%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET921+1 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n010.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 05:26:40 AM UTC 2025
% Result : Theorem 0.12s 0.39s
% Output : Proof 0.22s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : SET921+1 : TPTP v9.0.0. Released v3.2.0.
% 0.07/0.13 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n010.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Fri Jun 20 05:52:26 EDT 2025
% 0.12/0.35 % CPUTime :
% 0.12/0.39 % SZS status Theorem
% 0.12/0.39 % SZS output start Proof
% 0.12/0.39 tff(in_type, type, (
% 0.12/0.39 in: ( $i * $i ) > $o)).
% 0.12/0.39 tff(set_difference_type, type, (
% 0.12/0.39 set_difference: ( $i * $i ) > $i)).
% 0.12/0.39 tff(singleton_type, type, (
% 0.12/0.39 singleton: $i > $i)).
% 0.12/0.39 tff(tptp_fun_C_4_type, type, (
% 0.12/0.39 tptp_fun_C_4: $i)).
% 0.12/0.39 tff(tptp_fun_B_5_type, type, (
% 0.12/0.39 tptp_fun_B_5: $i)).
% 0.12/0.39 tff(tptp_fun_A_6_type, type, (
% 0.12/0.39 tptp_fun_A_6: $i)).
% 0.12/0.39 tff(tptp_fun_D_1_type, type, (
% 0.12/0.39 tptp_fun_D_1: ( $i * $i * $i ) > $i)).
% 0.12/0.39 tff(tptp_fun_C_0_type, type, (
% 0.12/0.39 tptp_fun_C_0: ( $i * $i ) > $i)).
% 0.12/0.39 tff(1,assumption,(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4)))), introduced(assumption)).
% 0.12/0.39 tff(2,plain,
% 0.12/0.39 (^[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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(C, B, A), C)))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(3,plain,
% 0.12/0.39 (![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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(C, B, A), C))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[2])).
% 0.12/0.39 tff(4,plain,
% 0.12/0.39 (![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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(C, B, A), C))))))),
% 0.12/0.39 inference(pull_quant,[status(thm)],[])).
% 0.12/0.39 tff(5,plain,
% 0.12/0.39 (^[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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(C, B, A), C))))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(6,plain,
% 0.12/0.39 (![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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(C, B, A), C))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[5])).
% 0.12/0.39 tff(7,plain,
% 0.12/0.39 (![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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(C, B, A), C))))))),
% 0.12/0.39 inference(transitivity,[status(thm)],[6, 4])).
% 0.12/0.39 tff(8,plain,
% 0.12/0.39 (![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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(C, B, A), C))))))),
% 0.12/0.39 inference(transitivity,[status(thm)],[7, 3])).
% 0.12/0.39 tff(9,plain,
% 0.12/0.39 (^[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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(C, B, A), C)))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(10,plain,
% 0.12/0.39 (![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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(C, B, A), C))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[9])).
% 0.12/0.39 tff(11,plain,
% 0.12/0.39 (![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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(C, B, A), C))))))),
% 0.12/0.39 inference(transitivity,[status(thm)],[10, 8])).
% 0.12/0.39 tff(12,plain,
% 0.12/0.39 (^[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_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) & (~in(tptp_fun_D_1(C, B, A), B)))))) <=> ((C = set_difference(A, B)) | (((~in(tptp_fun_D_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(C, B, A), C)))), ((((~(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_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) & (~in(tptp_fun_D_1(C, B, A), B))))))) <=> (((~(C = set_difference(A, B))) | ![D: $i] : (~(((~in(D, A)) | in(D, B)) <=> in(D, C)))) & ((C = set_difference(A, B)) | (((~in(tptp_fun_D_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(C, B, A), C)))))), 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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(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_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) & (~in(tptp_fun_D_1(C, B, A), B))))))) <=> (~((~((~(C = set_difference(A, B))) | ![D: $i] : (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(C, B, A), C))))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(13,plain,
% 0.12/0.39 (![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_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) & (~in(tptp_fun_D_1(C, B, A), B))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(C = set_difference(A, B))) | ![D: $i] : (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(C, B, A), C))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[12])).
% 0.12/0.39 tff(14,plain,
% 0.12/0.39 (![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)))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(15,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')).
% 0.12/0.39 tff(16,plain,
% 0.12/0.39 (![A: $i, B: $i, C: $i] : ((C = set_difference(A, B)) <=> ![D: $i] : (in(D, C) <=> (in(D, A) & (~in(D, B)))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[15, 14])).
% 0.12/0.39 tff(17,plain,(
% 0.12/0.39 ![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_1(C, B, A), C) <=> (in(tptp_fun_D_1(C, B, A), A) & (~in(tptp_fun_D_1(C, B, A), B)))))))),
% 0.12/0.39 inference(skolemize,[status(sab)],[16])).
% 0.12/0.39 tff(18,plain,
% 0.12/0.39 (![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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(C, B, A), C))))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[17, 13])).
% 0.12/0.39 tff(19,plain,
% 0.12/0.39 (![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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(C, B, A), C))))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[18, 11])).
% 0.12/0.39 tff(20,plain,
% 0.12/0.39 (((~![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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(C, B, A), C))))))) | (~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4)))))) <=> ((~![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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(C, B, A), C))))))) | (~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4))))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(21,plain,
% 0.12/0.39 ((~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4))))) <=> (~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4)))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(22,plain,
% 0.12/0.39 (((((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4)))) | $false) <=> (((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(23,plain,
% 0.12/0.39 ((~$true) <=> $false),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(24,plain,
% 0.12/0.39 (($true | (((~in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), B!5)) | in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), singleton(C!4))) <=> in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), set_difference(B!5, singleton(C!4))))) <=> $true),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(25,plain,
% 0.12/0.39 ((set_difference(B!5, singleton(C!4)) = set_difference(B!5, singleton(C!4))) <=> $true),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(26,plain,
% 0.12/0.39 (((set_difference(B!5, singleton(C!4)) = set_difference(B!5, singleton(C!4))) | (((~in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), B!5)) | in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), singleton(C!4))) <=> in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), set_difference(B!5, singleton(C!4))))) <=> ($true | (((~in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), B!5)) | in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), singleton(C!4))) <=> in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), set_difference(B!5, singleton(C!4)))))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[25])).
% 0.12/0.40 tff(27,plain,
% 0.12/0.40 (((set_difference(B!5, singleton(C!4)) = set_difference(B!5, singleton(C!4))) | (((~in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), B!5)) | in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), singleton(C!4))) <=> in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), set_difference(B!5, singleton(C!4))))) <=> $true),
% 0.12/0.40 inference(transitivity,[status(thm)],[26, 24])).
% 0.12/0.40 tff(28,plain,
% 0.12/0.40 ((~((set_difference(B!5, singleton(C!4)) = set_difference(B!5, singleton(C!4))) | (((~in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), B!5)) | in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), singleton(C!4))) <=> in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), set_difference(B!5, singleton(C!4)))))) <=> (~$true)),
% 0.12/0.40 inference(monotonicity,[status(thm)],[27])).
% 0.12/0.40 tff(29,plain,
% 0.12/0.40 ((~((set_difference(B!5, singleton(C!4)) = set_difference(B!5, singleton(C!4))) | (((~in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), B!5)) | in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), singleton(C!4))) <=> in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), set_difference(B!5, singleton(C!4)))))) <=> $false),
% 0.12/0.40 inference(transitivity,[status(thm)],[28, 23])).
% 0.12/0.40 tff(30,plain,
% 0.12/0.40 ((~(~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4)))))) <=> (((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4))))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(31,plain,
% 0.12/0.40 (($false | (~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4)))))) <=> (~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4)))))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(32,plain,
% 0.12/0.40 ((~(set_difference(B!5, singleton(C!4)) = set_difference(B!5, singleton(C!4)))) <=> (~$true)),
% 0.12/0.40 inference(monotonicity,[status(thm)],[25])).
% 0.12/0.40 tff(33,plain,
% 0.12/0.40 ((~(set_difference(B!5, singleton(C!4)) = set_difference(B!5, singleton(C!4)))) <=> $false),
% 0.12/0.40 inference(transitivity,[status(thm)],[32, 23])).
% 0.12/0.40 tff(34,plain,
% 0.12/0.40 (((~(set_difference(B!5, singleton(C!4)) = set_difference(B!5, singleton(C!4)))) | (~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4)))))) <=> ($false | (~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4))))))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[33, 21])).
% 0.12/0.40 tff(35,plain,
% 0.12/0.40 (((~(set_difference(B!5, singleton(C!4)) = set_difference(B!5, singleton(C!4)))) | (~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4)))))) <=> (~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4)))))),
% 0.12/0.40 inference(transitivity,[status(thm)],[34, 31])).
% 0.12/0.40 tff(36,plain,
% 0.12/0.40 ((~((~(set_difference(B!5, singleton(C!4)) = set_difference(B!5, singleton(C!4)))) | (~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4))))))) <=> (~(~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4))))))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[35])).
% 0.12/0.40 tff(37,plain,
% 0.12/0.40 ((~((~(set_difference(B!5, singleton(C!4)) = set_difference(B!5, singleton(C!4)))) | (~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4))))))) <=> (((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4))))),
% 0.12/0.40 inference(transitivity,[status(thm)],[36, 30])).
% 0.12/0.40 tff(38,plain,
% 0.12/0.40 (((~((~(set_difference(B!5, singleton(C!4)) = set_difference(B!5, singleton(C!4)))) | (~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4))))))) | (~((set_difference(B!5, singleton(C!4)) = set_difference(B!5, singleton(C!4))) | (((~in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), B!5)) | in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), singleton(C!4))) <=> in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), set_difference(B!5, singleton(C!4))))))) <=> ((((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4)))) | $false)),
% 0.12/0.40 inference(monotonicity,[status(thm)],[37, 29])).
% 0.12/0.40 tff(39,plain,
% 0.12/0.40 (((~((~(set_difference(B!5, singleton(C!4)) = set_difference(B!5, singleton(C!4)))) | (~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4))))))) | (~((set_difference(B!5, singleton(C!4)) = set_difference(B!5, singleton(C!4))) | (((~in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), B!5)) | in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), singleton(C!4))) <=> in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), set_difference(B!5, singleton(C!4))))))) <=> (((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4))))),
% 0.12/0.40 inference(transitivity,[status(thm)],[38, 22])).
% 0.12/0.40 tff(40,plain,
% 0.12/0.40 ((~((~((~(set_difference(B!5, singleton(C!4)) = set_difference(B!5, singleton(C!4)))) | (~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4))))))) | (~((set_difference(B!5, singleton(C!4)) = set_difference(B!5, singleton(C!4))) | (((~in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), B!5)) | in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), singleton(C!4))) <=> in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), set_difference(B!5, singleton(C!4)))))))) <=> (~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4)))))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[39])).
% 0.12/0.40 tff(41,plain,
% 0.12/0.40 ((~((~((~(set_difference(B!5, singleton(C!4)) = set_difference(B!5, singleton(C!4)))) | (~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4))))))) | (~((set_difference(B!5, singleton(C!4)) = set_difference(B!5, singleton(C!4))) | (((~in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), B!5)) | in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), singleton(C!4))) <=> in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), set_difference(B!5, singleton(C!4)))))))) <=> (~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4)))))),
% 0.12/0.40 inference(transitivity,[status(thm)],[40, 21])).
% 0.12/0.40 tff(42,plain,
% 0.12/0.40 (((~![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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(C, B, A), C))))))) | (~((~((~(set_difference(B!5, singleton(C!4)) = set_difference(B!5, singleton(C!4)))) | (~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4))))))) | (~((set_difference(B!5, singleton(C!4)) = set_difference(B!5, singleton(C!4))) | (((~in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), B!5)) | in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), singleton(C!4))) <=> in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), set_difference(B!5, singleton(C!4))))))))) <=> ((~![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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(C, B, A), C))))))) | (~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4))))))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[41])).
% 0.12/0.40 tff(43,plain,
% 0.12/0.40 (((~![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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(C, B, A), C))))))) | (~((~((~(set_difference(B!5, singleton(C!4)) = set_difference(B!5, singleton(C!4)))) | (~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4))))))) | (~((set_difference(B!5, singleton(C!4)) = set_difference(B!5, singleton(C!4))) | (((~in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), B!5)) | in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), singleton(C!4))) <=> in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), set_difference(B!5, singleton(C!4))))))))) <=> ((~![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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(C, B, A), C))))))) | (~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4))))))),
% 0.12/0.40 inference(transitivity,[status(thm)],[42, 20])).
% 0.12/0.40 tff(44,plain,
% 0.12/0.40 ((~![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_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(C, B, A), C))))))) | (~((~((~(set_difference(B!5, singleton(C!4)) = set_difference(B!5, singleton(C!4)))) | (~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4))))))) | (~((set_difference(B!5, singleton(C!4)) = set_difference(B!5, singleton(C!4))) | (((~in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), B!5)) | in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), singleton(C!4))) <=> in(tptp_fun_D_1(set_difference(B!5, singleton(C!4)), singleton(C!4), B!5), set_difference(B!5, singleton(C!4))))))))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(45,plain,
% 0.12/0.40 ((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_difference(A, B))) | (~(((~in(D, A)) | in(D, B)) <=> in(D, C))))) | (~((C = set_difference(A, B)) | (((~in(tptp_fun_D_1(C, B, A), A)) | in(tptp_fun_D_1(C, B, A), B)) <=> in(tptp_fun_D_1(C, B, A), C))))))) | (~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4)))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[44, 43])).
% 0.12/0.40 tff(46,plain,
% 0.12/0.40 ($false),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[45, 19, 1])).
% 0.12/0.40 tff(47,plain,(~(((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4))))), inference(lemma,lemma(discharge,[]))).
% 0.12/0.40 tff(48,assumption,(~(in(A!6, singleton(C!4)) <=> (A!6 = C!4))), introduced(assumption)).
% 0.12/0.40 tff(49,plain,
% 0.12/0.40 (^[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))))))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(50,plain,
% 0.12/0.40 (![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))))))) <=> ![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)))))))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[49])).
% 0.12/0.40 tff(51,plain,
% 0.12/0.40 (![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))))))) <=> ![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)))))))),
% 0.12/0.40 inference(pull_quant,[status(thm)],[])).
% 0.12/0.40 tff(52,plain,
% 0.12/0.40 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(pull_quant(((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) <=> ![C: $i] : ((~(B = singleton(A))) | (in(C, B) <=> (C = A)))), ((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) <=> (~![C: $i] : ((~(B = singleton(A))) | (in(C, B) <=> (C = A)))))), pull_quant((~![C: $i] : ((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) <=> ?[C: $i] : (~((~(B = singleton(A))) | (in(C, B) <=> (C = A))))), ((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) <=> ?[C: $i] : (~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))))), (((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))) <=> (?[C: $i] : (~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))), pull_quant((?[C: $i] : (~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))) <=> ?[C: $i] : ((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))), (((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))) <=> ?[C: $i] : ((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))), ((~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> (~?[C: $i] : ((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))))), pull_quant((~?[C: $i] : ((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))), ((~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(53,plain,
% 0.12/0.40 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i] : ![C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[52])).
% 0.12/0.40 tff(54,plain,
% 0.12/0.40 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.40 inference(transitivity,[status(thm)],[53, 51])).
% 0.12/0.40 tff(55,plain,
% 0.12/0.40 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.40 inference(transitivity,[status(thm)],[54, 50])).
% 0.12/0.40 tff(56,plain,
% 0.12/0.40 (^[A: $i, B: $i] : rewrite((~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.41 tff(57,plain,
% 0.12/0.41 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.41 inference(quant_intro,[status(thm)],[56])).
% 0.12/0.41 tff(58,plain,
% 0.12/0.41 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.41 inference(transitivity,[status(thm)],[57, 55])).
% 0.12/0.41 tff(59,plain,
% 0.12/0.41 (^[A: $i, B: $i] : trans(monotonicity(rewrite(((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) <=> ((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))), trans(monotonicity(rewrite((~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))) <=> (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))), (((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))) <=> ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))), rewrite(((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))) <=> ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))), (((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))) <=> ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))), ((((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))) <=> (((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))), rewrite((((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))) <=> (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))), ((((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))) <=> (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))))),
% 0.12/0.41 inference(bind,[status(th)],[])).
% 0.12/0.41 tff(60,plain,
% 0.12/0.41 (![A: $i, B: $i] : (((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))) <=> ![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.41 inference(quant_intro,[status(thm)],[59])).
% 0.12/0.41 tff(61,plain,
% 0.12/0.41 (![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A))) <=> ![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(62,axiom,(![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','d1_tarski')).
% 0.12/0.41 tff(63,plain,
% 0.12/0.41 (![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[62, 61])).
% 0.12/0.41 tff(64,plain,(
% 0.12/0.41 ![A: $i, B: $i] : (((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))),
% 0.12/0.41 inference(skolemize,[status(sab)],[63])).
% 0.12/0.41 tff(65,plain,
% 0.12/0.41 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[64, 60])).
% 0.12/0.41 tff(66,plain,
% 0.12/0.41 (![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)))))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[65, 58])).
% 0.12/0.41 tff(67,plain,
% 0.12/0.41 (((~![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)))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(68,plain,
% 0.12/0.41 ((~(~(in(A!6, singleton(C!4)) <=> (A!6 = C!4)))) <=> (in(A!6, singleton(C!4)) <=> (A!6 = C!4))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(69,plain,
% 0.12/0.41 (((~(in(A!6, singleton(C!4)) <=> (A!6 = C!4))) | $false) <=> (~(in(A!6, singleton(C!4)) <=> (A!6 = C!4)))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(70,plain,
% 0.12/0.41 (($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),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(71,plain,
% 0.12/0.41 ((~(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)))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(72,plain,
% 0.12/0.41 ((singleton(C!4) = singleton(C!4)) <=> $true),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(73,plain,
% 0.12/0.41 (((singleton(C!4) = singleton(C!4)) | (~(in(tptp_fun_C_0(singleton(C!4), C!4), singleton(C!4)) <=> (tptp_fun_C_0(singleton(C!4), C!4) = C!4)))) <=> ($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))))),
% 0.12/0.41 inference(monotonicity,[status(thm)],[72, 71])).
% 0.12/0.41 tff(74,plain,
% 0.12/0.41 (((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)))) <=> $true),
% 0.12/0.41 inference(transitivity,[status(thm)],[73, 70])).
% 0.12/0.41 tff(75,plain,
% 0.12/0.41 ((~((singleton(C!4) = singleton(C!4)) | (~(in(tptp_fun_C_0(singleton(C!4), C!4), singleton(C!4)) <=> (tptp_fun_C_0(singleton(C!4), C!4) = C!4))))) <=> (~$true)),
% 0.12/0.41 inference(monotonicity,[status(thm)],[74])).
% 0.12/0.41 tff(76,plain,
% 0.12/0.41 ((~((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))))) <=> $false),
% 0.12/0.41 inference(transitivity,[status(thm)],[75, 23])).
% 0.12/0.41 tff(77,plain,
% 0.12/0.41 ((~(in(A!6, singleton(C!4)) <=> (A!6 = C!4))) <=> (~(in(A!6, singleton(C!4)) <=> (A!6 = C!4)))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(78,plain,
% 0.12/0.41 (($false | (in(A!6, singleton(C!4)) <=> (A!6 = C!4))) <=> (in(A!6, singleton(C!4)) <=> (A!6 = C!4))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(79,plain,
% 0.12/0.41 ((~(singleton(C!4) = singleton(C!4))) <=> (~$true)),
% 0.12/0.41 inference(monotonicity,[status(thm)],[72])).
% 0.12/0.41 tff(80,plain,
% 0.12/0.41 ((~(singleton(C!4) = singleton(C!4))) <=> $false),
% 0.12/0.41 inference(transitivity,[status(thm)],[79, 23])).
% 0.12/0.41 tff(81,plain,
% 0.12/0.41 (((~(singleton(C!4) = singleton(C!4))) | (in(A!6, singleton(C!4)) <=> (A!6 = C!4))) <=> ($false | (in(A!6, singleton(C!4)) <=> (A!6 = C!4)))),
% 0.12/0.41 inference(monotonicity,[status(thm)],[80])).
% 0.12/0.41 tff(82,plain,
% 0.12/0.41 (((~(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))),
% 0.12/0.41 inference(transitivity,[status(thm)],[81, 78])).
% 0.12/0.41 tff(83,plain,
% 0.12/0.41 ((~((~(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)))),
% 0.12/0.41 inference(monotonicity,[status(thm)],[82])).
% 0.12/0.41 tff(84,plain,
% 0.12/0.41 ((~((~(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)))),
% 0.12/0.41 inference(transitivity,[status(thm)],[83, 77])).
% 0.12/0.41 tff(85,plain,
% 0.12/0.41 (((~((~(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)),
% 0.22/0.41 inference(monotonicity,[status(thm)],[84, 76])).
% 0.22/0.41 tff(86,plain,
% 0.22/0.41 (((~((~(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)))),
% 0.22/0.41 inference(transitivity,[status(thm)],[85, 69])).
% 0.22/0.41 tff(87,plain,
% 0.22/0.41 ((~((~((~(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))))),
% 0.22/0.41 inference(monotonicity,[status(thm)],[86])).
% 0.22/0.41 tff(88,plain,
% 0.22/0.41 ((~((~((~(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))),
% 0.22/0.41 inference(transitivity,[status(thm)],[87, 68])).
% 0.22/0.41 tff(89,plain,
% 0.22/0.41 (((~![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)))),
% 0.22/0.41 inference(monotonicity,[status(thm)],[88])).
% 0.22/0.41 tff(90,plain,
% 0.22/0.41 (((~![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)))),
% 0.22/0.41 inference(transitivity,[status(thm)],[89, 67])).
% 0.22/0.41 tff(91,plain,
% 0.22/0.41 ((~![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)))))))),
% 0.22/0.41 inference(quant_inst,[status(thm)],[])).
% 0.22/0.41 tff(92,plain,
% 0.22/0.41 ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(A!6, singleton(C!4)) <=> (A!6 = C!4))),
% 0.22/0.41 inference(modus_ponens,[status(thm)],[91, 90])).
% 0.22/0.41 tff(93,plain,
% 0.22/0.41 ($false),
% 0.22/0.41 inference(unit_resolution,[status(thm)],[92, 66, 48])).
% 0.22/0.41 tff(94,plain,(in(A!6, singleton(C!4)) <=> (A!6 = C!4)), inference(lemma,lemma(discharge,[]))).
% 0.22/0.41 tff(95,assumption,(~(A!6 = C!4)), introduced(assumption)).
% 0.22/0.41 tff(96,plain,
% 0.22/0.41 ((~(in(A!6, singleton(C!4)) <=> (A!6 = C!4))) | (~in(A!6, singleton(C!4))) | (A!6 = C!4)),
% 0.22/0.41 inference(tautology,[status(thm)],[])).
% 0.22/0.41 tff(97,plain,
% 0.22/0.41 (~in(A!6, singleton(C!4))),
% 0.22/0.41 inference(unit_resolution,[status(thm)],[96, 95, 94])).
% 0.22/0.42 tff(98,assumption,(~in(A!6, singleton(C!4))), introduced(assumption)).
% 0.22/0.42 tff(99,assumption,(~in(A!6, set_difference(B!5, singleton(C!4)))), introduced(assumption)).
% 0.22/0.42 tff(100,plain,
% 0.22/0.42 ((((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4)))) | ((~in(A!6, B!5)) | in(A!6, singleton(C!4))) | in(A!6, set_difference(B!5, singleton(C!4)))),
% 0.22/0.42 inference(tautology,[status(thm)],[])).
% 0.22/0.42 tff(101,plain,
% 0.22/0.42 ((~in(A!6, B!5)) | in(A!6, singleton(C!4))),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[100, 99, 47])).
% 0.22/0.42 tff(102,plain,
% 0.22/0.42 ((~(~(((~in(A!6, B!5)) | (A!6 = C!4)) <=> in(A!6, set_difference(B!5, singleton(C!4)))))) <=> (((~in(A!6, B!5)) | (A!6 = C!4)) <=> in(A!6, set_difference(B!5, singleton(C!4))))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(103,plain,
% 0.22/0.42 ((~(((~in(A!6, B!5)) | (A!6 = C!4)) <=> in(A!6, set_difference(B!5, singleton(C!4))))) <=> (~(((~in(A!6, B!5)) | (A!6 = C!4)) <=> in(A!6, set_difference(B!5, singleton(C!4)))))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(104,plain,
% 0.22/0.42 ((in(A!6, set_difference(B!5, singleton(C!4))) <=> (~((~in(A!6, B!5)) | (A!6 = C!4)))) <=> (~(((~in(A!6, B!5)) | (A!6 = C!4)) <=> in(A!6, set_difference(B!5, singleton(C!4)))))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(105,plain,
% 0.22/0.42 ((in(A!6, B!5) & (~(A!6 = C!4))) <=> (~((~in(A!6, B!5)) | (A!6 = C!4)))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(106,plain,
% 0.22/0.42 ((in(A!6, set_difference(B!5, singleton(C!4))) <=> (in(A!6, B!5) & (~(A!6 = C!4)))) <=> (in(A!6, set_difference(B!5, singleton(C!4))) <=> (~((~in(A!6, B!5)) | (A!6 = C!4))))),
% 0.22/0.42 inference(monotonicity,[status(thm)],[105])).
% 0.22/0.42 tff(107,plain,
% 0.22/0.42 ((in(A!6, set_difference(B!5, singleton(C!4))) <=> (in(A!6, B!5) & (~(A!6 = C!4)))) <=> (~(((~in(A!6, B!5)) | (A!6 = C!4)) <=> in(A!6, set_difference(B!5, singleton(C!4)))))),
% 0.22/0.42 inference(transitivity,[status(thm)],[106, 104])).
% 0.22/0.42 tff(108,plain,
% 0.22/0.42 ((in(A!6, set_difference(B!5, singleton(C!4))) <=> (in(A!6, B!5) & (~(A!6 = C!4)))) <=> (~(((~in(A!6, B!5)) | (A!6 = C!4)) <=> in(A!6, set_difference(B!5, singleton(C!4)))))),
% 0.22/0.42 inference(transitivity,[status(thm)],[107, 103])).
% 0.22/0.42 tff(109,plain,
% 0.22/0.42 ((~(in(A!6, set_difference(B!5, singleton(C!4))) <=> (in(A!6, B!5) & (~(A!6 = C!4))))) <=> (~(~(((~in(A!6, B!5)) | (A!6 = C!4)) <=> in(A!6, set_difference(B!5, singleton(C!4))))))),
% 0.22/0.42 inference(monotonicity,[status(thm)],[108])).
% 0.22/0.42 tff(110,plain,
% 0.22/0.42 ((~(in(A!6, set_difference(B!5, singleton(C!4))) <=> (in(A!6, B!5) & (~(A!6 = C!4))))) <=> (((~in(A!6, B!5)) | (A!6 = C!4)) <=> in(A!6, set_difference(B!5, singleton(C!4))))),
% 0.22/0.42 inference(transitivity,[status(thm)],[109, 102])).
% 0.22/0.42 tff(111,plain,
% 0.22/0.42 ((~(in(A!6, set_difference(B!5, singleton(C!4))) <=> (in(A!6, B!5) & (~(A!6 = C!4))))) <=> (~(in(A!6, set_difference(B!5, singleton(C!4))) <=> (in(A!6, B!5) & (~(A!6 = C!4)))))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(112,plain,
% 0.22/0.42 ((~![A: $i, B: $i, C: $i] : (in(A, set_difference(B, singleton(C))) <=> (in(A, B) & (~(A = C))))) <=> (~![A: $i, B: $i, C: $i] : (in(A, set_difference(B, singleton(C))) <=> (in(A, B) & (~(A = C)))))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(113,axiom,(~![A: $i, B: $i, C: $i] : (in(A, set_difference(B, singleton(C))) <=> (in(A, B) & (~(A = C))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t64_zfmisc_1')).
% 0.22/0.42 tff(114,plain,
% 0.22/0.42 (~![A: $i, B: $i, C: $i] : (in(A, set_difference(B, singleton(C))) <=> (in(A, B) & (~(A = C))))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[113, 112])).
% 0.22/0.42 tff(115,plain,
% 0.22/0.42 (~![A: $i, B: $i, C: $i] : (in(A, set_difference(B, singleton(C))) <=> (in(A, B) & (~(A = C))))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[114, 112])).
% 0.22/0.42 tff(116,plain,
% 0.22/0.42 (~![A: $i, B: $i, C: $i] : (in(A, set_difference(B, singleton(C))) <=> (in(A, B) & (~(A = C))))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[115, 112])).
% 0.22/0.42 tff(117,plain,
% 0.22/0.42 (~![A: $i, B: $i, C: $i] : (in(A, set_difference(B, singleton(C))) <=> (in(A, B) & (~(A = C))))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[116, 112])).
% 0.22/0.42 tff(118,plain,(
% 0.22/0.42 ~(in(A!6, set_difference(B!5, singleton(C!4))) <=> (in(A!6, B!5) & (~(A!6 = C!4))))),
% 0.22/0.42 inference(skolemize,[status(sab)],[117])).
% 0.22/0.42 tff(119,plain,
% 0.22/0.42 (~(in(A!6, set_difference(B!5, singleton(C!4))) <=> (in(A!6, B!5) & (~(A!6 = C!4))))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[118, 111])).
% 0.22/0.42 tff(120,plain,
% 0.22/0.42 (((~in(A!6, B!5)) | (A!6 = C!4)) <=> in(A!6, set_difference(B!5, singleton(C!4)))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[119, 110])).
% 0.22/0.42 tff(121,plain,
% 0.22/0.42 ((~((~in(A!6, B!5)) | (A!6 = C!4))) | in(A!6, set_difference(B!5, singleton(C!4))) | (~(((~in(A!6, B!5)) | (A!6 = C!4)) <=> in(A!6, set_difference(B!5, singleton(C!4)))))),
% 0.22/0.42 inference(tautology,[status(thm)],[])).
% 0.22/0.42 tff(122,plain,
% 0.22/0.42 ((~((~in(A!6, B!5)) | (A!6 = C!4))) | in(A!6, set_difference(B!5, singleton(C!4)))),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[121, 120])).
% 0.22/0.42 tff(123,plain,
% 0.22/0.42 (~((~in(A!6, B!5)) | (A!6 = C!4))),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[122, 99])).
% 0.22/0.42 tff(124,plain,
% 0.22/0.42 (((~in(A!6, B!5)) | (A!6 = C!4)) | in(A!6, B!5)),
% 0.22/0.42 inference(tautology,[status(thm)],[])).
% 0.22/0.42 tff(125,plain,
% 0.22/0.42 (in(A!6, B!5)),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[124, 123])).
% 0.22/0.42 tff(126,plain,
% 0.22/0.42 ((~((~in(A!6, B!5)) | in(A!6, singleton(C!4)))) | (~in(A!6, B!5)) | in(A!6, singleton(C!4))),
% 0.22/0.42 inference(tautology,[status(thm)],[])).
% 0.22/0.42 tff(127,plain,
% 0.22/0.42 ($false),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[126, 125, 101, 98])).
% 0.22/0.42 tff(128,plain,(in(A!6, set_difference(B!5, singleton(C!4))) | in(A!6, singleton(C!4))), inference(lemma,lemma(discharge,[]))).
% 0.22/0.42 tff(129,plain,
% 0.22/0.42 (in(A!6, set_difference(B!5, singleton(C!4)))),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[128, 97])).
% 0.22/0.42 tff(130,plain,
% 0.22/0.42 ((((~in(A!6, B!5)) | in(A!6, singleton(C!4))) <=> in(A!6, set_difference(B!5, singleton(C!4)))) | (~((~in(A!6, B!5)) | in(A!6, singleton(C!4)))) | (~in(A!6, set_difference(B!5, singleton(C!4))))),
% 0.22/0.42 inference(tautology,[status(thm)],[])).
% 0.22/0.42 tff(131,plain,
% 0.22/0.42 (~((~in(A!6, B!5)) | in(A!6, singleton(C!4)))),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[130, 129, 47])).
% 0.22/0.42 tff(132,plain,
% 0.22/0.42 (((~in(A!6, B!5)) | (A!6 = C!4)) | (~in(A!6, set_difference(B!5, singleton(C!4)))) | (~(((~in(A!6, B!5)) | (A!6 = C!4)) <=> in(A!6, set_difference(B!5, singleton(C!4)))))),
% 0.22/0.42 inference(tautology,[status(thm)],[])).
% 0.22/0.42 tff(133,plain,
% 0.22/0.42 (((~in(A!6, B!5)) | (A!6 = C!4)) | (~in(A!6, set_difference(B!5, singleton(C!4))))),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[132, 120])).
% 0.22/0.42 tff(134,plain,
% 0.22/0.42 ((~in(A!6, B!5)) | (A!6 = C!4)),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[133, 129])).
% 0.22/0.42 tff(135,plain,
% 0.22/0.42 ((~((~in(A!6, B!5)) | (A!6 = C!4))) | (~in(A!6, B!5)) | (A!6 = C!4)),
% 0.22/0.42 inference(tautology,[status(thm)],[])).
% 0.22/0.42 tff(136,plain,
% 0.22/0.42 (~in(A!6, B!5)),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[135, 134, 95])).
% 0.22/0.42 tff(137,plain,
% 0.22/0.42 (((~in(A!6, B!5)) | in(A!6, singleton(C!4))) | in(A!6, B!5)),
% 0.22/0.42 inference(tautology,[status(thm)],[])).
% 0.22/0.42 tff(138,plain,
% 0.22/0.42 ($false),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[137, 136, 131])).
% 0.22/0.42 tff(139,plain,(A!6 = C!4), inference(lemma,lemma(discharge,[]))).
% 0.22/0.42 tff(140,plain,
% 0.22/0.42 (((~in(A!6, B!5)) | (A!6 = C!4)) | (~(A!6 = C!4))),
% 0.22/0.42 inference(tautology,[status(thm)],[])).
% 0.22/0.42 tff(141,plain,
% 0.22/0.42 ((~in(A!6, B!5)) | (A!6 = C!4)),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[140, 139])).
% 0.22/0.42 tff(142,plain,
% 0.22/0.42 (in(A!6, set_difference(B!5, singleton(C!4)))),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[122, 141])).
% 0.22/0.42 tff(143,plain,
% 0.22/0.42 ((~(in(A!6, singleton(C!4)) <=> (A!6 = C!4))) | in(A!6, singleton(C!4)) | (~(A!6 = C!4))),
% 0.22/0.42 inference(tautology,[status(thm)],[])).
% 0.22/0.42 tff(144,plain,
% 0.22/0.42 (in(A!6, singleton(C!4))),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[143, 139, 94])).
% 0.22/0.42 tff(145,plain,
% 0.22/0.42 (((~in(A!6, B!5)) | in(A!6, singleton(C!4))) | (~in(A!6, singleton(C!4)))),
% 0.22/0.42 inference(tautology,[status(thm)],[])).
% 0.22/0.42 tff(146,plain,
% 0.22/0.42 ((~in(A!6, B!5)) | in(A!6, singleton(C!4))),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[145, 144])).
% 0.22/0.42 tff(147,plain,
% 0.22/0.42 ($false),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[130, 146, 142, 47])).
% 0.22/0.42 % SZS output end Proof
% 0.22/0.42 % E exiting
%------------------------------------------------------------------------------