 inference(monotonicity,[status(thm)],[87, 86])).
% 0.19/0.39 tff(89,plain,
% 0.19/0.39 (((~((~(singleton(C!4) = singleton(C!4))) | (in(C!4, singleton(C!4)) <=> (C!4 = C!4)))) | (~((singleton(C!4) = singleton(C!4)) | (~(in(tptp_fun_C_0(singleton(C!4), C!4), singleton(C!4)) <=> (tptp_fun_C_0(singleton(C!4), C!4) = C!4)))))) <=> (~in(C!4, singleton(C!4)))),
% 0.19/0.39 inference(transitivity,[status(thm)],[88, 78])).
% 0.19/0.39 tff(90,plain,
% 0.19/0.39 ((~((~((~(singleton(C!4) = singleton(C!4))) | (in(C!4, singleton(C!4)) <=> (C!4 = C!4)))) | (~((singleton(C!4) = singleton(C!4)) | (~(in(tptp_fun_C_0(singleton(C!4), C!4), singleton(C!4)) <=> (tptp_fun_C_0(singleton(C!4), C!4) = C!4))))))) <=> (~(~in(C!4, singleton(C!4))))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[89])).
% 0.19/0.39 tff(91,plain,
% 0.19/0.39 ((~((~((~(singleton(C!4) = singleton(C!4))) | (in(C!4, singleton(C!4)) <=> (C!4 = C!4)))) | (~((singleton(C!4) = singleton(C!4)) | (~(in(tptp_fun_C_0(singleton(C!4), C!4), singleton(C!4)) <=> (tptp_fun_C_0(singleton(C!4), C!4) = C!4))))))) <=> in(C!4, singleton(C!4))),
% 0.19/0.39 inference(transitivity,[status(thm)],[90, 77])).
% 0.19/0.39 tff(92,plain,
% 0.19/0.39 (((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (~((~((~(singleton(C!4) = singleton(C!4))) | (in(C!4, singleton(C!4)) <=> (C!4 = C!4)))) | (~((singleton(C!4) = singleton(C!4)) | (~(in(tptp_fun_C_0(singleton(C!4), C!4), singleton(C!4)) <=> (tptp_fun_C_0(singleton(C!4), C!4) = C!4)))))))) <=> ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | in(C!4, singleton(C!4)))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[91])).
% 0.19/0.39 tff(93,plain,
% 0.19/0.39 (((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (~((~((~(singleton(C!4) = singleton(C!4))) | (in(C!4, singleton(C!4)) <=> (C!4 = C!4)))) | (~((singleton(C!4) = singleton(C!4)) | (~(in(tptp_fun_C_0(singleton(C!4), C!4), singleton(C!4)) <=> (tptp_fun_C_0(singleton(C!4), C!4) = C!4)))))))) <=> ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | in(C!4, singleton(C!4)))),
% 0.19/0.39 inference(transitivity,[status(thm)],[92, 76])).
% 0.19/0.39 tff(94,plain,
% 0.19/0.39 ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (~((~((~(singleton(C!4) = singleton(C!4))) | (in(C!4, singleton(C!4)) <=> (C!4 = C!4)))) | (~((singleton(C!4) = singleton(C!4)) | (~(in(tptp_fun_C_0(singleton(C!4), C!4), singleton(C!4)) <=> (tptp_fun_C_0(singleton(C!4), C!4) = C!4)))))))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(95,plain,
% 0.19/0.39 ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | in(C!4, singleton(C!4))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[94, 93])).
% 0.19/0.39 tff(96,plain,
% 0.19/0.39 (in(C!4, singleton(C!4))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[95, 75])).
% 0.19/0.39 tff(97,plain,
% 0.19/0.39 ($false),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[96, 57])).
% 0.19/0.39 tff(98,plain,(in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.39 tff(99,plain,
% 0.19/0.39 ((((~in(B!5, D!3)) | (~in(A!6, singleton(C!4)))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3))) | (~((~in(B!5, D!3)) | (~in(A!6, singleton(C!4))))) | (~in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)))),
% 0.19/0.39 inference(tautology,[status(thm)],[])).
% 0.19/0.39 tff(100,plain,
% 0.19/0.39 ((~((~in(B!5, D!3)) | (~in(A!6, singleton(C!4))))) | (~in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[99, 49])).
% 0.19/0.39 tff(101,plain,
% 0.19/0.39 (~((~in(B!5, D!3)) | (~in(A!6, singleton(C!4))))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[100, 98])).
% 0.19/0.39 tff(102,plain,
% 0.19/0.39 (((~in(B!5, D!3)) | (~in(A!6, singleton(C!4)))) | in(A!6, singleton(C!4))),
% 0.19/0.39 inference(tautology,[status(thm)],[])).
% 0.19/0.39 tff(103,plain,
% 0.19/0.39 (in(A!6, singleton(C!4))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[102, 101])).
% 0.19/0.39 tff(104,plain,
% 0.19/0.39 (((~(A!6 = C!4)) | (~in(B!5, D!3))) | (~in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3))) | (~(((~(A!6 = C!4)) | (~in(B!5, D!3))) <=> in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3))))),
% 0.19/0.39 inference(tautology,[status(thm)],[])).
% 0.19/0.39 tff(105,plain,
% 0.19/0.39 (((~(A!6 = C!4)) | (~in(B!5, D!3))) | (~in(ordered_pair(A!6, B!5), cartesian_product2(singleton(C!4), D!3)))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[104, 20])).
% 0.19/0.39 tff(106,plain,
% 0.19/0.39 ((~(A!6 = C!4)) | (~in(B!5, D!3))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[105, 98])).
% 0.19/0.39 tff(107,plain,
% 0.19/0.39 (((~in(B!5, D!3)) | (~in(A!6, singleton(C!4)))) | in(B!5, D!3)),
% 0.19/0.39 inference(tautology,[status(thm)],[])).
% 0.19/0.39 tff(108,plain,
% 0.19/0.39 (in(B!5, D!3)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[107, 101])).
% 0.19/0.39 tff(109,plain,
% 0.19/0.39 ((~((~(A!6 = C!4)) | (~in(B!5, D!3)))) | (~(A!6 = C!4)) | (~in(B!5, D!3))),
% 0.19/0.39 inference(tautology,[status(thm)],[])).
% 0.19/0.39 tff(110,plain,
% 0.19/0.39 (~(A!6 = C!4)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[109, 108, 106])).
% 0.19/0.40 tff(111,plain,
% 0.19/0.40 ((~(in(A!6, singleton(C!4)) <=> (A!6 = C!4))) | (~in(A!6, singleton(C!4))) | (A!6 = C!4)),
% 0.19/0.40 inference(tautology,[status(thm)],[])).
% 0.19/0.40 tff(112,plain,
% 0.19/0.40 (~(in(A!6, singleton(C!4)) <=> (A!6 = C!4))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[111, 110, 103])).
% 0.19/0.40 tff(113,plain,
% 0.19/0.40 (((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(A!6, singleton(C!4)) <=> (A!6 = C!4))) <=> ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(A!6, singleton(C!4)) <=> (A!6 = C!4)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(114,plain,
% 0.19/0.40 ((~(~(in(A!6, singleton(C!4)) <=> (A!6 = C!4)))) <=> (in(A!6, singleton(C!4)) <=> (A!6 = C!4))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(115,plain,
% 0.19/0.40 (((~(in(A!6, singleton(C!4)) <=> (A!6 = C!4))) | $false) <=> (~(in(A!6, singleton(C!4)) <=> (A!6 = C!4)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(116,plain,
% 0.19/0.40 ((~(in(A!6, singleton(C!4)) <=> (A!6 = C!4))) <=> (~(in(A!6, singleton(C!4)) <=> (A!6 = C!4)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(117,plain,
% 0.19/0.40 (($false | (in(A!6, singleton(C!4)) <=> (A!6 = C!4))) <=> (in(A!6, singleton(C!4)) <=> (A!6 = C!4))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(118,plain,
% 0.19/0.40 ((~(singleton(C!4) = singleton(C!4))) <=> (~$true)),
% 0.19/0.40 inference(monotonicity,[status(thm)],[82])).
% 0.19/0.40 tff(119,plain,
% 0.19/0.40 ((~(singleton(C!4) = singleton(C!4))) <=> $false),
% 0.19/0.40 inference(transitivity,[status(thm)],[118, 79])).
% 0.19/0.40 tff(120,plain,
% 0.19/0.40 (((~(singleton(C!4) = singleton(C!4))) | (in(A!6, singleton(C!4)) <=> (A!6 = C!4))) <=> ($false | (in(A!6, singleton(C!4)) <=> (A!6 = C!4)))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[119])).
% 0.19/0.40 tff(121,plain,
% 0.19/0.40 (((~(singleton(C!4) = singleton(C!4))) | (in(A!6, singleton(C!4)) <=> (A!6 = C!4))) <=> (in(A!6, singleton(C!4)) <=> (A!6 = C!4))),
% 0.19/0.40 inference(transitivity,[status(thm)],[120, 117])).
% 0.19/0.40 tff(122,plain,
% 0.19/0.40 ((~((~(singleton(C!4) = singleton(C!4))) | (in(A!6, singleton(C!4)) <=> (A!6 = C!4)))) <=> (~(in(A!6, singleton(C!4)) <=> (A!6 = C!4)))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[121])).
% 0.19/0.40 tff(123,plain,
% 0.19/0.40 ((~((~(singleton(C!4) = singleton(C!4))) | (in(A!6, singleton(C!4)) <=> (A!6 = C!4)))) <=> (~(in(A!6, singleton(C!4)) <=> (A!6 = C!4)))),
% 0.19/0.40 inference(transitivity,[status(thm)],[122, 116])).
% 0.19/0.40 tff(124,plain,
% 0.19/0.40 (((~((~(singleton(C!4) = singleton(C!4))) | (in(A!6, singleton(C!4)) <=> (A!6 = C!4)))) | (~((singleton(C!4) = singleton(C!4)) | (~(in(tptp_fun_C_0(singleton(C!4), C!4), singleton(C!4)) <=> (tptp_fun_C_0(singleton(C!4), C!4) = C!4)))))) <=> ((~(in(A!6, singleton(C!4)) <=> (A!6 = C!4))) | $false)),
% 0.19/0.40 inference(monotonicity,[status(thm)],[123, 86])).
% 0.19/0.40 tff(125,plain,
% 0.19/0.40 (((~((~(singleton(C!4) = singleton(C!4))) | (in(A!6, singleton(C!4)) <=> (A!6 = C!4)))) | (~((singleton(C!4) = singleton(C!4)) | (~(in(tptp_fun_C_0(singleton(C!4), C!4), singleton(C!4)) <=> (tptp_fun_C_0(singleton(C!4), C!4) = C!4)))))) <=> (~(in(A!6, singleton(C!4)) <=> (A!6 = C!4)))),
% 0.19/0.40 inference(transitivity,[status(thm)],[124, 115])).
% 0.19/0.40 tff(126,plain,
% 0.19/0.40 ((~((~((~(singleton(C!4) = singleton(C!4))) | (in(A!6, singleton(C!4)) <=> (A!6 = C!4)))) | (~((singleton(C!4) = singleton(C!4)) | (~(in(tptp_fun_C_0(singleton(C!4), C!4), singleton(C!4)) <=> (tptp_fun_C_0(singleton(C!4), C!4) = C!4))))))) <=> (~(~(in(A!6, singleton(C!4)) <=> (A!6 = C!4))))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[125])).
% 0.19/0.40 tff(127,plain,
% 0.19/0.40 ((~((~((~(singleton(C!4) = singleton(C!4))) | (in(A!6, singleton(C!4)) <=> (A!6 = C!4)))) | (~((singleton(C!4) = singleton(C!4)) | (~(in(tptp_fun_C_0(singleton(C!4), C!4), singleton(C!4)) <=> (tptp_fun_C_0(singleton(C!4), C!4) = C!4))))))) <=> (in(A!6, singleton(C!4)) <=> (A!6 = C!4))),
% 0.19/0.40 inference(transitivity,[status(thm)],[126, 114])).
% 0.19/0.40 tff(128,plain,
% 0.19/0.40 (((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (~((~((~(singleton(C!4) = singleton(C!4))) | (in(A!6, singleton(C!4)) <=> (A!6 = C!4)))) | (~((singleton(C!4) = singleton(C!4)) | (~(in(tptp_fun_C_0(singleton(C!4), C!4), singleton(C!4)) <=> (tptp_fun_C_0(singleton(C!4), C!4) = C!4)))))))) <=> ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(A!6, singleton(C!4)) <=> (A!6 = C!4)))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[127])).
% 0.19/0.40 tff(129,plain,
% 0.19/0.40 (((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (~((~((~(singleton(C!4) = singleton(C!4))) | (in(A!6, singleton(C!4)) <=> (A!6 = C!4)))) | (~((singleton(C!4) = singleton(C!4)) | (~(in(tptp_fun_C_0(singleton(C!4), C!4), singleton(C!4)) <=> (tptp_fun_C_0(singleton(C!4), C!4) = C!4)))))))) <=> ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(A!6, singleton(C!4)) <=> (A!6 = C!4)))),
% 0.19/0.40 inference(transitivity,[status(thm)],[128, 113])).
% 0.19/0.40 tff(130,plain,
% 0.19/0.40 ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (~((~((~(singleton(C!4) = singleton(C!4))) | (in(A!6, singleton(C!4)) <=> (A!6 = C!4)))) | (~((singleton(C!4) = singleton(C!4)) | (~(in(tptp_fun_C_0(singleton(C!4), C!4), singleton(C!4)) <=> (tptp_fun_C_0(singleton(C!4), C!4) = C!4)))))))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(131,plain,
% 0.19/0.40 ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(A!6, singleton(C!4)) <=> (A!6 = C!4))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[130, 129])).
% 0.19/0.40 tff(132,plain,
% 0.19/0.40 ($false),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[131, 75, 112])).
% 0.19/0.40 % SZS output end Proof
% 0.19/0.40 % E exiting
%------------------------------------------------------------------------------