%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET909+1 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n001.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 05:26:38 AM UTC 2025
% Result : Theorem 0.19s 0.39s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SET909+1 : TPTP v9.0.0. Released v3.2.0.
% 0.00/0.12 % Command : run_E %s %d THM
% 0.11/0.33 % Computer : n001.cluster.edu
% 0.11/0.33 % Model : x86_64 x86_64
% 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.33 % Memory : 8042.1875MB
% 0.11/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.34 % CPULimit : 300
% 0.11/0.34 % WCLimit : 300
% 0.11/0.34 % DateTime : Fri Jun 20 05:54:35 EDT 2025
% 0.11/0.34 % CPUTime :
% 0.19/0.39 % SZS status Theorem
% 0.19/0.39 % SZS output start Proof
% 0.19/0.39 tff(tptp_fun_A_7_type, type, (
% 0.19/0.39 tptp_fun_A_7: $i)).
% 0.19/0.39 tff(tptp_fun_D_1_type, type, (
% 0.19/0.39 tptp_fun_D_1: ( $i * $i * $i ) > $i)).
% 0.19/0.39 tff(tptp_fun_B_6_type, type, (
% 0.19/0.39 tptp_fun_B_6: $i)).
% 0.19/0.39 tff(unordered_pair_type, type, (
% 0.19/0.39 unordered_pair: ( $i * $i ) > $i)).
% 0.19/0.39 tff(in_type, type, (
% 0.19/0.39 in: ( $i * $i ) > $o)).
% 0.19/0.39 tff(tptp_fun_C_5_type, type, (
% 0.19/0.39 tptp_fun_C_5: $i)).
% 0.19/0.39 tff(empty_set_type, type, (
% 0.19/0.39 empty_set: $i)).
% 0.19/0.39 tff(set_union2_type, type, (
% 0.19/0.39 set_union2: ( $i * $i ) > $i)).
% 0.19/0.39 tff(tptp_fun_D_2_type, type, (
% 0.19/0.39 tptp_fun_D_2: ( $i * $i * $i ) > $i)).
% 0.19/0.39 tff(tptp_fun_B_0_type, type, (
% 0.19/0.39 tptp_fun_B_0: $i > $i)).
% 0.19/0.39 tff(1,plain,
% 0.19/0.39 (^[A: $i, B: $i, C: $i, D: $i] : refl((~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))))))) <=> (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))))))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(2,plain,
% 0.19/0.39 (![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))))))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))))))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[1])).
% 0.19/0.39 tff(3,plain,
% 0.19/0.39 (![A: $i, B: $i, C: $i] : ![D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))))))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))))))),
% 0.19/0.39 inference(pull_quant,[status(thm)],[])).
% 0.19/0.39 tff(4,plain,
% 0.19/0.39 (^[A: $i, B: $i, C: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(pull_quant(((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) <=> ![D: $i] : ((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))), ((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) <=> (~![D: $i] : ((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))))), pull_quant((~![D: $i] : ((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) <=> ?[D: $i] : (~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B)))))), ((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) <=> ?[D: $i] : (~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))))), (((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))))) <=> (?[D: $i] : (~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))))))), pull_quant((?[D: $i] : (~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))))) <=> ?[D: $i] : ((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))))))), (((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))))) <=> ?[D: $i] : ((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))))))), ((~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))))))) <=> (~?[D: $i] : ((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))))))))), pull_quant((~?[D: $i] : ((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))))))) <=> ![D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))))))), ((~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))))))) <=> ![D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))))))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(5,plain,
% 0.19/0.39 (![A: $i, B: $i, C: $i] : (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))))))) <=> ![A: $i, B: $i, C: $i] : ![D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))))))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[4])).
% 0.19/0.39 tff(6,plain,
% 0.19/0.39 (![A: $i, B: $i, C: $i] : (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))))))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))))))),
% 0.19/0.39 inference(transitivity,[status(thm)],[5, 3])).
% 0.19/0.39 tff(7,plain,
% 0.19/0.39 (![A: $i, B: $i, C: $i] : (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))))))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))))))),
% 0.19/0.39 inference(transitivity,[status(thm)],[6, 2])).
% 0.19/0.39 tff(8,plain,
% 0.19/0.39 (^[A: $i, B: $i, C: $i] : rewrite((~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))))))) <=> (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))))))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(9,plain,
% 0.19/0.39 (![A: $i, B: $i, C: $i] : (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))))))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[8])).
% 0.19/0.39 tff(10,plain,
% 0.19/0.39 (![A: $i, B: $i, C: $i] : (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))))))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))))))),
% 0.19/0.39 inference(transitivity,[status(thm)],[9, 7])).
% 0.19/0.39 tff(11,plain,
% 0.19/0.39 (^[A: $i, B: $i, C: $i] : trans(monotonicity(rewrite(((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) <=> ((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))), trans(monotonicity(trans(monotonicity(rewrite((in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))) <=> (in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))), ((~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))) <=> (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))))), rewrite((~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))) <=> (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))), ((~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))) <=> (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))))), (((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))) <=> ((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))))), rewrite(((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))) <=> ((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))))), (((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))) <=> ((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))))), ((((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) & ((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))))) <=> (((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) & ((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))))))), rewrite((((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) & ((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))))) <=> (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))))))), ((((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) & ((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))))) <=> (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))))))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(12,plain,
% 0.19/0.40 (![A: $i, B: $i, C: $i] : (((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) & ((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B)))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))))))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[11])).
% 0.19/0.40 tff(13,plain,
% 0.19/0.40 (![A: $i, B: $i, C: $i] : ((C = set_union2(A, B)) <=> ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) <=> ![A: $i, B: $i, C: $i] : ((C = set_union2(A, B)) <=> ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(14,axiom,(![A: $i, B: $i, C: $i] : ((C = set_union2(A, B)) <=> ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','d2_xboole_0')).
% 0.19/0.40 tff(15,plain,
% 0.19/0.40 (![A: $i, B: $i, C: $i] : ((C = set_union2(A, B)) <=> ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[14, 13])).
% 0.19/0.40 tff(16,plain,(
% 0.19/0.40 ![A: $i, B: $i, C: $i] : (((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B)))) & ((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))))),
% 0.19/0.40 inference(skolemize,[status(sab)],[15])).
% 0.19/0.40 tff(17,plain,
% 0.19/0.40 (![A: $i, B: $i, C: $i] : (~((~((~(C = set_union2(A, B))) | ![D: $i] : (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[16, 12])).
% 0.19/0.40 tff(18,plain,
% 0.19/0.40 (![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[17, 10])).
% 0.19/0.40 tff(19,plain,
% 0.19/0.40 ((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = set_union2(A, B))) | (in(D, C) <=> (in(D, A) | in(D, B))))) | (~((C = set_union2(A, B)) | (~(in(tptp_fun_D_2(C, B, A), C) <=> (in(tptp_fun_D_2(C, B, A), A) | in(tptp_fun_D_2(C, B, A), B))))))))) | (~((~((~(empty_set = set_union2(C!5, unordered_pair(B!6, A!7)))) | (in(B!6, empty_set) <=> (in(B!6, C!5) | in(B!6, unordered_pair(B!6, A!7)))))) | (~((empty_set = set_union2(C!5, unordered_pair(B!6, A!7))) | (~(in(tptp_fun_D_2(empty_set, unordered_pair(B!6, A!7), C!5), empty_set) <=> (in(tptp_fun_D_2(empty_set, unordered_pair(B!6, A!7), C!5), C!5) | in(tptp_fun_D_2(empty_set, unordered_pair(B!6, A!7), C!5), unordered_pair(B!6, A!7)))))))))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(20,plain,
% 0.19/0.40 (~((~((~(empty_set = set_union2(C!5, unordered_pair(B!6, A!7)))) | (in(B!6, empty_set) <=> (in(B!6, C!5) | in(B!6, unordered_pair(B!6, A!7)))))) | (~((empty_set = set_union2(C!5, unordered_pair(B!6, A!7))) | (~(in(tptp_fun_D_2(empty_set, unordered_pair(B!6, A!7), C!5), empty_set) <=> (in(tptp_fun_D_2(empty_set, unordered_pair(B!6, A!7), C!5), C!5) | in(tptp_fun_D_2(empty_set, unordered_pair(B!6, A!7), C!5), unordered_pair(B!6, A!7))))))))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[19, 18])).
% 0.19/0.40 tff(21,plain,
% 0.19/0.40 (((~((~(empty_set = set_union2(C!5, unordered_pair(B!6, A!7)))) | (in(B!6, empty_set) <=> (in(B!6, C!5) | in(B!6, unordered_pair(B!6, A!7)))))) | (~((empty_set = set_union2(C!5, unordered_pair(B!6, A!7))) | (~(in(tptp_fun_D_2(empty_set, unordered_pair(B!6, A!7), C!5), empty_set) <=> (in(tptp_fun_D_2(empty_set, unordered_pair(B!6, A!7), C!5), C!5) | in(tptp_fun_D_2(empty_set, unordered_pair(B!6, A!7), C!5), unordered_pair(B!6, A!7)))))))) | ((~(empty_set = set_union2(C!5, unordered_pair(B!6, A!7)))) | (in(B!6, empty_set) <=> (in(B!6, C!5) | in(B!6, unordered_pair(B!6, A!7)))))),
% 0.19/0.40 inference(tautology,[status(thm)],[])).
% 0.19/0.40 tff(22,plain,
% 0.19/0.40 ((~(empty_set = set_union2(C!5, unordered_pair(B!6, A!7)))) | (in(B!6, empty_set) <=> (in(B!6, C!5) | in(B!6, unordered_pair(B!6, A!7))))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[21, 20])).
% 0.19/0.40 tff(23,plain,
% 0.19/0.40 (^[A: $i, B: $i] : refl((unordered_pair(A, B) = unordered_pair(B, A)) <=> (unordered_pair(A, B) = unordered_pair(B, A)))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(24,plain,
% 0.19/0.40 (![A: $i, B: $i] : (unordered_pair(A, B) = unordered_pair(B, A)) <=> ![A: $i, B: $i] : (unordered_pair(A, B) = unordered_pair(B, A))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[23])).
% 0.19/0.40 tff(25,plain,
% 0.19/0.40 (![A: $i, B: $i] : (unordered_pair(A, B) = unordered_pair(B, A)) <=> ![A: $i, B: $i] : (unordered_pair(A, B) = unordered_pair(B, A))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(26,axiom,(![A: $i, B: $i] : (unordered_pair(A, B) = unordered_pair(B, A))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','commutativity_k2_tarski')).
% 0.19/0.40 tff(27,plain,
% 0.19/0.40 (![A: $i, B: $i] : (unordered_pair(A, B) = unordered_pair(B, A))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[26, 25])).
% 0.19/0.40 tff(28,plain,(
% 0.19/0.40 ![A: $i, B: $i] : (unordered_pair(A, B) = unordered_pair(B, A))),
% 0.19/0.40 inference(skolemize,[status(sab)],[27])).
% 0.19/0.40 tff(29,plain,
% 0.19/0.40 (![A: $i, B: $i] : (unordered_pair(A, B) = unordered_pair(B, A))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[28, 24])).
% 0.19/0.40 tff(30,plain,
% 0.19/0.40 ((~![A: $i, B: $i] : (unordered_pair(A, B) = unordered_pair(B, A))) | (unordered_pair(A!7, B!6) = unordered_pair(B!6, A!7))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(31,plain,
% 0.19/0.40 (unordered_pair(A!7, B!6) = unordered_pair(B!6, A!7)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[30, 29])).
% 0.19/0.40 tff(32,plain,
% 0.19/0.40 (unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6)),
% 0.19/0.40 inference(symmetry,[status(thm)],[31])).
% 0.19/0.40 tff(33,plain,
% 0.19/0.40 (set_union2(C!5, unordered_pair(B!6, A!7)) = set_union2(C!5, unordered_pair(A!7, B!6))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[32])).
% 0.19/0.40 tff(34,plain,
% 0.19/0.40 (set_union2(C!5, unordered_pair(A!7, B!6)) = set_union2(C!5, unordered_pair(B!6, A!7))),
% 0.19/0.40 inference(symmetry,[status(thm)],[33])).
% 0.19/0.40 tff(35,plain,
% 0.19/0.40 (^[A: $i, B: $i] : refl((set_union2(A, B) = set_union2(B, A)) <=> (set_union2(A, B) = set_union2(B, A)))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(36,plain,
% 0.19/0.40 (![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A)) <=> ![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[35])).
% 0.19/0.40 tff(37,plain,
% 0.19/0.40 (![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A)) <=> ![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(38,axiom,(![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','commutativity_k2_xboole_0')).
% 0.19/0.40 tff(39,plain,
% 0.19/0.40 (![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[38, 37])).
% 0.19/0.40 tff(40,plain,(
% 0.19/0.40 ![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))),
% 0.19/0.40 inference(skolemize,[status(sab)],[39])).
% 0.19/0.40 tff(41,plain,
% 0.19/0.40 (![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[40, 36])).
% 0.19/0.40 tff(42,plain,
% 0.19/0.40 ((~![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))) | (set_union2(unordered_pair(A!7, B!6), C!5) = set_union2(C!5, unordered_pair(A!7, B!6)))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(43,plain,
% 0.19/0.40 (set_union2(unordered_pair(A!7, B!6), C!5) = set_union2(C!5, unordered_pair(A!7, B!6))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[42, 41])).
% 0.19/0.40 tff(44,plain,
% 0.19/0.40 ((~![A: $i, B: $i, C: $i] : (~(set_union2(unordered_pair(A, B), C) = empty_set))) <=> (~![A: $i, B: $i, C: $i] : (~(set_union2(unordered_pair(A, B), C) = empty_set)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(45,axiom,(~![A: $i, B: $i, C: $i] : (~(set_union2(unordered_pair(A, B), C) = empty_set))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t50_zfmisc_1')).
% 0.19/0.40 tff(46,plain,
% 0.19/0.40 (~![A: $i, B: $i, C: $i] : (~(set_union2(unordered_pair(A, B), C) = empty_set))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[45, 44])).
% 0.19/0.40 tff(47,plain,
% 0.19/0.40 (~![A: $i, B: $i, C: $i] : (~(set_union2(unordered_pair(A, B), C) = empty_set))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[46, 44])).
% 0.19/0.40 tff(48,plain,
% 0.19/0.40 (~![A: $i, B: $i, C: $i] : (~(set_union2(unordered_pair(A, B), C) = empty_set))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[47, 44])).
% 0.19/0.40 tff(49,plain,
% 0.19/0.40 (~![A: $i, B: $i, C: $i] : (~(set_union2(unordered_pair(A, B), C) = empty_set))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[48, 44])).
% 0.19/0.40 tff(50,plain,(
% 0.19/0.40 set_union2(unordered_pair(A!7, B!6), C!5) = empty_set),
% 0.19/0.40 inference(skolemize,[status(sab)],[49])).
% 0.19/0.40 tff(51,plain,
% 0.19/0.40 (empty_set = set_union2(unordered_pair(A!7, B!6), C!5)),
% 0.19/0.40 inference(symmetry,[status(thm)],[50])).
% 0.19/0.40 tff(52,plain,
% 0.19/0.40 (empty_set = set_union2(C!5, unordered_pair(B!6, A!7))),
% 0.19/0.40 inference(transitivity,[status(thm)],[51, 43, 34])).
% 0.19/0.40 tff(53,plain,
% 0.19/0.40 ((~((~(empty_set = set_union2(C!5, unordered_pair(B!6, A!7)))) | (in(B!6, empty_set) <=> (in(B!6, C!5) | in(B!6, unordered_pair(B!6, A!7)))))) | (~(empty_set = set_union2(C!5, unordered_pair(B!6, A!7)))) | (in(B!6, empty_set) <=> (in(B!6, C!5) | in(B!6, unordered_pair(B!6, A!7))))),
% 0.19/0.40 inference(tautology,[status(thm)],[])).
% 0.19/0.40 tff(54,plain,
% 0.19/0.40 ((~((~(empty_set = set_union2(C!5, unordered_pair(B!6, A!7)))) | (in(B!6, empty_set) <=> (in(B!6, C!5) | in(B!6, unordered_pair(B!6, A!7)))))) | (in(B!6, empty_set) <=> (in(B!6, C!5) | in(B!6, unordered_pair(B!6, A!7))))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[53, 52])).
% 0.19/0.40 tff(55,plain,
% 0.19/0.40 (in(B!6, empty_set) <=> (in(B!6, C!5) | in(B!6, unordered_pair(B!6, A!7)))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[54, 22])).
% 0.19/0.40 tff(56,plain,
% 0.19/0.40 (^[A: $i, B: $i] : refl((~((~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A))))) <=> (~((~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A))))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(57,plain,
% 0.19/0.40 (![A: $i, B: $i] : (~((~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A))))) <=> ![A: $i, B: $i] : (~((~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[56])).
% 0.19/0.40 tff(58,plain,
% 0.19/0.40 (![A: $i] : ![B: $i] : (~((~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A))))) <=> ![A: $i, B: $i] : (~((~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))))),
% 0.19/0.40 inference(pull_quant,[status(thm)],[])).
% 0.19/0.40 tff(59,plain,
% 0.19/0.40 (^[A: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(pull_quant(((~(A = empty_set)) | ![B: $i] : (~in(B, A))) <=> ![B: $i] : ((~(A = empty_set)) | (~in(B, A)))), ((~((~(A = empty_set)) | ![B: $i] : (~in(B, A)))) <=> (~![B: $i] : ((~(A = empty_set)) | (~in(B, A)))))), pull_quant((~![B: $i] : ((~(A = empty_set)) | (~in(B, A)))) <=> ?[B: $i] : (~((~(A = empty_set)) | (~in(B, A))))), ((~((~(A = empty_set)) | ![B: $i] : (~in(B, A)))) <=> ?[B: $i] : (~((~(A = empty_set)) | (~in(B, A)))))), (((~((~(A = empty_set)) | ![B: $i] : (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))) <=> (?[B: $i] : (~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))))), pull_quant((?[B: $i] : (~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))) <=> ?[B: $i] : ((~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A))))), (((~((~(A = empty_set)) | ![B: $i] : (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))) <=> ?[B: $i] : ((~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))))), ((~((~((~(A = empty_set)) | ![B: $i] : (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A))))) <=> (~?[B: $i] : ((~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A))))))), pull_quant((~?[B: $i] : ((~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A))))) <=> ![B: $i] : (~((~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))))), ((~((~((~(A = empty_set)) | ![B: $i] : (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A))))) <=> ![B: $i] : (~((~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(60,plain,
% 0.19/0.40 (![A: $i] : (~((~((~(A = empty_set)) | ![B: $i] : (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A))))) <=> ![A: $i] : ![B: $i] : (~((~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[59])).
% 0.19/0.40 tff(61,plain,
% 0.19/0.40 (![A: $i] : (~((~((~(A = empty_set)) | ![B: $i] : (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A))))) <=> ![A: $i, B: $i] : (~((~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))))),
% 0.19/0.40 inference(transitivity,[status(thm)],[60, 58])).
% 0.19/0.40 tff(62,plain,
% 0.19/0.40 (![A: $i] : (~((~((~(A = empty_set)) | ![B: $i] : (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A))))) <=> ![A: $i, B: $i] : (~((~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))))),
% 0.19/0.40 inference(transitivity,[status(thm)],[61, 57])).
% 0.19/0.40 tff(63,plain,
% 0.19/0.40 (^[A: $i] : rewrite((~((~((~(A = empty_set)) | ![B: $i] : (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A))))) <=> (~((~((~(A = empty_set)) | ![B: $i] : (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A))))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(64,plain,
% 0.19/0.40 (![A: $i] : (~((~((~(A = empty_set)) | ![B: $i] : (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A))))) <=> ![A: $i] : (~((~((~(A = empty_set)) | ![B: $i] : (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[63])).
% 0.19/0.40 tff(65,plain,
% 0.19/0.40 (![A: $i] : (~((~((~(A = empty_set)) | ![B: $i] : (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A))))) <=> ![A: $i, B: $i] : (~((~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))))),
% 0.19/0.40 inference(transitivity,[status(thm)],[64, 62])).
% 0.19/0.40 tff(66,plain,
% 0.19/0.40 (^[A: $i] : trans(monotonicity(rewrite(((~(A = empty_set)) | ![B: $i] : (~in(B, A))) <=> ((~(A = empty_set)) | ![B: $i] : (~in(B, A)))), ((((~(A = empty_set)) | ![B: $i] : (~in(B, A))) & ((A = empty_set) | in(tptp_fun_B_0(A), A))) <=> (((~(A = empty_set)) | ![B: $i] : (~in(B, A))) & ((A = empty_set) | in(tptp_fun_B_0(A), A))))), rewrite((((~(A = empty_set)) | ![B: $i] : (~in(B, A))) & ((A = empty_set) | in(tptp_fun_B_0(A), A))) <=> (~((~((~(A = empty_set)) | ![B: $i] : (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))))), ((((~(A = empty_set)) | ![B: $i] : (~in(B, A))) & ((A = empty_set) | in(tptp_fun_B_0(A), A))) <=> (~((~((~(A = empty_set)) | ![B: $i] : (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(67,plain,
% 0.19/0.40 (![A: $i] : (((~(A = empty_set)) | ![B: $i] : (~in(B, A))) & ((A = empty_set) | in(tptp_fun_B_0(A), A))) <=> ![A: $i] : (~((~((~(A = empty_set)) | ![B: $i] : (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[66])).
% 0.19/0.40 tff(68,plain,
% 0.19/0.40 (![A: $i] : ((A = empty_set) <=> ![B: $i] : (~in(B, A))) <=> ![A: $i] : ((A = empty_set) <=> ![B: $i] : (~in(B, A)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(69,axiom,(![A: $i] : ((A = empty_set) <=> ![B: $i] : (~in(B, A)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','d1_xboole_0')).
% 0.19/0.40 tff(70,plain,
% 0.19/0.40 (![A: $i] : ((A = empty_set) <=> ![B: $i] : (~in(B, A)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[69, 68])).
% 0.19/0.40 tff(71,plain,(
% 0.19/0.40 ![A: $i] : (((~(A = empty_set)) | ![B: $i] : (~in(B, A))) & ((A = empty_set) | in(tptp_fun_B_0(A), A)))),
% 0.19/0.40 inference(skolemize,[status(sab)],[70])).
% 0.19/0.40 tff(72,plain,
% 0.19/0.40 (![A: $i] : (~((~((~(A = empty_set)) | ![B: $i] : (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[71, 67])).
% 0.19/0.40 tff(73,plain,
% 0.19/0.40 (![A: $i, B: $i] : (~((~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[72, 65])).
% 0.19/0.40 tff(74,plain,
% 0.19/0.40 (((~![A: $i, B: $i] : (~((~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))))) | (~in(B!6, empty_set))) <=> ((~![A: $i, B: $i] : (~((~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))))) | (~in(B!6, empty_set)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(75,plain,
% 0.19/0.41 ((~((~((~(empty_set = empty_set)) | (~in(B!6, empty_set)))) | (~((empty_set = empty_set) | in(tptp_fun_B_0(empty_set), empty_set))))) <=> (~in(B!6, empty_set))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(76,plain,
% 0.19/0.41 (((~![A: $i, B: $i] : (~((~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))))) | (~((~((~(empty_set = empty_set)) | (~in(B!6, empty_set)))) | (~((empty_set = empty_set) | in(tptp_fun_B_0(empty_set), empty_set)))))) <=> ((~![A: $i, B: $i] : (~((~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))))) | (~in(B!6, empty_set)))),
% 0.19/0.41 inference(monotonicity,[status(thm)],[75])).
% 0.19/0.41 tff(77,plain,
% 0.19/0.41 (((~![A: $i, B: $i] : (~((~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))))) | (~((~((~(empty_set = empty_set)) | (~in(B!6, empty_set)))) | (~((empty_set = empty_set) | in(tptp_fun_B_0(empty_set), empty_set)))))) <=> ((~![A: $i, B: $i] : (~((~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))))) | (~in(B!6, empty_set)))),
% 0.19/0.41 inference(transitivity,[status(thm)],[76, 74])).
% 0.19/0.41 tff(78,plain,
% 0.19/0.41 ((~![A: $i, B: $i] : (~((~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))))) | (~((~((~(empty_set = empty_set)) | (~in(B!6, empty_set)))) | (~((empty_set = empty_set) | in(tptp_fun_B_0(empty_set), empty_set)))))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(79,plain,
% 0.19/0.41 ((~![A: $i, B: $i] : (~((~((~(A = empty_set)) | (~in(B, A)))) | (~((A = empty_set) | in(tptp_fun_B_0(A), A)))))) | (~in(B!6, empty_set))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[78, 77])).
% 0.19/0.41 tff(80,plain,
% 0.19/0.41 (~in(B!6, empty_set)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[79, 73])).
% 0.19/0.41 tff(81,plain,
% 0.19/0.41 ((~(in(B!6, empty_set) <=> (in(B!6, C!5) | in(B!6, unordered_pair(B!6, A!7))))) | in(B!6, empty_set) | (~(in(B!6, C!5) | in(B!6, unordered_pair(B!6, A!7))))),
% 0.19/0.41 inference(tautology,[status(thm)],[])).
% 0.19/0.41 tff(82,plain,
% 0.19/0.41 ((~(in(B!6, empty_set) <=> (in(B!6, C!5) | in(B!6, unordered_pair(B!6, A!7))))) | (~(in(B!6, C!5) | in(B!6, unordered_pair(B!6, A!7))))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[81, 80])).
% 0.19/0.41 tff(83,plain,
% 0.19/0.41 (~(in(B!6, C!5) | in(B!6, unordered_pair(B!6, A!7)))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[82, 55])).
% 0.19/0.41 tff(84,plain,
% 0.19/0.41 ((in(B!6, C!5) | in(B!6, unordered_pair(B!6, A!7))) | (~in(B!6, unordered_pair(B!6, A!7)))),
% 0.19/0.41 inference(tautology,[status(thm)],[])).
% 0.19/0.41 tff(85,plain,
% 0.19/0.41 (~in(B!6, unordered_pair(B!6, A!7))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[84, 83])).
% 0.19/0.41 tff(86,plain,
% 0.19/0.41 ((~![A: $i, B: $i] : (unordered_pair(A, B) = unordered_pair(B, A))) | (unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(87,plain,
% 0.19/0.41 (unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[86, 29])).
% 0.19/0.41 tff(88,plain,
% 0.19/0.41 ((~((~(unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6))) | in(B!6, unordered_pair(B!6, A!7)))) | (~(unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6))) | in(B!6, unordered_pair(B!6, A!7))),
% 0.19/0.41 inference(tautology,[status(thm)],[])).
% 0.19/0.41 tff(89,plain,
% 0.19/0.41 (~((~(unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6))) | in(B!6, unordered_pair(B!6, A!7)))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[88, 87, 85])).
% 0.19/0.41 tff(90,plain,
% 0.19/0.41 (((~((~(unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6))) | in(B!6, unordered_pair(B!6, A!7)))) | (~((unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6)) | (~(in(tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7), unordered_pair(B!6, A!7)) <=> ((tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7) = B!6) | (tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7) = A!7))))))) | ((~(unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6))) | in(B!6, unordered_pair(B!6, A!7)))),
% 0.19/0.41 inference(tautology,[status(thm)],[])).
% 0.19/0.41 tff(91,plain,
% 0.19/0.41 ((~((~(unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6))) | in(B!6, unordered_pair(B!6, A!7)))) | (~((unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6)) | (~(in(tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7), unordered_pair(B!6, A!7)) <=> ((tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7) = B!6) | (tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7) = A!7))))))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[90, 89])).
% 0.19/0.41 tff(92,plain,
% 0.19/0.41 (^[A: $i, B: $i, C: $i, D: $i] : refl((~((~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A)))))))) <=> (~((~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A)))))))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(93,plain,
% 0.19/0.41 (![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A)))))))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[92])).
% 0.19/0.41 tff(94,plain,
% 0.19/0.41 (![A: $i, B: $i, C: $i] : ![D: $i] : (~((~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A)))))))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))))),
% 0.19/0.41 inference(pull_quant,[status(thm)],[])).
% 0.19/0.41 tff(95,plain,
% 0.19/0.41 (^[A: $i, B: $i, C: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(pull_quant(((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A)))) <=> ![D: $i] : ((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))), ((~((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A))))) <=> (~![D: $i] : ((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))))), pull_quant((~![D: $i] : ((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) <=> ?[D: $i] : (~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A)))))), ((~((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A))))) <=> ?[D: $i] : (~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))))), (((~((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))) <=> (?[D: $i] : (~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))))), pull_quant((?[D: $i] : (~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))) <=> ?[D: $i] : ((~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A)))))))), (((~((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))) <=> ?[D: $i] : ((~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))))), ((~((~((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A)))))))) <=> (~?[D: $i] : ((~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A)))))))))), pull_quant((~?[D: $i] : ((~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A)))))))) <=> ![D: $i] : (~((~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))))), ((~((~((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A)))))))) <=> ![D: $i] : (~((~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(96,plain,
% 0.19/0.41 (![A: $i, B: $i, C: $i] : (~((~((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A)))))))) <=> ![A: $i, B: $i, C: $i] : ![D: $i] : (~((~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[95])).
% 0.19/0.41 tff(97,plain,
% 0.19/0.41 (![A: $i, B: $i, C: $i] : (~((~((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A)))))))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))))),
% 0.19/0.41 inference(transitivity,[status(thm)],[96, 94])).
% 0.19/0.41 tff(98,plain,
% 0.19/0.41 (![A: $i, B: $i, C: $i] : (~((~((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A)))))))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))))),
% 0.19/0.41 inference(transitivity,[status(thm)],[97, 93])).
% 0.19/0.41 tff(99,plain,
% 0.19/0.41 (^[A: $i, B: $i, C: $i] : rewrite((~((~((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A)))))))) <=> (~((~((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A)))))))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(100,plain,
% 0.19/0.41 (![A: $i, B: $i, C: $i] : (~((~((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A)))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[99])).
% 0.19/0.41 tff(101,plain,
% 0.19/0.41 (![A: $i, B: $i, C: $i] : (~((~((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A)))))))) <=> ![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))))),
% 0.19/0.41 inference(transitivity,[status(thm)],[100, 98])).
% 0.19/0.41 tff(102,plain,
% 0.19/0.41 (^[A: $i, B: $i, C: $i] : trans(monotonicity(rewrite(((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A)))) <=> ((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A))))), trans(monotonicity(rewrite((~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A)))) <=> (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))), (((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))) <=> ((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))), rewrite(((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))) <=> ((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A)))))), (((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))) <=> ((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))), ((((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A)))) & ((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A)))))) <=> (((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A)))) & ((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A)))))))), rewrite((((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A)))) & ((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A)))))) <=> (~((~((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))))), ((((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A)))) & ((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A)))))) <=> (~((~((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(103,plain,
% 0.19/0.41 (![A: $i, B: $i, C: $i] : (((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A)))) & ((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A)))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[102])).
% 0.19/0.42 tff(104,plain,
% 0.19/0.42 (![A: $i, B: $i, C: $i] : ((C = unordered_pair(A, B)) <=> ![D: $i] : (in(D, C) <=> ((D = B) | (D = A)))) <=> ![A: $i, B: $i, C: $i] : ((C = unordered_pair(A, B)) <=> ![D: $i] : (in(D, C) <=> ((D = B) | (D = A))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(105,plain,
% 0.19/0.42 (^[A: $i, B: $i, C: $i] : rewrite(((C = unordered_pair(A, B)) <=> ![D: $i] : (in(D, C) <=> ((D = A) | (D = B)))) <=> ((C = unordered_pair(A, B)) <=> ![D: $i] : (in(D, C) <=> ((D = B) | (D = A)))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(106,plain,
% 0.19/0.42 (![A: $i, B: $i, C: $i] : ((C = unordered_pair(A, B)) <=> ![D: $i] : (in(D, C) <=> ((D = A) | (D = B)))) <=> ![A: $i, B: $i, C: $i] : ((C = unordered_pair(A, B)) <=> ![D: $i] : (in(D, C) <=> ((D = B) | (D = A))))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[105])).
% 0.19/0.42 tff(107,axiom,(![A: $i, B: $i, C: $i] : ((C = unordered_pair(A, B)) <=> ![D: $i] : (in(D, C) <=> ((D = A) | (D = B))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','d2_tarski')).
% 0.19/0.42 tff(108,plain,
% 0.19/0.42 (![A: $i, B: $i, C: $i] : ((C = unordered_pair(A, B)) <=> ![D: $i] : (in(D, C) <=> ((D = B) | (D = A))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[107, 106])).
% 0.19/0.42 tff(109,plain,
% 0.19/0.42 (![A: $i, B: $i, C: $i] : ((C = unordered_pair(A, B)) <=> ![D: $i] : (in(D, C) <=> ((D = B) | (D = A))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[108, 104])).
% 0.19/0.42 tff(110,plain,(
% 0.19/0.42 ![A: $i, B: $i, C: $i] : (((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A)))) & ((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))),
% 0.19/0.42 inference(skolemize,[status(sab)],[109])).
% 0.19/0.42 tff(111,plain,
% 0.19/0.42 (![A: $i, B: $i, C: $i] : (~((~((~(C = unordered_pair(A, B))) | ![D: $i] : (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[110, 103])).
% 0.19/0.42 tff(112,plain,
% 0.19/0.42 (![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[111, 101])).
% 0.19/0.42 tff(113,plain,
% 0.19/0.42 (((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))))) | (~((~((~(unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6))) | in(B!6, unordered_pair(B!6, A!7)))) | (~((unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6)) | (~(in(tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7), unordered_pair(B!6, A!7)) <=> ((tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7) = B!6) | (tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7) = A!7))))))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))))) | (~((~((~(unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6))) | in(B!6, unordered_pair(B!6, A!7)))) | (~((unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6)) | (~(in(tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7), unordered_pair(B!6, A!7)) <=> ((tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7) = B!6) | (tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7) = A!7)))))))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(114,plain,
% 0.19/0.42 ((~((~((~(unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6))) | (in(B!6, unordered_pair(B!6, A!7)) <=> ((B!6 = B!6) | (B!6 = A!7))))) | (~((unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6)) | (~(in(tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7), unordered_pair(B!6, A!7)) <=> ((tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7) = B!6) | (tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7) = A!7)))))))) <=> (~((~((~(unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6))) | in(B!6, unordered_pair(B!6, A!7)))) | (~((unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6)) | (~(in(tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7), unordered_pair(B!6, A!7)) <=> ((tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7) = B!6) | (tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7) = A!7))))))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(115,plain,
% 0.19/0.42 (((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))))) | (~((~((~(unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6))) | (in(B!6, unordered_pair(B!6, A!7)) <=> ((B!6 = B!6) | (B!6 = A!7))))) | (~((unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6)) | (~(in(tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7), unordered_pair(B!6, A!7)) <=> ((tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7) = B!6) | (tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7) = A!7))))))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))))) | (~((~((~(unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6))) | in(B!6, unordered_pair(B!6, A!7)))) | (~((unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6)) | (~(in(tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7), unordered_pair(B!6, A!7)) <=> ((tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7) = B!6) | (tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7) = A!7)))))))))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[114])).
% 0.19/0.42 tff(116,plain,
% 0.19/0.42 (((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))))) | (~((~((~(unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6))) | (in(B!6, unordered_pair(B!6, A!7)) <=> ((B!6 = B!6) | (B!6 = A!7))))) | (~((unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6)) | (~(in(tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7), unordered_pair(B!6, A!7)) <=> ((tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7) = B!6) | (tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7) = A!7))))))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))))) | (~((~((~(unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6))) | in(B!6, unordered_pair(B!6, A!7)))) | (~((unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6)) | (~(in(tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7), unordered_pair(B!6, A!7)) <=> ((tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7) = B!6) | (tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7) = A!7)))))))))),
% 0.19/0.42 inference(transitivity,[status(thm)],[115, 113])).
% 0.19/0.42 tff(117,plain,
% 0.19/0.42 ((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))))) | (~((~((~(unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6))) | (in(B!6, unordered_pair(B!6, A!7)) <=> ((B!6 = B!6) | (B!6 = A!7))))) | (~((unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6)) | (~(in(tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7), unordered_pair(B!6, A!7)) <=> ((tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7) = B!6) | (tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7) = A!7))))))))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(118,plain,
% 0.19/0.42 ((~![A: $i, B: $i, C: $i, D: $i] : (~((~((~(C = unordered_pair(A, B))) | (in(D, C) <=> ((D = B) | (D = A))))) | (~((C = unordered_pair(A, B)) | (~(in(tptp_fun_D_1(C, B, A), C) <=> ((tptp_fun_D_1(C, B, A) = B) | (tptp_fun_D_1(C, B, A) = A))))))))) | (~((~((~(unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6))) | in(B!6, unordered_pair(B!6, A!7)))) | (~((unordered_pair(B!6, A!7) = unordered_pair(A!7, B!6)) | (~(in(tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7), unordered_pair(B!6, A!7)) <=> ((tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7) = B!6) | (tptp_fun_D_1(unordered_pair(B!6, A!7), B!6, A!7) = A!7))))))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[117, 116])).
% 0.19/0.42 tff(119,plain,
% 0.19/0.42 ($false),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[118, 112, 91])).
% 0.19/0.42 % SZS output end Proof
% 0.19/0.42 % E exiting
%------------------------------------------------------------------------------