%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SEU236+3 : 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 05:27:21 AM UTC 2025
% Result : Theorem 0.12s 0.39s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : SEU236+3 : TPTP v9.0.0. Released v3.2.0.
% 0.06/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n001.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Jun 20 06:31:20 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.12/0.39 % SZS status Theorem
% 0.12/0.39 % SZS output start Proof
% 0.12/0.39 tff(element_type, type, (
% 0.12/0.39 element: ( $i * $i ) > $o)).
% 0.12/0.39 tff(powerset_type, type, (
% 0.12/0.39 powerset: $i > $i)).
% 0.12/0.39 tff(tptp_fun_B_18_type, type, (
% 0.12/0.39 tptp_fun_B_18: $i)).
% 0.12/0.39 tff(set_union2_type, type, (
% 0.12/0.39 set_union2: ( $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_A_17_type, type, (
% 0.12/0.39 tptp_fun_A_17: $i)).
% 0.12/0.39 tff(subset_type, type, (
% 0.12/0.39 subset: ( $i * $i ) > $o)).
% 0.12/0.39 tff(succ_type, type, (
% 0.12/0.39 succ: $i > $i)).
% 0.12/0.39 tff(ordinal_subset_type, type, (
% 0.12/0.39 ordinal_subset: ( $i * $i ) > $o)).
% 0.12/0.39 tff(in_type, type, (
% 0.12/0.39 in: ( $i * $i ) > $o)).
% 0.12/0.39 tff(tptp_fun_C_2_type, type, (
% 0.12/0.39 tptp_fun_C_2: ( $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(ordinal_type, type, (
% 0.12/0.39 ordinal: $i > $o)).
% 0.12/0.39 tff(epsilon_transitive_type, type, (
% 0.12/0.39 epsilon_transitive: $i > $o)).
% 0.12/0.39 tff(tptp_fun_B_1_type, type, (
% 0.12/0.39 tptp_fun_B_1: $i > $i)).
% 0.12/0.39 tff(epsilon_connected_type, type, (
% 0.12/0.39 epsilon_connected: $i > $o)).
% 0.12/0.39 tff(empty_type, type, (
% 0.12/0.39 empty: $i > $o)).
% 0.12/0.39 tff(1,plain,
% 0.12/0.39 (^[A: $i] : refl((succ(A) = set_union2(A, singleton(A))) <=> (succ(A) = set_union2(A, singleton(A))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(2,plain,
% 0.12/0.39 (![A: $i] : (succ(A) = set_union2(A, singleton(A))) <=> ![A: $i] : (succ(A) = set_union2(A, singleton(A)))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[1])).
% 0.12/0.39 tff(3,plain,
% 0.12/0.39 (![A: $i] : (succ(A) = set_union2(A, singleton(A))) <=> ![A: $i] : (succ(A) = set_union2(A, singleton(A)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(4,axiom,(![A: $i] : (succ(A) = set_union2(A, singleton(A)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','d1_ordinal1')).
% 0.12/0.39 tff(5,plain,
% 0.12/0.39 (![A: $i] : (succ(A) = set_union2(A, singleton(A)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.12/0.39 tff(6,plain,(
% 0.12/0.39 ![A: $i] : (succ(A) = set_union2(A, singleton(A)))),
% 0.12/0.39 inference(skolemize,[status(sab)],[5])).
% 0.12/0.39 tff(7,plain,
% 0.12/0.39 (![A: $i] : (succ(A) = set_union2(A, singleton(A)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.12/0.39 tff(8,plain,
% 0.12/0.39 ((~![A: $i] : (succ(A) = set_union2(A, singleton(A)))) | (succ(A!17) = set_union2(A!17, singleton(A!17)))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(9,plain,
% 0.12/0.39 (succ(A!17) = set_union2(A!17, singleton(A!17))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.12/0.39 tff(10,plain,
% 0.12/0.39 (set_union2(A!17, singleton(A!17)) = succ(A!17)),
% 0.12/0.39 inference(symmetry,[status(thm)],[9])).
% 0.12/0.39 tff(11,plain,
% 0.12/0.39 (subset(set_union2(A!17, singleton(A!17)), B!18) <=> subset(succ(A!17), B!18)),
% 0.12/0.39 inference(monotonicity,[status(thm)],[10])).
% 0.12/0.39 tff(12,plain,
% 0.12/0.39 (subset(succ(A!17), B!18) <=> subset(set_union2(A!17, singleton(A!17)), B!18)),
% 0.12/0.39 inference(symmetry,[status(thm)],[11])).
% 0.12/0.39 tff(13,plain,
% 0.12/0.39 (^[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))))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(14,plain,
% 0.12/0.39 (![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![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)))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[13])).
% 0.12/0.39 tff(15,plain,
% 0.12/0.39 (![A: $i, B: $i] : ![C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![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)))))))),
% 0.12/0.39 inference(pull_quant,[status(thm)],[])).
% 0.12/0.39 tff(16,plain,
% 0.12/0.39 (^[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)))))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(17,plain,
% 0.12/0.39 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i] : ![C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[16])).
% 0.12/0.39 tff(18,plain,
% 0.12/0.39 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.39 inference(transitivity,[status(thm)],[17, 15])).
% 0.12/0.39 tff(19,plain,
% 0.12/0.39 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.39 inference(transitivity,[status(thm)],[18, 14])).
% 0.12/0.39 tff(20,plain,
% 0.12/0.39 (^[A: $i, B: $i] : rewrite((~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(21,plain,
% 0.12/0.39 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[20])).
% 0.12/0.39 tff(22,plain,
% 0.12/0.39 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))) <=> ![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.39 inference(transitivity,[status(thm)],[21, 19])).
% 0.12/0.39 tff(23,plain,
% 0.12/0.39 (^[A: $i, B: $i] : trans(monotonicity(rewrite(((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) <=> ((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))), trans(monotonicity(rewrite((~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))) <=> (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))), (((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))) <=> ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))), rewrite(((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))) <=> ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))), (((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))) <=> ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))), ((((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))) <=> (((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))))), rewrite((((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))) <=> (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))), ((((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))) <=> (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(24,plain,
% 0.12/0.39 (![A: $i, B: $i] : (((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A))))) <=> ![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[23])).
% 0.12/0.39 tff(25,plain,
% 0.12/0.39 (![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A))) <=> ![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(26,axiom,(![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','d1_tarski')).
% 0.12/0.39 tff(27,plain,
% 0.12/0.39 (![A: $i, B: $i] : ((B = singleton(A)) <=> ![C: $i] : (in(C, B) <=> (C = A)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[26, 25])).
% 0.12/0.39 tff(28,plain,(
% 0.12/0.39 ![A: $i, B: $i] : (((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A))) & ((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))),
% 0.12/0.39 inference(skolemize,[status(sab)],[27])).
% 0.12/0.39 tff(29,plain,
% 0.12/0.39 (![A: $i, B: $i] : (~((~((~(B = singleton(A))) | ![C: $i] : (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[28, 24])).
% 0.12/0.39 tff(30,plain,
% 0.12/0.39 (![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[29, 22])).
% 0.12/0.39 tff(31,plain,
% 0.12/0.39 (((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17))) <=> ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(32,plain,
% 0.12/0.39 ((~(~(in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17)))) <=> (in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(33,plain,
% 0.12/0.39 (((~(in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17))) | $false) <=> (~(in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(34,plain,
% 0.12/0.39 ((~$true) <=> $false),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(35,plain,
% 0.12/0.39 (($true | (~(in(tptp_fun_C_0(singleton(A!17), A!17), singleton(A!17)) <=> (tptp_fun_C_0(singleton(A!17), A!17) = A!17)))) <=> $true),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(36,plain,
% 0.12/0.39 ((~(in(tptp_fun_C_0(singleton(A!17), A!17), singleton(A!17)) <=> (tptp_fun_C_0(singleton(A!17), A!17) = A!17))) <=> (~(in(tptp_fun_C_0(singleton(A!17), A!17), singleton(A!17)) <=> (tptp_fun_C_0(singleton(A!17), A!17) = A!17)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(37,plain,
% 0.12/0.39 ((singleton(A!17) = singleton(A!17)) <=> $true),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(38,plain,
% 0.12/0.39 (((singleton(A!17) = singleton(A!17)) | (~(in(tptp_fun_C_0(singleton(A!17), A!17), singleton(A!17)) <=> (tptp_fun_C_0(singleton(A!17), A!17) = A!17)))) <=> ($true | (~(in(tptp_fun_C_0(singleton(A!17), A!17), singleton(A!17)) <=> (tptp_fun_C_0(singleton(A!17), A!17) = A!17))))),
% 0.12/0.39 inference(monotonicity,[status(thm)],[37, 36])).
% 0.12/0.39 tff(39,plain,
% 0.12/0.39 (((singleton(A!17) = singleton(A!17)) | (~(in(tptp_fun_C_0(singleton(A!17), A!17), singleton(A!17)) <=> (tptp_fun_C_0(singleton(A!17), A!17) = A!17)))) <=> $true),
% 0.12/0.39 inference(transitivity,[status(thm)],[38, 35])).
% 0.12/0.39 tff(40,plain,
% 0.12/0.39 ((~((singleton(A!17) = singleton(A!17)) | (~(in(tptp_fun_C_0(singleton(A!17), A!17), singleton(A!17)) <=> (tptp_fun_C_0(singleton(A!17), A!17) = A!17))))) <=> (~$true)),
% 0.12/0.39 inference(monotonicity,[status(thm)],[39])).
% 0.12/0.39 tff(41,plain,
% 0.12/0.39 ((~((singleton(A!17) = singleton(A!17)) | (~(in(tptp_fun_C_0(singleton(A!17), A!17), singleton(A!17)) <=> (tptp_fun_C_0(singleton(A!17), A!17) = A!17))))) <=> $false),
% 0.12/0.39 inference(transitivity,[status(thm)],[40, 34])).
% 0.12/0.39 tff(42,plain,
% 0.12/0.39 ((~(in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17))) <=> (~(in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(43,plain,
% 0.12/0.39 (($false | (in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17))) <=> (in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(44,plain,
% 0.12/0.39 ((~(singleton(A!17) = singleton(A!17))) <=> (~$true)),
% 0.12/0.39 inference(monotonicity,[status(thm)],[37])).
% 0.12/0.39 tff(45,plain,
% 0.12/0.39 ((~(singleton(A!17) = singleton(A!17))) <=> $false),
% 0.12/0.39 inference(transitivity,[status(thm)],[44, 34])).
% 0.12/0.39 tff(46,plain,
% 0.12/0.39 (((~(singleton(A!17) = singleton(A!17))) | (in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17))) <=> ($false | (in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17)))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[45])).
% 0.12/0.40 tff(47,plain,
% 0.12/0.40 (((~(singleton(A!17) = singleton(A!17))) | (in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17))) <=> (in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17))),
% 0.12/0.40 inference(transitivity,[status(thm)],[46, 43])).
% 0.12/0.40 tff(48,plain,
% 0.12/0.40 ((~((~(singleton(A!17) = singleton(A!17))) | (in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17)))) <=> (~(in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17)))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[47])).
% 0.12/0.40 tff(49,plain,
% 0.12/0.40 ((~((~(singleton(A!17) = singleton(A!17))) | (in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17)))) <=> (~(in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17)))),
% 0.12/0.40 inference(transitivity,[status(thm)],[48, 42])).
% 0.12/0.40 tff(50,plain,
% 0.12/0.40 (((~((~(singleton(A!17) = singleton(A!17))) | (in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17)))) | (~((singleton(A!17) = singleton(A!17)) | (~(in(tptp_fun_C_0(singleton(A!17), A!17), singleton(A!17)) <=> (tptp_fun_C_0(singleton(A!17), A!17) = A!17)))))) <=> ((~(in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17))) | $false)),
% 0.12/0.40 inference(monotonicity,[status(thm)],[49, 41])).
% 0.12/0.40 tff(51,plain,
% 0.12/0.40 (((~((~(singleton(A!17) = singleton(A!17))) | (in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17)))) | (~((singleton(A!17) = singleton(A!17)) | (~(in(tptp_fun_C_0(singleton(A!17), A!17), singleton(A!17)) <=> (tptp_fun_C_0(singleton(A!17), A!17) = A!17)))))) <=> (~(in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17)))),
% 0.12/0.40 inference(transitivity,[status(thm)],[50, 33])).
% 0.12/0.40 tff(52,plain,
% 0.12/0.40 ((~((~((~(singleton(A!17) = singleton(A!17))) | (in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17)))) | (~((singleton(A!17) = singleton(A!17)) | (~(in(tptp_fun_C_0(singleton(A!17), A!17), singleton(A!17)) <=> (tptp_fun_C_0(singleton(A!17), A!17) = A!17))))))) <=> (~(~(in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17))))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[51])).
% 0.12/0.40 tff(53,plain,
% 0.12/0.40 ((~((~((~(singleton(A!17) = singleton(A!17))) | (in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17)))) | (~((singleton(A!17) = singleton(A!17)) | (~(in(tptp_fun_C_0(singleton(A!17), A!17), singleton(A!17)) <=> (tptp_fun_C_0(singleton(A!17), A!17) = A!17))))))) <=> (in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17))),
% 0.12/0.40 inference(transitivity,[status(thm)],[52, 32])).
% 0.12/0.40 tff(54,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)))))))) | (~((~((~(singleton(A!17) = singleton(A!17))) | (in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17)))) | (~((singleton(A!17) = singleton(A!17)) | (~(in(tptp_fun_C_0(singleton(A!17), A!17), singleton(A!17)) <=> (tptp_fun_C_0(singleton(A!17), A!17) = A!17)))))))) <=> ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17)))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[53])).
% 0.12/0.40 tff(55,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)))))))) | (~((~((~(singleton(A!17) = singleton(A!17))) | (in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17)))) | (~((singleton(A!17) = singleton(A!17)) | (~(in(tptp_fun_C_0(singleton(A!17), A!17), singleton(A!17)) <=> (tptp_fun_C_0(singleton(A!17), A!17) = A!17)))))))) <=> ((~![A: $i, B: $i, C: $i] : (~((~((~(B = singleton(A))) | (in(C, B) <=> (C = A)))) | (~((B = singleton(A)) | (~(in(tptp_fun_C_0(B, A), B) <=> (tptp_fun_C_0(B, A) = A)))))))) | (in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17)))),
% 0.12/0.40 inference(transitivity,[status(thm)],[54, 31])).
% 0.12/0.40 tff(56,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)))))))) | (~((~((~(singleton(A!17) = singleton(A!17))) | (in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17)))) | (~((singleton(A!17) = singleton(A!17)) | (~(in(tptp_fun_C_0(singleton(A!17), A!17), singleton(A!17)) <=> (tptp_fun_C_0(singleton(A!17), A!17) = A!17)))))))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(57,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)))))))) | (in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[56, 55])).
% 0.12/0.40 tff(58,plain,
% 0.12/0.40 (in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17)),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[57, 30])).
% 0.12/0.40 tff(59,plain,
% 0.12/0.40 (^[A: $i, B: $i] : refl((~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_2(B, A), A)) | in(tptp_fun_C_2(B, A), B))))))) <=> (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_2(B, A), A)) | in(tptp_fun_C_2(B, A), B))))))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(60,plain,
% 0.12/0.40 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_2(B, A), A)) | in(tptp_fun_C_2(B, A), B))))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_2(B, A), A)) | in(tptp_fun_C_2(B, A), B)))))))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[59])).
% 0.12/0.40 tff(61,plain,
% 0.12/0.40 (^[A: $i, B: $i] : rewrite((~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_2(B, A), A)) | in(tptp_fun_C_2(B, A), B))))))) <=> (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_2(B, A), A)) | in(tptp_fun_C_2(B, A), B))))))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(62,plain,
% 0.12/0.40 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_2(B, A), A)) | in(tptp_fun_C_2(B, A), B))))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_2(B, A), A)) | in(tptp_fun_C_2(B, A), B)))))))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[61])).
% 0.12/0.40 tff(63,plain,
% 0.12/0.40 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_2(B, A), A)) | in(tptp_fun_C_2(B, A), B))))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_2(B, A), A)) | in(tptp_fun_C_2(B, A), B)))))))),
% 0.12/0.40 inference(transitivity,[status(thm)],[62, 60])).
% 0.12/0.40 tff(64,plain,
% 0.12/0.40 (^[A: $i, B: $i] : trans(monotonicity(rewrite(((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) <=> ((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))), rewrite((subset(A, B) | (~((~in(tptp_fun_C_2(B, A), A)) | in(tptp_fun_C_2(B, A), B)))) <=> (subset(A, B) | (~((~in(tptp_fun_C_2(B, A), A)) | in(tptp_fun_C_2(B, A), B))))), ((((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) & (subset(A, B) | (~((~in(tptp_fun_C_2(B, A), A)) | in(tptp_fun_C_2(B, A), B))))) <=> (((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) & (subset(A, B) | (~((~in(tptp_fun_C_2(B, A), A)) | in(tptp_fun_C_2(B, A), B))))))), rewrite((((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) & (subset(A, B) | (~((~in(tptp_fun_C_2(B, A), A)) | in(tptp_fun_C_2(B, A), B))))) <=> (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_2(B, A), A)) | in(tptp_fun_C_2(B, A), B)))))))), ((((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) & (subset(A, B) | (~((~in(tptp_fun_C_2(B, A), A)) | in(tptp_fun_C_2(B, A), B))))) <=> (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_2(B, A), A)) | in(tptp_fun_C_2(B, A), B)))))))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(65,plain,
% 0.12/0.40 (![A: $i, B: $i] : (((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) & (subset(A, B) | (~((~in(tptp_fun_C_2(B, A), A)) | in(tptp_fun_C_2(B, A), B))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_2(B, A), A)) | in(tptp_fun_C_2(B, A), B)))))))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[64])).
% 0.12/0.40 tff(66,plain,
% 0.12/0.40 (![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : ((~in(C, A)) | in(C, B))) <=> ![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : ((~in(C, A)) | in(C, B)))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(67,plain,
% 0.12/0.40 (^[A: $i, B: $i] : rewrite((subset(A, B) <=> ![C: $i] : (in(C, A) => in(C, B))) <=> (subset(A, B) <=> ![C: $i] : ((~in(C, A)) | in(C, B))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(68,plain,
% 0.12/0.40 (![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : (in(C, A) => in(C, B))) <=> ![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : ((~in(C, A)) | in(C, B)))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[67])).
% 0.12/0.40 tff(69,axiom,(![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : (in(C, A) => in(C, B)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','d3_tarski')).
% 0.12/0.40 tff(70,plain,
% 0.12/0.40 (![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : ((~in(C, A)) | in(C, B)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[69, 68])).
% 0.12/0.40 tff(71,plain,
% 0.12/0.40 (![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : ((~in(C, A)) | in(C, B)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[70, 66])).
% 0.12/0.40 tff(72,plain,(
% 0.12/0.40 ![A: $i, B: $i] : (((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) & (subset(A, B) | (~((~in(tptp_fun_C_2(B, A), A)) | in(tptp_fun_C_2(B, A), B)))))),
% 0.12/0.40 inference(skolemize,[status(sab)],[71])).
% 0.12/0.40 tff(73,plain,
% 0.12/0.40 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_2(B, A), A)) | in(tptp_fun_C_2(B, A), B)))))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[72, 65])).
% 0.12/0.40 tff(74,plain,
% 0.12/0.40 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_2(B, A), A)) | in(tptp_fun_C_2(B, A), B)))))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[73, 63])).
% 0.12/0.40 tff(75,plain,
% 0.12/0.40 ((~![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_2(B, A), A)) | in(tptp_fun_C_2(B, A), B)))))))) | (~((~((~subset(singleton(A!17), B!18)) | ![C: $i] : ((~in(C, singleton(A!17))) | in(C, B!18)))) | (~(subset(singleton(A!17), B!18) | (~((~in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17))) | in(tptp_fun_C_2(B!18, singleton(A!17)), B!18)))))))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(76,plain,
% 0.12/0.40 (~((~((~subset(singleton(A!17), B!18)) | ![C: $i] : ((~in(C, singleton(A!17))) | in(C, B!18)))) | (~(subset(singleton(A!17), B!18) | (~((~in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17))) | in(tptp_fun_C_2(B!18, singleton(A!17)), B!18))))))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[75, 74])).
% 0.19/0.41 tff(77,plain,
% 0.19/0.41 (((~((~subset(singleton(A!17), B!18)) | ![C: $i] : ((~in(C, singleton(A!17))) | in(C, B!18)))) | (~(subset(singleton(A!17), B!18) | (~((~in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17))) | in(tptp_fun_C_2(B!18, singleton(A!17)), B!18)))))) | (subset(singleton(A!17), B!18) | (~((~in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17))) | in(tptp_fun_C_2(B!18, singleton(A!17)), B!18))))),
% 0.19/0.41 inference(tautology,[status(thm)],[])).
% 0.19/0.41 tff(78,plain,
% 0.19/0.41 (subset(singleton(A!17), B!18) | (~((~in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17))) | in(tptp_fun_C_2(B!18, singleton(A!17)), B!18)))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[77, 76])).
% 0.19/0.41 tff(79,assumption,(~ordinal_subset(succ(A!17), B!18)), introduced(assumption)).
% 0.19/0.41 tff(80,plain,
% 0.19/0.41 ((~![A: $i] : ((~ordinal(A)) | ![B: $i] : ((~ordinal(B)) | (in(A, B) <=> ordinal_subset(succ(A), B))))) <=> (~![A: $i] : ((~ordinal(A)) | ![B: $i] : ((~ordinal(B)) | (in(A, B) <=> ordinal_subset(succ(A), B)))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(81,plain,
% 0.19/0.41 ((~![A: $i] : (ordinal(A) => ![B: $i] : (ordinal(B) => (in(A, B) <=> ordinal_subset(succ(A), B))))) <=> (~![A: $i] : ((~ordinal(A)) | ![B: $i] : ((~ordinal(B)) | (in(A, B) <=> ordinal_subset(succ(A), B)))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(82,axiom,(~![A: $i] : (ordinal(A) => ![B: $i] : (ordinal(B) => (in(A, B) <=> ordinal_subset(succ(A), B))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t33_ordinal1')).
% 0.19/0.41 tff(83,plain,
% 0.19/0.41 (~![A: $i] : ((~ordinal(A)) | ![B: $i] : ((~ordinal(B)) | (in(A, B) <=> ordinal_subset(succ(A), B))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[82, 81])).
% 0.19/0.41 tff(84,plain,
% 0.19/0.41 (~![A: $i] : ((~ordinal(A)) | ![B: $i] : ((~ordinal(B)) | (in(A, B) <=> ordinal_subset(succ(A), B))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[83, 80])).
% 0.19/0.41 tff(85,plain,
% 0.19/0.41 (~![A: $i] : ((~ordinal(A)) | ![B: $i] : ((~ordinal(B)) | (in(A, B) <=> ordinal_subset(succ(A), B))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[84, 80])).
% 0.19/0.41 tff(86,plain,
% 0.19/0.41 (~![A: $i] : ((~ordinal(A)) | ![B: $i] : ((~ordinal(B)) | (in(A, B) <=> ordinal_subset(succ(A), B))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[85, 80])).
% 0.19/0.41 tff(87,plain,
% 0.19/0.41 (~((~ordinal(B!18)) | (in(A!17, B!18) <=> ordinal_subset(succ(A!17), B!18)))),
% 0.19/0.41 inference(and_elim,[status(thm)],[86])).
% 0.19/0.41 tff(88,plain,
% 0.19/0.41 (~(in(A!17, B!18) <=> ordinal_subset(succ(A!17), B!18))),
% 0.19/0.41 inference(or_elim,[status(thm)],[87])).
% 0.19/0.41 tff(89,plain,
% 0.19/0.41 ((in(A!17, B!18) <=> ordinal_subset(succ(A!17), B!18)) | in(A!17, B!18) | ordinal_subset(succ(A!17), B!18)),
% 0.19/0.41 inference(tautology,[status(thm)],[])).
% 0.19/0.41 tff(90,plain,
% 0.19/0.41 (in(A!17, B!18) | ordinal_subset(succ(A!17), B!18)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[89, 88])).
% 0.19/0.41 tff(91,plain,
% 0.19/0.41 (in(A!17, B!18)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[90, 79])).
% 0.19/0.41 tff(92,plain,
% 0.19/0.41 (^[A: $i] : refl((~((~((~epsilon_transitive(A)) | ![B: $i] : ((~in(B, A)) | subset(B, A)))) | (~(epsilon_transitive(A) | (~((~in(tptp_fun_B_1(A), A)) | subset(tptp_fun_B_1(A), A))))))) <=> (~((~((~epsilon_transitive(A)) | ![B: $i] : ((~in(B, A)) | subset(B, A)))) | (~(epsilon_transitive(A) | (~((~in(tptp_fun_B_1(A), A)) | subset(tptp_fun_B_1(A), A))))))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(93,plain,
% 0.19/0.41 (![A: $i] : (~((~((~epsilon_transitive(A)) | ![B: $i] : ((~in(B, A)) | subset(B, A)))) | (~(epsilon_transitive(A) | (~((~in(tptp_fun_B_1(A), A)) | subset(tptp_fun_B_1(A), A))))))) <=> ![A: $i] : (~((~((~epsilon_transitive(A)) | ![B: $i] : ((~in(B, A)) | subset(B, A)))) | (~(epsilon_transitive(A) | (~((~in(tptp_fun_B_1(A), A)) | subset(tptp_fun_B_1(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] : rewrite((~((~((~epsilon_transitive(A)) | ![B: $i] : ((~in(B, A)) | subset(B, A)))) | (~(epsilon_transitive(A) | (~((~in(tptp_fun_B_1(A), A)) | subset(tptp_fun_B_1(A), A))))))) <=> (~((~((~epsilon_transitive(A)) | ![B: $i] : ((~in(B, A)) | subset(B, A)))) | (~(epsilon_transitive(A) | (~((~in(tptp_fun_B_1(A), A)) | subset(tptp_fun_B_1(A), A))))))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(95,plain,
% 0.19/0.41 (![A: $i] : (~((~((~epsilon_transitive(A)) | ![B: $i] : ((~in(B, A)) | subset(B, A)))) | (~(epsilon_transitive(A) | (~((~in(tptp_fun_B_1(A), A)) | subset(tptp_fun_B_1(A), A))))))) <=> ![A: $i] : (~((~((~epsilon_transitive(A)) | ![B: $i] : ((~in(B, A)) | subset(B, A)))) | (~(epsilon_transitive(A) | (~((~in(tptp_fun_B_1(A), A)) | subset(tptp_fun_B_1(A), A)))))))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[94])).
% 0.19/0.41 tff(96,plain,
% 0.19/0.41 (![A: $i] : (~((~((~epsilon_transitive(A)) | ![B: $i] : ((~in(B, A)) | subset(B, A)))) | (~(epsilon_transitive(A) | (~((~in(tptp_fun_B_1(A), A)) | subset(tptp_fun_B_1(A), A))))))) <=> ![A: $i] : (~((~((~epsilon_transitive(A)) | ![B: $i] : ((~in(B, A)) | subset(B, A)))) | (~(epsilon_transitive(A) | (~((~in(tptp_fun_B_1(A), A)) | subset(tptp_fun_B_1(A), A)))))))),
% 0.19/0.41 inference(transitivity,[status(thm)],[95, 93])).
% 0.19/0.41 tff(97,plain,
% 0.19/0.41 (^[A: $i] : trans(monotonicity(rewrite(((~epsilon_transitive(A)) | ![B: $i] : ((~in(B, A)) | subset(B, A))) <=> ((~epsilon_transitive(A)) | ![B: $i] : ((~in(B, A)) | subset(B, A)))), ((((~epsilon_transitive(A)) | ![B: $i] : ((~in(B, A)) | subset(B, A))) & (epsilon_transitive(A) | (~((~in(tptp_fun_B_1(A), A)) | subset(tptp_fun_B_1(A), A))))) <=> (((~epsilon_transitive(A)) | ![B: $i] : ((~in(B, A)) | subset(B, A))) & (epsilon_transitive(A) | (~((~in(tptp_fun_B_1(A), A)) | subset(tptp_fun_B_1(A), A))))))), rewrite((((~epsilon_transitive(A)) | ![B: $i] : ((~in(B, A)) | subset(B, A))) & (epsilon_transitive(A) | (~((~in(tptp_fun_B_1(A), A)) | subset(tptp_fun_B_1(A), A))))) <=> (~((~((~epsilon_transitive(A)) | ![B: $i] : ((~in(B, A)) | subset(B, A)))) | (~(epsilon_transitive(A) | (~((~in(tptp_fun_B_1(A), A)) | subset(tptp_fun_B_1(A), A)))))))), ((((~epsilon_transitive(A)) | ![B: $i] : ((~in(B, A)) | subset(B, A))) & (epsilon_transitive(A) | (~((~in(tptp_fun_B_1(A), A)) | subset(tptp_fun_B_1(A), A))))) <=> (~((~((~epsilon_transitive(A)) | ![B: $i] : ((~in(B, A)) | subset(B, A)))) | (~(epsilon_transitive(A) | (~((~in(tptp_fun_B_1(A), A)) | subset(tptp_fun_B_1(A), A)))))))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(98,plain,
% 0.19/0.41 (![A: $i] : (((~epsilon_transitive(A)) | ![B: $i] : ((~in(B, A)) | subset(B, A))) & (epsilon_transitive(A) | (~((~in(tptp_fun_B_1(A), A)) | subset(tptp_fun_B_1(A), A))))) <=> ![A: $i] : (~((~((~epsilon_transitive(A)) | ![B: $i] : ((~in(B, A)) | subset(B, A)))) | (~(epsilon_transitive(A) | (~((~in(tptp_fun_B_1(A), A)) | subset(tptp_fun_B_1(A), A)))))))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[97])).
% 0.19/0.41 tff(99,plain,
% 0.19/0.41 (![A: $i] : (epsilon_transitive(A) <=> ![B: $i] : ((~in(B, A)) | subset(B, A))) <=> ![A: $i] : (epsilon_transitive(A) <=> ![B: $i] : ((~in(B, A)) | subset(B, A)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(100,plain,
% 0.19/0.41 (^[A: $i] : rewrite((epsilon_transitive(A) <=> ![B: $i] : (in(B, A) => subset(B, A))) <=> (epsilon_transitive(A) <=> ![B: $i] : ((~in(B, A)) | subset(B, A))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(101,plain,
% 0.19/0.41 (![A: $i] : (epsilon_transitive(A) <=> ![B: $i] : (in(B, A) => subset(B, A))) <=> ![A: $i] : (epsilon_transitive(A) <=> ![B: $i] : ((~in(B, A)) | subset(B, A)))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[100])).
% 0.19/0.41 tff(102,axiom,(![A: $i] : (epsilon_transitive(A) <=> ![B: $i] : (in(B, A) => subset(B, A)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','d2_ordinal1')).
% 0.19/0.41 tff(103,plain,
% 0.19/0.41 (![A: $i] : (epsilon_transitive(A) <=> ![B: $i] : ((~in(B, A)) | subset(B, A)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[102, 101])).
% 0.19/0.41 tff(104,plain,
% 0.19/0.41 (![A: $i] : (epsilon_transitive(A) <=> ![B: $i] : ((~in(B, A)) | subset(B, A)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[103, 99])).
% 0.19/0.41 tff(105,plain,(
% 0.19/0.41 ![A: $i] : (((~epsilon_transitive(A)) | ![B: $i] : ((~in(B, A)) | subset(B, A))) & (epsilon_transitive(A) | (~((~in(tptp_fun_B_1(A), A)) | subset(tptp_fun_B_1(A), A)))))),
% 0.19/0.41 inference(skolemize,[status(sab)],[104])).
% 0.19/0.41 tff(106,plain,
% 0.19/0.41 (![A: $i] : (~((~((~epsilon_transitive(A)) | ![B: $i] : ((~in(B, A)) | subset(B, A)))) | (~(epsilon_transitive(A) | (~((~in(tptp_fun_B_1(A), A)) | subset(tptp_fun_B_1(A), A)))))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[105, 98])).
% 0.19/0.41 tff(107,plain,
% 0.19/0.41 (![A: $i] : (~((~((~epsilon_transitive(A)) | ![B: $i] : ((~in(B, A)) | subset(B, A)))) | (~(epsilon_transitive(A) | (~((~in(tptp_fun_B_1(A), A)) | subset(tptp_fun_B_1(A), A)))))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[106, 96])).
% 0.19/0.41 tff(108,plain,
% 0.19/0.41 ((~![A: $i] : (~((~((~epsilon_transitive(A)) | ![B: $i] : ((~in(B, A)) | subset(B, A)))) | (~(epsilon_transitive(A) | (~((~in(tptp_fun_B_1(A), A)) | subset(tptp_fun_B_1(A), A)))))))) | (~((~((~epsilon_transitive(B!18)) | ![B: $i] : ((~in(B, B!18)) | subset(B, B!18)))) | (~(epsilon_transitive(B!18) | (~((~in(tptp_fun_B_1(B!18), B!18)) | subset(tptp_fun_B_1(B!18), B!18)))))))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(109,plain,
% 0.19/0.41 (~((~((~epsilon_transitive(B!18)) | ![B: $i] : ((~in(B, B!18)) | subset(B, B!18)))) | (~(epsilon_transitive(B!18) | (~((~in(tptp_fun_B_1(B!18), B!18)) | subset(tptp_fun_B_1(B!18), B!18))))))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[108, 107])).
% 0.19/0.41 tff(110,plain,
% 0.19/0.41 (((~((~epsilon_transitive(B!18)) | ![B: $i] : ((~in(B, B!18)) | subset(B, B!18)))) | (~(epsilon_transitive(B!18) | (~((~in(tptp_fun_B_1(B!18), B!18)) | subset(tptp_fun_B_1(B!18), B!18)))))) | ((~epsilon_transitive(B!18)) | ![B: $i] : ((~in(B, B!18)) | subset(B, B!18)))),
% 0.19/0.41 inference(tautology,[status(thm)],[])).
% 0.19/0.41 tff(111,plain,
% 0.19/0.41 ((~epsilon_transitive(B!18)) | ![B: $i] : ((~in(B, B!18)) | subset(B, B!18))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[110, 109])).
% 0.19/0.41 tff(112,plain,
% 0.19/0.41 (ordinal(B!18)),
% 0.19/0.41 inference(or_elim,[status(thm)],[87])).
% 0.19/0.41 tff(113,plain,
% 0.19/0.41 (^[A: $i] : refl(((~ordinal(A)) | (~((~epsilon_transitive(A)) | (~epsilon_connected(A))))) <=> ((~ordinal(A)) | (~((~epsilon_transitive(A)) | (~epsilon_connected(A))))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(114,plain,
% 0.19/0.41 (![A: $i] : ((~ordinal(A)) | (~((~epsilon_transitive(A)) | (~epsilon_connected(A))))) <=> ![A: $i] : ((~ordinal(A)) | (~((~epsilon_transitive(A)) | (~epsilon_connected(A)))))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[113])).
% 0.19/0.41 tff(115,plain,
% 0.19/0.41 (^[A: $i] : rewrite(((~ordinal(A)) | (epsilon_transitive(A) & epsilon_connected(A))) <=> ((~ordinal(A)) | (~((~epsilon_transitive(A)) | (~epsilon_connected(A))))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(116,plain,
% 0.19/0.41 (![A: $i] : ((~ordinal(A)) | (epsilon_transitive(A) & epsilon_connected(A))) <=> ![A: $i] : ((~ordinal(A)) | (~((~epsilon_transitive(A)) | (~epsilon_connected(A)))))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[115])).
% 0.19/0.41 tff(117,plain,
% 0.19/0.41 (![A: $i] : ((~ordinal(A)) | (epsilon_transitive(A) & epsilon_connected(A))) <=> ![A: $i] : ((~ordinal(A)) | (epsilon_transitive(A) & epsilon_connected(A)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(118,plain,
% 0.19/0.41 (^[A: $i] : rewrite((ordinal(A) => (epsilon_transitive(A) & epsilon_connected(A))) <=> ((~ordinal(A)) | (epsilon_transitive(A) & epsilon_connected(A))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(119,plain,
% 0.19/0.41 (![A: $i] : (ordinal(A) => (epsilon_transitive(A) & epsilon_connected(A))) <=> ![A: $i] : ((~ordinal(A)) | (epsilon_transitive(A) & epsilon_connected(A)))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[118])).
% 0.19/0.41 tff(120,axiom,(![A: $i] : (ordinal(A) => (epsilon_transitive(A) & epsilon_connected(A)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','cc1_ordinal1')).
% 0.19/0.41 tff(121,plain,
% 0.19/0.41 (![A: $i] : ((~ordinal(A)) | (epsilon_transitive(A) & epsilon_connected(A)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[120, 119])).
% 0.19/0.41 tff(122,plain,
% 0.19/0.41 (![A: $i] : ((~ordinal(A)) | (epsilon_transitive(A) & epsilon_connected(A)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[121, 117])).
% 0.19/0.41 tff(123,plain,(
% 0.19/0.41 ![A: $i] : ((~ordinal(A)) | (epsilon_transitive(A) & epsilon_connected(A)))),
% 0.19/0.41 inference(skolemize,[status(sab)],[122])).
% 0.19/0.41 tff(124,plain,
% 0.19/0.41 (![A: $i] : ((~ordinal(A)) | (~((~epsilon_transitive(A)) | (~epsilon_connected(A)))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[123, 116])).
% 0.19/0.41 tff(125,plain,
% 0.19/0.41 (![A: $i] : ((~ordinal(A)) | (~((~epsilon_transitive(A)) | (~epsilon_connected(A)))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[124, 114])).
% 0.19/0.41 tff(126,plain,
% 0.19/0.41 (((~![A: $i] : ((~ordinal(A)) | (~((~epsilon_transitive(A)) | (~epsilon_connected(A)))))) | ((~ordinal(B!18)) | (~((~epsilon_transitive(B!18)) | (~epsilon_connected(B!18)))))) <=> ((~![A: $i] : ((~ordinal(A)) | (~((~epsilon_transitive(A)) | (~epsilon_connected(A)))))) | (~ordinal(B!18)) | (~((~epsilon_transitive(B!18)) | (~epsilon_connected(B!18)))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(127,plain,
% 0.19/0.41 ((~![A: $i] : ((~ordinal(A)) | (~((~epsilon_transitive(A)) | (~epsilon_connected(A)))))) | ((~ordinal(B!18)) | (~((~epsilon_transitive(B!18)) | (~epsilon_connected(B!18)))))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(128,plain,
% 0.19/0.41 ((~![A: $i] : ((~ordinal(A)) | (~((~epsilon_transitive(A)) | (~epsilon_connected(A)))))) | (~ordinal(B!18)) | (~((~epsilon_transitive(B!18)) | (~epsilon_connected(B!18))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[127, 126])).
% 0.19/0.41 tff(129,plain,
% 0.19/0.41 (~((~epsilon_transitive(B!18)) | (~epsilon_connected(B!18)))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[128, 125, 112])).
% 0.19/0.41 tff(130,plain,
% 0.19/0.41 (((~epsilon_transitive(B!18)) | (~epsilon_connected(B!18))) | epsilon_transitive(B!18)),
% 0.19/0.41 inference(tautology,[status(thm)],[])).
% 0.19/0.41 tff(131,plain,
% 0.19/0.41 (epsilon_transitive(B!18)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[130, 129])).
% 0.19/0.41 tff(132,plain,
% 0.19/0.41 ((~((~epsilon_transitive(B!18)) | ![B: $i] : ((~in(B, B!18)) | subset(B, B!18)))) | (~epsilon_transitive(B!18)) | ![B: $i] : ((~in(B, B!18)) | subset(B, B!18))),
% 0.19/0.41 inference(tautology,[status(thm)],[])).
% 0.19/0.41 tff(133,plain,
% 0.19/0.41 ((~((~epsilon_transitive(B!18)) | ![B: $i] : ((~in(B, B!18)) | subset(B, B!18)))) | ![B: $i] : ((~in(B, B!18)) | subset(B, B!18))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[132, 131])).
% 0.19/0.41 tff(134,plain,
% 0.19/0.41 (![B: $i] : ((~in(B, B!18)) | subset(B, B!18))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[133, 111])).
% 0.19/0.41 tff(135,plain,
% 0.19/0.41 (((~![B: $i] : ((~in(B, B!18)) | subset(B, B!18))) | ((~in(A!17, B!18)) | subset(A!17, B!18))) <=> ((~![B: $i] : ((~in(B, B!18)) | subset(B, B!18))) | (~in(A!17, B!18)) | subset(A!17, B!18))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(136,plain,
% 0.19/0.41 ((~![B: $i] : ((~in(B, B!18)) | subset(B, B!18))) | ((~in(A!17, B!18)) | subset(A!17, B!18))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(137,plain,
% 0.19/0.41 ((~![B: $i] : ((~in(B, B!18)) | subset(B, B!18))) | (~in(A!17, B!18)) | subset(A!17, B!18)),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[136, 135])).
% 0.19/0.41 tff(138,plain,
% 0.19/0.41 ((~in(A!17, B!18)) | subset(A!17, B!18)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[137, 134])).
% 0.19/0.41 tff(139,plain,
% 0.19/0.41 (subset(A!17, B!18)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[138, 91])).
% 0.19/0.41 tff(140,plain,
% 0.19/0.41 (^[A: $i, B: $i] : refl((set_union2(A, B) = set_union2(B, A)) <=> (set_union2(A, B) = set_union2(B, A)))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(141,plain,
% 0.19/0.41 (![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A)) <=> ![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[140])).
% 0.19/0.41 tff(142,plain,
% 0.19/0.41 (![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A)) <=> ![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(143,axiom,(![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','commutativity_k2_xboole_0')).
% 0.19/0.41 tff(144,plain,
% 0.19/0.41 (![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[143, 142])).
% 0.19/0.41 tff(145,plain,(
% 0.19/0.41 ![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))),
% 0.19/0.41 inference(skolemize,[status(sab)],[144])).
% 0.19/0.41 tff(146,plain,
% 0.19/0.41 (![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[145, 141])).
% 0.19/0.41 tff(147,plain,
% 0.19/0.41 ((~![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))) | (set_union2(A!17, singleton(A!17)) = set_union2(singleton(A!17), A!17))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(148,plain,
% 0.19/0.41 (set_union2(A!17, singleton(A!17)) = set_union2(singleton(A!17), A!17)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[147, 146])).
% 0.19/0.41 tff(149,plain,
% 0.19/0.41 (set_union2(singleton(A!17), A!17) = set_union2(A!17, singleton(A!17))),
% 0.19/0.41 inference(symmetry,[status(thm)],[148])).
% 0.19/0.41 tff(150,plain,
% 0.19/0.41 (set_union2(singleton(A!17), A!17) = succ(A!17)),
% 0.19/0.41 inference(transitivity,[status(thm)],[149, 10])).
% 0.19/0.41 tff(151,plain,
% 0.19/0.41 (subset(set_union2(singleton(A!17), A!17), B!18) <=> subset(succ(A!17), B!18)),
% 0.19/0.41 inference(monotonicity,[status(thm)],[150])).
% 0.19/0.41 tff(152,plain,
% 0.19/0.41 (subset(succ(A!17), B!18) <=> subset(set_union2(singleton(A!17), A!17), B!18)),
% 0.19/0.41 inference(symmetry,[status(thm)],[151])).
% 0.19/0.41 tff(153,plain,
% 0.19/0.41 ((~subset(succ(A!17), B!18)) <=> (~subset(set_union2(singleton(A!17), A!17), B!18))),
% 0.19/0.41 inference(monotonicity,[status(thm)],[152])).
% 0.19/0.41 tff(154,plain,
% 0.19/0.41 (ordinal(A!17)),
% 0.19/0.41 inference(and_elim,[status(thm)],[86])).
% 0.19/0.41 tff(155,plain,
% 0.19/0.41 (^[A: $i] : refl(((~ordinal(A)) | (~(empty(succ(A)) | (~epsilon_transitive(succ(A))) | (~epsilon_connected(succ(A))) | (~ordinal(succ(A)))))) <=> ((~ordinal(A)) | (~(empty(succ(A)) | (~epsilon_transitive(succ(A))) | (~epsilon_connected(succ(A))) | (~ordinal(succ(A)))))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(156,plain,
% 0.19/0.41 (![A: $i] : ((~ordinal(A)) | (~(empty(succ(A)) | (~epsilon_transitive(succ(A))) | (~epsilon_connected(succ(A))) | (~ordinal(succ(A)))))) <=> ![A: $i] : ((~ordinal(A)) | (~(empty(succ(A)) | (~epsilon_transitive(succ(A))) | (~epsilon_connected(succ(A))) | (~ordinal(succ(A))))))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[155])).
% 0.19/0.41 tff(157,plain,
% 0.19/0.41 (^[A: $i] : rewrite(((~ordinal(A)) | ((~empty(succ(A))) & epsilon_transitive(succ(A)) & epsilon_connected(succ(A)) & ordinal(succ(A)))) <=> ((~ordinal(A)) | (~(empty(succ(A)) | (~epsilon_transitive(succ(A))) | (~epsilon_connected(succ(A))) | (~ordinal(succ(A)))))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(158,plain,
% 0.19/0.41 (![A: $i] : ((~ordinal(A)) | ((~empty(succ(A))) & epsilon_transitive(succ(A)) & epsilon_connected(succ(A)) & ordinal(succ(A)))) <=> ![A: $i] : ((~ordinal(A)) | (~(empty(succ(A)) | (~epsilon_transitive(succ(A))) | (~epsilon_connected(succ(A))) | (~ordinal(succ(A))))))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[157])).
% 0.19/0.41 tff(159,plain,
% 0.19/0.41 (![A: $i] : ((~ordinal(A)) | ((~empty(succ(A))) & epsilon_transitive(succ(A)) & epsilon_connected(succ(A)) & ordinal(succ(A)))) <=> ![A: $i] : ((~ordinal(A)) | ((~empty(succ(A))) & epsilon_transitive(succ(A)) & epsilon_connected(succ(A)) & ordinal(succ(A))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(160,plain,
% 0.19/0.41 (^[A: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~empty(succ(A))) & epsilon_transitive(succ(A))) & epsilon_connected(succ(A))) <=> ((~empty(succ(A))) & epsilon_transitive(succ(A)) & epsilon_connected(succ(A)))), (((((~empty(succ(A))) & epsilon_transitive(succ(A))) & epsilon_connected(succ(A))) & ordinal(succ(A))) <=> (((~empty(succ(A))) & epsilon_transitive(succ(A)) & epsilon_connected(succ(A))) & ordinal(succ(A))))), rewrite((((~empty(succ(A))) & epsilon_transitive(succ(A)) & epsilon_connected(succ(A))) & ordinal(succ(A))) <=> ((~empty(succ(A))) & epsilon_transitive(succ(A)) & epsilon_connected(succ(A)) & ordinal(succ(A)))), (((((~empty(succ(A))) & epsilon_transitive(succ(A))) & epsilon_connected(succ(A))) & ordinal(succ(A))) <=> ((~empty(succ(A))) & epsilon_transitive(succ(A)) & epsilon_connected(succ(A)) & ordinal(succ(A))))), ((ordinal(A) => ((((~empty(succ(A))) & epsilon_transitive(succ(A))) & epsilon_connected(succ(A))) & ordinal(succ(A)))) <=> (ordinal(A) => ((~empty(succ(A))) & epsilon_transitive(succ(A)) & epsilon_connected(succ(A)) & ordinal(succ(A)))))), rewrite((ordinal(A) => ((~empty(succ(A))) & epsilon_transitive(succ(A)) & epsilon_connected(succ(A)) & ordinal(succ(A)))) <=> ((~ordinal(A)) | ((~empty(succ(A))) & epsilon_transitive(succ(A)) & epsilon_connected(succ(A)) & ordinal(succ(A))))), ((ordinal(A) => ((((~empty(succ(A))) & epsilon_transitive(succ(A))) & epsilon_connected(succ(A))) & ordinal(succ(A)))) <=> ((~ordinal(A)) | ((~empty(succ(A))) & epsilon_transitive(succ(A)) & epsilon_connected(succ(A)) & ordinal(succ(A))))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(161,plain,
% 0.19/0.41 (![A: $i] : (ordinal(A) => ((((~empty(succ(A))) & epsilon_transitive(succ(A))) & epsilon_connected(succ(A))) & ordinal(succ(A)))) <=> ![A: $i] : ((~ordinal(A)) | ((~empty(succ(A))) & epsilon_transitive(succ(A)) & epsilon_connected(succ(A)) & ordinal(succ(A))))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[160])).
% 0.19/0.41 tff(162,axiom,(![A: $i] : (ordinal(A) => ((((~empty(succ(A))) & epsilon_transitive(succ(A))) & epsilon_connected(succ(A))) & ordinal(succ(A))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','fc3_ordinal1')).
% 0.19/0.41 tff(163,plain,
% 0.19/0.41 (![A: $i] : ((~ordinal(A)) | ((~empty(succ(A))) & epsilon_transitive(succ(A)) & epsilon_connected(succ(A)) & ordinal(succ(A))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[162, 161])).
% 0.19/0.41 tff(164,plain,
% 0.19/0.41 (![A: $i] : ((~ordinal(A)) | ((~empty(succ(A))) & epsilon_transitive(succ(A)) & epsilon_connected(succ(A)) & ordinal(succ(A))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[163, 159])).
% 0.19/0.41 tff(165,plain,(
% 0.19/0.41 ![A: $i] : ((~ordinal(A)) | ((~empty(succ(A))) & epsilon_transitive(succ(A)) & epsilon_connected(succ(A)) & ordinal(succ(A))))),
% 0.19/0.41 inference(skolemize,[status(sab)],[164])).
% 0.19/0.41 tff(166,plain,
% 0.19/0.41 (![A: $i] : ((~ordinal(A)) | (~(empty(succ(A)) | (~epsilon_transitive(succ(A))) | (~epsilon_connected(succ(A))) | (~ordinal(succ(A))))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[165, 158])).
% 0.19/0.41 tff(167,plain,
% 0.19/0.41 (![A: $i] : ((~ordinal(A)) | (~(empty(succ(A)) | (~epsilon_transitive(succ(A))) | (~epsilon_connected(succ(A))) | (~ordinal(succ(A))))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[166, 156])).
% 0.19/0.41 tff(168,plain,
% 0.19/0.41 (((~![A: $i] : ((~ordinal(A)) | (~(empty(succ(A)) | (~epsilon_transitive(succ(A))) | (~epsilon_connected(succ(A))) | (~ordinal(succ(A))))))) | ((~ordinal(A!17)) | (~((~ordinal(succ(A!17))) | empty(succ(A!17)) | (~epsilon_transitive(succ(A!17))) | (~epsilon_connected(succ(A!17))))))) <=> ((~![A: $i] : ((~ordinal(A)) | (~(empty(succ(A)) | (~epsilon_transitive(succ(A))) | (~epsilon_connected(succ(A))) | (~ordinal(succ(A))))))) | (~ordinal(A!17)) | (~((~ordinal(succ(A!17))) | empty(succ(A!17)) | (~epsilon_transitive(succ(A!17))) | (~epsilon_connected(succ(A!17))))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(169,plain,
% 0.19/0.41 (((~ordinal(A!17)) | (~(empty(succ(A!17)) | (~epsilon_transitive(succ(A!17))) | (~epsilon_connected(succ(A!17))) | (~ordinal(succ(A!17)))))) <=> ((~ordinal(A!17)) | (~((~ordinal(succ(A!17))) | empty(succ(A!17)) | (~epsilon_transitive(succ(A!17))) | (~epsilon_connected(succ(A!17))))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(170,plain,
% 0.19/0.41 (((~![A: $i] : ((~ordinal(A)) | (~(empty(succ(A)) | (~epsilon_transitive(succ(A))) | (~epsilon_connected(succ(A))) | (~ordinal(succ(A))))))) | ((~ordinal(A!17)) | (~(empty(succ(A!17)) | (~epsilon_transitive(succ(A!17))) | (~epsilon_connected(succ(A!17))) | (~ordinal(succ(A!17))))))) <=> ((~![A: $i] : ((~ordinal(A)) | (~(empty(succ(A)) | (~epsilon_transitive(succ(A))) | (~epsilon_connected(succ(A))) | (~ordinal(succ(A))))))) | ((~ordinal(A!17)) | (~((~ordinal(succ(A!17))) | empty(succ(A!17)) | (~epsilon_transitive(succ(A!17))) | (~epsilon_connected(succ(A!17)))))))),
% 0.19/0.41 inference(monotonicity,[status(thm)],[169])).
% 0.19/0.41 tff(171,plain,
% 0.19/0.41 (((~![A: $i] : ((~ordinal(A)) | (~(empty(succ(A)) | (~epsilon_transitive(succ(A))) | (~epsilon_connected(succ(A))) | (~ordinal(succ(A))))))) | ((~ordinal(A!17)) | (~(empty(succ(A!17)) | (~epsilon_transitive(succ(A!17))) | (~epsilon_connected(succ(A!17))) | (~ordinal(succ(A!17))))))) <=> ((~![A: $i] : ((~ordinal(A)) | (~(empty(succ(A)) | (~epsilon_transitive(succ(A))) | (~epsilon_connected(succ(A))) | (~ordinal(succ(A))))))) | (~ordinal(A!17)) | (~((~ordinal(succ(A!17))) | empty(succ(A!17)) | (~epsilon_transitive(succ(A!17))) | (~epsilon_connected(succ(A!17))))))),
% 0.19/0.41 inference(transitivity,[status(thm)],[170, 168])).
% 0.19/0.41 tff(172,plain,
% 0.19/0.41 ((~![A: $i] : ((~ordinal(A)) | (~(empty(succ(A)) | (~epsilon_transitive(succ(A))) | (~epsilon_connected(succ(A))) | (~ordinal(succ(A))))))) | ((~ordinal(A!17)) | (~(empty(succ(A!17)) | (~epsilon_transitive(succ(A!17))) | (~epsilon_connected(succ(A!17))) | (~ordinal(succ(A!17))))))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(173,plain,
% 0.19/0.42 ((~![A: $i] : ((~ordinal(A)) | (~(empty(succ(A)) | (~epsilon_transitive(succ(A))) | (~epsilon_connected(succ(A))) | (~ordinal(succ(A))))))) | (~ordinal(A!17)) | (~((~ordinal(succ(A!17))) | empty(succ(A!17)) | (~epsilon_transitive(succ(A!17))) | (~epsilon_connected(succ(A!17)))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[172, 171])).
% 0.19/0.42 tff(174,plain,
% 0.19/0.42 (~((~ordinal(succ(A!17))) | empty(succ(A!17)) | (~epsilon_transitive(succ(A!17))) | (~epsilon_connected(succ(A!17))))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[173, 167, 154])).
% 0.19/0.42 tff(175,plain,
% 0.19/0.42 (((~ordinal(succ(A!17))) | empty(succ(A!17)) | (~epsilon_transitive(succ(A!17))) | (~epsilon_connected(succ(A!17)))) | ordinal(succ(A!17))),
% 0.19/0.42 inference(tautology,[status(thm)],[])).
% 0.19/0.42 tff(176,plain,
% 0.19/0.42 (ordinal(succ(A!17))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[175, 174])).
% 0.19/0.42 tff(177,plain,
% 0.19/0.42 (^[A: $i, B: $i] : refl(((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A))) <=> ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(178,plain,
% 0.19/0.42 (![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A))) <=> ![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[177])).
% 0.19/0.42 tff(179,plain,
% 0.19/0.42 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(rewrite((ordinal(A) & ordinal(B)) <=> (~((~ordinal(B)) | (~ordinal(A))))), ((~(ordinal(A) & ordinal(B))) <=> (~(~((~ordinal(B)) | (~ordinal(A))))))), rewrite((~(~((~ordinal(B)) | (~ordinal(A))))) <=> ((~ordinal(B)) | (~ordinal(A)))), ((~(ordinal(A) & ordinal(B))) <=> ((~ordinal(B)) | (~ordinal(A))))), (((~(ordinal(A) & ordinal(B))) | (ordinal_subset(A, B) <=> subset(A, B))) <=> (((~ordinal(B)) | (~ordinal(A))) | (ordinal_subset(A, B) <=> subset(A, B))))), rewrite((((~ordinal(B)) | (~ordinal(A))) | (ordinal_subset(A, B) <=> subset(A, B))) <=> ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))), (((~(ordinal(A) & ordinal(B))) | (ordinal_subset(A, B) <=> subset(A, B))) <=> ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(180,plain,
% 0.19/0.42 (![A: $i, B: $i] : ((~(ordinal(A) & ordinal(B))) | (ordinal_subset(A, B) <=> subset(A, B))) <=> ![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[179])).
% 0.19/0.42 tff(181,plain,
% 0.19/0.42 (![A: $i, B: $i] : ((~(ordinal(A) & ordinal(B))) | (ordinal_subset(A, B) <=> subset(A, B))) <=> ![A: $i, B: $i] : ((~(ordinal(A) & ordinal(B))) | (ordinal_subset(A, B) <=> subset(A, B)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(182,plain,
% 0.19/0.42 (^[A: $i, B: $i] : rewrite(((ordinal(A) & ordinal(B)) => (ordinal_subset(A, B) <=> subset(A, B))) <=> ((~(ordinal(A) & ordinal(B))) | (ordinal_subset(A, B) <=> subset(A, B))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(183,plain,
% 0.19/0.42 (![A: $i, B: $i] : ((ordinal(A) & ordinal(B)) => (ordinal_subset(A, B) <=> subset(A, B))) <=> ![A: $i, B: $i] : ((~(ordinal(A) & ordinal(B))) | (ordinal_subset(A, B) <=> subset(A, B)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[182])).
% 0.19/0.42 tff(184,axiom,(![A: $i, B: $i] : ((ordinal(A) & ordinal(B)) => (ordinal_subset(A, B) <=> subset(A, B)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','redefinition_r1_ordinal1')).
% 0.19/0.42 tff(185,plain,
% 0.19/0.42 (![A: $i, B: $i] : ((~(ordinal(A) & ordinal(B))) | (ordinal_subset(A, B) <=> subset(A, B)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[184, 183])).
% 0.19/0.42 tff(186,plain,
% 0.19/0.42 (![A: $i, B: $i] : ((~(ordinal(A) & ordinal(B))) | (ordinal_subset(A, B) <=> subset(A, B)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[185, 181])).
% 0.19/0.42 tff(187,plain,(
% 0.19/0.42 ![A: $i, B: $i] : ((~(ordinal(A) & ordinal(B))) | (ordinal_subset(A, B) <=> subset(A, B)))),
% 0.19/0.42 inference(skolemize,[status(sab)],[186])).
% 0.19/0.42 tff(188,plain,
% 0.19/0.42 (![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[187, 180])).
% 0.19/0.42 tff(189,plain,
% 0.19/0.42 (![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[188, 178])).
% 0.19/0.42 tff(190,plain,
% 0.19/0.42 (((~![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))) | ((~ordinal(B!18)) | (ordinal_subset(succ(A!17), B!18) <=> subset(succ(A!17), B!18)) | (~ordinal(succ(A!17))))) <=> ((~![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))) | (~ordinal(B!18)) | (ordinal_subset(succ(A!17), B!18) <=> subset(succ(A!17), B!18)) | (~ordinal(succ(A!17))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(191,plain,
% 0.19/0.42 (((ordinal_subset(succ(A!17), B!18) <=> subset(succ(A!17), B!18)) | (~ordinal(B!18)) | (~ordinal(succ(A!17)))) <=> ((~ordinal(B!18)) | (ordinal_subset(succ(A!17), B!18) <=> subset(succ(A!17), B!18)) | (~ordinal(succ(A!17))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(192,plain,
% 0.19/0.42 (((~![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))) | ((ordinal_subset(succ(A!17), B!18) <=> subset(succ(A!17), B!18)) | (~ordinal(B!18)) | (~ordinal(succ(A!17))))) <=> ((~![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))) | ((~ordinal(B!18)) | (ordinal_subset(succ(A!17), B!18) <=> subset(succ(A!17), B!18)) | (~ordinal(succ(A!17)))))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[191])).
% 0.19/0.42 tff(193,plain,
% 0.19/0.42 (((~![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))) | ((ordinal_subset(succ(A!17), B!18) <=> subset(succ(A!17), B!18)) | (~ordinal(B!18)) | (~ordinal(succ(A!17))))) <=> ((~![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))) | (~ordinal(B!18)) | (ordinal_subset(succ(A!17), B!18) <=> subset(succ(A!17), B!18)) | (~ordinal(succ(A!17))))),
% 0.19/0.42 inference(transitivity,[status(thm)],[192, 190])).
% 0.19/0.42 tff(194,plain,
% 0.19/0.42 ((~![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))) | ((ordinal_subset(succ(A!17), B!18) <=> subset(succ(A!17), B!18)) | (~ordinal(B!18)) | (~ordinal(succ(A!17))))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(195,plain,
% 0.19/0.42 ((~![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))) | (~ordinal(B!18)) | (ordinal_subset(succ(A!17), B!18) <=> subset(succ(A!17), B!18)) | (~ordinal(succ(A!17)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[194, 193])).
% 0.19/0.42 tff(196,plain,
% 0.19/0.42 ((ordinal_subset(succ(A!17), B!18) <=> subset(succ(A!17), B!18)) | (~ordinal(succ(A!17)))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[195, 189, 112])).
% 0.19/0.42 tff(197,plain,
% 0.19/0.42 (ordinal_subset(succ(A!17), B!18) <=> subset(succ(A!17), B!18)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[196, 176])).
% 0.19/0.42 tff(198,plain,
% 0.19/0.42 ((~(ordinal_subset(succ(A!17), B!18) <=> subset(succ(A!17), B!18))) | ordinal_subset(succ(A!17), B!18) | (~subset(succ(A!17), B!18))),
% 0.19/0.42 inference(tautology,[status(thm)],[])).
% 0.19/0.42 tff(199,plain,
% 0.19/0.42 (ordinal_subset(succ(A!17), B!18) | (~subset(succ(A!17), B!18))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[198, 197])).
% 0.19/0.42 tff(200,plain,
% 0.19/0.42 (~subset(succ(A!17), B!18)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[199, 79])).
% 0.19/0.42 tff(201,plain,
% 0.19/0.42 (~subset(set_union2(singleton(A!17), A!17), B!18)),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[200, 153])).
% 0.19/0.42 tff(202,plain,
% 0.19/0.42 (^[A: $i, B: $i, C: $i] : refl((subset(set_union2(A, C), B) | (~subset(A, B)) | (~subset(C, B))) <=> (subset(set_union2(A, C), B) | (~subset(A, B)) | (~subset(C, B))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(203,plain,
% 0.19/0.42 (![A: $i, B: $i, C: $i] : (subset(set_union2(A, C), B) | (~subset(A, B)) | (~subset(C, B))) <=> ![A: $i, B: $i, C: $i] : (subset(set_union2(A, C), B) | (~subset(A, B)) | (~subset(C, B)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[202])).
% 0.19/0.42 tff(204,plain,
% 0.19/0.42 (^[A: $i, B: $i, C: $i] : trans(monotonicity(trans(monotonicity(rewrite((subset(A, B) & subset(C, B)) <=> (~((~subset(A, B)) | (~subset(C, B))))), ((~(subset(A, B) & subset(C, B))) <=> (~(~((~subset(A, B)) | (~subset(C, B))))))), rewrite((~(~((~subset(A, B)) | (~subset(C, B))))) <=> ((~subset(A, B)) | (~subset(C, B)))), ((~(subset(A, B) & subset(C, B))) <=> ((~subset(A, B)) | (~subset(C, B))))), (((~(subset(A, B) & subset(C, B))) | subset(set_union2(A, C), B)) <=> (((~subset(A, B)) | (~subset(C, B))) | subset(set_union2(A, C), B)))), rewrite((((~subset(A, B)) | (~subset(C, B))) | subset(set_union2(A, C), B)) <=> (subset(set_union2(A, C), B) | (~subset(A, B)) | (~subset(C, B)))), (((~(subset(A, B) & subset(C, B))) | subset(set_union2(A, C), B)) <=> (subset(set_union2(A, C), B) | (~subset(A, B)) | (~subset(C, B)))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(205,plain,
% 0.19/0.42 (![A: $i, B: $i, C: $i] : ((~(subset(A, B) & subset(C, B))) | subset(set_union2(A, C), B)) <=> ![A: $i, B: $i, C: $i] : (subset(set_union2(A, C), B) | (~subset(A, B)) | (~subset(C, B)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[204])).
% 0.19/0.42 tff(206,plain,
% 0.19/0.42 (![A: $i, B: $i, C: $i] : ((~(subset(A, B) & subset(C, B))) | subset(set_union2(A, C), B)) <=> ![A: $i, B: $i, C: $i] : ((~(subset(A, B) & subset(C, B))) | subset(set_union2(A, C), B))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(207,plain,
% 0.19/0.42 (^[A: $i, B: $i, C: $i] : rewrite(((subset(A, B) & subset(C, B)) => subset(set_union2(A, C), B)) <=> ((~(subset(A, B) & subset(C, B))) | subset(set_union2(A, C), B)))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(208,plain,
% 0.19/0.42 (![A: $i, B: $i, C: $i] : ((subset(A, B) & subset(C, B)) => subset(set_union2(A, C), B)) <=> ![A: $i, B: $i, C: $i] : ((~(subset(A, B) & subset(C, B))) | subset(set_union2(A, C), B))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[207])).
% 0.19/0.42 tff(209,axiom,(![A: $i, B: $i, C: $i] : ((subset(A, B) & subset(C, B)) => subset(set_union2(A, C), B))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t8_xboole_1')).
% 0.19/0.42 tff(210,plain,
% 0.19/0.42 (![A: $i, B: $i, C: $i] : ((~(subset(A, B) & subset(C, B))) | subset(set_union2(A, C), B))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[209, 208])).
% 0.19/0.42 tff(211,plain,
% 0.19/0.42 (![A: $i, B: $i, C: $i] : ((~(subset(A, B) & subset(C, B))) | subset(set_union2(A, C), B))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[210, 206])).
% 0.19/0.42 tff(212,plain,(
% 0.19/0.42 ![A: $i, B: $i, C: $i] : ((~(subset(A, B) & subset(C, B))) | subset(set_union2(A, C), B))),
% 0.19/0.42 inference(skolemize,[status(sab)],[211])).
% 0.19/0.42 tff(213,plain,
% 0.19/0.42 (![A: $i, B: $i, C: $i] : (subset(set_union2(A, C), B) | (~subset(A, B)) | (~subset(C, B)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[212, 205])).
% 0.19/0.42 tff(214,plain,
% 0.19/0.42 (![A: $i, B: $i, C: $i] : (subset(set_union2(A, C), B) | (~subset(A, B)) | (~subset(C, B)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[213, 203])).
% 0.19/0.42 tff(215,plain,
% 0.19/0.42 (((~![A: $i, B: $i, C: $i] : (subset(set_union2(A, C), B) | (~subset(A, B)) | (~subset(C, B)))) | ((~subset(A!17, B!18)) | (~subset(singleton(A!17), B!18)) | subset(set_union2(singleton(A!17), A!17), B!18))) <=> ((~![A: $i, B: $i, C: $i] : (subset(set_union2(A, C), B) | (~subset(A, B)) | (~subset(C, B)))) | (~subset(A!17, B!18)) | (~subset(singleton(A!17), B!18)) | subset(set_union2(singleton(A!17), A!17), B!18))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(216,plain,
% 0.19/0.42 ((subset(set_union2(singleton(A!17), A!17), B!18) | (~subset(singleton(A!17), B!18)) | (~subset(A!17, B!18))) <=> ((~subset(A!17, B!18)) | (~subset(singleton(A!17), B!18)) | subset(set_union2(singleton(A!17), A!17), B!18))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(217,plain,
% 0.19/0.42 (((~![A: $i, B: $i, C: $i] : (subset(set_union2(A, C), B) | (~subset(A, B)) | (~subset(C, B)))) | (subset(set_union2(singleton(A!17), A!17), B!18) | (~subset(singleton(A!17), B!18)) | (~subset(A!17, B!18)))) <=> ((~![A: $i, B: $i, C: $i] : (subset(set_union2(A, C), B) | (~subset(A, B)) | (~subset(C, B)))) | ((~subset(A!17, B!18)) | (~subset(singleton(A!17), B!18)) | subset(set_union2(singleton(A!17), A!17), B!18)))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[216])).
% 0.19/0.42 tff(218,plain,
% 0.19/0.42 (((~![A: $i, B: $i, C: $i] : (subset(set_union2(A, C), B) | (~subset(A, B)) | (~subset(C, B)))) | (subset(set_union2(singleton(A!17), A!17), B!18) | (~subset(singleton(A!17), B!18)) | (~subset(A!17, B!18)))) <=> ((~![A: $i, B: $i, C: $i] : (subset(set_union2(A, C), B) | (~subset(A, B)) | (~subset(C, B)))) | (~subset(A!17, B!18)) | (~subset(singleton(A!17), B!18)) | subset(set_union2(singleton(A!17), A!17), B!18))),
% 0.19/0.42 inference(transitivity,[status(thm)],[217, 215])).
% 0.19/0.42 tff(219,plain,
% 0.19/0.42 ((~![A: $i, B: $i, C: $i] : (subset(set_union2(A, C), B) | (~subset(A, B)) | (~subset(C, B)))) | (subset(set_union2(singleton(A!17), A!17), B!18) | (~subset(singleton(A!17), B!18)) | (~subset(A!17, B!18)))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(220,plain,
% 0.19/0.42 ((~![A: $i, B: $i, C: $i] : (subset(set_union2(A, C), B) | (~subset(A, B)) | (~subset(C, B)))) | (~subset(A!17, B!18)) | (~subset(singleton(A!17), B!18)) | subset(set_union2(singleton(A!17), A!17), B!18)),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[219, 218])).
% 0.19/0.42 tff(221,plain,
% 0.19/0.42 ((~subset(A!17, B!18)) | (~subset(singleton(A!17), B!18)) | subset(set_union2(singleton(A!17), A!17), B!18)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[220, 214])).
% 0.19/0.42 tff(222,plain,
% 0.19/0.42 (~subset(singleton(A!17), B!18)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[221, 201, 139])).
% 0.19/0.42 tff(223,plain,
% 0.19/0.42 ((~(subset(singleton(A!17), B!18) | (~((~in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17))) | in(tptp_fun_C_2(B!18, singleton(A!17)), B!18))))) | subset(singleton(A!17), B!18) | (~((~in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17))) | in(tptp_fun_C_2(B!18, singleton(A!17)), B!18)))),
% 0.19/0.42 inference(tautology,[status(thm)],[])).
% 0.19/0.42 tff(224,plain,
% 0.19/0.42 ((~(subset(singleton(A!17), B!18) | (~((~in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17))) | in(tptp_fun_C_2(B!18, singleton(A!17)), B!18))))) | (~((~in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17))) | in(tptp_fun_C_2(B!18, singleton(A!17)), B!18)))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[223, 222])).
% 0.19/0.42 tff(225,plain,
% 0.19/0.42 (~((~in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17))) | in(tptp_fun_C_2(B!18, singleton(A!17)), B!18))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[224, 78])).
% 0.19/0.42 tff(226,plain,
% 0.19/0.42 (((~in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17))) | in(tptp_fun_C_2(B!18, singleton(A!17)), B!18)) | in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17))),
% 0.19/0.42 inference(tautology,[status(thm)],[])).
% 0.19/0.42 tff(227,plain,
% 0.19/0.42 (in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[226, 225])).
% 0.19/0.42 tff(228,plain,
% 0.19/0.42 ((~(in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17))) | (~in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17))) | (tptp_fun_C_2(B!18, singleton(A!17)) = A!17)),
% 0.19/0.42 inference(tautology,[status(thm)],[])).
% 0.19/0.42 tff(229,plain,
% 0.19/0.42 ((~(in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17)) <=> (tptp_fun_C_2(B!18, singleton(A!17)) = A!17))) | (tptp_fun_C_2(B!18, singleton(A!17)) = A!17)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[228, 227])).
% 0.19/0.42 tff(230,plain,
% 0.19/0.42 (tptp_fun_C_2(B!18, singleton(A!17)) = A!17),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[229, 58])).
% 0.19/0.42 tff(231,plain,
% 0.19/0.42 (in(tptp_fun_C_2(B!18, singleton(A!17)), B!18) <=> in(A!17, B!18)),
% 0.19/0.42 inference(monotonicity,[status(thm)],[230])).
% 0.19/0.42 tff(232,plain,
% 0.19/0.42 (in(A!17, B!18) <=> in(tptp_fun_C_2(B!18, singleton(A!17)), B!18)),
% 0.19/0.42 inference(symmetry,[status(thm)],[231])).
% 0.19/0.42 tff(233,plain,
% 0.19/0.42 (in(tptp_fun_C_2(B!18, singleton(A!17)), B!18)),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[91, 232])).
% 0.19/0.42 tff(234,plain,
% 0.19/0.42 (((~in(tptp_fun_C_2(B!18, singleton(A!17)), singleton(A!17))) | in(tptp_fun_C_2(B!18, singleton(A!17)), B!18)) | (~in(tptp_fun_C_2(B!18, singleton(A!17)), B!18))),
% 0.19/0.42 inference(tautology,[status(thm)],[])).
% 0.19/0.42 tff(235,plain,
% 0.19/0.42 (~in(tptp_fun_C_2(B!18, singleton(A!17)), B!18)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[234, 225])).
% 0.19/0.43 tff(236,plain,
% 0.19/0.43 ($false),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[235, 233])).
% 0.19/0.43 tff(237,plain,(ordinal_subset(succ(A!17), B!18)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.43 tff(238,plain,
% 0.19/0.43 ((~(ordinal_subset(succ(A!17), B!18) <=> subset(succ(A!17), B!18))) | (~ordinal_subset(succ(A!17), B!18)) | subset(succ(A!17), B!18)),
% 0.19/0.43 inference(tautology,[status(thm)],[])).
% 0.19/0.43 tff(239,plain,
% 0.19/0.43 ((~ordinal_subset(succ(A!17), B!18)) | subset(succ(A!17), B!18)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[238, 197])).
% 0.19/0.43 tff(240,plain,
% 0.19/0.43 (subset(succ(A!17), B!18)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[239, 237])).
% 0.19/0.43 tff(241,plain,
% 0.19/0.43 (subset(set_union2(A!17, singleton(A!17)), B!18)),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[240, 12])).
% 0.19/0.43 tff(242,plain,
% 0.19/0.43 (^[A: $i, B: $i] : refl((element(A, powerset(B)) <=> subset(A, B)) <=> (element(A, powerset(B)) <=> subset(A, B)))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(243,plain,
% 0.19/0.43 (![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B)) <=> ![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[242])).
% 0.19/0.43 tff(244,plain,
% 0.19/0.43 (![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B)) <=> ![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(245,axiom,(![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t3_subset')).
% 0.19/0.43 tff(246,plain,
% 0.19/0.43 (![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[245, 244])).
% 0.19/0.43 tff(247,plain,(
% 0.19/0.43 ![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B))),
% 0.19/0.43 inference(skolemize,[status(sab)],[246])).
% 0.19/0.43 tff(248,plain,
% 0.19/0.43 (![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[247, 243])).
% 0.19/0.43 tff(249,plain,
% 0.19/0.43 ((~![A: $i, B: $i] : (element(A, powerset(B)) <=> subset(A, B))) | (element(set_union2(A!17, singleton(A!17)), powerset(B!18)) <=> subset(set_union2(A!17, singleton(A!17)), B!18))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(250,plain,
% 0.19/0.43 (element(set_union2(A!17, singleton(A!17)), powerset(B!18)) <=> subset(set_union2(A!17, singleton(A!17)), B!18)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[249, 248])).
% 0.19/0.43 tff(251,plain,
% 0.19/0.43 ((~(element(set_union2(A!17, singleton(A!17)), powerset(B!18)) <=> subset(set_union2(A!17, singleton(A!17)), B!18))) | element(set_union2(A!17, singleton(A!17)), powerset(B!18)) | (~subset(set_union2(A!17, singleton(A!17)), B!18))),
% 0.19/0.43 inference(tautology,[status(thm)],[])).
% 0.19/0.43 tff(252,plain,
% 0.19/0.43 (element(set_union2(A!17, singleton(A!17)), powerset(B!18)) | (~subset(set_union2(A!17, singleton(A!17)), B!18))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[251, 250])).
% 0.19/0.43 tff(253,plain,
% 0.19/0.43 (element(set_union2(A!17, singleton(A!17)), powerset(B!18))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[252, 241])).
% 0.19/0.43 tff(254,plain,
% 0.19/0.43 (in(A!17, set_union2(A!17, singleton(A!17))) <=> in(A!17, succ(A!17))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[10])).
% 0.19/0.43 tff(255,plain,
% 0.19/0.43 (in(A!17, succ(A!17)) <=> in(A!17, set_union2(A!17, singleton(A!17)))),
% 0.19/0.43 inference(symmetry,[status(thm)],[254])).
% 0.19/0.43 tff(256,plain,
% 0.19/0.43 (^[A: $i] : refl(in(A, succ(A)) <=> in(A, succ(A)))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(257,plain,
% 0.19/0.43 (![A: $i] : in(A, succ(A)) <=> ![A: $i] : in(A, succ(A))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[256])).
% 0.19/0.43 tff(258,plain,
% 0.19/0.43 (![A: $i] : in(A, succ(A)) <=> ![A: $i] : in(A, succ(A))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(259,axiom,(![A: $i] : in(A, succ(A))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t10_ordinal1')).
% 0.19/0.43 tff(260,plain,
% 0.19/0.43 (![A: $i] : in(A, succ(A))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[259, 258])).
% 0.19/0.43 tff(261,plain,(
% 0.19/0.43 ![A: $i] : in(A, succ(A))),
% 0.19/0.43 inference(skolemize,[status(sab)],[260])).
% 0.19/0.43 tff(262,plain,
% 0.19/0.43 (![A: $i] : in(A, succ(A))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[261, 257])).
% 0.19/0.43 tff(263,plain,
% 0.19/0.43 ((~![A: $i] : in(A, succ(A))) | in(A!17, succ(A!17))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(264,plain,
% 0.19/0.43 (in(A!17, succ(A!17))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[263, 262])).
% 0.19/0.43 tff(265,plain,
% 0.19/0.43 (in(A!17, set_union2(A!17, singleton(A!17)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[264, 255])).
% 0.19/0.43 tff(266,plain,
% 0.19/0.43 ((in(A!17, B!18) <=> ordinal_subset(succ(A!17), B!18)) | (~in(A!17, B!18)) | (~ordinal_subset(succ(A!17), B!18))),
% 0.19/0.43 inference(tautology,[status(thm)],[])).
% 0.19/0.43 tff(267,plain,
% 0.19/0.43 ((~in(A!17, B!18)) | (~ordinal_subset(succ(A!17), B!18))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[266, 88])).
% 0.19/0.43 tff(268,plain,
% 0.19/0.43 (~in(A!17, B!18)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[267, 237])).
% 0.19/0.43 tff(269,plain,
% 0.19/0.43 (^[A: $i, B: $i, C: $i] : refl(((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C)))) <=> ((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C)))))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(270,plain,
% 0.19/0.43 (![A: $i, B: $i, C: $i] : ((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C)))) <=> ![A: $i, B: $i, C: $i] : ((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C))))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[269])).
% 0.19/0.43 tff(271,plain,
% 0.19/0.43 (^[A: $i, B: $i, C: $i] : trans(monotonicity(rewrite((in(A, B) & element(B, powerset(C)) & empty(C)) <=> (~((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C)))))), ((~(in(A, B) & element(B, powerset(C)) & empty(C))) <=> (~(~((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C)))))))), rewrite((~(~((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C)))))) <=> ((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C))))), ((~(in(A, B) & element(B, powerset(C)) & empty(C))) <=> ((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C))))))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(272,plain,
% 0.19/0.43 (![A: $i, B: $i, C: $i] : (~(in(A, B) & element(B, powerset(C)) & empty(C))) <=> ![A: $i, B: $i, C: $i] : ((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C))))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[271])).
% 0.19/0.43 tff(273,plain,
% 0.19/0.43 (![A: $i, B: $i, C: $i] : (~(in(A, B) & element(B, powerset(C)) & empty(C))) <=> ![A: $i, B: $i, C: $i] : (~(in(A, B) & element(B, powerset(C)) & empty(C)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(274,plain,
% 0.19/0.43 (^[A: $i, B: $i, C: $i] : rewrite((~((in(A, B) & element(B, powerset(C))) & empty(C))) <=> (~(in(A, B) & element(B, powerset(C)) & empty(C))))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(275,plain,
% 0.19/0.43 (![A: $i, B: $i, C: $i] : (~((in(A, B) & element(B, powerset(C))) & empty(C))) <=> ![A: $i, B: $i, C: $i] : (~(in(A, B) & element(B, powerset(C)) & empty(C)))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[274])).
% 0.19/0.43 tff(276,axiom,(![A: $i, B: $i, C: $i] : (~((in(A, B) & element(B, powerset(C))) & empty(C)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t5_subset')).
% 0.19/0.43 tff(277,plain,
% 0.19/0.43 (![A: $i, B: $i, C: $i] : (~(in(A, B) & element(B, powerset(C)) & empty(C)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[276, 275])).
% 0.19/0.43 tff(278,plain,
% 0.19/0.43 (![A: $i, B: $i, C: $i] : (~(in(A, B) & element(B, powerset(C)) & empty(C)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[277, 273])).
% 0.19/0.43 tff(279,plain,(
% 0.19/0.43 ![A: $i, B: $i, C: $i] : (~(in(A, B) & element(B, powerset(C)) & empty(C)))),
% 0.19/0.43 inference(skolemize,[status(sab)],[278])).
% 0.19/0.43 tff(280,plain,
% 0.19/0.43 (![A: $i, B: $i, C: $i] : ((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C))))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[279, 272])).
% 0.19/0.43 tff(281,plain,
% 0.19/0.43 (![A: $i, B: $i, C: $i] : ((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C))))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[280, 270])).
% 0.19/0.43 tff(282,plain,
% 0.19/0.43 (((~![A: $i, B: $i, C: $i] : ((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C))))) | ((~empty(B!18)) | (~in(A!17, set_union2(A!17, singleton(A!17)))) | (~element(set_union2(A!17, singleton(A!17)), powerset(B!18))))) <=> ((~![A: $i, B: $i, C: $i] : ((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C))))) | (~empty(B!18)) | (~in(A!17, set_union2(A!17, singleton(A!17)))) | (~element(set_union2(A!17, singleton(A!17)), powerset(B!18))))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(283,plain,
% 0.19/0.43 ((~![A: $i, B: $i, C: $i] : ((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C))))) | ((~empty(B!18)) | (~in(A!17, set_union2(A!17, singleton(A!17)))) | (~element(set_union2(A!17, singleton(A!17)), powerset(B!18))))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(284,plain,
% 0.19/0.43 ((~![A: $i, B: $i, C: $i] : ((~empty(C)) | (~in(A, B)) | (~element(B, powerset(C))))) | (~empty(B!18)) | (~in(A!17, set_union2(A!17, singleton(A!17)))) | (~element(set_union2(A!17, singleton(A!17)), powerset(B!18)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[283, 282])).
% 0.19/0.43 tff(285,plain,
% 0.19/0.43 ((~empty(B!18)) | (~element(set_union2(A!17, singleton(A!17)), powerset(B!18)))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[284, 281, 265])).
% 0.19/0.43 tff(286,plain,
% 0.19/0.43 (~empty(B!18)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[285, 253])).
% 0.19/0.43 tff(287,plain,
% 0.19/0.43 (^[A: $i, B: $i] : refl((empty(B) | in(A, B) | (~element(A, B))) <=> (empty(B) | in(A, B) | (~element(A, B))))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(288,plain,
% 0.19/0.43 (![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B))) <=> ![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[287])).
% 0.19/0.43 tff(289,plain,
% 0.19/0.43 (![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B))) <=> ![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(290,plain,
% 0.19/0.43 (^[A: $i, B: $i] : rewrite((element(A, B) => (empty(B) | in(A, B))) <=> (empty(B) | in(A, B) | (~element(A, B))))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(291,plain,
% 0.19/0.43 (![A: $i, B: $i] : (element(A, B) => (empty(B) | in(A, B))) <=> ![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[290])).
% 0.19/0.43 tff(292,axiom,(![A: $i, B: $i] : (element(A, B) => (empty(B) | in(A, B)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t2_subset')).
% 0.19/0.43 tff(293,plain,
% 0.19/0.43 (![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[292, 291])).
% 0.19/0.43 tff(294,plain,
% 0.19/0.43 (![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[293, 289])).
% 0.19/0.43 tff(295,plain,(
% 0.19/0.43 ![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))),
% 0.19/0.43 inference(skolemize,[status(sab)],[294])).
% 0.19/0.43 tff(296,plain,
% 0.19/0.43 (![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[295, 288])).
% 0.19/0.43 tff(297,plain,
% 0.19/0.43 (((~![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))) | (in(A!17, B!18) | empty(B!18) | (~element(A!17, B!18)))) <=> ((~![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))) | in(A!17, B!18) | empty(B!18) | (~element(A!17, B!18)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(298,plain,
% 0.19/0.43 ((empty(B!18) | in(A!17, B!18) | (~element(A!17, B!18))) <=> (in(A!17, B!18) | empty(B!18) | (~element(A!17, B!18)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(299,plain,
% 0.19/0.43 (((~![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))) | (empty(B!18) | in(A!17, B!18) | (~element(A!17, B!18)))) <=> ((~![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))) | (in(A!17, B!18) | empty(B!18) | (~element(A!17, B!18))))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[298])).
% 0.19/0.43 tff(300,plain,
% 0.19/0.43 (((~![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))) | (empty(B!18) | in(A!17, B!18) | (~element(A!17, B!18)))) <=> ((~![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))) | in(A!17, B!18) | empty(B!18) | (~element(A!17, B!18)))),
% 0.19/0.43 inference(transitivity,[status(thm)],[299, 297])).
% 0.19/0.43 tff(301,plain,
% 0.19/0.43 ((~![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))) | (empty(B!18) | in(A!17, B!18) | (~element(A!17, B!18)))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(302,plain,
% 0.19/0.43 ((~![A: $i, B: $i] : (empty(B) | in(A, B) | (~element(A, B)))) | in(A!17, B!18) | empty(B!18) | (~element(A!17, B!18))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[301, 300])).
% 0.19/0.43 tff(303,plain,
% 0.19/0.43 (in(A!17, B!18) | empty(B!18) | (~element(A!17, B!18))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[302, 296])).
% 0.19/0.43 tff(304,plain,
% 0.19/0.43 (~element(A!17, B!18)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[303, 286, 268])).
% 0.19/0.43 tff(305,plain,
% 0.19/0.43 (^[A: $i, B: $i, C: $i] : refl((element(A, C) | (~in(A, B)) | (~element(B, powerset(C)))) <=> (element(A, C) | (~in(A, B)) | (~element(B, powerset(C)))))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(306,plain,
% 0.19/0.43 (![A: $i, B: $i, C: $i] : (element(A, C) | (~in(A, B)) | (~element(B, powerset(C)))) <=> ![A: $i, B: $i, C: $i] : (element(A, C) | (~in(A, B)) | (~element(B, powerset(C))))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[305])).
% 0.19/0.43 tff(307,plain,
% 0.19/0.43 (^[A: $i, B: $i, C: $i] : trans(monotonicity(trans(monotonicity(rewrite((in(A, B) & element(B, powerset(C))) <=> (~((~in(A, B)) | (~element(B, powerset(C)))))), ((~(in(A, B) & element(B, powerset(C)))) <=> (~(~((~in(A, B)) | (~element(B, powerset(C)))))))), rewrite((~(~((~in(A, B)) | (~element(B, powerset(C)))))) <=> ((~in(A, B)) | (~element(B, powerset(C))))), ((~(in(A, B) & element(B, powerset(C)))) <=> ((~in(A, B)) | (~element(B, powerset(C)))))), (((~(in(A, B) & element(B, powerset(C)))) | element(A, C)) <=> (((~in(A, B)) | (~element(B, powerset(C)))) | element(A, C)))), rewrite((((~in(A, B)) | (~element(B, powerset(C)))) | element(A, C)) <=> (element(A, C) | (~in(A, B)) | (~element(B, powerset(C))))), (((~(in(A, B) & element(B, powerset(C)))) | element(A, C)) <=> (element(A, C) | (~in(A, B)) | (~element(B, powerset(C))))))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(308,plain,
% 0.19/0.43 (![A: $i, B: $i, C: $i] : ((~(in(A, B) & element(B, powerset(C)))) | element(A, C)) <=> ![A: $i, B: $i, C: $i] : (element(A, C) | (~in(A, B)) | (~element(B, powerset(C))))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[307])).
% 0.19/0.43 tff(309,plain,
% 0.19/0.43 (![A: $i, B: $i, C: $i] : ((~(in(A, B) & element(B, powerset(C)))) | element(A, C)) <=> ![A: $i, B: $i, C: $i] : ((~(in(A, B) & element(B, powerset(C)))) | element(A, C))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(310,plain,
% 0.19/0.43 (^[A: $i, B: $i, C: $i] : rewrite(((in(A, B) & element(B, powerset(C))) => element(A, C)) <=> ((~(in(A, B) & element(B, powerset(C)))) | element(A, C)))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(311,plain,
% 0.19/0.43 (![A: $i, B: $i, C: $i] : ((in(A, B) & element(B, powerset(C))) => element(A, C)) <=> ![A: $i, B: $i, C: $i] : ((~(in(A, B) & element(B, powerset(C)))) | element(A, C))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[310])).
% 0.19/0.43 tff(312,axiom,(![A: $i, B: $i, C: $i] : ((in(A, B) & element(B, powerset(C))) => element(A, C))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t4_subset')).
% 0.19/0.43 tff(313,plain,
% 0.19/0.43 (![A: $i, B: $i, C: $i] : ((~(in(A, B) & element(B, powerset(C)))) | element(A, C))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[312, 311])).
% 0.19/0.43 tff(314,plain,
% 0.19/0.43 (![A: $i, B: $i, C: $i] : ((~(in(A, B) & element(B, powerset(C)))) | element(A, C))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[313, 309])).
% 0.19/0.43 tff(315,plain,(
% 0.19/0.43 ![A: $i, B: $i, C: $i] : ((~(in(A, B) & element(B, powerset(C)))) | element(A, C))),
% 0.19/0.43 inference(skolemize,[status(sab)],[314])).
% 0.19/0.43 tff(316,plain,
% 0.19/0.43 (![A: $i, B: $i, C: $i] : (element(A, C) | (~in(A, B)) | (~element(B, powerset(C))))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[315, 308])).
% 0.19/0.43 tff(317,plain,
% 0.19/0.43 (![A: $i, B: $i, C: $i] : (element(A, C) | (~in(A, B)) | (~element(B, powerset(C))))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[316, 306])).
% 0.19/0.43 tff(318,plain,
% 0.19/0.43 (((~![A: $i, B: $i, C: $i] : (element(A, C) | (~in(A, B)) | (~element(B, powerset(C))))) | (element(A!17, B!18) | (~in(A!17, set_union2(A!17, singleton(A!17)))) | (~element(set_union2(A!17, singleton(A!17)), powerset(B!18))))) <=> ((~![A: $i, B: $i, C: $i] : (element(A, C) | (~in(A, B)) | (~element(B, powerset(C))))) | element(A!17, B!18) | (~in(A!17, set_union2(A!17, singleton(A!17)))) | (~element(set_union2(A!17, singleton(A!17)), powerset(B!18))))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(319,plain,
% 0.19/0.43 ((~![A: $i, B: $i, C: $i] : (element(A, C) | (~in(A, B)) | (~element(B, powerset(C))))) | (element(A!17, B!18) | (~in(A!17, set_union2(A!17, singleton(A!17)))) | (~element(set_union2(A!17, singleton(A!17)), powerset(B!18))))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(320,plain,
% 0.19/0.43 ((~![A: $i, B: $i, C: $i] : (element(A, C) | (~in(A, B)) | (~element(B, powerset(C))))) | element(A!17, B!18) | (~in(A!17, set_union2(A!17, singleton(A!17)))) | (~element(set_union2(A!17, singleton(A!17)), powerset(B!18)))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[319, 318])).
% 0.19/0.44 tff(321,plain,
% 0.19/0.44 ($false),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[320, 317, 304, 265, 253])).
% 0.19/0.44 % SZS output end Proof
% 0.19/0.44 % E exiting
%------------------------------------------------------------------------------