%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SEU337+1 : TPTP v9.0.0. Released v3.3.0.
% 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 05:27:38 AM UTC 2025
% Result : Theorem 1.17s 1.37s
% Output : Proof 1.23s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU337+1 : TPTP v9.0.0. Released v3.3.0.
% 0.00/0.12 % Command : run_E %s %d THM
% 0.11/0.33 % Computer : n001.cluster.edu
% 0.11/0.33 % Model : x86_64 x86_64
% 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.33 % Memory : 8042.1875MB
% 0.11/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.33 % CPULimit : 300
% 0.11/0.33 % WCLimit : 300
% 0.11/0.33 % DateTime : Fri Jun 20 06:52:35 EDT 2025
% 0.11/0.34 % CPUTime :
% 1.17/1.37 % SZS status Theorem
% 1.17/1.37 % SZS output start Proof
% 1.17/1.37 tff(element_type, type, (
% 1.17/1.37 element: ( $i * $i ) > $o)).
% 1.17/1.37 tff(powerset_type, type, (
% 1.17/1.37 powerset: $i > $i)).
% 1.17/1.37 tff(the_carrier_type, type, (
% 1.17/1.37 the_carrier: $i > $i)).
% 1.17/1.37 tff(tptp_fun_A_9_type, type, (
% 1.17/1.37 tptp_fun_A_9: $i)).
% 1.17/1.37 tff(subset_complement_type, type, (
% 1.17/1.37 subset_complement: ( $i * $i ) > $i)).
% 1.17/1.37 tff(tptp_fun_C_0_type, type, (
% 1.17/1.37 tptp_fun_C_0: ( $i * $i ) > $i)).
% 1.17/1.37 tff(tptp_fun_B_10_type, type, (
% 1.17/1.37 tptp_fun_B_10: $i)).
% 1.17/1.37 tff(in_type, type, (
% 1.17/1.37 in: ( $i * $i ) > $o)).
% 1.17/1.37 tff(complements_of_subsets_type, type, (
% 1.17/1.37 complements_of_subsets: ( $i * $i ) > $i)).
% 1.17/1.37 tff(tptp_fun_D_2_type, type, (
% 1.17/1.37 tptp_fun_D_2: ( $i * $i * $i ) > $i)).
% 1.17/1.37 tff(closed_subsets_type, type, (
% 1.17/1.37 closed_subsets: ( $i * $i ) > $o)).
% 1.17/1.37 tff(open_subsets_type, type, (
% 1.17/1.37 open_subsets: ( $i * $i ) > $o)).
% 1.17/1.37 tff(top_str_type, type, (
% 1.17/1.37 top_str: $i > $o)).
% 1.17/1.37 tff(open_subset_type, type, (
% 1.17/1.37 open_subset: ( $i * $i ) > $o)).
% 1.17/1.37 tff(tptp_fun_C_1_type, type, (
% 1.17/1.37 tptp_fun_C_1: ( $i * $i ) > $i)).
% 1.17/1.37 tff(closed_subset_type, type, (
% 1.17/1.37 closed_subset: ( $i * $i ) > $o)).
% 1.17/1.37 tff(1,plain,
% 1.17/1.37 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> closed_subsets(complements_of_subsets(the_carrier(A), B), A))))) <=> (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> closed_subsets(complements_of_subsets(the_carrier(A), B), A)))))),
% 1.17/1.37 inference(rewrite,[status(thm)],[])).
% 1.17/1.37 tff(2,plain,
% 1.17/1.37 ((~![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (open_subsets(B, A) <=> closed_subsets(complements_of_subsets(the_carrier(A), B), A))))) <=> (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> closed_subsets(complements_of_subsets(the_carrier(A), B), A)))))),
% 1.17/1.37 inference(rewrite,[status(thm)],[])).
% 1.17/1.37 tff(3,axiom,(~![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (open_subsets(B, A) <=> closed_subsets(complements_of_subsets(the_carrier(A), B), A))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t17_tops_2')).
% 1.17/1.37 tff(4,plain,
% 1.17/1.37 (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> closed_subsets(complements_of_subsets(the_carrier(A), B), A))))),
% 1.17/1.37 inference(modus_ponens,[status(thm)],[3, 2])).
% 1.17/1.37 tff(5,plain,
% 1.17/1.37 (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> closed_subsets(complements_of_subsets(the_carrier(A), B), A))))),
% 1.17/1.37 inference(modus_ponens,[status(thm)],[4, 1])).
% 1.17/1.37 tff(6,plain,
% 1.17/1.37 (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> closed_subsets(complements_of_subsets(the_carrier(A), B), A))))),
% 1.17/1.37 inference(modus_ponens,[status(thm)],[5, 1])).
% 1.17/1.37 tff(7,plain,
% 1.17/1.37 (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> closed_subsets(complements_of_subsets(the_carrier(A), B), A))))),
% 1.17/1.37 inference(modus_ponens,[status(thm)],[6, 1])).
% 1.17/1.37 tff(8,plain,
% 1.17/1.37 (~((~element(B!10, powerset(powerset(the_carrier(A!9))))) | (open_subsets(B!10, A!9) <=> closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)))),
% 1.17/1.37 inference(and_elim,[status(thm)],[7])).
% 1.17/1.37 tff(9,plain,
% 1.17/1.37 (element(B!10, powerset(powerset(the_carrier(A!9))))),
% 1.17/1.37 inference(or_elim,[status(thm)],[8])).
% 1.17/1.37 tff(10,plain,
% 1.17/1.37 (^[A: $i, B: $i] : refl(((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A)))) <=> ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A)))))),
% 1.17/1.37 inference(bind,[status(th)],[])).
% 1.17/1.37 tff(11,plain,
% 1.17/1.37 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A)))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 1.17/1.37 inference(quant_intro,[status(thm)],[10])).
% 1.17/1.37 tff(12,plain,
% 1.17/1.37 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A)))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 1.17/1.37 inference(rewrite,[status(thm)],[])).
% 1.17/1.37 tff(13,plain,
% 1.17/1.37 (^[A: $i, B: $i] : rewrite((element(B, powerset(powerset(A))) => element(complements_of_subsets(A, B), powerset(powerset(A)))) <=> ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A)))))),
% 1.17/1.37 inference(bind,[status(th)],[])).
% 1.17/1.37 tff(14,plain,
% 1.17/1.37 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => element(complements_of_subsets(A, B), powerset(powerset(A)))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 1.17/1.37 inference(quant_intro,[status(thm)],[13])).
% 1.17/1.37 tff(15,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => element(complements_of_subsets(A, B), powerset(powerset(A))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','dt_k7_setfam_1')).
% 1.17/1.37 tff(16,plain,
% 1.17/1.37 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 1.17/1.37 inference(modus_ponens,[status(thm)],[15, 14])).
% 1.17/1.37 tff(17,plain,
% 1.17/1.37 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 1.17/1.37 inference(modus_ponens,[status(thm)],[16, 12])).
% 1.17/1.37 tff(18,plain,(
% 1.17/1.37 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 1.17/1.37 inference(skolemize,[status(sab)],[17])).
% 1.17/1.37 tff(19,plain,
% 1.17/1.37 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 1.17/1.37 inference(modus_ponens,[status(thm)],[18, 11])).
% 1.17/1.37 tff(20,plain,
% 1.17/1.37 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9)))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | (~element(B!10, powerset(powerset(the_carrier(A!9))))) | element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9)))))),
% 1.17/1.37 inference(rewrite,[status(thm)],[])).
% 1.17/1.37 tff(21,plain,
% 1.17/1.37 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9)))))),
% 1.17/1.37 inference(quant_inst,[status(thm)],[])).
% 1.17/1.37 tff(22,plain,
% 1.17/1.37 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | (~element(B!10, powerset(powerset(the_carrier(A!9))))) | element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))),
% 1.17/1.37 inference(modus_ponens,[status(thm)],[21, 20])).
% 1.17/1.37 tff(23,plain,
% 1.17/1.37 (element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))),
% 1.17/1.37 inference(unit_resolution,[status(thm)],[22, 19, 9])).
% 1.17/1.37 tff(24,plain,
% 1.17/1.37 (^[A: $i, B: $i] : refl(((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B)))))))))) <=> ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B)))))))))))),
% 1.17/1.37 inference(bind,[status(th)],[])).
% 1.17/1.37 tff(25,plain,
% 1.17/1.37 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B)))))))))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))),
% 1.17/1.37 inference(quant_intro,[status(thm)],[24])).
% 1.17/1.37 tff(26,plain,
% 1.17/1.37 (^[A: $i, B: $i] : rewrite(((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B)))))))))) <=> ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B)))))))))))),
% 1.17/1.37 inference(bind,[status(th)],[])).
% 1.17/1.37 tff(27,plain,
% 1.17/1.37 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B)))))))))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))),
% 1.17/1.37 inference(quant_intro,[status(thm)],[26])).
% 1.17/1.37 tff(28,plain,
% 1.17/1.37 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B)))))))))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))),
% 1.17/1.37 inference(transitivity,[status(thm)],[27, 25])).
% 1.17/1.37 tff(29,plain,
% 1.17/1.37 (^[A: $i, B: $i] : rewrite(((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B)))) & ((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B)))))))) <=> ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B)))))))))))),
% 1.17/1.37 inference(bind,[status(th)],[])).
% 1.17/1.37 tff(30,plain,
% 1.17/1.37 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B)))) & ((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B)))))))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))),
% 1.17/1.37 inference(quant_intro,[status(thm)],[29])).
% 1.17/1.37 tff(31,plain,
% 1.17/1.37 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B)))))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))))),
% 1.17/1.37 inference(rewrite,[status(thm)],[])).
% 1.17/1.37 tff(32,plain,
% 1.17/1.37 (^[A: $i, B: $i] : trans(monotonicity(quant_intro(proof_bind(^[C: $i] : trans(monotonicity(rewrite(((C = complements_of_subsets(A, B)) <=> ![D: $i] : (element(D, powerset(A)) => (in(D, C) <=> in(subset_complement(A, D), B)))) <=> ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))), ((element(C, powerset(powerset(A))) => ((C = complements_of_subsets(A, B)) <=> ![D: $i] : (element(D, powerset(A)) => (in(D, C) <=> in(subset_complement(A, D), B))))) <=> (element(C, powerset(powerset(A))) => ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))))), rewrite((element(C, powerset(powerset(A))) => ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) <=> ((~element(C, powerset(powerset(A)))) | ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B)))))), ((element(C, powerset(powerset(A))) => ((C = complements_of_subsets(A, B)) <=> ![D: $i] : (element(D, powerset(A)) => (in(D, C) <=> in(subset_complement(A, D), B))))) <=> ((~element(C, powerset(powerset(A)))) | ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B)))))))), (![C: $i] : (element(C, powerset(powerset(A))) => ((C = complements_of_subsets(A, B)) <=> ![D: $i] : (element(D, powerset(A)) => (in(D, C) <=> in(subset_complement(A, D), B))))) <=> ![C: $i] : ((~element(C, powerset(powerset(A)))) | ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))))), ((element(B, powerset(powerset(A))) => ![C: $i] : (element(C, powerset(powerset(A))) => ((C = complements_of_subsets(A, B)) <=> ![D: $i] : (element(D, powerset(A)) => (in(D, C) <=> in(subset_complement(A, D), B)))))) <=> (element(B, powerset(powerset(A))) => ![C: $i] : ((~element(C, powerset(powerset(A)))) | ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B)))))))), rewrite((element(B, powerset(powerset(A))) => ![C: $i] : ((~element(C, powerset(powerset(A)))) | ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B)))))) <=> ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))))), ((element(B, powerset(powerset(A))) => ![C: $i] : (element(C, powerset(powerset(A))) => ((C = complements_of_subsets(A, B)) <=> ![D: $i] : (element(D, powerset(A)) => (in(D, C) <=> in(subset_complement(A, D), B)))))) <=> ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))))))),
% 1.17/1.37 inference(bind,[status(th)],[])).
% 1.17/1.37 tff(33,plain,
% 1.17/1.37 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => ![C: $i] : (element(C, powerset(powerset(A))) => ((C = complements_of_subsets(A, B)) <=> ![D: $i] : (element(D, powerset(A)) => (in(D, C) <=> in(subset_complement(A, D), B)))))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))))),
% 1.17/1.37 inference(quant_intro,[status(thm)],[32])).
% 1.17/1.37 tff(34,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => ![C: $i] : (element(C, powerset(powerset(A))) => ((C = complements_of_subsets(A, B)) <=> ![D: $i] : (element(D, powerset(A)) => (in(D, C) <=> in(subset_complement(A, D), B))))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d8_setfam_1')).
% 1.17/1.37 tff(35,plain,
% 1.17/1.37 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))))),
% 1.17/1.37 inference(modus_ponens,[status(thm)],[34, 33])).
% 1.17/1.37 tff(36,plain,
% 1.17/1.37 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | ((C = complements_of_subsets(A, B)) <=> ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))))),
% 1.17/1.37 inference(modus_ponens,[status(thm)],[35, 31])).
% 1.17/1.37 tff(37,plain,(
% 1.17/1.37 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B)))) & ((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))),
% 1.17/1.37 inference(skolemize,[status(sab)],[36])).
% 1.17/1.37 tff(38,plain,
% 1.17/1.37 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))),
% 1.17/1.37 inference(modus_ponens,[status(thm)],[37, 30])).
% 1.17/1.37 tff(39,plain,
% 1.17/1.37 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))),
% 1.17/1.37 inference(modus_ponens,[status(thm)],[38, 28])).
% 1.17/1.37 tff(40,plain,
% 1.17/1.37 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((~((~element(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))) | (C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))))))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))) | (~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((~((~element(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))) | (C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))))))))),
% 1.17/1.37 inference(rewrite,[status(thm)],[])).
% 1.17/1.37 tff(41,plain,
% 1.17/1.37 (((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))) | (~((~element(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))))))))) <=> ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((~((~element(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))) | (C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))))))))),
% 1.17/1.37 inference(rewrite,[status(thm)],[])).
% 1.17/1.37 tff(42,plain,
% 1.17/1.37 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))) | (~((~element(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10)))))))))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((~((~element(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))) | (C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))))))))))),
% 1.17/1.37 inference(monotonicity,[status(thm)],[41])).
% 1.17/1.37 tff(43,plain,
% 1.17/1.37 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))) | (~((~element(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10)))))))))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))) | (~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((~((~element(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))) | (C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))))))))),
% 1.17/1.37 inference(transitivity,[status(thm)],[42, 40])).
% 1.17/1.37 tff(44,plain,
% 1.17/1.37 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))) | (~((~element(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10)))))))))))),
% 1.17/1.37 inference(quant_inst,[status(thm)],[])).
% 1.17/1.37 tff(45,plain,
% 1.17/1.37 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))) | (~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((~((~element(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))) | (C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))))))))),
% 1.17/1.38 inference(modus_ponens,[status(thm)],[44, 43])).
% 1.17/1.38 tff(46,plain,
% 1.17/1.38 (![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((~((~element(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))) | (C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))))))))),
% 1.17/1.38 inference(unit_resolution,[status(thm)],[45, 39, 23])).
% 1.17/1.38 tff(47,plain,
% 1.17/1.38 (((~![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((~((~element(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))) | (C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))))))))) | ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | (~((~((~(B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((~((~element(tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), B!10) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))) | (B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))))))))) <=> ((~![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((~((~element(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))) | (C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))))))))) | (~element(B!10, powerset(powerset(the_carrier(A!9))))) | (~((~((~(B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((~((~element(tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), B!10) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))) | (B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))))))))),
% 1.17/1.38 inference(rewrite,[status(thm)],[])).
% 1.17/1.38 tff(48,plain,
% 1.17/1.38 ((~![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((~((~element(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))) | (C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))))))))) | ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | (~((~((~(B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((~((~element(tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), B!10) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))) | (B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))))))))),
% 1.17/1.38 inference(quant_inst,[status(thm)],[])).
% 1.17/1.38 tff(49,plain,
% 1.17/1.38 ((~![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((~((~element(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))) | (C = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))))))))) | (~element(B!10, powerset(powerset(the_carrier(A!9))))) | (~((~((~(B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((~((~element(tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), B!10) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))) | (B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))))))),
% 1.17/1.38 inference(modus_ponens,[status(thm)],[48, 47])).
% 1.17/1.38 tff(50,plain,
% 1.17/1.38 (~((~((~(B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((~((~element(tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), B!10) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))) | (B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))))))),
% 1.17/1.38 inference(unit_resolution,[status(thm)],[49, 9, 46])).
% 1.17/1.38 tff(51,plain,
% 1.17/1.38 (((~((~(B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~((~((~element(tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9)), B!10) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(B!10, complements_of_subsets(the_carrier(A!9), B!10), the_carrier(A!9))), complements_of_subsets(the_carrier(A!9), B!10))))) | (B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))))) | ((~(B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))),
% 1.17/1.38 inference(tautology,[status(thm)],[])).
% 1.17/1.38 tff(52,plain,
% 1.17/1.38 ((~(B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10))))),
% 1.17/1.38 inference(unit_resolution,[status(thm)],[51, 50])).
% 1.17/1.38 tff(53,plain,
% 1.17/1.38 (^[A: $i, B: $i] : refl(((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B)) <=> ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B)))),
% 1.17/1.38 inference(bind,[status(th)],[])).
% 1.17/1.38 tff(54,plain,
% 1.17/1.38 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B)) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 1.17/1.38 inference(quant_intro,[status(thm)],[53])).
% 1.17/1.38 tff(55,plain,
% 1.17/1.38 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B)) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 1.17/1.38 inference(rewrite,[status(thm)],[])).
% 1.17/1.38 tff(56,plain,
% 1.17/1.38 (^[A: $i, B: $i] : rewrite((element(B, powerset(powerset(A))) => (complements_of_subsets(A, complements_of_subsets(A, B)) = B)) <=> ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B)))),
% 1.17/1.38 inference(bind,[status(th)],[])).
% 1.17/1.38 tff(57,plain,
% 1.17/1.38 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => (complements_of_subsets(A, complements_of_subsets(A, B)) = B)) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 1.17/1.38 inference(quant_intro,[status(thm)],[56])).
% 1.17/1.38 tff(58,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => (complements_of_subsets(A, complements_of_subsets(A, B)) = B))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','involutiveness_k7_setfam_1')).
% 1.17/1.38 tff(59,plain,
% 1.17/1.38 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 1.17/1.38 inference(modus_ponens,[status(thm)],[58, 57])).
% 1.17/1.38 tff(60,plain,
% 1.17/1.38 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 1.17/1.38 inference(modus_ponens,[status(thm)],[59, 55])).
% 1.17/1.38 tff(61,plain,(
% 1.17/1.38 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 1.17/1.38 inference(skolemize,[status(sab)],[60])).
% 1.17/1.38 tff(62,plain,
% 1.17/1.38 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 1.17/1.38 inference(modus_ponens,[status(thm)],[61, 54])).
% 1.17/1.38 tff(63,plain,
% 1.17/1.38 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | (complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)) = B!10))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | (~element(B!10, powerset(powerset(the_carrier(A!9))))) | (complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)) = B!10))),
% 1.17/1.38 inference(rewrite,[status(thm)],[])).
% 1.17/1.38 tff(64,plain,
% 1.17/1.38 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | (complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)) = B!10))),
% 1.17/1.38 inference(quant_inst,[status(thm)],[])).
% 1.17/1.38 tff(65,plain,
% 1.17/1.38 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | (~element(B!10, powerset(powerset(the_carrier(A!9))))) | (complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)) = B!10)),
% 1.17/1.38 inference(modus_ponens,[status(thm)],[64, 63])).
% 1.17/1.38 tff(66,plain,
% 1.17/1.38 (complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)) = B!10),
% 1.17/1.38 inference(unit_resolution,[status(thm)],[65, 62, 9])).
% 1.17/1.38 tff(67,plain,
% 1.17/1.38 (B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10))),
% 1.17/1.38 inference(symmetry,[status(thm)],[66])).
% 1.17/1.38 tff(68,plain,
% 1.17/1.38 ((~((~(B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | (~(B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10))))),
% 1.17/1.38 inference(tautology,[status(thm)],[])).
% 1.17/1.38 tff(69,plain,
% 1.17/1.38 ((~((~(B!10 = complements_of_subsets(the_carrier(A!9), complements_of_subsets(the_carrier(A!9), B!10)))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10)))))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10))))),
% 1.17/1.38 inference(unit_resolution,[status(thm)],[68, 67])).
% 1.17/1.38 tff(70,plain,
% 1.17/1.38 (![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10))))),
% 1.17/1.38 inference(unit_resolution,[status(thm)],[69, 52])).
% 1.17/1.38 tff(71,assumption,(~closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)), introduced(assumption)).
% 1.17/1.38 tff(72,plain,
% 1.17/1.38 (top_str(A!9)),
% 1.17/1.38 inference(and_elim,[status(thm)],[7])).
% 1.17/1.38 tff(73,plain,
% 1.17/1.38 (^[A: $i] : refl(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A)))))))))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A)))))))))))))),
% 1.17/1.38 inference(bind,[status(th)],[])).
% 1.17/1.38 tff(74,plain,
% 1.17/1.38 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A)))))))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))),
% 1.17/1.38 inference(quant_intro,[status(thm)],[73])).
% 1.17/1.38 tff(75,plain,
% 1.17/1.38 (^[A: $i] : rewrite(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A)))))))))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A)))))))))))))),
% 1.17/1.38 inference(bind,[status(th)],[])).
% 1.17/1.38 tff(76,plain,
% 1.17/1.38 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A)))))))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))),
% 1.17/1.38 inference(quant_intro,[status(thm)],[75])).
% 1.17/1.38 tff(77,plain,
% 1.17/1.38 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A)))))))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))),
% 1.17/1.38 inference(transitivity,[status(thm)],[76, 74])).
% 1.17/1.38 tff(78,plain,
% 1.17/1.38 (^[A: $i] : rewrite(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) & (closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A)))))))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A)))))))))))))),
% 1.17/1.38 inference(bind,[status(th)],[])).
% 1.17/1.38 tff(79,plain,
% 1.17/1.38 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) & (closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A)))))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))),
% 1.17/1.38 inference(quant_intro,[status(thm)],[78])).
% 1.17/1.38 tff(80,plain,
% 1.17/1.38 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))),
% 1.17/1.38 inference(rewrite,[status(thm)],[])).
% 1.17/1.38 tff(81,plain,
% 1.17/1.38 (^[A: $i] : trans(monotonicity(quant_intro(proof_bind(^[B: $i] : trans(monotonicity(rewrite((closed_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A)))) <=> (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))), ((element(B, powerset(powerset(the_carrier(A)))) => (closed_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A))))) <=> (element(B, powerset(powerset(the_carrier(A)))) => (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))), rewrite((element(B, powerset(powerset(the_carrier(A)))) => (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) <=> ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))))), ((element(B, powerset(powerset(the_carrier(A)))) => (closed_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A))))) <=> ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))))))), (![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (closed_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A))))) <=> ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))), ((top_str(A) => ![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (closed_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A)))))) <=> (top_str(A) => ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))))))), rewrite((top_str(A) => ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))), ((top_str(A) => ![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (closed_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A)))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))))),
% 1.17/1.38 inference(bind,[status(th)],[])).
% 1.17/1.38 tff(82,plain,
% 1.17/1.38 (![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (closed_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A)))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))),
% 1.17/1.38 inference(quant_intro,[status(thm)],[81])).
% 1.17/1.38 tff(83,axiom,(![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (closed_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => closed_subset(C, A))))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d2_tops_2')).
% 1.17/1.38 tff(84,plain,
% 1.17/1.38 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))),
% 1.17/1.38 inference(modus_ponens,[status(thm)],[83, 82])).
% 1.17/1.38 tff(85,plain,
% 1.17/1.38 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (closed_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))),
% 1.17/1.38 inference(modus_ponens,[status(thm)],[84, 80])).
% 1.17/1.38 tff(86,plain,(
% 1.17/1.38 ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) & (closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))),
% 1.17/1.38 inference(skolemize,[status(sab)],[85])).
% 1.17/1.38 tff(87,plain,
% 1.17/1.38 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))),
% 1.17/1.38 inference(modus_ponens,[status(thm)],[86, 79])).
% 1.17/1.38 tff(88,plain,
% 1.17/1.38 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))),
% 1.17/1.38 inference(modus_ponens,[status(thm)],[87, 77])).
% 1.17/1.38 tff(89,plain,
% 1.17/1.38 (((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B, A!9) | (~((~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9)))) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~in(tptp_fun_C_1(B, A!9), B))))))))))) <=> ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))) | (~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B, A!9) | (~((~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9)))) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~in(tptp_fun_C_1(B, A!9), B))))))))))),
% 1.17/1.38 inference(rewrite,[status(thm)],[])).
% 1.17/1.38 tff(90,plain,
% 1.17/1.38 (((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A!9) | (~element(C, powerset(the_carrier(A!9))))))) | (~(closed_subsets(B, A!9) | (~((~in(tptp_fun_C_1(B, A!9), B)) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9)))))))))))) <=> ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B, A!9) | (~((~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9)))) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~in(tptp_fun_C_1(B, A!9), B))))))))))),
% 1.17/1.38 inference(rewrite,[status(thm)],[])).
% 1.17/1.38 tff(91,plain,
% 1.17/1.38 (((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A!9) | (~element(C, powerset(the_carrier(A!9))))))) | (~(closed_subsets(B, A!9) | (~((~in(tptp_fun_C_1(B, A!9), B)) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9))))))))))))) <=> ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B, A!9) | (~((~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9)))) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~in(tptp_fun_C_1(B, A!9), B)))))))))))),
% 1.17/1.38 inference(monotonicity,[status(thm)],[90])).
% 1.17/1.38 tff(92,plain,
% 1.17/1.38 (((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A!9) | (~element(C, powerset(the_carrier(A!9))))))) | (~(closed_subsets(B, A!9) | (~((~in(tptp_fun_C_1(B, A!9), B)) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9))))))))))))) <=> ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))) | (~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B, A!9) | (~((~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9)))) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~in(tptp_fun_C_1(B, A!9), B))))))))))),
% 1.17/1.38 inference(transitivity,[status(thm)],[91, 89])).
% 1.17/1.38 tff(93,plain,
% 1.17/1.38 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A!9) | (~element(C, powerset(the_carrier(A!9))))))) | (~(closed_subsets(B, A!9) | (~((~in(tptp_fun_C_1(B, A!9), B)) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9))))))))))))),
% 1.17/1.38 inference(quant_inst,[status(thm)],[])).
% 1.17/1.38 tff(94,plain,
% 1.17/1.38 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~closed_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | closed_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(closed_subsets(B, A) | (~((~in(tptp_fun_C_1(B, A), B)) | closed_subset(tptp_fun_C_1(B, A), A) | (~element(tptp_fun_C_1(B, A), powerset(the_carrier(A))))))))))))) | (~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B, A!9) | (~((~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9)))) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~in(tptp_fun_C_1(B, A!9), B)))))))))),
% 1.17/1.38 inference(modus_ponens,[status(thm)],[93, 92])).
% 1.17/1.38 tff(95,plain,
% 1.17/1.38 (![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B, A!9) | (~((~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9)))) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~in(tptp_fun_C_1(B, A!9), B)))))))))),
% 1.17/1.38 inference(unit_resolution,[status(thm)],[94, 88, 72])).
% 1.17/1.38 tff(96,plain,
% 1.17/1.38 (((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B, A!9) | (~((~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9)))) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~in(tptp_fun_C_1(B, A!9), B)))))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10)))))) | (~(closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))))))))))) <=> ((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B, A!9) | (~((~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9)))) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~in(tptp_fun_C_1(B, A!9), B)))))))))) | (~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10)))))) | (~(closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))))))))))),
% 1.17/1.38 inference(rewrite,[status(thm)],[])).
% 1.17/1.38 tff(97,plain,
% 1.17/1.38 (((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~in(C, complements_of_subsets(the_carrier(A!9), B!10))) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10)))))))))) <=> ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10)))))) | (~(closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))))))))))),
% 1.17/1.38 inference(rewrite,[status(thm)],[])).
% 1.17/1.39 tff(98,plain,
% 1.17/1.39 (((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B, A!9) | (~((~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9)))) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~in(tptp_fun_C_1(B, A!9), B)))))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~in(C, complements_of_subsets(the_carrier(A!9), B!10))) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))))))))))) <=> ((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B, A!9) | (~((~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9)))) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~in(tptp_fun_C_1(B, A!9), B)))))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10)))))) | (~(closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9))))))))))))),
% 1.17/1.39 inference(monotonicity,[status(thm)],[97])).
% 1.17/1.39 tff(99,plain,
% 1.17/1.39 (((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B, A!9) | (~((~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9)))) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~in(tptp_fun_C_1(B, A!9), B)))))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~in(C, complements_of_subsets(the_carrier(A!9), B!10))) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))))))))))) <=> ((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B, A!9) | (~((~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9)))) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~in(tptp_fun_C_1(B, A!9), B)))))))))) | (~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10)))))) | (~(closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))))))))))),
% 1.17/1.39 inference(transitivity,[status(thm)],[98, 96])).
% 1.17/1.39 tff(100,plain,
% 1.17/1.39 ((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B, A!9) | (~((~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9)))) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~in(tptp_fun_C_1(B, A!9), B)))))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~in(C, complements_of_subsets(the_carrier(A!9), B!10))) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))))))))))),
% 1.17/1.39 inference(quant_inst,[status(thm)],[])).
% 1.17/1.39 tff(101,plain,
% 1.17/1.39 ((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9)))) | (~(closed_subsets(B, A!9) | (~((~element(tptp_fun_C_1(B, A!9), powerset(the_carrier(A!9)))) | closed_subset(tptp_fun_C_1(B, A!9), A!9) | (~in(tptp_fun_C_1(B, A!9), B)))))))))) | (~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10)))))) | (~(closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9))))))))))),
% 1.17/1.39 inference(modus_ponens,[status(thm)],[100, 99])).
% 1.17/1.39 tff(102,plain,
% 1.17/1.39 (~((~((~closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10)))))) | (~(closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))))))))),
% 1.17/1.39 inference(unit_resolution,[status(thm)],[101, 95, 23])).
% 1.17/1.39 tff(103,plain,
% 1.17/1.39 (((~((~closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10)))))) | (~(closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9))))))))) | (closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))))))),
% 1.17/1.39 inference(tautology,[status(thm)],[])).
% 1.17/1.39 tff(104,plain,
% 1.17/1.39 (closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9))))))),
% 1.17/1.39 inference(unit_resolution,[status(thm)],[103, 102])).
% 1.17/1.39 tff(105,plain,
% 1.17/1.39 ((~(closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))))))) | closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9))))))),
% 1.17/1.39 inference(tautology,[status(thm)],[])).
% 1.17/1.39 tff(106,plain,
% 1.17/1.39 (closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9))))))),
% 1.17/1.39 inference(unit_resolution,[status(thm)],[105, 104])).
% 1.17/1.39 tff(107,plain,
% 1.17/1.39 (~((~in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))))),
% 1.17/1.39 inference(unit_resolution,[status(thm)],[106, 71])).
% 1.17/1.39 tff(108,plain,
% 1.17/1.39 (((~in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9))))) | element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))),
% 1.17/1.39 inference(tautology,[status(thm)],[])).
% 1.17/1.39 tff(109,plain,
% 1.17/1.39 (element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))),
% 1.17/1.39 inference(unit_resolution,[status(thm)],[108, 107])).
% 1.17/1.39 tff(110,plain,
% 1.17/1.39 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))) | ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10)))) | (C = complements_of_subsets(the_carrier(A!9), B!10))))))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))) | (~element(B!10, powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10)))) | (C = complements_of_subsets(the_carrier(A!9), B!10))))))))),
% 1.17/1.39 inference(rewrite,[status(thm)],[])).
% 1.17/1.39 tff(111,plain,
% 1.17/1.39 (((~element(B!10, powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((C = complements_of_subsets(the_carrier(A!9), B!10)) | (~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10)))))))))) <=> ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10)))) | (C = complements_of_subsets(the_carrier(A!9), B!10))))))))),
% 1.17/1.39 inference(rewrite,[status(thm)],[])).
% 1.17/1.39 tff(112,plain,
% 1.17/1.39 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))) | ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((C = complements_of_subsets(the_carrier(A!9), B!10)) | (~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10))))))))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))) | ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10)))) | (C = complements_of_subsets(the_carrier(A!9), B!10)))))))))),
% 1.23/1.39 inference(monotonicity,[status(thm)],[111])).
% 1.23/1.39 tff(113,plain,
% 1.23/1.39 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))) | ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((C = complements_of_subsets(the_carrier(A!9), B!10)) | (~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10))))))))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))) | (~element(B!10, powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10)))) | (C = complements_of_subsets(the_carrier(A!9), B!10))))))))),
% 1.23/1.39 inference(transitivity,[status(thm)],[112, 110])).
% 1.23/1.39 tff(114,plain,
% 1.23/1.39 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))) | ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((C = complements_of_subsets(the_carrier(A!9), B!10)) | (~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10))))))))))),
% 1.23/1.39 inference(quant_inst,[status(thm)],[])).
% 1.23/1.39 tff(115,plain,
% 1.23/1.39 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | ![C: $i] : ((~element(C, powerset(powerset(A)))) | (~((~((~(C = complements_of_subsets(A, B))) | ![D: $i] : ((~element(D, powerset(A))) | (in(D, C) <=> in(subset_complement(A, D), B))))) | (~((C = complements_of_subsets(A, B)) | (~((~element(tptp_fun_D_2(C, B, A), powerset(A))) | (in(tptp_fun_D_2(C, B, A), C) <=> in(subset_complement(A, tptp_fun_D_2(C, B, A)), B))))))))))) | (~element(B!10, powerset(powerset(the_carrier(A!9))))) | ![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10)))) | (C = complements_of_subsets(the_carrier(A!9), B!10)))))))),
% 1.23/1.39 inference(modus_ponens,[status(thm)],[114, 113])).
% 1.23/1.39 tff(116,plain,
% 1.23/1.39 (![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10)))) | (C = complements_of_subsets(the_carrier(A!9), B!10)))))))),
% 1.23/1.39 inference(unit_resolution,[status(thm)],[115, 39, 9])).
% 1.23/1.39 tff(117,plain,
% 1.23/1.39 (((~![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10)))) | (C = complements_of_subsets(the_carrier(A!9), B!10)))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) <=> ((~![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10)))) | (C = complements_of_subsets(the_carrier(A!9), B!10)))))))) | (~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10))))),
% 1.23/1.39 inference(rewrite,[status(thm)],[])).
% 1.23/1.39 tff(118,plain,
% 1.23/1.39 (((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~(complements_of_subsets(the_carrier(A!9), B!10) = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((~((~element(tptp_fun_D_2(complements_of_subsets(the_carrier(A!9), B!10), B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(complements_of_subsets(the_carrier(A!9), B!10), B!10, the_carrier(A!9)), complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(complements_of_subsets(the_carrier(A!9), B!10), B!10, the_carrier(A!9))), B!10)))) | (complements_of_subsets(the_carrier(A!9), B!10) = complements_of_subsets(the_carrier(A!9), B!10))))))) <=> ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10))))),
% 1.23/1.39 inference(rewrite,[status(thm)],[])).
% 1.23/1.39 tff(119,plain,
% 1.23/1.39 (((~![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10)))) | (C = complements_of_subsets(the_carrier(A!9), B!10)))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~(complements_of_subsets(the_carrier(A!9), B!10) = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((~((~element(tptp_fun_D_2(complements_of_subsets(the_carrier(A!9), B!10), B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(complements_of_subsets(the_carrier(A!9), B!10), B!10, the_carrier(A!9)), complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(complements_of_subsets(the_carrier(A!9), B!10), B!10, the_carrier(A!9))), B!10)))) | (complements_of_subsets(the_carrier(A!9), B!10) = complements_of_subsets(the_carrier(A!9), B!10)))))))) <=> ((~![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10)))) | (C = complements_of_subsets(the_carrier(A!9), B!10)))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10)))))),
% 1.23/1.39 inference(monotonicity,[status(thm)],[118])).
% 1.23/1.39 tff(120,plain,
% 1.23/1.39 (((~![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10)))) | (C = complements_of_subsets(the_carrier(A!9), B!10)))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~(complements_of_subsets(the_carrier(A!9), B!10) = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((~((~element(tptp_fun_D_2(complements_of_subsets(the_carrier(A!9), B!10), B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(complements_of_subsets(the_carrier(A!9), B!10), B!10, the_carrier(A!9)), complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(complements_of_subsets(the_carrier(A!9), B!10), B!10, the_carrier(A!9))), B!10)))) | (complements_of_subsets(the_carrier(A!9), B!10) = complements_of_subsets(the_carrier(A!9), B!10)))))))) <=> ((~![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10)))) | (C = complements_of_subsets(the_carrier(A!9), B!10)))))))) | (~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10))))),
% 1.23/1.39 inference(transitivity,[status(thm)],[119, 117])).
% 1.23/1.39 tff(121,plain,
% 1.23/1.39 ((~![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10)))) | (C = complements_of_subsets(the_carrier(A!9), B!10)))))))) | ((~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | (~((~((~(complements_of_subsets(the_carrier(A!9), B!10) = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((~((~element(tptp_fun_D_2(complements_of_subsets(the_carrier(A!9), B!10), B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(complements_of_subsets(the_carrier(A!9), B!10), B!10, the_carrier(A!9)), complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(complements_of_subsets(the_carrier(A!9), B!10), B!10, the_carrier(A!9))), B!10)))) | (complements_of_subsets(the_carrier(A!9), B!10) = complements_of_subsets(the_carrier(A!9), B!10)))))))),
% 1.23/1.39 inference(quant_inst,[status(thm)],[])).
% 1.23/1.39 tff(122,plain,
% 1.23/1.39 ((~![C: $i] : ((~element(C, powerset(powerset(the_carrier(A!9))))) | (~((~((~(C = complements_of_subsets(the_carrier(A!9), B!10))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, C) <=> in(subset_complement(the_carrier(A!9), D), B!10))))) | (~((~((~element(tptp_fun_D_2(C, B!10, the_carrier(A!9)), powerset(the_carrier(A!9)))) | (in(tptp_fun_D_2(C, B!10, the_carrier(A!9)), C) <=> in(subset_complement(the_carrier(A!9), tptp_fun_D_2(C, B!10, the_carrier(A!9))), B!10)))) | (C = complements_of_subsets(the_carrier(A!9), B!10)))))))) | (~element(complements_of_subsets(the_carrier(A!9), B!10), powerset(powerset(the_carrier(A!9))))) | ![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10)))),
% 1.23/1.39 inference(modus_ponens,[status(thm)],[121, 120])).
% 1.23/1.39 tff(123,plain,
% 1.23/1.39 (![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10)))),
% 1.23/1.39 inference(unit_resolution,[status(thm)],[122, 116, 23])).
% 1.23/1.39 tff(124,plain,
% 1.23/1.39 (((~![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10)))) | ((~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | (in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)))) <=> ((~![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10)))) | (~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | (in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)))),
% 1.23/1.40 inference(rewrite,[status(thm)],[])).
% 1.23/1.40 tff(125,plain,
% 1.23/1.40 ((~![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10)))) | ((~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | (in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)))),
% 1.23/1.40 inference(quant_inst,[status(thm)],[])).
% 1.23/1.40 tff(126,plain,
% 1.23/1.40 ((~![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), D), B!10)))) | (~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | (in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10))),
% 1.23/1.40 inference(modus_ponens,[status(thm)],[125, 124])).
% 1.23/1.40 tff(127,plain,
% 1.23/1.40 (in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)),
% 1.23/1.40 inference(unit_resolution,[status(thm)],[126, 123, 109])).
% 1.23/1.40 tff(128,plain,
% 1.23/1.40 (((~in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9))))) | in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))),
% 1.23/1.40 inference(tautology,[status(thm)],[])).
% 1.23/1.40 tff(129,plain,
% 1.23/1.40 (in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))),
% 1.23/1.40 inference(unit_resolution,[status(thm)],[128, 107])).
% 1.23/1.40 tff(130,plain,
% 1.23/1.40 ((~(in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10))) | (~in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | in(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)),
% 1.23/1.40 inference(tautology,[status(thm)],[])).
% 1.23/1.40 tff(131,plain,
% 1.23/1.40 ((~(in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10)) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10))) | in(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)),
% 1.23/1.40 inference(unit_resolution,[status(thm)],[130, 129])).
% 1.23/1.40 tff(132,plain,
% 1.23/1.40 (in(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)),
% 1.23/1.40 inference(unit_resolution,[status(thm)],[131, 127])).
% 1.23/1.40 tff(133,plain,
% 1.23/1.40 (^[A: $i] : refl(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))))),
% 1.23/1.40 inference(bind,[status(th)],[])).
% 1.23/1.40 tff(134,plain,
% 1.23/1.40 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 1.23/1.40 inference(quant_intro,[status(thm)],[133])).
% 1.23/1.40 tff(135,plain,
% 1.23/1.40 (^[A: $i] : rewrite(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))))),
% 1.23/1.40 inference(bind,[status(th)],[])).
% 1.23/1.40 tff(136,plain,
% 1.23/1.40 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 1.23/1.40 inference(quant_intro,[status(thm)],[135])).
% 1.23/1.40 tff(137,plain,
% 1.23/1.40 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 1.23/1.40 inference(transitivity,[status(thm)],[136, 134])).
% 1.23/1.40 tff(138,plain,
% 1.23/1.40 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 1.23/1.40 inference(rewrite,[status(thm)],[])).
% 1.23/1.40 tff(139,plain,
% 1.23/1.40 (^[A: $i] : trans(monotonicity(quant_intro(proof_bind(^[B: $i] : rewrite((element(B, powerset(the_carrier(A))) => (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))) <=> ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))), (![B: $i] : (element(B, powerset(the_carrier(A))) => (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))) <=> ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))), ((top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> (top_str(A) => ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))))), rewrite((top_str(A) => ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))), ((top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))))),
% 1.23/1.40 inference(bind,[status(th)],[])).
% 1.23/1.40 tff(140,plain,
% 1.23/1.40 (![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 1.23/1.40 inference(quant_intro,[status(thm)],[139])).
% 1.23/1.40 tff(141,axiom,(![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t29_tops_1')).
% 1.23/1.40 tff(142,plain,
% 1.23/1.40 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 1.23/1.40 inference(modus_ponens,[status(thm)],[141, 140])).
% 1.23/1.40 tff(143,plain,
% 1.23/1.40 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 1.23/1.40 inference(modus_ponens,[status(thm)],[142, 138])).
% 1.23/1.40 tff(144,plain,(
% 1.23/1.40 ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 1.23/1.40 inference(skolemize,[status(sab)],[143])).
% 1.23/1.40 tff(145,plain,
% 1.23/1.40 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 1.23/1.40 inference(modus_ponens,[status(thm)],[144, 137])).
% 1.23/1.40 tff(146,plain,
% 1.23/1.40 (((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (closed_subset(B, A!9) <=> open_subset(subset_complement(the_carrier(A!9), B), A!9))))) <=> ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))) | (~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (closed_subset(B, A!9) <=> open_subset(subset_complement(the_carrier(A!9), B), A!9))))),
% 1.23/1.40 inference(rewrite,[status(thm)],[])).
% 1.23/1.40 tff(147,plain,
% 1.23/1.40 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (closed_subset(B, A!9) <=> open_subset(subset_complement(the_carrier(A!9), B), A!9))))),
% 1.23/1.40 inference(quant_inst,[status(thm)],[])).
% 1.23/1.40 tff(148,plain,
% 1.23/1.40 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))) | (~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (closed_subset(B, A!9) <=> open_subset(subset_complement(the_carrier(A!9), B), A!9)))),
% 1.23/1.40 inference(modus_ponens,[status(thm)],[147, 146])).
% 1.23/1.40 tff(149,plain,
% 1.23/1.40 (![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (closed_subset(B, A!9) <=> open_subset(subset_complement(the_carrier(A!9), B), A!9)))),
% 1.23/1.40 inference(unit_resolution,[status(thm)],[148, 72, 145])).
% 1.23/1.40 tff(150,plain,
% 1.23/1.40 (((~![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (closed_subset(B, A!9) <=> open_subset(subset_complement(the_carrier(A!9), B), A!9)))) | ((~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | (closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) <=> open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9)))) <=> ((~![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (closed_subset(B, A!9) <=> open_subset(subset_complement(the_carrier(A!9), B), A!9)))) | (~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | (closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) <=> open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9)))),
% 1.23/1.40 inference(rewrite,[status(thm)],[])).
% 1.23/1.40 tff(151,plain,
% 1.23/1.40 ((~![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (closed_subset(B, A!9) <=> open_subset(subset_complement(the_carrier(A!9), B), A!9)))) | ((~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | (closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) <=> open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9)))),
% 1.23/1.40 inference(quant_inst,[status(thm)],[])).
% 1.23/1.40 tff(152,plain,
% 1.23/1.40 ((~![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (closed_subset(B, A!9) <=> open_subset(subset_complement(the_carrier(A!9), B), A!9)))) | (~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | (closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) <=> open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9))),
% 1.23/1.40 inference(modus_ponens,[status(thm)],[151, 150])).
% 1.23/1.40 tff(153,plain,
% 1.23/1.40 (closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) <=> open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9)),
% 1.23/1.40 inference(unit_resolution,[status(thm)],[152, 149, 109])).
% 1.23/1.40 tff(154,plain,
% 1.23/1.40 (((~in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9))))) | (~closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9))),
% 1.23/1.40 inference(tautology,[status(thm)],[])).
% 1.23/1.40 tff(155,plain,
% 1.23/1.40 (~closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9)),
% 1.23/1.40 inference(unit_resolution,[status(thm)],[154, 107])).
% 1.23/1.40 tff(156,plain,
% 1.23/1.40 ((~(closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) <=> open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9))) | closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9))),
% 1.23/1.40 inference(tautology,[status(thm)],[])).
% 1.23/1.40 tff(157,plain,
% 1.23/1.40 ((~(closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) <=> open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9))) | (~open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9))),
% 1.23/1.40 inference(unit_resolution,[status(thm)],[156, 155])).
% 1.23/1.40 tff(158,plain,
% 1.23/1.40 (~open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9)),
% 1.23/1.40 inference(unit_resolution,[status(thm)],[157, 153])).
% 1.23/1.40 tff(159,plain,
% 1.23/1.40 (~(open_subsets(B!10, A!9) <=> closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9))),
% 1.23/1.40 inference(or_elim,[status(thm)],[8])).
% 1.23/1.40 tff(160,plain,
% 1.23/1.40 ((open_subsets(B!10, A!9) <=> closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | open_subsets(B!10, A!9) | closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)),
% 1.23/1.40 inference(tautology,[status(thm)],[])).
% 1.23/1.40 tff(161,plain,
% 1.23/1.40 (open_subsets(B!10, A!9) | closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)),
% 1.23/1.40 inference(unit_resolution,[status(thm)],[160, 159])).
% 1.23/1.40 tff(162,plain,
% 1.23/1.40 (open_subsets(B!10, A!9)),
% 1.23/1.40 inference(unit_resolution,[status(thm)],[161, 71])).
% 1.23/1.40 tff(163,plain,
% 1.23/1.40 (^[A: $i] : refl(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A)))))))))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A)))))))))))))),
% 1.23/1.40 inference(bind,[status(th)],[])).
% 1.23/1.40 tff(164,plain,
% 1.23/1.40 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A)))))))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))),
% 1.23/1.40 inference(quant_intro,[status(thm)],[163])).
% 1.23/1.40 tff(165,plain,
% 1.23/1.40 (^[A: $i] : rewrite(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A)))))))))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A)))))))))))))),
% 1.23/1.40 inference(bind,[status(th)],[])).
% 1.23/1.40 tff(166,plain,
% 1.23/1.40 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A)))))))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))),
% 1.23/1.40 inference(quant_intro,[status(thm)],[165])).
% 1.23/1.40 tff(167,plain,
% 1.23/1.40 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A)))))))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))),
% 1.23/1.40 inference(transitivity,[status(thm)],[166, 164])).
% 1.23/1.40 tff(168,plain,
% 1.23/1.40 (^[A: $i] : rewrite(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) & (open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A)))))))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A)))))))))))))),
% 1.23/1.40 inference(bind,[status(th)],[])).
% 1.23/1.40 tff(169,plain,
% 1.23/1.40 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) & (open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A)))))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))),
% 1.23/1.40 inference(quant_intro,[status(thm)],[168])).
% 1.23/1.40 tff(170,plain,
% 1.23/1.40 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A)))))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))),
% 1.23/1.40 inference(rewrite,[status(thm)],[])).
% 1.23/1.40 tff(171,plain,
% 1.23/1.40 (^[A: $i] : trans(monotonicity(quant_intro(proof_bind(^[B: $i] : trans(monotonicity(rewrite((open_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => open_subset(C, A)))) <=> (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))), ((element(B, powerset(powerset(the_carrier(A)))) => (open_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => open_subset(C, A))))) <=> (element(B, powerset(powerset(the_carrier(A)))) => (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))), rewrite((element(B, powerset(powerset(the_carrier(A)))) => (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) <=> ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A)))))))), ((element(B, powerset(powerset(the_carrier(A)))) => (open_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => open_subset(C, A))))) <=> ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A)))))))))), (![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (open_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => open_subset(C, A))))) <=> ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))), ((top_str(A) => ![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (open_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => open_subset(C, A)))))) <=> (top_str(A) => ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A)))))))))), rewrite((top_str(A) => ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A)))))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))), ((top_str(A) => ![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (open_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => open_subset(C, A)))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))))),
% 1.23/1.40 inference(bind,[status(th)],[])).
% 1.23/1.40 tff(172,plain,
% 1.23/1.40 (![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (open_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => open_subset(C, A)))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))),
% 1.23/1.40 inference(quant_intro,[status(thm)],[171])).
% 1.23/1.40 tff(173,axiom,(![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(powerset(the_carrier(A)))) => (open_subsets(B, A) <=> ![C: $i] : (element(C, powerset(the_carrier(A))) => (in(C, B) => open_subset(C, A))))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d1_tops_2')).
% 1.23/1.40 tff(174,plain,
% 1.23/1.40 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))),
% 1.23/1.40 inference(modus_ponens,[status(thm)],[173, 172])).
% 1.23/1.40 tff(175,plain,
% 1.23/1.40 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (open_subsets(B, A) <=> ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))))),
% 1.23/1.40 inference(modus_ponens,[status(thm)],[174, 170])).
% 1.23/1.40 tff(176,plain,(
% 1.23/1.40 ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A)))))) & (open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))),
% 1.23/1.40 inference(skolemize,[status(sab)],[175])).
% 1.23/1.40 tff(177,plain,
% 1.23/1.40 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))),
% 1.23/1.40 inference(modus_ponens,[status(thm)],[176, 169])).
% 1.23/1.40 tff(178,plain,
% 1.23/1.40 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))),
% 1.23/1.40 inference(modus_ponens,[status(thm)],[177, 167])).
% 1.23/1.40 tff(179,plain,
% 1.23/1.40 (((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~in(tptp_fun_C_0(B, A!9), B))))))))))) <=> ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))) | (~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~in(tptp_fun_C_0(B, A!9), B))))))))))),
% 1.23/1.40 inference(rewrite,[status(thm)],[])).
% 1.23/1.40 tff(180,plain,
% 1.23/1.40 (((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A!9) | (~element(C, powerset(the_carrier(A!9))))))) | (~(open_subsets(B, A!9) | (~((~in(tptp_fun_C_0(B, A!9), B)) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))))))))))) <=> ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~in(tptp_fun_C_0(B, A!9), B))))))))))),
% 1.23/1.40 inference(rewrite,[status(thm)],[])).
% 1.23/1.40 tff(181,plain,
% 1.23/1.40 (((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A!9) | (~element(C, powerset(the_carrier(A!9))))))) | (~(open_subsets(B, A!9) | (~((~in(tptp_fun_C_0(B, A!9), B)) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9))))))))))))) <=> ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~in(tptp_fun_C_0(B, A!9), B)))))))))))),
% 1.23/1.40 inference(monotonicity,[status(thm)],[180])).
% 1.23/1.40 tff(182,plain,
% 1.23/1.40 (((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A!9) | (~element(C, powerset(the_carrier(A!9))))))) | (~(open_subsets(B, A!9) | (~((~in(tptp_fun_C_0(B, A!9), B)) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9))))))))))))) <=> ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))) | (~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~in(tptp_fun_C_0(B, A!9), B))))))))))),
% 1.23/1.40 inference(transitivity,[status(thm)],[181, 179])).
% 1.23/1.40 tff(183,plain,
% 1.23/1.40 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A!9) | (~element(C, powerset(the_carrier(A!9))))))) | (~(open_subsets(B, A!9) | (~((~in(tptp_fun_C_0(B, A!9), B)) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9))))))))))))),
% 1.23/1.40 inference(quant_inst,[status(thm)],[])).
% 1.23/1.40 tff(184,plain,
% 1.23/1.40 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A))))) | (~((~((~open_subsets(B, A)) | ![C: $i] : ((~in(C, B)) | open_subset(C, A) | (~element(C, powerset(the_carrier(A))))))) | (~(open_subsets(B, A) | (~((~in(tptp_fun_C_0(B, A), B)) | open_subset(tptp_fun_C_0(B, A), A) | (~element(tptp_fun_C_0(B, A), powerset(the_carrier(A))))))))))))) | (~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~in(tptp_fun_C_0(B, A!9), B)))))))))),
% 1.23/1.40 inference(modus_ponens,[status(thm)],[183, 182])).
% 1.23/1.40 tff(185,plain,
% 1.23/1.40 (![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~in(tptp_fun_C_0(B, A!9), B)))))))))),
% 1.23/1.40 inference(unit_resolution,[status(thm)],[184, 178, 72])).
% 1.23/1.40 tff(186,plain,
% 1.23/1.40 (((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~in(tptp_fun_C_0(B, A!9), B)))))))))) | ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B!10, A!9) | (~((~in(tptp_fun_C_0(B!10, A!9), B!10)) | open_subset(tptp_fun_C_0(B!10, A!9), A!9) | (~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9)))))))))))) <=> ((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~in(tptp_fun_C_0(B, A!9), B)))))))))) | (~element(B!10, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B!10, A!9) | (~((~in(tptp_fun_C_0(B!10, A!9), B!10)) | open_subset(tptp_fun_C_0(B!10, A!9), A!9) | (~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9)))))))))))),
% 1.23/1.41 inference(rewrite,[status(thm)],[])).
% 1.23/1.41 tff(187,plain,
% 1.23/1.41 (((~element(B!10, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B!10, A!9) | (~((~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9)))) | open_subset(tptp_fun_C_0(B!10, A!9), A!9) | (~in(tptp_fun_C_0(B!10, A!9), B!10))))))))) <=> ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B!10, A!9) | (~((~in(tptp_fun_C_0(B!10, A!9), B!10)) | open_subset(tptp_fun_C_0(B!10, A!9), A!9) | (~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9)))))))))))),
% 1.23/1.41 inference(rewrite,[status(thm)],[])).
% 1.23/1.41 tff(188,plain,
% 1.23/1.41 (((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~in(tptp_fun_C_0(B, A!9), B)))))))))) | ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B!10, A!9) | (~((~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9)))) | open_subset(tptp_fun_C_0(B!10, A!9), A!9) | (~in(tptp_fun_C_0(B!10, A!9), B!10)))))))))) <=> ((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~in(tptp_fun_C_0(B, A!9), B)))))))))) | ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B!10, A!9) | (~((~in(tptp_fun_C_0(B!10, A!9), B!10)) | open_subset(tptp_fun_C_0(B!10, A!9), A!9) | (~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9))))))))))))),
% 1.23/1.41 inference(monotonicity,[status(thm)],[187])).
% 1.23/1.41 tff(189,plain,
% 1.23/1.41 (((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~in(tptp_fun_C_0(B, A!9), B)))))))))) | ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B!10, A!9) | (~((~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9)))) | open_subset(tptp_fun_C_0(B!10, A!9), A!9) | (~in(tptp_fun_C_0(B!10, A!9), B!10)))))))))) <=> ((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~in(tptp_fun_C_0(B, A!9), B)))))))))) | (~element(B!10, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B!10, A!9) | (~((~in(tptp_fun_C_0(B!10, A!9), B!10)) | open_subset(tptp_fun_C_0(B!10, A!9), A!9) | (~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9)))))))))))),
% 1.23/1.41 inference(transitivity,[status(thm)],[188, 186])).
% 1.23/1.41 tff(190,plain,
% 1.23/1.41 ((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~in(tptp_fun_C_0(B, A!9), B)))))))))) | ((~element(B!10, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B!10, A!9) | (~((~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9)))) | open_subset(tptp_fun_C_0(B!10, A!9), A!9) | (~in(tptp_fun_C_0(B!10, A!9), B!10)))))))))),
% 1.23/1.41 inference(quant_inst,[status(thm)],[])).
% 1.23/1.41 tff(191,plain,
% 1.23/1.41 ((~![B: $i] : ((~element(B, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B, A!9)) | ![C: $i] : ((~in(C, B)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B, A!9) | (~((~element(tptp_fun_C_0(B, A!9), powerset(the_carrier(A!9)))) | open_subset(tptp_fun_C_0(B, A!9), A!9) | (~in(tptp_fun_C_0(B, A!9), B)))))))))) | (~element(B!10, powerset(powerset(the_carrier(A!9))))) | (~((~((~open_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B!10, A!9) | (~((~in(tptp_fun_C_0(B!10, A!9), B!10)) | open_subset(tptp_fun_C_0(B!10, A!9), A!9) | (~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9))))))))))),
% 1.23/1.41 inference(modus_ponens,[status(thm)],[190, 189])).
% 1.23/1.41 tff(192,plain,
% 1.23/1.41 (~((~((~open_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B!10, A!9) | (~((~in(tptp_fun_C_0(B!10, A!9), B!10)) | open_subset(tptp_fun_C_0(B!10, A!9), A!9) | (~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9)))))))))),
% 1.23/1.41 inference(unit_resolution,[status(thm)],[191, 9, 185])).
% 1.23/1.41 tff(193,plain,
% 1.23/1.41 (((~((~open_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B!10, A!9) | (~((~in(tptp_fun_C_0(B!10, A!9), B!10)) | open_subset(tptp_fun_C_0(B!10, A!9), A!9) | (~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9))))))))) | ((~open_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))),
% 1.23/1.41 inference(tautology,[status(thm)],[])).
% 1.23/1.41 tff(194,plain,
% 1.23/1.41 ((~open_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9))),
% 1.23/1.41 inference(unit_resolution,[status(thm)],[193, 192])).
% 1.23/1.41 tff(195,plain,
% 1.23/1.41 ((~((~open_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~open_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9))),
% 1.23/1.41 inference(tautology,[status(thm)],[])).
% 1.23/1.41 tff(196,plain,
% 1.23/1.41 ((~open_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9))),
% 1.23/1.41 inference(unit_resolution,[status(thm)],[195, 194])).
% 1.23/1.41 tff(197,plain,
% 1.23/1.41 (![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9))),
% 1.23/1.41 inference(unit_resolution,[status(thm)],[196, 162])).
% 1.23/1.41 tff(198,plain,
% 1.23/1.41 (((~![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9))) | (open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9)))))) <=> ((~![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9))) | open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9)))))),
% 1.23/1.41 inference(rewrite,[status(thm)],[])).
% 1.23/1.41 tff(199,plain,
% 1.23/1.41 (((~in(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9)))) | open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9)) <=> (open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9)))))),
% 1.23/1.41 inference(rewrite,[status(thm)],[])).
% 1.23/1.41 tff(200,plain,
% 1.23/1.41 (((~![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9))) | ((~in(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9)))) | open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9))) <=> ((~![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9))) | (open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9))))))),
% 1.23/1.41 inference(monotonicity,[status(thm)],[199])).
% 1.23/1.41 tff(201,plain,
% 1.23/1.41 (((~![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9))) | ((~in(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9)))) | open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9))) <=> ((~![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9))) | open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9)))))),
% 1.23/1.41 inference(transitivity,[status(thm)],[200, 198])).
% 1.23/1.41 tff(202,plain,
% 1.23/1.41 ((~![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9))) | ((~in(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9)))) | open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9))),
% 1.23/1.41 inference(quant_inst,[status(thm)],[])).
% 1.23/1.41 tff(203,plain,
% 1.23/1.41 ((~![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9))) | open_subset(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), B!10)) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9))))),
% 1.23/1.41 inference(modus_ponens,[status(thm)],[202, 201])).
% 1.23/1.41 tff(204,plain,
% 1.23/1.41 (~element(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9)))),
% 1.23/1.41 inference(unit_resolution,[status(thm)],[203, 197, 158, 132])).
% 1.23/1.41 tff(205,plain,
% 1.23/1.41 (^[A: $i, B: $i] : refl(((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A))) <=> ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A))))),
% 1.23/1.41 inference(bind,[status(th)],[])).
% 1.23/1.41 tff(206,plain,
% 1.23/1.41 (![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 1.23/1.41 inference(quant_intro,[status(thm)],[205])).
% 1.23/1.41 tff(207,plain,
% 1.23/1.41 (![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 1.23/1.41 inference(rewrite,[status(thm)],[])).
% 1.23/1.41 tff(208,plain,
% 1.23/1.41 (^[A: $i, B: $i] : rewrite((element(B, powerset(A)) => element(subset_complement(A, B), powerset(A))) <=> ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A))))),
% 1.23/1.41 inference(bind,[status(th)],[])).
% 1.23/1.41 tff(209,plain,
% 1.23/1.41 (![A: $i, B: $i] : (element(B, powerset(A)) => element(subset_complement(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 1.23/1.41 inference(quant_intro,[status(thm)],[208])).
% 1.23/1.41 tff(210,axiom,(![A: $i, B: $i] : (element(B, powerset(A)) => element(subset_complement(A, B), powerset(A)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','dt_k3_subset_1')).
% 1.23/1.41 tff(211,plain,
% 1.23/1.41 (![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 1.23/1.41 inference(modus_ponens,[status(thm)],[210, 209])).
% 1.23/1.41 tff(212,plain,
% 1.23/1.41 (![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 1.23/1.41 inference(modus_ponens,[status(thm)],[211, 207])).
% 1.23/1.41 tff(213,plain,(
% 1.23/1.41 ![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 1.23/1.41 inference(skolemize,[status(sab)],[212])).
% 1.23/1.41 tff(214,plain,
% 1.23/1.41 (![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 1.23/1.41 inference(modus_ponens,[status(thm)],[213, 206])).
% 1.23/1.41 tff(215,plain,
% 1.23/1.41 (((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | ((~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | element(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | (~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | element(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9))))),
% 1.23/1.41 inference(rewrite,[status(thm)],[])).
% 1.23/1.41 tff(216,plain,
% 1.23/1.41 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | ((~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | element(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9))))),
% 1.23/1.41 inference(quant_inst,[status(thm)],[])).
% 1.23/1.41 tff(217,plain,
% 1.23/1.41 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | (~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9)))) | element(subset_complement(the_carrier(A!9), tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9)), powerset(the_carrier(A!9)))),
% 1.23/1.41 inference(modus_ponens,[status(thm)],[216, 215])).
% 1.23/1.41 tff(218,plain,
% 1.23/1.41 ($false),
% 1.23/1.41 inference(unit_resolution,[status(thm)],[217, 214, 109, 204])).
% 1.23/1.41 tff(219,plain,(closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)), inference(lemma,lemma(discharge,[]))).
% 1.23/1.41 tff(220,plain,
% 1.23/1.41 ((open_subsets(B!10, A!9) <=> closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | (~open_subsets(B!10, A!9)) | (~closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9))),
% 1.23/1.41 inference(tautology,[status(thm)],[])).
% 1.23/1.41 tff(221,plain,
% 1.23/1.41 ((~open_subsets(B!10, A!9)) | (~closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9))),
% 1.23/1.41 inference(unit_resolution,[status(thm)],[220, 159])).
% 1.23/1.41 tff(222,plain,
% 1.23/1.41 (~open_subsets(B!10, A!9)),
% 1.23/1.41 inference(unit_resolution,[status(thm)],[221, 219])).
% 1.23/1.41 tff(223,plain,
% 1.23/1.41 (((~((~open_subsets(B!10, A!9)) | ![C: $i] : ((~in(C, B!10)) | (~element(C, powerset(the_carrier(A!9)))) | open_subset(C, A!9)))) | (~(open_subsets(B!10, A!9) | (~((~in(tptp_fun_C_0(B!10, A!9), B!10)) | open_subset(tptp_fun_C_0(B!10, A!9), A!9) | (~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9))))))))) | (open_subsets(B!10, A!9) | (~((~in(tptp_fun_C_0(B!10, A!9), B!10)) | open_subset(tptp_fun_C_0(B!10, A!9), A!9) | (~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9)))))))),
% 1.23/1.41 inference(tautology,[status(thm)],[])).
% 1.23/1.41 tff(224,plain,
% 1.23/1.41 (open_subsets(B!10, A!9) | (~((~in(tptp_fun_C_0(B!10, A!9), B!10)) | open_subset(tptp_fun_C_0(B!10, A!9), A!9) | (~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9))))))),
% 1.23/1.41 inference(unit_resolution,[status(thm)],[223, 192])).
% 1.23/1.41 tff(225,plain,
% 1.23/1.41 ((~(open_subsets(B!10, A!9) | (~((~in(tptp_fun_C_0(B!10, A!9), B!10)) | open_subset(tptp_fun_C_0(B!10, A!9), A!9) | (~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9)))))))) | open_subsets(B!10, A!9) | (~((~in(tptp_fun_C_0(B!10, A!9), B!10)) | open_subset(tptp_fun_C_0(B!10, A!9), A!9) | (~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9))))))),
% 1.23/1.41 inference(tautology,[status(thm)],[])).
% 1.23/1.41 tff(226,plain,
% 1.23/1.41 (open_subsets(B!10, A!9) | (~((~in(tptp_fun_C_0(B!10, A!9), B!10)) | open_subset(tptp_fun_C_0(B!10, A!9), A!9) | (~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9))))))),
% 1.23/1.41 inference(unit_resolution,[status(thm)],[225, 224])).
% 1.23/1.41 tff(227,plain,
% 1.23/1.41 (~((~in(tptp_fun_C_0(B!10, A!9), B!10)) | open_subset(tptp_fun_C_0(B!10, A!9), A!9) | (~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9)))))),
% 1.23/1.41 inference(unit_resolution,[status(thm)],[226, 222])).
% 1.23/1.41 tff(228,plain,
% 1.23/1.41 (((~in(tptp_fun_C_0(B!10, A!9), B!10)) | open_subset(tptp_fun_C_0(B!10, A!9), A!9) | (~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9))))) | element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9)))),
% 1.23/1.41 inference(tautology,[status(thm)],[])).
% 1.23/1.41 tff(229,plain,
% 1.23/1.41 (element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9)))),
% 1.23/1.41 inference(unit_resolution,[status(thm)],[228, 227])).
% 1.23/1.41 tff(230,plain,
% 1.23/1.41 (((~![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10))))) | ((~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9)))) | (in(tptp_fun_C_0(B!10, A!9), B!10) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))))) <=> ((~![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10))))) | (~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9)))) | (in(tptp_fun_C_0(B!10, A!9), B!10) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))))),
% 1.23/1.41 inference(rewrite,[status(thm)],[])).
% 1.23/1.41 tff(231,plain,
% 1.23/1.41 ((~![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10))))) | ((~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9)))) | (in(tptp_fun_C_0(B!10, A!9), B!10) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))))),
% 1.23/1.41 inference(quant_inst,[status(thm)],[])).
% 1.23/1.41 tff(232,plain,
% 1.23/1.41 ((~![D: $i] : ((~element(D, powerset(the_carrier(A!9)))) | (in(D, B!10) <=> in(subset_complement(the_carrier(A!9), D), complements_of_subsets(the_carrier(A!9), B!10))))) | (~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9)))) | (in(tptp_fun_C_0(B!10, A!9), B!10) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10)))),
% 1.23/1.41 inference(modus_ponens,[status(thm)],[231, 230])).
% 1.23/1.41 tff(233,plain,
% 1.23/1.41 (in(tptp_fun_C_0(B!10, A!9), B!10) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))),
% 1.23/1.41 inference(unit_resolution,[status(thm)],[232, 229, 70])).
% 1.23/1.41 tff(234,plain,
% 1.23/1.41 (((~in(tptp_fun_C_0(B!10, A!9), B!10)) | open_subset(tptp_fun_C_0(B!10, A!9), A!9) | (~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9))))) | in(tptp_fun_C_0(B!10, A!9), B!10)),
% 1.23/1.41 inference(tautology,[status(thm)],[])).
% 1.23/1.41 tff(235,plain,
% 1.23/1.41 (in(tptp_fun_C_0(B!10, A!9), B!10)),
% 1.23/1.41 inference(unit_resolution,[status(thm)],[234, 227])).
% 1.23/1.41 tff(236,plain,
% 1.23/1.41 ((~(in(tptp_fun_C_0(B!10, A!9), B!10) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10)))) | (~in(tptp_fun_C_0(B!10, A!9), B!10)) | in(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))),
% 1.23/1.41 inference(tautology,[status(thm)],[])).
% 1.23/1.41 tff(237,plain,
% 1.23/1.41 ((~(in(tptp_fun_C_0(B!10, A!9), B!10) <=> in(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10)))) | in(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))),
% 1.23/1.41 inference(unit_resolution,[status(thm)],[236, 235])).
% 1.23/1.41 tff(238,plain,
% 1.23/1.41 (in(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))),
% 1.23/1.41 inference(unit_resolution,[status(thm)],[237, 233])).
% 1.23/1.41 tff(239,plain,
% 1.23/1.41 (^[A: $i] : refl(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))))),
% 1.23/1.41 inference(bind,[status(th)],[])).
% 1.23/1.41 tff(240,plain,
% 1.23/1.41 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))),
% 1.23/1.41 inference(quant_intro,[status(thm)],[239])).
% 1.23/1.41 tff(241,plain,
% 1.23/1.41 (^[A: $i] : rewrite(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))))),
% 1.23/1.41 inference(bind,[status(th)],[])).
% 1.23/1.41 tff(242,plain,
% 1.23/1.41 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))),
% 1.23/1.41 inference(quant_intro,[status(thm)],[241])).
% 1.23/1.41 tff(243,plain,
% 1.23/1.41 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))),
% 1.23/1.41 inference(transitivity,[status(thm)],[242, 240])).
% 1.23/1.41 tff(244,plain,
% 1.23/1.41 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))),
% 1.23/1.41 inference(rewrite,[status(thm)],[])).
% 1.23/1.41 tff(245,plain,
% 1.23/1.41 (^[A: $i] : trans(monotonicity(quant_intro(proof_bind(^[B: $i] : rewrite((element(B, powerset(the_carrier(A))) => (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))) <=> ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))), (![B: $i] : (element(B, powerset(the_carrier(A))) => (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))) <=> ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))), ((top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))) <=> (top_str(A) => ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))))), rewrite((top_str(A) => ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))), ((top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))))),
% 1.23/1.41 inference(bind,[status(th)],[])).
% 1.23/1.41 tff(246,plain,
% 1.23/1.41 (![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))),
% 1.23/1.41 inference(quant_intro,[status(thm)],[245])).
% 1.23/1.41 tff(247,axiom,(![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t30_tops_1')).
% 1.23/1.41 tff(248,plain,
% 1.23/1.41 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))),
% 1.23/1.41 inference(modus_ponens,[status(thm)],[247, 246])).
% 1.23/1.41 tff(249,plain,
% 1.23/1.41 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))),
% 1.23/1.41 inference(modus_ponens,[status(thm)],[248, 244])).
% 1.23/1.41 tff(250,plain,(
% 1.23/1.41 ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))),
% 1.23/1.41 inference(skolemize,[status(sab)],[249])).
% 1.23/1.41 tff(251,plain,
% 1.23/1.41 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))),
% 1.23/1.41 inference(modus_ponens,[status(thm)],[250, 243])).
% 1.23/1.41 tff(252,plain,
% 1.23/1.41 (((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (open_subset(B, A!9) <=> closed_subset(subset_complement(the_carrier(A!9), B), A!9))))) <=> ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))) | (~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (open_subset(B, A!9) <=> closed_subset(subset_complement(the_carrier(A!9), B), A!9))))),
% 1.23/1.41 inference(rewrite,[status(thm)],[])).
% 1.23/1.41 tff(253,plain,
% 1.23/1.41 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))) | ((~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (open_subset(B, A!9) <=> closed_subset(subset_complement(the_carrier(A!9), B), A!9))))),
% 1.23/1.41 inference(quant_inst,[status(thm)],[])).
% 1.23/1.41 tff(254,plain,
% 1.23/1.41 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))) | (~top_str(A!9)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (open_subset(B, A!9) <=> closed_subset(subset_complement(the_carrier(A!9), B), A!9)))),
% 1.23/1.41 inference(modus_ponens,[status(thm)],[253, 252])).
% 1.23/1.41 tff(255,plain,
% 1.23/1.41 (![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (open_subset(B, A!9) <=> closed_subset(subset_complement(the_carrier(A!9), B), A!9)))),
% 1.23/1.41 inference(unit_resolution,[status(thm)],[254, 72, 251])).
% 1.23/1.41 tff(256,plain,
% 1.23/1.41 (((~![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (open_subset(B, A!9) <=> closed_subset(subset_complement(the_carrier(A!9), B), A!9)))) | ((~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9)))) | (open_subset(tptp_fun_C_0(B!10, A!9), A!9) <=> closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), A!9)))) <=> ((~![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (open_subset(B, A!9) <=> closed_subset(subset_complement(the_carrier(A!9), B), A!9)))) | (~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9)))) | (open_subset(tptp_fun_C_0(B!10, A!9), A!9) <=> closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), A!9)))),
% 1.23/1.41 inference(rewrite,[status(thm)],[])).
% 1.23/1.41 tff(257,plain,
% 1.23/1.41 ((~![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (open_subset(B, A!9) <=> closed_subset(subset_complement(the_carrier(A!9), B), A!9)))) | ((~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9)))) | (open_subset(tptp_fun_C_0(B!10, A!9), A!9) <=> closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), A!9)))),
% 1.23/1.41 inference(quant_inst,[status(thm)],[])).
% 1.23/1.41 tff(258,plain,
% 1.23/1.41 ((~![B: $i] : ((~element(B, powerset(the_carrier(A!9)))) | (open_subset(B, A!9) <=> closed_subset(subset_complement(the_carrier(A!9), B), A!9)))) | (~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9)))) | (open_subset(tptp_fun_C_0(B!10, A!9), A!9) <=> closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), A!9))),
% 1.23/1.41 inference(modus_ponens,[status(thm)],[257, 256])).
% 1.23/1.41 tff(259,plain,
% 1.23/1.41 (open_subset(tptp_fun_C_0(B!10, A!9), A!9) <=> closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), A!9)),
% 1.23/1.41 inference(unit_resolution,[status(thm)],[258, 255, 229])).
% 1.23/1.41 tff(260,plain,
% 1.23/1.41 (((~in(tptp_fun_C_0(B!10, A!9), B!10)) | open_subset(tptp_fun_C_0(B!10, A!9), A!9) | (~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9))))) | (~open_subset(tptp_fun_C_0(B!10, A!9), A!9))),
% 1.23/1.41 inference(tautology,[status(thm)],[])).
% 1.23/1.41 tff(261,plain,
% 1.23/1.41 (~open_subset(tptp_fun_C_0(B!10, A!9), A!9)),
% 1.23/1.41 inference(unit_resolution,[status(thm)],[260, 227])).
% 1.23/1.41 tff(262,plain,
% 1.23/1.41 ((~(open_subset(tptp_fun_C_0(B!10, A!9), A!9) <=> closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), A!9))) | open_subset(tptp_fun_C_0(B!10, A!9), A!9) | (~closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), A!9))),
% 1.23/1.41 inference(tautology,[status(thm)],[])).
% 1.23/1.41 tff(263,plain,
% 1.23/1.41 ((~(open_subset(tptp_fun_C_0(B!10, A!9), A!9) <=> closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), A!9))) | (~closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), A!9))),
% 1.23/1.41 inference(unit_resolution,[status(thm)],[262, 261])).
% 1.23/1.41 tff(264,plain,
% 1.23/1.41 (~closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), A!9)),
% 1.23/1.41 inference(unit_resolution,[status(thm)],[263, 259])).
% 1.23/1.41 tff(265,plain,
% 1.23/1.41 (((~((~closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10)))))) | (~(closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9) | (~((~in(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), complements_of_subsets(the_carrier(A!9), B!10))) | closed_subset(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), A!9) | (~element(tptp_fun_C_1(complements_of_subsets(the_carrier(A!9), B!10), A!9), powerset(the_carrier(A!9))))))))) | ((~closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10)))))),
% 1.23/1.41 inference(tautology,[status(thm)],[])).
% 1.23/1.41 tff(266,plain,
% 1.23/1.41 ((~closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10))))),
% 1.23/1.41 inference(unit_resolution,[status(thm)],[265, 102])).
% 1.23/1.41 tff(267,plain,
% 1.23/1.41 ((~((~closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10)))))) | (~closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10))))),
% 1.23/1.42 inference(tautology,[status(thm)],[])).
% 1.23/1.42 tff(268,plain,
% 1.23/1.42 ((~closed_subsets(complements_of_subsets(the_carrier(A!9), B!10), A!9)) | ![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10))))),
% 1.23/1.42 inference(unit_resolution,[status(thm)],[267, 266])).
% 1.23/1.42 tff(269,plain,
% 1.23/1.42 (![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10))))),
% 1.23/1.42 inference(unit_resolution,[status(thm)],[268, 219])).
% 1.23/1.42 tff(270,plain,
% 1.23/1.42 (((~![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10))))) | (closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), powerset(the_carrier(A!9)))))) <=> ((~![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10))))) | closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), powerset(the_carrier(A!9)))))),
% 1.23/1.42 inference(rewrite,[status(thm)],[])).
% 1.23/1.42 tff(271,plain,
% 1.23/1.42 (((~element(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), powerset(the_carrier(A!9)))) | closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10)))) <=> (closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), powerset(the_carrier(A!9)))))),
% 1.23/1.42 inference(rewrite,[status(thm)],[])).
% 1.23/1.42 tff(272,plain,
% 1.23/1.42 (((~![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10))))) | ((~element(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), powerset(the_carrier(A!9)))) | closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))))) <=> ((~![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10))))) | (closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), powerset(the_carrier(A!9))))))),
% 1.23/1.42 inference(monotonicity,[status(thm)],[271])).
% 1.23/1.42 tff(273,plain,
% 1.23/1.42 (((~![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10))))) | ((~element(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), powerset(the_carrier(A!9)))) | closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))))) <=> ((~![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10))))) | closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), powerset(the_carrier(A!9)))))),
% 1.23/1.42 inference(transitivity,[status(thm)],[272, 270])).
% 1.23/1.42 tff(274,plain,
% 1.23/1.42 ((~![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10))))) | ((~element(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), powerset(the_carrier(A!9)))) | closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))))),
% 1.23/1.42 inference(quant_inst,[status(thm)],[])).
% 1.23/1.42 tff(275,plain,
% 1.23/1.42 ((~![C: $i] : ((~element(C, powerset(the_carrier(A!9)))) | closed_subset(C, A!9) | (~in(C, complements_of_subsets(the_carrier(A!9), B!10))))) | closed_subset(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), A!9) | (~in(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), complements_of_subsets(the_carrier(A!9), B!10))) | (~element(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), powerset(the_carrier(A!9))))),
% 1.23/1.42 inference(modus_ponens,[status(thm)],[274, 273])).
% 1.23/1.42 tff(276,plain,
% 1.23/1.42 (~element(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), powerset(the_carrier(A!9)))),
% 1.23/1.42 inference(unit_resolution,[status(thm)],[275, 269, 264, 238])).
% 1.23/1.42 tff(277,plain,
% 1.23/1.42 (((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | ((~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9)))) | element(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), powerset(the_carrier(A!9))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | (~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9)))) | element(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), powerset(the_carrier(A!9))))),
% 1.23/1.42 inference(rewrite,[status(thm)],[])).
% 1.23/1.42 tff(278,plain,
% 1.23/1.42 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | ((~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9)))) | element(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), powerset(the_carrier(A!9))))),
% 1.23/1.42 inference(quant_inst,[status(thm)],[])).
% 1.23/1.42 tff(279,plain,
% 1.23/1.42 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | (~element(tptp_fun_C_0(B!10, A!9), powerset(the_carrier(A!9)))) | element(subset_complement(the_carrier(A!9), tptp_fun_C_0(B!10, A!9)), powerset(the_carrier(A!9)))),
% 1.23/1.42 inference(modus_ponens,[status(thm)],[278, 277])).
% 1.23/1.42 tff(280,plain,
% 1.23/1.42 ($false),
% 1.23/1.42 inference(unit_resolution,[status(thm)],[279, 214, 229, 276])).
% 1.23/1.42 % SZS output end Proof
% 1.23/1.43 % E exiting
%------------------------------------------------------------------------------