%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SEU327+2 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n014.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 05:27:37 AM UTC 2025
% Result : Theorem 1.54s 1.71s
% Output : Proof 1.54s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : SEU327+2 : TPTP v9.0.0. Released v3.3.0.
% 0.07/0.12 % Command : run_E %s %d THM
% 0.12/0.32 % Computer : n014.cluster.edu
% 0.12/0.32 % Model : x86_64 x86_64
% 0.12/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.32 % Memory : 8042.1875MB
% 0.12/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Jun 20 06:45:09 EDT 2025
% 0.12/0.33 % CPUTime :
% 1.54/1.71 % SZS status Theorem
% 1.54/1.71 % SZS output start Proof
% 1.54/1.71 tff(subset_complement_type, type, (
% 1.54/1.71 subset_complement: ( $i * $i ) > $i)).
% 1.54/1.71 tff(union_of_subsets_type, type, (
% 1.54/1.71 union_of_subsets: ( $i * $i ) > $i)).
% 1.54/1.71 tff(tptp_fun_B_218_type, type, (
% 1.54/1.71 tptp_fun_B_218: $i)).
% 1.54/1.71 tff(tptp_fun_A_219_type, type, (
% 1.54/1.71 tptp_fun_A_219: $i)).
% 1.54/1.71 tff(meet_of_subsets_type, type, (
% 1.54/1.71 meet_of_subsets: ( $i * $i ) > $i)).
% 1.54/1.71 tff(complements_of_subsets_type, type, (
% 1.54/1.71 complements_of_subsets: ( $i * $i ) > $i)).
% 1.54/1.71 tff(element_type, type, (
% 1.54/1.71 element: ( $i * $i ) > $o)).
% 1.54/1.71 tff(powerset_type, type, (
% 1.54/1.71 powerset: $i > $i)).
% 1.54/1.71 tff(empty_set_type, type, (
% 1.54/1.71 empty_set: $i)).
% 1.54/1.71 tff(subset_difference_type, type, (
% 1.54/1.71 subset_difference: ( $i * $i * $i ) > $i)).
% 1.54/1.71 tff(cast_to_subset_type, type, (
% 1.54/1.71 cast_to_subset: $i > $i)).
% 1.54/1.71 tff(set_difference_type, type, (
% 1.54/1.71 set_difference: ( $i * $i ) > $i)).
% 1.54/1.71 tff(1,plain,
% 1.54/1.71 ((~((B!218 = empty_set) | (meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)) = subset_complement(A!219, union_of_subsets(A!219, B!218))) | (~element(B!218, powerset(powerset(A!219)))))) <=> (~((B!218 = empty_set) | (meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)) = subset_complement(A!219, union_of_subsets(A!219, B!218))) | (~element(B!218, powerset(powerset(A!219))))))),
% 1.54/1.71 inference(rewrite,[status(thm)],[])).
% 1.54/1.71 tff(2,plain,
% 1.54/1.71 ((~![A: $i, B: $i] : ((B = empty_set) | (meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) <=> (~![A: $i, B: $i] : ((B = empty_set) | (meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (~element(B, powerset(powerset(A))))))),
% 1.54/1.71 inference(rewrite,[status(thm)],[])).
% 1.54/1.71 tff(3,plain,
% 1.54/1.71 ((~![A: $i, B: $i] : (element(B, powerset(powerset(A))) => ((~(B = empty_set)) => (meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B)))))) <=> (~![A: $i, B: $i] : ((B = empty_set) | (meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (~element(B, powerset(powerset(A))))))),
% 1.54/1.71 inference(rewrite,[status(thm)],[])).
% 1.54/1.71 tff(4,axiom,(~![A: $i, B: $i] : (element(B, powerset(powerset(A))) => ((~(B = empty_set)) => (meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B)))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t11_tops_2')).
% 1.54/1.71 tff(5,plain,
% 1.54/1.71 (~![A: $i, B: $i] : ((B = empty_set) | (meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[4, 3])).
% 1.54/1.71 tff(6,plain,
% 1.54/1.71 (~![A: $i, B: $i] : ((B = empty_set) | (meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[5, 2])).
% 1.54/1.71 tff(7,plain,
% 1.54/1.71 (~![A: $i, B: $i] : ((B = empty_set) | (meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[6, 2])).
% 1.54/1.71 tff(8,plain,
% 1.54/1.71 (~![A: $i, B: $i] : ((B = empty_set) | (meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[7, 2])).
% 1.54/1.71 tff(9,plain,(
% 1.54/1.71 ~((B!218 = empty_set) | (meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)) = subset_complement(A!219, union_of_subsets(A!219, B!218))) | (~element(B!218, powerset(powerset(A!219)))))),
% 1.54/1.71 inference(skolemize,[status(sab)],[8])).
% 1.54/1.71 tff(10,plain,
% 1.54/1.71 (~((B!218 = empty_set) | (meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)) = subset_complement(A!219, union_of_subsets(A!219, B!218))) | (~element(B!218, powerset(powerset(A!219)))))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[9, 1])).
% 1.54/1.71 tff(11,plain,
% 1.54/1.71 (element(B!218, powerset(powerset(A!219)))),
% 1.54/1.71 inference(or_elim,[status(thm)],[10])).
% 1.54/1.71 tff(12,plain,
% 1.54/1.71 (^[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)))),
% 1.54/1.71 inference(bind,[status(th)],[])).
% 1.54/1.71 tff(13,plain,
% 1.54/1.71 (![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))),
% 1.54/1.71 inference(quant_intro,[status(thm)],[12])).
% 1.54/1.71 tff(14,plain,
% 1.54/1.71 (![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))),
% 1.54/1.71 inference(rewrite,[status(thm)],[])).
% 1.54/1.71 tff(15,plain,
% 1.54/1.71 (^[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)))),
% 1.54/1.71 inference(bind,[status(th)],[])).
% 1.54/1.71 tff(16,plain,
% 1.54/1.71 (![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))),
% 1.54/1.71 inference(quant_intro,[status(thm)],[15])).
% 1.54/1.71 tff(17,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => (complements_of_subsets(A, complements_of_subsets(A, B)) = B))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','involutiveness_k7_setfam_1')).
% 1.54/1.71 tff(18,plain,
% 1.54/1.71 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[17, 16])).
% 1.54/1.71 tff(19,plain,
% 1.54/1.71 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[18, 14])).
% 1.54/1.71 tff(20,plain,(
% 1.54/1.71 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 1.54/1.71 inference(skolemize,[status(sab)],[19])).
% 1.54/1.71 tff(21,plain,
% 1.54/1.71 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[20, 13])).
% 1.54/1.71 tff(22,plain,
% 1.54/1.71 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | ((~element(B!218, powerset(powerset(A!219)))) | (complements_of_subsets(A!219, complements_of_subsets(A!219, B!218)) = B!218))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | (~element(B!218, powerset(powerset(A!219)))) | (complements_of_subsets(A!219, complements_of_subsets(A!219, B!218)) = B!218))),
% 1.54/1.71 inference(rewrite,[status(thm)],[])).
% 1.54/1.71 tff(23,plain,
% 1.54/1.71 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | ((~element(B!218, powerset(powerset(A!219)))) | (complements_of_subsets(A!219, complements_of_subsets(A!219, B!218)) = B!218))),
% 1.54/1.71 inference(quant_inst,[status(thm)],[])).
% 1.54/1.71 tff(24,plain,
% 1.54/1.71 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | (~element(B!218, powerset(powerset(A!219)))) | (complements_of_subsets(A!219, complements_of_subsets(A!219, B!218)) = B!218)),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[23, 22])).
% 1.54/1.71 tff(25,plain,
% 1.54/1.71 (complements_of_subsets(A!219, complements_of_subsets(A!219, B!218)) = B!218),
% 1.54/1.71 inference(unit_resolution,[status(thm)],[24, 21, 11])).
% 1.54/1.71 tff(26,plain,
% 1.54/1.71 (union_of_subsets(A!219, complements_of_subsets(A!219, complements_of_subsets(A!219, B!218))) = union_of_subsets(A!219, B!218)),
% 1.54/1.71 inference(monotonicity,[status(thm)],[25])).
% 1.54/1.71 tff(27,plain,
% 1.54/1.71 (^[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)))))),
% 1.54/1.71 inference(bind,[status(th)],[])).
% 1.54/1.71 tff(28,plain,
% 1.54/1.71 (![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))))),
% 1.54/1.71 inference(quant_intro,[status(thm)],[27])).
% 1.54/1.71 tff(29,plain,
% 1.54/1.71 (![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))))),
% 1.54/1.71 inference(rewrite,[status(thm)],[])).
% 1.54/1.71 tff(30,plain,
% 1.54/1.71 (^[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)))))),
% 1.54/1.71 inference(bind,[status(th)],[])).
% 1.54/1.71 tff(31,plain,
% 1.54/1.71 (![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))))),
% 1.54/1.71 inference(quant_intro,[status(thm)],[30])).
% 1.54/1.71 tff(32,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => element(complements_of_subsets(A, B), powerset(powerset(A))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','dt_k7_setfam_1')).
% 1.54/1.71 tff(33,plain,
% 1.54/1.71 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[32, 31])).
% 1.54/1.71 tff(34,plain,
% 1.54/1.71 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[33, 29])).
% 1.54/1.71 tff(35,plain,(
% 1.54/1.71 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 1.54/1.71 inference(skolemize,[status(sab)],[34])).
% 1.54/1.71 tff(36,plain,
% 1.54/1.71 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[35, 28])).
% 1.54/1.71 tff(37,plain,
% 1.54/1.71 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | ((~element(B!218, powerset(powerset(A!219)))) | element(complements_of_subsets(A!219, B!218), powerset(powerset(A!219))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | (~element(B!218, powerset(powerset(A!219)))) | element(complements_of_subsets(A!219, B!218), powerset(powerset(A!219))))),
% 1.54/1.71 inference(rewrite,[status(thm)],[])).
% 1.54/1.71 tff(38,plain,
% 1.54/1.71 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | ((~element(B!218, powerset(powerset(A!219)))) | element(complements_of_subsets(A!219, B!218), powerset(powerset(A!219))))),
% 1.54/1.71 inference(quant_inst,[status(thm)],[])).
% 1.54/1.71 tff(39,plain,
% 1.54/1.71 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | (~element(B!218, powerset(powerset(A!219)))) | element(complements_of_subsets(A!219, B!218), powerset(powerset(A!219)))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[38, 37])).
% 1.54/1.71 tff(40,plain,
% 1.54/1.71 (element(complements_of_subsets(A!219, B!218), powerset(powerset(A!219)))),
% 1.54/1.71 inference(unit_resolution,[status(thm)],[39, 36, 11])).
% 1.54/1.71 tff(41,plain,
% 1.54/1.71 (^[A: $i, B: $i] : refl(((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A))))) <=> ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A))))))),
% 1.54/1.71 inference(bind,[status(th)],[])).
% 1.54/1.71 tff(42,plain,
% 1.54/1.71 (![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A))))) <=> ![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))),
% 1.54/1.71 inference(quant_intro,[status(thm)],[41])).
% 1.54/1.71 tff(43,plain,
% 1.54/1.71 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(rewrite(((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set)) <=> (~((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set))))), ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) <=> (~(~((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set))))))), rewrite((~(~((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set))))) <=> ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)))), ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) <=> ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set))))), (((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A))))) <=> (((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A))))))), rewrite((((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A))))) <=> ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))), (((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A))))) <=> ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))))),
% 1.54/1.71 inference(bind,[status(th)],[])).
% 1.54/1.71 tff(44,plain,
% 1.54/1.71 (![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A))))) <=> ![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))),
% 1.54/1.71 inference(quant_intro,[status(thm)],[43])).
% 1.54/1.71 tff(45,plain,
% 1.54/1.71 (![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A))))) <=> ![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A)))))),
% 1.54/1.71 inference(rewrite,[status(thm)],[])).
% 1.54/1.71 tff(46,plain,
% 1.54/1.71 (^[A: $i, B: $i] : rewrite((element(B, powerset(powerset(A))) => (~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set)))) <=> ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A))))))),
% 1.54/1.71 inference(bind,[status(th)],[])).
% 1.54/1.71 tff(47,plain,
% 1.54/1.71 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => (~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set)))) <=> ![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A)))))),
% 1.54/1.71 inference(quant_intro,[status(thm)],[46])).
% 1.54/1.71 tff(48,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => (~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t46_setfam_1')).
% 1.54/1.71 tff(49,plain,
% 1.54/1.71 (![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A)))))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[48, 47])).
% 1.54/1.71 tff(50,plain,
% 1.54/1.71 (![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A)))))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[49, 45])).
% 1.54/1.71 tff(51,plain,(
% 1.54/1.71 ![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A)))))),
% 1.54/1.71 inference(skolemize,[status(sab)],[50])).
% 1.54/1.71 tff(52,plain,
% 1.54/1.71 (![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[51, 44])).
% 1.54/1.71 tff(53,plain,
% 1.54/1.71 (![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[52, 42])).
% 1.54/1.71 tff(54,plain,
% 1.54/1.71 (~(B!218 = empty_set)),
% 1.54/1.71 inference(or_elim,[status(thm)],[10])).
% 1.54/1.71 tff(55,plain,
% 1.54/1.71 (((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | ((B!218 = empty_set) | (~element(B!218, powerset(powerset(A!219)))) | (~(complements_of_subsets(A!219, B!218) = empty_set)))) <=> ((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | (B!218 = empty_set) | (~element(B!218, powerset(powerset(A!219)))) | (~(complements_of_subsets(A!219, B!218) = empty_set)))),
% 1.54/1.71 inference(rewrite,[status(thm)],[])).
% 1.54/1.71 tff(56,plain,
% 1.54/1.71 (((B!218 = empty_set) | (~(complements_of_subsets(A!219, B!218) = empty_set)) | (~element(B!218, powerset(powerset(A!219))))) <=> ((B!218 = empty_set) | (~element(B!218, powerset(powerset(A!219)))) | (~(complements_of_subsets(A!219, B!218) = empty_set)))),
% 1.54/1.71 inference(rewrite,[status(thm)],[])).
% 1.54/1.71 tff(57,plain,
% 1.54/1.71 (((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | ((B!218 = empty_set) | (~(complements_of_subsets(A!219, B!218) = empty_set)) | (~element(B!218, powerset(powerset(A!219)))))) <=> ((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | ((B!218 = empty_set) | (~element(B!218, powerset(powerset(A!219)))) | (~(complements_of_subsets(A!219, B!218) = empty_set))))),
% 1.54/1.71 inference(monotonicity,[status(thm)],[56])).
% 1.54/1.71 tff(58,plain,
% 1.54/1.71 (((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | ((B!218 = empty_set) | (~(complements_of_subsets(A!219, B!218) = empty_set)) | (~element(B!218, powerset(powerset(A!219)))))) <=> ((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | (B!218 = empty_set) | (~element(B!218, powerset(powerset(A!219)))) | (~(complements_of_subsets(A!219, B!218) = empty_set)))),
% 1.54/1.71 inference(transitivity,[status(thm)],[57, 55])).
% 1.54/1.71 tff(59,plain,
% 1.54/1.71 ((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | ((B!218 = empty_set) | (~(complements_of_subsets(A!219, B!218) = empty_set)) | (~element(B!218, powerset(powerset(A!219)))))),
% 1.54/1.71 inference(quant_inst,[status(thm)],[])).
% 1.54/1.71 tff(60,plain,
% 1.54/1.71 ((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | (B!218 = empty_set) | (~element(B!218, powerset(powerset(A!219)))) | (~(complements_of_subsets(A!219, B!218) = empty_set))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[59, 58])).
% 1.54/1.71 tff(61,plain,
% 1.54/1.71 (~(complements_of_subsets(A!219, B!218) = empty_set)),
% 1.54/1.71 inference(unit_resolution,[status(thm)],[60, 54, 11, 53])).
% 1.54/1.71 tff(62,plain,
% 1.54/1.71 (^[A: $i, B: $i] : refl(((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A))))) <=> ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A))))))),
% 1.54/1.71 inference(bind,[status(th)],[])).
% 1.54/1.71 tff(63,plain,
% 1.54/1.71 (![A: $i, B: $i] : ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A))))) <=> ![A: $i, B: $i] : ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 1.54/1.71 inference(quant_intro,[status(thm)],[62])).
% 1.54/1.71 tff(64,plain,
% 1.54/1.71 (![A: $i, B: $i] : ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A))))) <=> ![A: $i, B: $i] : ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 1.54/1.71 inference(rewrite,[status(thm)],[])).
% 1.54/1.71 tff(65,plain,
% 1.54/1.71 (^[A: $i, B: $i] : trans(monotonicity(rewrite(((~(B = empty_set)) => (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B)))) <=> ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))))), ((element(B, powerset(powerset(A))) => ((~(B = empty_set)) => (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))))) <=> (element(B, powerset(powerset(A))) => ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))))))), rewrite((element(B, powerset(powerset(A))) => ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))))) <=> ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))), ((element(B, powerset(powerset(A))) => ((~(B = empty_set)) => (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))))) <=> ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))))),
% 1.54/1.71 inference(bind,[status(th)],[])).
% 1.54/1.71 tff(66,plain,
% 1.54/1.71 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => ((~(B = empty_set)) => (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))))) <=> ![A: $i, B: $i] : ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 1.54/1.71 inference(quant_intro,[status(thm)],[65])).
% 1.54/1.71 tff(67,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => ((~(B = empty_set)) => (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B)))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t48_setfam_1')).
% 1.54/1.71 tff(68,plain,
% 1.54/1.71 (![A: $i, B: $i] : ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[67, 66])).
% 1.54/1.71 tff(69,plain,
% 1.54/1.71 (![A: $i, B: $i] : ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[68, 64])).
% 1.54/1.71 tff(70,plain,(
% 1.54/1.71 ![A: $i, B: $i] : ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 1.54/1.71 inference(skolemize,[status(sab)],[69])).
% 1.54/1.71 tff(71,plain,
% 1.54/1.71 (![A: $i, B: $i] : ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[70, 63])).
% 1.54/1.71 tff(72,plain,
% 1.54/1.71 (((~![A: $i, B: $i] : ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) | ((complements_of_subsets(A!219, B!218) = empty_set) | (union_of_subsets(A!219, complements_of_subsets(A!219, complements_of_subsets(A!219, B!218))) = subset_difference(A!219, cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))) | (~element(complements_of_subsets(A!219, B!218), powerset(powerset(A!219)))))) <=> ((~![A: $i, B: $i] : ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) | (complements_of_subsets(A!219, B!218) = empty_set) | (union_of_subsets(A!219, complements_of_subsets(A!219, complements_of_subsets(A!219, B!218))) = subset_difference(A!219, cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))) | (~element(complements_of_subsets(A!219, B!218), powerset(powerset(A!219)))))),
% 1.54/1.71 inference(rewrite,[status(thm)],[])).
% 1.54/1.71 tff(73,plain,
% 1.54/1.71 ((~![A: $i, B: $i] : ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) | ((complements_of_subsets(A!219, B!218) = empty_set) | (union_of_subsets(A!219, complements_of_subsets(A!219, complements_of_subsets(A!219, B!218))) = subset_difference(A!219, cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))) | (~element(complements_of_subsets(A!219, B!218), powerset(powerset(A!219)))))),
% 1.54/1.71 inference(quant_inst,[status(thm)],[])).
% 1.54/1.71 tff(74,plain,
% 1.54/1.71 ((~![A: $i, B: $i] : ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) | (complements_of_subsets(A!219, B!218) = empty_set) | (union_of_subsets(A!219, complements_of_subsets(A!219, complements_of_subsets(A!219, B!218))) = subset_difference(A!219, cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))) | (~element(complements_of_subsets(A!219, B!218), powerset(powerset(A!219))))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[73, 72])).
% 1.54/1.71 tff(75,plain,
% 1.54/1.71 (union_of_subsets(A!219, complements_of_subsets(A!219, complements_of_subsets(A!219, B!218))) = subset_difference(A!219, cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))),
% 1.54/1.71 inference(unit_resolution,[status(thm)],[74, 71, 61, 40])).
% 1.54/1.71 tff(76,plain,
% 1.54/1.71 (subset_difference(A!219, cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))) = union_of_subsets(A!219, complements_of_subsets(A!219, complements_of_subsets(A!219, B!218)))),
% 1.54/1.71 inference(symmetry,[status(thm)],[75])).
% 1.54/1.71 tff(77,plain,
% 1.54/1.71 (^[A: $i, B: $i] : refl(((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A))) <=> ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A))))),
% 1.54/1.71 inference(bind,[status(th)],[])).
% 1.54/1.71 tff(78,plain,
% 1.54/1.71 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))),
% 1.54/1.71 inference(quant_intro,[status(thm)],[77])).
% 1.54/1.71 tff(79,plain,
% 1.54/1.71 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))),
% 1.54/1.71 inference(rewrite,[status(thm)],[])).
% 1.54/1.71 tff(80,plain,
% 1.54/1.71 (^[A: $i, B: $i] : rewrite((element(B, powerset(powerset(A))) => element(meet_of_subsets(A, B), powerset(A))) <=> ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A))))),
% 1.54/1.71 inference(bind,[status(th)],[])).
% 1.54/1.71 tff(81,plain,
% 1.54/1.71 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => element(meet_of_subsets(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))),
% 1.54/1.71 inference(quant_intro,[status(thm)],[80])).
% 1.54/1.71 tff(82,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => element(meet_of_subsets(A, B), powerset(A)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','dt_k6_setfam_1')).
% 1.54/1.71 tff(83,plain,
% 1.54/1.71 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[82, 81])).
% 1.54/1.71 tff(84,plain,
% 1.54/1.71 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[83, 79])).
% 1.54/1.71 tff(85,plain,(
% 1.54/1.71 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))),
% 1.54/1.71 inference(skolemize,[status(sab)],[84])).
% 1.54/1.71 tff(86,plain,
% 1.54/1.71 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[85, 78])).
% 1.54/1.71 tff(87,plain,
% 1.54/1.71 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))) | ((~element(complements_of_subsets(A!219, B!218), powerset(powerset(A!219)))) | element(meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)), powerset(A!219)))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))) | (~element(complements_of_subsets(A!219, B!218), powerset(powerset(A!219)))) | element(meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)), powerset(A!219)))),
% 1.54/1.71 inference(rewrite,[status(thm)],[])).
% 1.54/1.71 tff(88,plain,
% 1.54/1.71 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))) | ((~element(complements_of_subsets(A!219, B!218), powerset(powerset(A!219)))) | element(meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)), powerset(A!219)))),
% 1.54/1.71 inference(quant_inst,[status(thm)],[])).
% 1.54/1.71 tff(89,plain,
% 1.54/1.71 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))) | (~element(complements_of_subsets(A!219, B!218), powerset(powerset(A!219)))) | element(meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)), powerset(A!219))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[88, 87])).
% 1.54/1.71 tff(90,plain,
% 1.54/1.71 (element(meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)), powerset(A!219))),
% 1.54/1.71 inference(unit_resolution,[status(thm)],[89, 86, 40])).
% 1.54/1.71 tff(91,plain,
% 1.54/1.71 (^[A: $i] : refl(element(cast_to_subset(A), powerset(A)) <=> element(cast_to_subset(A), powerset(A)))),
% 1.54/1.71 inference(bind,[status(th)],[])).
% 1.54/1.71 tff(92,plain,
% 1.54/1.71 (![A: $i] : element(cast_to_subset(A), powerset(A)) <=> ![A: $i] : element(cast_to_subset(A), powerset(A))),
% 1.54/1.71 inference(quant_intro,[status(thm)],[91])).
% 1.54/1.71 tff(93,plain,
% 1.54/1.71 (![A: $i] : element(cast_to_subset(A), powerset(A)) <=> ![A: $i] : element(cast_to_subset(A), powerset(A))),
% 1.54/1.71 inference(rewrite,[status(thm)],[])).
% 1.54/1.71 tff(94,axiom,(![A: $i] : element(cast_to_subset(A), powerset(A))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','dt_k2_subset_1')).
% 1.54/1.71 tff(95,plain,
% 1.54/1.71 (![A: $i] : element(cast_to_subset(A), powerset(A))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[94, 93])).
% 1.54/1.71 tff(96,plain,(
% 1.54/1.71 ![A: $i] : element(cast_to_subset(A), powerset(A))),
% 1.54/1.71 inference(skolemize,[status(sab)],[95])).
% 1.54/1.71 tff(97,plain,
% 1.54/1.71 (![A: $i] : element(cast_to_subset(A), powerset(A))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[96, 92])).
% 1.54/1.71 tff(98,plain,
% 1.54/1.71 ((~![A: $i] : element(cast_to_subset(A), powerset(A))) | element(cast_to_subset(A!219), powerset(A!219))),
% 1.54/1.71 inference(quant_inst,[status(thm)],[])).
% 1.54/1.71 tff(99,plain,
% 1.54/1.71 (element(cast_to_subset(A!219), powerset(A!219))),
% 1.54/1.71 inference(unit_resolution,[status(thm)],[98, 97])).
% 1.54/1.71 tff(100,plain,
% 1.54/1.71 (^[A: $i, B: $i, C: $i] : refl(((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A)))) <=> ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A)))))),
% 1.54/1.71 inference(bind,[status(th)],[])).
% 1.54/1.71 tff(101,plain,
% 1.54/1.71 (![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A)))) <=> ![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))),
% 1.54/1.71 inference(quant_intro,[status(thm)],[100])).
% 1.54/1.71 tff(102,plain,
% 1.54/1.71 (^[A: $i, B: $i, C: $i] : trans(monotonicity(trans(monotonicity(rewrite((element(B, powerset(A)) & element(C, powerset(A))) <=> (~((~element(C, powerset(A))) | (~element(B, powerset(A)))))), ((~(element(B, powerset(A)) & element(C, powerset(A)))) <=> (~(~((~element(C, powerset(A))) | (~element(B, powerset(A)))))))), rewrite((~(~((~element(C, powerset(A))) | (~element(B, powerset(A)))))) <=> ((~element(C, powerset(A))) | (~element(B, powerset(A))))), ((~(element(B, powerset(A)) & element(C, powerset(A)))) <=> ((~element(C, powerset(A))) | (~element(B, powerset(A)))))), (((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C))) <=> (((~element(C, powerset(A))) | (~element(B, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C))))), rewrite((((~element(C, powerset(A))) | (~element(B, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C))) <=> ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))), (((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C))) <=> ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))))),
% 1.54/1.71 inference(bind,[status(th)],[])).
% 1.54/1.71 tff(103,plain,
% 1.54/1.71 (![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C))) <=> ![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))),
% 1.54/1.71 inference(quant_intro,[status(thm)],[102])).
% 1.54/1.71 tff(104,plain,
% 1.54/1.71 (![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C))) <=> ![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C)))),
% 1.54/1.71 inference(rewrite,[status(thm)],[])).
% 1.54/1.71 tff(105,plain,
% 1.54/1.71 (^[A: $i, B: $i, C: $i] : rewrite(((element(B, powerset(A)) & element(C, powerset(A))) => (subset_difference(A, B, C) = set_difference(B, C))) <=> ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C))))),
% 1.54/1.71 inference(bind,[status(th)],[])).
% 1.54/1.71 tff(106,plain,
% 1.54/1.71 (![A: $i, B: $i, C: $i] : ((element(B, powerset(A)) & element(C, powerset(A))) => (subset_difference(A, B, C) = set_difference(B, C))) <=> ![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C)))),
% 1.54/1.71 inference(quant_intro,[status(thm)],[105])).
% 1.54/1.71 tff(107,axiom,(![A: $i, B: $i, C: $i] : ((element(B, powerset(A)) & element(C, powerset(A))) => (subset_difference(A, B, C) = set_difference(B, C)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','redefinition_k6_subset_1')).
% 1.54/1.71 tff(108,plain,
% 1.54/1.71 (![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C)))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[107, 106])).
% 1.54/1.71 tff(109,plain,
% 1.54/1.71 (![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C)))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[108, 104])).
% 1.54/1.71 tff(110,plain,(
% 1.54/1.71 ![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C)))),
% 1.54/1.71 inference(skolemize,[status(sab)],[109])).
% 1.54/1.71 tff(111,plain,
% 1.54/1.71 (![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[110, 103])).
% 1.54/1.71 tff(112,plain,
% 1.54/1.71 (![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))),
% 1.54/1.71 inference(modus_ponens,[status(thm)],[111, 101])).
% 1.54/1.71 tff(113,plain,
% 1.54/1.71 (((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | ((~element(cast_to_subset(A!219), powerset(A!219))) | (~element(meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)), powerset(A!219))) | (subset_difference(A!219, cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))) = set_difference(cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))))) <=> ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | (~element(cast_to_subset(A!219), powerset(A!219))) | (~element(meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)), powerset(A!219))) | (subset_difference(A!219, cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))) = set_difference(cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))))),
% 1.54/1.71 inference(rewrite,[status(thm)],[])).
% 1.54/1.71 tff(114,plain,
% 1.54/1.71 (((subset_difference(A!219, cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))) = set_difference(cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))) | (~element(meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)), powerset(A!219))) | (~element(cast_to_subset(A!219), powerset(A!219)))) <=> ((~element(cast_to_subset(A!219), powerset(A!219))) | (~element(meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)), powerset(A!219))) | (subset_difference(A!219, cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))) = set_difference(cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))))),
% 1.54/1.71 inference(rewrite,[status(thm)],[])).
% 1.54/1.71 tff(115,plain,
% 1.54/1.71 (((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | ((subset_difference(A!219, cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))) = set_difference(cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))) | (~element(meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)), powerset(A!219))) | (~element(cast_to_subset(A!219), powerset(A!219))))) <=> ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | ((~element(cast_to_subset(A!219), powerset(A!219))) | (~element(meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)), powerset(A!219))) | (subset_difference(A!219, cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))) = set_difference(cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))))))),
% 1.54/1.71 inference(monotonicity,[status(thm)],[114])).
% 1.54/1.71 tff(116,plain,
% 1.54/1.71 (((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | ((subset_difference(A!219, cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))) = set_difference(cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))) | (~element(meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)), powerset(A!219))) | (~element(cast_to_subset(A!219), powerset(A!219))))) <=> ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | (~element(cast_to_subset(A!219), powerset(A!219))) | (~element(meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)), powerset(A!219))) | (subset_difference(A!219, cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))) = set_difference(cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))))),
% 1.54/1.71 inference(transitivity,[status(thm)],[115, 113])).
% 1.54/1.71 tff(117,plain,
% 1.54/1.71 ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | ((subset_difference(A!219, cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))) = set_difference(cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))) | (~element(meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)), powerset(A!219))) | (~element(cast_to_subset(A!219), powerset(A!219))))),
% 1.54/1.72 inference(quant_inst,[status(thm)],[])).
% 1.54/1.72 tff(118,plain,
% 1.54/1.72 ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | (~element(cast_to_subset(A!219), powerset(A!219))) | (~element(meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)), powerset(A!219))) | (subset_difference(A!219, cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))) = set_difference(cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))))),
% 1.54/1.72 inference(modus_ponens,[status(thm)],[117, 116])).
% 1.54/1.72 tff(119,plain,
% 1.54/1.72 (subset_difference(A!219, cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))) = set_difference(cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))),
% 1.54/1.72 inference(unit_resolution,[status(thm)],[118, 112, 99, 90])).
% 1.54/1.72 tff(120,plain,
% 1.54/1.72 (set_difference(cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))) = subset_difference(A!219, cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))),
% 1.54/1.72 inference(symmetry,[status(thm)],[119])).
% 1.54/1.72 tff(121,plain,
% 1.54/1.72 (^[A: $i] : refl((cast_to_subset(A) = A) <=> (cast_to_subset(A) = A))),
% 1.54/1.72 inference(bind,[status(th)],[])).
% 1.54/1.72 tff(122,plain,
% 1.54/1.72 (![A: $i] : (cast_to_subset(A) = A) <=> ![A: $i] : (cast_to_subset(A) = A)),
% 1.54/1.72 inference(quant_intro,[status(thm)],[121])).
% 1.54/1.72 tff(123,plain,
% 1.54/1.72 (![A: $i] : (cast_to_subset(A) = A) <=> ![A: $i] : (cast_to_subset(A) = A)),
% 1.54/1.72 inference(rewrite,[status(thm)],[])).
% 1.54/1.72 tff(124,axiom,(![A: $i] : (cast_to_subset(A) = A)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','d4_subset_1')).
% 1.54/1.72 tff(125,plain,
% 1.54/1.72 (![A: $i] : (cast_to_subset(A) = A)),
% 1.54/1.72 inference(modus_ponens,[status(thm)],[124, 123])).
% 1.54/1.72 tff(126,plain,(
% 1.54/1.72 ![A: $i] : (cast_to_subset(A) = A)),
% 1.54/1.72 inference(skolemize,[status(sab)],[125])).
% 1.54/1.72 tff(127,plain,
% 1.54/1.72 (![A: $i] : (cast_to_subset(A) = A)),
% 1.54/1.72 inference(modus_ponens,[status(thm)],[126, 122])).
% 1.54/1.72 tff(128,plain,
% 1.54/1.72 ((~![A: $i] : (cast_to_subset(A) = A)) | (cast_to_subset(A!219) = A!219)),
% 1.54/1.72 inference(quant_inst,[status(thm)],[])).
% 1.54/1.72 tff(129,plain,
% 1.54/1.72 (cast_to_subset(A!219) = A!219),
% 1.54/1.72 inference(unit_resolution,[status(thm)],[128, 127])).
% 1.54/1.72 tff(130,plain,
% 1.54/1.72 (A!219 = cast_to_subset(A!219)),
% 1.54/1.72 inference(symmetry,[status(thm)],[129])).
% 1.54/1.72 tff(131,plain,
% 1.54/1.72 (set_difference(A!219, meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))) = set_difference(cast_to_subset(A!219), meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))),
% 1.54/1.72 inference(monotonicity,[status(thm)],[130])).
% 1.54/1.72 tff(132,plain,
% 1.54/1.72 (^[A: $i, B: $i] : refl(((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B))) <=> ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B))))),
% 1.54/1.72 inference(bind,[status(th)],[])).
% 1.54/1.72 tff(133,plain,
% 1.54/1.72 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B))) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 1.54/1.72 inference(quant_intro,[status(thm)],[132])).
% 1.54/1.72 tff(134,plain,
% 1.54/1.72 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B))) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 1.54/1.72 inference(rewrite,[status(thm)],[])).
% 1.54/1.72 tff(135,plain,
% 1.54/1.72 (^[A: $i, B: $i] : rewrite((element(B, powerset(A)) => (subset_complement(A, B) = set_difference(A, B))) <=> ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B))))),
% 1.54/1.72 inference(bind,[status(th)],[])).
% 1.54/1.72 tff(136,plain,
% 1.54/1.72 (![A: $i, B: $i] : (element(B, powerset(A)) => (subset_complement(A, B) = set_difference(A, B))) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 1.54/1.72 inference(quant_intro,[status(thm)],[135])).
% 1.54/1.72 tff(137,axiom,(![A: $i, B: $i] : (element(B, powerset(A)) => (subset_complement(A, B) = set_difference(A, B)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','d5_subset_1')).
% 1.54/1.72 tff(138,plain,
% 1.54/1.72 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 1.54/1.72 inference(modus_ponens,[status(thm)],[137, 136])).
% 1.54/1.72 tff(139,plain,
% 1.54/1.72 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 1.54/1.72 inference(modus_ponens,[status(thm)],[138, 134])).
% 1.54/1.72 tff(140,plain,(
% 1.54/1.72 ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 1.54/1.72 inference(skolemize,[status(sab)],[139])).
% 1.54/1.72 tff(141,plain,
% 1.54/1.72 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 1.54/1.72 inference(modus_ponens,[status(thm)],[140, 133])).
% 1.54/1.72 tff(142,plain,
% 1.54/1.72 (((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | ((~element(meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)), powerset(A!219))) | (subset_complement(A!219, meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))) = set_difference(A!219, meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | (~element(meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)), powerset(A!219))) | (subset_complement(A!219, meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))) = set_difference(A!219, meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))))),
% 1.54/1.72 inference(rewrite,[status(thm)],[])).
% 1.54/1.72 tff(143,plain,
% 1.54/1.72 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | ((~element(meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)), powerset(A!219))) | (subset_complement(A!219, meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))) = set_difference(A!219, meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))))),
% 1.54/1.72 inference(quant_inst,[status(thm)],[])).
% 1.54/1.72 tff(144,plain,
% 1.54/1.72 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | (~element(meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)), powerset(A!219))) | (subset_complement(A!219, meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))) = set_difference(A!219, meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))))),
% 1.54/1.72 inference(modus_ponens,[status(thm)],[143, 142])).
% 1.54/1.72 tff(145,plain,
% 1.54/1.72 (subset_complement(A!219, meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))) = set_difference(A!219, meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))),
% 1.54/1.72 inference(unit_resolution,[status(thm)],[144, 141, 90])).
% 1.54/1.72 tff(146,plain,
% 1.54/1.72 (subset_complement(A!219, meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))) = union_of_subsets(A!219, B!218)),
% 1.54/1.72 inference(transitivity,[status(thm)],[145, 131, 120, 76, 26])).
% 1.54/1.72 tff(147,plain,
% 1.54/1.72 (subset_complement(A!219, subset_complement(A!219, meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))) = subset_complement(A!219, union_of_subsets(A!219, B!218))),
% 1.54/1.72 inference(monotonicity,[status(thm)],[146])).
% 1.54/1.72 tff(148,plain,
% 1.54/1.72 (^[A: $i, B: $i] : refl(((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B)) <=> ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B)))),
% 1.54/1.72 inference(bind,[status(th)],[])).
% 1.54/1.72 tff(149,plain,
% 1.54/1.72 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B)) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 1.54/1.72 inference(quant_intro,[status(thm)],[148])).
% 1.54/1.72 tff(150,plain,
% 1.54/1.72 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B)) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 1.54/1.72 inference(rewrite,[status(thm)],[])).
% 1.54/1.72 tff(151,plain,
% 1.54/1.72 (^[A: $i, B: $i] : rewrite((element(B, powerset(A)) => (subset_complement(A, subset_complement(A, B)) = B)) <=> ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B)))),
% 1.54/1.72 inference(bind,[status(th)],[])).
% 1.54/1.72 tff(152,plain,
% 1.54/1.72 (![A: $i, B: $i] : (element(B, powerset(A)) => (subset_complement(A, subset_complement(A, B)) = B)) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 1.54/1.72 inference(quant_intro,[status(thm)],[151])).
% 1.54/1.72 tff(153,axiom,(![A: $i, B: $i] : (element(B, powerset(A)) => (subset_complement(A, subset_complement(A, B)) = B))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','involutiveness_k3_subset_1')).
% 1.54/1.72 tff(154,plain,
% 1.54/1.72 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 1.54/1.72 inference(modus_ponens,[status(thm)],[153, 152])).
% 1.54/1.72 tff(155,plain,
% 1.54/1.72 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 1.54/1.72 inference(modus_ponens,[status(thm)],[154, 150])).
% 1.54/1.72 tff(156,plain,(
% 1.54/1.72 ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 1.54/1.72 inference(skolemize,[status(sab)],[155])).
% 1.54/1.72 tff(157,plain,
% 1.54/1.72 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 1.54/1.72 inference(modus_ponens,[status(thm)],[156, 149])).
% 1.54/1.72 tff(158,plain,
% 1.54/1.72 (((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))) | ((~element(meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)), powerset(A!219))) | (subset_complement(A!219, subset_complement(A!219, meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))) = meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))) | (~element(meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)), powerset(A!219))) | (subset_complement(A!219, subset_complement(A!219, meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))) = meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))))),
% 1.54/1.72 inference(rewrite,[status(thm)],[])).
% 1.54/1.72 tff(159,plain,
% 1.54/1.72 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))) | ((~element(meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)), powerset(A!219))) | (subset_complement(A!219, subset_complement(A!219, meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))) = meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))))),
% 1.54/1.72 inference(quant_inst,[status(thm)],[])).
% 1.54/1.72 tff(160,plain,
% 1.54/1.72 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))) | (~element(meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)), powerset(A!219))) | (subset_complement(A!219, subset_complement(A!219, meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))) = meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))),
% 1.54/1.72 inference(modus_ponens,[status(thm)],[159, 158])).
% 1.54/1.72 tff(161,plain,
% 1.54/1.72 (subset_complement(A!219, subset_complement(A!219, meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)))) = meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))),
% 1.54/1.72 inference(unit_resolution,[status(thm)],[160, 157, 90])).
% 1.54/1.72 tff(162,plain,
% 1.54/1.72 (meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)) = subset_complement(A!219, subset_complement(A!219, meet_of_subsets(A!219, complements_of_subsets(A!219, B!218))))),
% 1.54/1.72 inference(symmetry,[status(thm)],[161])).
% 1.54/1.72 tff(163,plain,
% 1.54/1.72 (meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)) = subset_complement(A!219, union_of_subsets(A!219, B!218))),
% 1.54/1.72 inference(transitivity,[status(thm)],[162, 147])).
% 1.54/1.72 tff(164,plain,
% 1.54/1.72 (~(meet_of_subsets(A!219, complements_of_subsets(A!219, B!218)) = subset_complement(A!219, union_of_subsets(A!219, B!218)))),
% 1.54/1.74 inference(or_elim,[status(thm)],[10])).
% 1.54/1.74 tff(165,plain,
% 1.54/1.74 ($false),
% 1.54/1.74 inference(unit_resolution,[status(thm)],[164, 163])).
% 1.54/1.74 % SZS output end Proof
% 1.60/1.75 % E exiting
%------------------------------------------------------------------------------