%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SEU176+2 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n002.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 05:27:09 AM UTC 2025
% Result : Theorem 0.21s 0.59s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : SEU176+2 : TPTP v9.0.0. Released v3.3.0.
% 0.07/0.12 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n002.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Jun 20 06:14:31 EDT 2025
% 0.13/0.34 % CPUTime :
% 0.21/0.59 % SZS status Theorem
% 0.21/0.59 % SZS output start Proof
% 0.21/0.59 tff(subset_difference_type, type, (
% 0.21/0.59 subset_difference: ( $i * $i * $i ) > $i)).
% 0.21/0.59 tff(meet_of_subsets_type, type, (
% 0.21/0.59 meet_of_subsets: ( $i * $i ) > $i)).
% 0.21/0.59 tff(tptp_fun_B_28_type, type, (
% 0.21/0.59 tptp_fun_B_28: $i)).
% 0.21/0.59 tff(tptp_fun_A_29_type, type, (
% 0.21/0.59 tptp_fun_A_29: $i)).
% 0.21/0.59 tff(cast_to_subset_type, type, (
% 0.21/0.59 cast_to_subset: $i > $i)).
% 0.21/0.59 tff(union_of_subsets_type, type, (
% 0.21/0.59 union_of_subsets: ( $i * $i ) > $i)).
% 0.21/0.59 tff(complements_of_subsets_type, type, (
% 0.21/0.59 complements_of_subsets: ( $i * $i ) > $i)).
% 0.21/0.59 tff(set_difference_type, type, (
% 0.21/0.59 set_difference: ( $i * $i ) > $i)).
% 0.21/0.59 tff(element_type, type, (
% 0.21/0.59 element: ( $i * $i ) > $o)).
% 0.21/0.59 tff(powerset_type, type, (
% 0.21/0.59 powerset: $i > $i)).
% 0.21/0.59 tff(empty_set_type, type, (
% 0.21/0.59 empty_set: $i)).
% 0.21/0.59 tff(set_meet_type, type, (
% 0.21/0.59 set_meet: $i > $i)).
% 0.21/0.59 tff(subset_complement_type, type, (
% 0.21/0.59 subset_complement: ( $i * $i ) > $i)).
% 0.21/0.59 tff(union_type, type, (
% 0.21/0.59 union: $i > $i)).
% 0.21/0.59 tff(1,plain,
% 0.21/0.59 (^[A: $i] : refl(element(cast_to_subset(A), powerset(A)) <=> element(cast_to_subset(A), powerset(A)))),
% 0.21/0.59 inference(bind,[status(th)],[])).
% 0.21/0.59 tff(2,plain,
% 0.21/0.59 (![A: $i] : element(cast_to_subset(A), powerset(A)) <=> ![A: $i] : element(cast_to_subset(A), powerset(A))),
% 0.21/0.59 inference(quant_intro,[status(thm)],[1])).
% 0.21/0.59 tff(3,plain,
% 0.21/0.59 (![A: $i] : element(cast_to_subset(A), powerset(A)) <=> ![A: $i] : element(cast_to_subset(A), powerset(A))),
% 0.21/0.59 inference(rewrite,[status(thm)],[])).
% 0.21/0.59 tff(4,axiom,(![A: $i] : element(cast_to_subset(A), powerset(A))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','dt_k2_subset_1')).
% 0.21/0.59 tff(5,plain,
% 0.21/0.59 (![A: $i] : element(cast_to_subset(A), powerset(A))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.21/0.59 tff(6,plain,(
% 0.21/0.59 ![A: $i] : element(cast_to_subset(A), powerset(A))),
% 0.21/0.59 inference(skolemize,[status(sab)],[5])).
% 0.21/0.59 tff(7,plain,
% 0.21/0.59 (![A: $i] : element(cast_to_subset(A), powerset(A))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.21/0.59 tff(8,plain,
% 0.21/0.59 ((~![A: $i] : element(cast_to_subset(A), powerset(A))) | element(cast_to_subset(A!29), powerset(A!29))),
% 0.21/0.59 inference(quant_inst,[status(thm)],[])).
% 0.21/0.59 tff(9,plain,
% 0.21/0.59 (element(cast_to_subset(A!29), powerset(A!29))),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.21/0.59 tff(10,plain,
% 0.21/0.59 ((~((B!28 = empty_set) | (union_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28))) | (~element(B!28, powerset(powerset(A!29)))))) <=> (~((B!28 = empty_set) | (union_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28))) | (~element(B!28, powerset(powerset(A!29))))))),
% 0.21/0.59 inference(rewrite,[status(thm)],[])).
% 0.21/0.59 tff(11,plain,
% 0.21/0.59 ((~![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))))))),
% 0.21/0.59 inference(rewrite,[status(thm)],[])).
% 0.21/0.59 tff(12,plain,
% 0.21/0.59 ((~![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))))))),
% 0.21/0.59 inference(rewrite,[status(thm)],[])).
% 0.21/0.59 tff(13,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')).
% 0.21/0.59 tff(14,plain,
% 0.21/0.59 (~![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)))))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[13, 12])).
% 0.21/0.59 tff(15,plain,
% 0.21/0.59 (~![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)))))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[14, 11])).
% 0.21/0.59 tff(16,plain,
% 0.21/0.59 (~![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)))))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[15, 11])).
% 0.21/0.59 tff(17,plain,
% 0.21/0.59 (~![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)))))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[16, 11])).
% 0.21/0.59 tff(18,plain,(
% 0.21/0.59 ~((B!28 = empty_set) | (union_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28))) | (~element(B!28, powerset(powerset(A!29)))))),
% 0.21/0.59 inference(skolemize,[status(sab)],[17])).
% 0.21/0.59 tff(19,plain,
% 0.21/0.59 (~((B!28 = empty_set) | (union_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28))) | (~element(B!28, powerset(powerset(A!29)))))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[18, 10])).
% 0.21/0.59 tff(20,plain,
% 0.21/0.59 (element(B!28, powerset(powerset(A!29)))),
% 0.21/0.59 inference(or_elim,[status(thm)],[19])).
% 0.21/0.59 tff(21,plain,
% 0.21/0.59 (^[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))))),
% 0.21/0.59 inference(bind,[status(th)],[])).
% 0.21/0.59 tff(22,plain,
% 0.21/0.59 (![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)))),
% 0.21/0.59 inference(quant_intro,[status(thm)],[21])).
% 0.21/0.59 tff(23,plain,
% 0.21/0.59 (![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)))),
% 0.21/0.59 inference(rewrite,[status(thm)],[])).
% 0.21/0.59 tff(24,plain,
% 0.21/0.59 (^[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))))),
% 0.21/0.59 inference(bind,[status(th)],[])).
% 0.21/0.59 tff(25,plain,
% 0.21/0.59 (![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)))),
% 0.21/0.59 inference(quant_intro,[status(thm)],[24])).
% 0.21/0.59 tff(26,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')).
% 0.21/0.59 tff(27,plain,
% 0.21/0.59 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[26, 25])).
% 0.21/0.59 tff(28,plain,
% 0.21/0.59 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[27, 23])).
% 0.21/0.59 tff(29,plain,(
% 0.21/0.59 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))),
% 0.21/0.59 inference(skolemize,[status(sab)],[28])).
% 0.21/0.59 tff(30,plain,
% 0.21/0.59 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[29, 22])).
% 0.21/0.59 tff(31,plain,
% 0.21/0.59 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))) | ((~element(B!28, powerset(powerset(A!29)))) | element(meet_of_subsets(A!29, B!28), powerset(A!29)))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))) | (~element(B!28, powerset(powerset(A!29)))) | element(meet_of_subsets(A!29, B!28), powerset(A!29)))),
% 0.21/0.59 inference(rewrite,[status(thm)],[])).
% 0.21/0.59 tff(32,plain,
% 0.21/0.59 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))) | ((~element(B!28, powerset(powerset(A!29)))) | element(meet_of_subsets(A!29, B!28), powerset(A!29)))),
% 0.21/0.59 inference(quant_inst,[status(thm)],[])).
% 0.21/0.59 tff(33,plain,
% 0.21/0.59 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))) | (~element(B!28, powerset(powerset(A!29)))) | element(meet_of_subsets(A!29, B!28), powerset(A!29))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[32, 31])).
% 0.21/0.59 tff(34,plain,
% 0.21/0.59 (element(meet_of_subsets(A!29, B!28), powerset(A!29))),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[33, 30, 20])).
% 0.21/0.59 tff(35,plain,
% 0.21/0.59 (^[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)))))),
% 0.21/0.59 inference(bind,[status(th)],[])).
% 0.21/0.59 tff(36,plain,
% 0.21/0.59 (![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))))),
% 0.21/0.59 inference(quant_intro,[status(thm)],[35])).
% 0.21/0.59 tff(37,plain,
% 0.21/0.59 (^[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))))))),
% 0.21/0.59 inference(bind,[status(th)],[])).
% 0.21/0.59 tff(38,plain,
% 0.21/0.59 (![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))))),
% 0.21/0.59 inference(quant_intro,[status(thm)],[37])).
% 0.21/0.59 tff(39,plain,
% 0.21/0.59 (![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)))),
% 0.21/0.59 inference(rewrite,[status(thm)],[])).
% 0.21/0.59 tff(40,plain,
% 0.21/0.59 (^[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))))),
% 0.21/0.59 inference(bind,[status(th)],[])).
% 0.21/0.59 tff(41,plain,
% 0.21/0.59 (![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)))),
% 0.21/0.59 inference(quant_intro,[status(thm)],[40])).
% 0.21/0.59 tff(42,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')).
% 0.21/0.59 tff(43,plain,
% 0.21/0.59 (![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C)))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[42, 41])).
% 0.21/0.59 tff(44,plain,
% 0.21/0.59 (![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C)))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[43, 39])).
% 0.21/0.59 tff(45,plain,(
% 0.21/0.59 ![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C)))),
% 0.21/0.59 inference(skolemize,[status(sab)],[44])).
% 0.21/0.59 tff(46,plain,
% 0.21/0.59 (![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[45, 38])).
% 0.21/0.59 tff(47,plain,
% 0.21/0.59 (![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[46, 36])).
% 0.21/0.59 tff(48,plain,
% 0.21/0.59 (((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | ((~element(meet_of_subsets(A!29, B!28), powerset(A!29))) | (~element(cast_to_subset(A!29), powerset(A!29))) | (subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28)) = set_difference(cast_to_subset(A!29), meet_of_subsets(A!29, B!28))))) <=> ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | (~element(meet_of_subsets(A!29, B!28), powerset(A!29))) | (~element(cast_to_subset(A!29), powerset(A!29))) | (subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28)) = set_difference(cast_to_subset(A!29), meet_of_subsets(A!29, B!28))))),
% 0.21/0.59 inference(rewrite,[status(thm)],[])).
% 0.21/0.59 tff(49,plain,
% 0.21/0.59 (((subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28)) = set_difference(cast_to_subset(A!29), meet_of_subsets(A!29, B!28))) | (~element(meet_of_subsets(A!29, B!28), powerset(A!29))) | (~element(cast_to_subset(A!29), powerset(A!29)))) <=> ((~element(meet_of_subsets(A!29, B!28), powerset(A!29))) | (~element(cast_to_subset(A!29), powerset(A!29))) | (subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28)) = set_difference(cast_to_subset(A!29), meet_of_subsets(A!29, B!28))))),
% 0.21/0.59 inference(rewrite,[status(thm)],[])).
% 0.21/0.59 tff(50,plain,
% 0.21/0.59 (((~![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!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28)) = set_difference(cast_to_subset(A!29), meet_of_subsets(A!29, B!28))) | (~element(meet_of_subsets(A!29, B!28), powerset(A!29))) | (~element(cast_to_subset(A!29), powerset(A!29))))) <=> ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | ((~element(meet_of_subsets(A!29, B!28), powerset(A!29))) | (~element(cast_to_subset(A!29), powerset(A!29))) | (subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28)) = set_difference(cast_to_subset(A!29), meet_of_subsets(A!29, B!28)))))),
% 0.21/0.59 inference(monotonicity,[status(thm)],[49])).
% 0.21/0.59 tff(51,plain,
% 0.21/0.59 (((~![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!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28)) = set_difference(cast_to_subset(A!29), meet_of_subsets(A!29, B!28))) | (~element(meet_of_subsets(A!29, B!28), powerset(A!29))) | (~element(cast_to_subset(A!29), powerset(A!29))))) <=> ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | (~element(meet_of_subsets(A!29, B!28), powerset(A!29))) | (~element(cast_to_subset(A!29), powerset(A!29))) | (subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28)) = set_difference(cast_to_subset(A!29), meet_of_subsets(A!29, B!28))))),
% 0.21/0.59 inference(transitivity,[status(thm)],[50, 48])).
% 0.21/0.59 tff(52,plain,
% 0.21/0.59 ((~![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!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28)) = set_difference(cast_to_subset(A!29), meet_of_subsets(A!29, B!28))) | (~element(meet_of_subsets(A!29, B!28), powerset(A!29))) | (~element(cast_to_subset(A!29), powerset(A!29))))),
% 0.21/0.59 inference(quant_inst,[status(thm)],[])).
% 0.21/0.59 tff(53,plain,
% 0.21/0.59 ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | (~element(meet_of_subsets(A!29, B!28), powerset(A!29))) | (~element(cast_to_subset(A!29), powerset(A!29))) | (subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28)) = set_difference(cast_to_subset(A!29), meet_of_subsets(A!29, B!28)))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[52, 51])).
% 0.21/0.59 tff(54,plain,
% 0.21/0.59 (subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28)) = set_difference(cast_to_subset(A!29), meet_of_subsets(A!29, B!28))),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[53, 47, 34, 9])).
% 0.21/0.59 tff(55,plain,
% 0.21/0.59 (set_difference(cast_to_subset(A!29), meet_of_subsets(A!29, B!28)) = subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28))),
% 0.21/0.59 inference(symmetry,[status(thm)],[54])).
% 0.21/0.59 tff(56,plain,
% 0.21/0.59 (^[A: $i, B: $i] : refl(((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B))) <=> ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B))))),
% 0.21/0.59 inference(bind,[status(th)],[])).
% 0.21/0.59 tff(57,plain,
% 0.21/0.59 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B)))),
% 0.21/0.59 inference(quant_intro,[status(thm)],[56])).
% 0.21/0.59 tff(58,plain,
% 0.21/0.59 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B)))),
% 0.21/0.59 inference(rewrite,[status(thm)],[])).
% 0.21/0.59 tff(59,plain,
% 0.21/0.59 (^[A: $i, B: $i] : rewrite((element(B, powerset(powerset(A))) => (meet_of_subsets(A, B) = set_meet(B))) <=> ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B))))),
% 0.21/0.59 inference(bind,[status(th)],[])).
% 0.21/0.59 tff(60,plain,
% 0.21/0.59 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => (meet_of_subsets(A, B) = set_meet(B))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B)))),
% 0.21/0.59 inference(quant_intro,[status(thm)],[59])).
% 0.21/0.59 tff(61,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => (meet_of_subsets(A, B) = set_meet(B)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','redefinition_k6_setfam_1')).
% 0.21/0.59 tff(62,plain,
% 0.21/0.59 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B)))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[61, 60])).
% 0.21/0.59 tff(63,plain,
% 0.21/0.59 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B)))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[62, 58])).
% 0.21/0.59 tff(64,plain,(
% 0.21/0.59 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B)))),
% 0.21/0.59 inference(skolemize,[status(sab)],[63])).
% 0.21/0.59 tff(65,plain,
% 0.21/0.59 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B)))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[64, 57])).
% 0.21/0.59 tff(66,plain,
% 0.21/0.59 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B)))) | ((~element(B!28, powerset(powerset(A!29)))) | (meet_of_subsets(A!29, B!28) = set_meet(B!28)))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B)))) | (~element(B!28, powerset(powerset(A!29)))) | (meet_of_subsets(A!29, B!28) = set_meet(B!28)))),
% 0.21/0.59 inference(rewrite,[status(thm)],[])).
% 0.21/0.59 tff(67,plain,
% 0.21/0.59 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B)))) | ((~element(B!28, powerset(powerset(A!29)))) | (meet_of_subsets(A!29, B!28) = set_meet(B!28)))),
% 0.21/0.59 inference(quant_inst,[status(thm)],[])).
% 0.21/0.59 tff(68,plain,
% 0.21/0.59 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B)))) | (~element(B!28, powerset(powerset(A!29)))) | (meet_of_subsets(A!29, B!28) = set_meet(B!28))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[67, 66])).
% 0.21/0.59 tff(69,plain,
% 0.21/0.59 (meet_of_subsets(A!29, B!28) = set_meet(B!28)),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[68, 65, 20])).
% 0.21/0.59 tff(70,plain,
% 0.21/0.59 (set_meet(B!28) = meet_of_subsets(A!29, B!28)),
% 0.21/0.59 inference(symmetry,[status(thm)],[69])).
% 0.21/0.59 tff(71,plain,
% 0.21/0.59 (^[A: $i] : refl((cast_to_subset(A) = A) <=> (cast_to_subset(A) = A))),
% 0.21/0.59 inference(bind,[status(th)],[])).
% 0.21/0.59 tff(72,plain,
% 0.21/0.59 (![A: $i] : (cast_to_subset(A) = A) <=> ![A: $i] : (cast_to_subset(A) = A)),
% 0.21/0.59 inference(quant_intro,[status(thm)],[71])).
% 0.21/0.59 tff(73,plain,
% 0.21/0.59 (![A: $i] : (cast_to_subset(A) = A) <=> ![A: $i] : (cast_to_subset(A) = A)),
% 0.21/0.59 inference(rewrite,[status(thm)],[])).
% 0.21/0.59 tff(74,axiom,(![A: $i] : (cast_to_subset(A) = A)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','d4_subset_1')).
% 0.21/0.59 tff(75,plain,
% 0.21/0.59 (![A: $i] : (cast_to_subset(A) = A)),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[74, 73])).
% 0.21/0.59 tff(76,plain,(
% 0.21/0.59 ![A: $i] : (cast_to_subset(A) = A)),
% 0.21/0.59 inference(skolemize,[status(sab)],[75])).
% 0.21/0.59 tff(77,plain,
% 0.21/0.59 (![A: $i] : (cast_to_subset(A) = A)),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[76, 72])).
% 0.21/0.59 tff(78,plain,
% 0.21/0.59 ((~![A: $i] : (cast_to_subset(A) = A)) | (cast_to_subset(A!29) = A!29)),
% 0.21/0.59 inference(quant_inst,[status(thm)],[])).
% 0.21/0.59 tff(79,plain,
% 0.21/0.59 (cast_to_subset(A!29) = A!29),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[78, 77])).
% 0.21/0.59 tff(80,plain,
% 0.21/0.59 (A!29 = cast_to_subset(A!29)),
% 0.21/0.59 inference(symmetry,[status(thm)],[79])).
% 0.21/0.59 tff(81,plain,
% 0.21/0.59 (set_difference(A!29, set_meet(B!28)) = set_difference(cast_to_subset(A!29), meet_of_subsets(A!29, B!28))),
% 0.21/0.59 inference(monotonicity,[status(thm)],[80, 70])).
% 0.21/0.59 tff(82,plain,
% 0.21/0.59 (element(set_meet(B!28), powerset(A!29)) <=> element(meet_of_subsets(A!29, B!28), powerset(A!29))),
% 0.21/0.59 inference(monotonicity,[status(thm)],[70])).
% 0.21/0.59 tff(83,plain,
% 0.21/0.59 (element(meet_of_subsets(A!29, B!28), powerset(A!29)) <=> element(set_meet(B!28), powerset(A!29))),
% 0.21/0.59 inference(symmetry,[status(thm)],[82])).
% 0.21/0.59 tff(84,plain,
% 0.21/0.59 (element(set_meet(B!28), powerset(A!29))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[34, 83])).
% 0.21/0.59 tff(85,plain,
% 0.21/0.59 (^[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))))),
% 0.21/0.59 inference(bind,[status(th)],[])).
% 0.21/0.59 tff(86,plain,
% 0.21/0.59 (![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)))),
% 0.21/0.59 inference(quant_intro,[status(thm)],[85])).
% 0.21/0.59 tff(87,plain,
% 0.21/0.59 (![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)))),
% 0.21/0.59 inference(rewrite,[status(thm)],[])).
% 0.21/0.59 tff(88,plain,
% 0.21/0.59 (^[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))))),
% 0.21/0.59 inference(bind,[status(th)],[])).
% 0.21/0.59 tff(89,plain,
% 0.21/0.59 (![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)))),
% 0.21/0.59 inference(quant_intro,[status(thm)],[88])).
% 0.21/0.59 tff(90,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')).
% 0.21/0.59 tff(91,plain,
% 0.21/0.59 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[90, 89])).
% 0.21/0.59 tff(92,plain,
% 0.21/0.59 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[91, 87])).
% 0.21/0.59 tff(93,plain,(
% 0.21/0.59 ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 0.21/0.59 inference(skolemize,[status(sab)],[92])).
% 0.21/0.59 tff(94,plain,
% 0.21/0.59 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[93, 86])).
% 0.21/0.59 tff(95,plain,
% 0.21/0.59 (((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | ((~element(set_meet(B!28), powerset(A!29))) | (subset_complement(A!29, set_meet(B!28)) = set_difference(A!29, set_meet(B!28))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | (~element(set_meet(B!28), powerset(A!29))) | (subset_complement(A!29, set_meet(B!28)) = set_difference(A!29, set_meet(B!28))))),
% 0.21/0.59 inference(rewrite,[status(thm)],[])).
% 0.21/0.59 tff(96,plain,
% 0.21/0.59 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | ((~element(set_meet(B!28), powerset(A!29))) | (subset_complement(A!29, set_meet(B!28)) = set_difference(A!29, set_meet(B!28))))),
% 0.21/0.59 inference(quant_inst,[status(thm)],[])).
% 0.21/0.59 tff(97,plain,
% 0.21/0.59 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | (~element(set_meet(B!28), powerset(A!29))) | (subset_complement(A!29, set_meet(B!28)) = set_difference(A!29, set_meet(B!28)))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[96, 95])).
% 0.21/0.59 tff(98,plain,
% 0.21/0.59 ((~element(set_meet(B!28), powerset(A!29))) | (subset_complement(A!29, set_meet(B!28)) = set_difference(A!29, set_meet(B!28)))),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[97, 94])).
% 0.21/0.59 tff(99,plain,
% 0.21/0.59 (subset_complement(A!29, set_meet(B!28)) = set_difference(A!29, set_meet(B!28))),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[98, 84])).
% 0.21/0.59 tff(100,plain,
% 0.21/0.59 (^[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)))),
% 0.21/0.59 inference(bind,[status(th)],[])).
% 0.21/0.59 tff(101,plain,
% 0.21/0.59 (![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))),
% 0.21/0.59 inference(quant_intro,[status(thm)],[100])).
% 0.21/0.59 tff(102,plain,
% 0.21/0.59 (![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))),
% 0.21/0.59 inference(rewrite,[status(thm)],[])).
% 0.21/0.59 tff(103,plain,
% 0.21/0.59 (^[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)))),
% 0.21/0.59 inference(bind,[status(th)],[])).
% 0.21/0.59 tff(104,plain,
% 0.21/0.59 (![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))),
% 0.21/0.59 inference(quant_intro,[status(thm)],[103])).
% 0.21/0.59 tff(105,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')).
% 0.21/0.60 tff(106,plain,
% 0.21/0.60 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[105, 104])).
% 0.21/0.60 tff(107,plain,
% 0.21/0.60 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[106, 102])).
% 0.21/0.60 tff(108,plain,(
% 0.21/0.60 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 0.21/0.60 inference(skolemize,[status(sab)],[107])).
% 0.21/0.60 tff(109,plain,
% 0.21/0.60 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[108, 101])).
% 0.21/0.60 tff(110,plain,
% 0.21/0.60 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | ((~element(B!28, powerset(powerset(A!29)))) | (complements_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = B!28))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | (~element(B!28, powerset(powerset(A!29)))) | (complements_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = B!28))),
% 0.21/0.60 inference(rewrite,[status(thm)],[])).
% 0.21/0.60 tff(111,plain,
% 0.21/0.60 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | ((~element(B!28, powerset(powerset(A!29)))) | (complements_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = B!28))),
% 0.21/0.60 inference(quant_inst,[status(thm)],[])).
% 0.21/0.60 tff(112,plain,
% 0.21/0.60 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | (~element(B!28, powerset(powerset(A!29)))) | (complements_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = B!28)),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[111, 110])).
% 0.21/0.60 tff(113,plain,
% 0.21/0.60 (complements_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = B!28),
% 0.21/0.60 inference(unit_resolution,[status(thm)],[112, 109, 20])).
% 0.21/0.60 tff(114,plain,
% 0.21/0.60 (meet_of_subsets(A!29, complements_of_subsets(A!29, complements_of_subsets(A!29, B!28))) = meet_of_subsets(A!29, B!28)),
% 0.21/0.60 inference(monotonicity,[status(thm)],[113])).
% 0.21/0.60 tff(115,plain,
% 0.21/0.60 (^[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)))))),
% 0.21/0.60 inference(bind,[status(th)],[])).
% 0.21/0.60 tff(116,plain,
% 0.21/0.60 (![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))))),
% 0.21/0.60 inference(quant_intro,[status(thm)],[115])).
% 0.21/0.60 tff(117,plain,
% 0.21/0.60 (![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))))),
% 0.21/0.60 inference(rewrite,[status(thm)],[])).
% 0.21/0.60 tff(118,plain,
% 0.21/0.60 (^[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)))))),
% 0.21/0.60 inference(bind,[status(th)],[])).
% 0.21/0.60 tff(119,plain,
% 0.21/0.60 (![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))))),
% 0.21/0.60 inference(quant_intro,[status(thm)],[118])).
% 0.21/0.60 tff(120,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')).
% 0.21/0.60 tff(121,plain,
% 0.21/0.60 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[120, 119])).
% 0.21/0.60 tff(122,plain,
% 0.21/0.60 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[121, 117])).
% 0.21/0.60 tff(123,plain,(
% 0.21/0.60 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 0.21/0.60 inference(skolemize,[status(sab)],[122])).
% 0.21/0.60 tff(124,plain,
% 0.21/0.60 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[123, 116])).
% 0.21/0.60 tff(125,plain,
% 0.21/0.60 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | ((~element(B!28, powerset(powerset(A!29)))) | element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | (~element(B!28, powerset(powerset(A!29)))) | element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29))))),
% 0.21/0.60 inference(rewrite,[status(thm)],[])).
% 0.21/0.60 tff(126,plain,
% 0.21/0.60 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | ((~element(B!28, powerset(powerset(A!29)))) | element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29))))),
% 0.21/0.60 inference(quant_inst,[status(thm)],[])).
% 0.21/0.60 tff(127,plain,
% 0.21/0.60 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | (~element(B!28, powerset(powerset(A!29)))) | element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29)))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[126, 125])).
% 0.21/0.60 tff(128,plain,
% 0.21/0.60 (element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29)))),
% 0.21/0.60 inference(unit_resolution,[status(thm)],[127, 124, 20])).
% 0.21/0.60 tff(129,plain,
% 0.21/0.60 (~(B!28 = empty_set)),
% 0.21/0.60 inference(or_elim,[status(thm)],[19])).
% 0.21/0.60 tff(130,plain,
% 0.21/0.60 (^[A: $i, B: $i] : refl(((B = empty_set) | (~element(B, powerset(powerset(A)))) | (~(complements_of_subsets(A, B) = empty_set))) <=> ((B = empty_set) | (~element(B, powerset(powerset(A)))) | (~(complements_of_subsets(A, B) = empty_set))))),
% 0.21/0.60 inference(bind,[status(th)],[])).
% 0.21/0.60 tff(131,plain,
% 0.21/0.60 (![A: $i, B: $i] : ((B = empty_set) | (~element(B, powerset(powerset(A)))) | (~(complements_of_subsets(A, B) = empty_set))) <=> ![A: $i, B: $i] : ((B = empty_set) | (~element(B, powerset(powerset(A)))) | (~(complements_of_subsets(A, B) = empty_set)))),
% 0.21/0.60 inference(quant_intro,[status(thm)],[130])).
% 0.21/0.60 tff(132,plain,
% 0.21/0.60 (^[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) | (~element(B, powerset(powerset(A)))) | (~(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) | (~element(B, powerset(powerset(A)))) | (~(complements_of_subsets(A, B) = empty_set)))))),
% 0.21/0.60 inference(bind,[status(th)],[])).
% 0.21/0.60 tff(133,plain,
% 0.21/0.60 (![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) | (~element(B, powerset(powerset(A)))) | (~(complements_of_subsets(A, B) = empty_set)))),
% 0.21/0.60 inference(quant_intro,[status(thm)],[132])).
% 0.21/0.60 tff(134,plain,
% 0.21/0.60 (![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)))))),
% 0.21/0.60 inference(rewrite,[status(thm)],[])).
% 0.21/0.60 tff(135,plain,
% 0.21/0.60 (^[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))))))),
% 0.21/0.60 inference(bind,[status(th)],[])).
% 0.21/0.60 tff(136,plain,
% 0.21/0.60 (![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)))))),
% 0.21/0.60 inference(quant_intro,[status(thm)],[135])).
% 0.21/0.60 tff(137,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')).
% 0.21/0.60 tff(138,plain,
% 0.21/0.60 (![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A)))))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[137, 136])).
% 0.21/0.60 tff(139,plain,
% 0.21/0.60 (![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A)))))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[138, 134])).
% 0.21/0.60 tff(140,plain,(
% 0.21/0.60 ![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A)))))),
% 0.21/0.60 inference(skolemize,[status(sab)],[139])).
% 0.21/0.60 tff(141,plain,
% 0.21/0.60 (![A: $i, B: $i] : ((B = empty_set) | (~element(B, powerset(powerset(A)))) | (~(complements_of_subsets(A, B) = empty_set)))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[140, 133])).
% 0.21/0.60 tff(142,plain,
% 0.21/0.60 (![A: $i, B: $i] : ((B = empty_set) | (~element(B, powerset(powerset(A)))) | (~(complements_of_subsets(A, B) = empty_set)))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[141, 131])).
% 0.21/0.60 tff(143,plain,
% 0.21/0.60 (((~![A: $i, B: $i] : ((B = empty_set) | (~element(B, powerset(powerset(A)))) | (~(complements_of_subsets(A, B) = empty_set)))) | ((B!28 = empty_set) | (~element(B!28, powerset(powerset(A!29)))) | (~(complements_of_subsets(A!29, B!28) = empty_set)))) <=> ((~![A: $i, B: $i] : ((B = empty_set) | (~element(B, powerset(powerset(A)))) | (~(complements_of_subsets(A, B) = empty_set)))) | (B!28 = empty_set) | (~element(B!28, powerset(powerset(A!29)))) | (~(complements_of_subsets(A!29, B!28) = empty_set)))),
% 0.21/0.60 inference(rewrite,[status(thm)],[])).
% 0.21/0.60 tff(144,plain,
% 0.21/0.60 ((~![A: $i, B: $i] : ((B = empty_set) | (~element(B, powerset(powerset(A)))) | (~(complements_of_subsets(A, B) = empty_set)))) | ((B!28 = empty_set) | (~element(B!28, powerset(powerset(A!29)))) | (~(complements_of_subsets(A!29, B!28) = empty_set)))),
% 0.21/0.60 inference(quant_inst,[status(thm)],[])).
% 0.21/0.60 tff(145,plain,
% 0.21/0.60 ((~![A: $i, B: $i] : ((B = empty_set) | (~element(B, powerset(powerset(A)))) | (~(complements_of_subsets(A, B) = empty_set)))) | (B!28 = empty_set) | (~element(B!28, powerset(powerset(A!29)))) | (~(complements_of_subsets(A!29, B!28) = empty_set))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[144, 143])).
% 0.21/0.60 tff(146,plain,
% 0.21/0.60 (~(complements_of_subsets(A!29, B!28) = empty_set)),
% 0.21/0.60 inference(unit_resolution,[status(thm)],[145, 142, 129, 20])).
% 0.21/0.60 tff(147,plain,
% 0.21/0.60 (^[A: $i, B: $i] : refl(((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A))))) <=> ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A))))))),
% 0.21/0.60 inference(bind,[status(th)],[])).
% 0.21/0.60 tff(148,plain,
% 0.21/0.60 (![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A))))) <=> ![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.21/0.60 inference(quant_intro,[status(thm)],[147])).
% 0.21/0.60 tff(149,plain,
% 0.21/0.60 (![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A))))) <=> ![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.21/0.60 inference(rewrite,[status(thm)],[])).
% 0.21/0.60 tff(150,plain,
% 0.21/0.60 (^[A: $i, B: $i] : trans(monotonicity(rewrite(((~(B = empty_set)) => (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B)))) <=> ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))))), ((element(B, powerset(powerset(A))) => ((~(B = empty_set)) => (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))))) <=> (element(B, powerset(powerset(A))) => ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))))))), rewrite((element(B, powerset(powerset(A))) => ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))))) <=> ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))), ((element(B, powerset(powerset(A))) => ((~(B = empty_set)) => (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))))) <=> ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))))),
% 0.21/0.60 inference(bind,[status(th)],[])).
% 0.21/0.60 tff(151,plain,
% 0.21/0.60 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => ((~(B = empty_set)) => (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))))) <=> ![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.21/0.60 inference(quant_intro,[status(thm)],[150])).
% 0.21/0.60 tff(152,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => ((~(B = empty_set)) => (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B)))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t47_setfam_1')).
% 0.21/0.60 tff(153,plain,
% 0.21/0.60 (![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[152, 151])).
% 0.21/0.60 tff(154,plain,
% 0.21/0.60 (![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[153, 149])).
% 0.21/0.60 tff(155,plain,(
% 0.21/0.60 ![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.21/0.60 inference(skolemize,[status(sab)],[154])).
% 0.21/0.60 tff(156,plain,
% 0.21/0.60 (![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[155, 148])).
% 0.21/0.60 tff(157,plain,
% 0.21/0.60 (((~![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) | ((complements_of_subsets(A!29, B!28) = empty_set) | (subset_difference(A!29, cast_to_subset(A!29), union_of_subsets(A!29, complements_of_subsets(A!29, B!28))) = meet_of_subsets(A!29, complements_of_subsets(A!29, complements_of_subsets(A!29, B!28)))) | (~element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29)))))) <=> ((~![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) | (complements_of_subsets(A!29, B!28) = empty_set) | (subset_difference(A!29, cast_to_subset(A!29), union_of_subsets(A!29, complements_of_subsets(A!29, B!28))) = meet_of_subsets(A!29, complements_of_subsets(A!29, complements_of_subsets(A!29, B!28)))) | (~element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29)))))),
% 0.21/0.60 inference(rewrite,[status(thm)],[])).
% 0.21/0.60 tff(158,plain,
% 0.21/0.60 ((~![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) | ((complements_of_subsets(A!29, B!28) = empty_set) | (subset_difference(A!29, cast_to_subset(A!29), union_of_subsets(A!29, complements_of_subsets(A!29, B!28))) = meet_of_subsets(A!29, complements_of_subsets(A!29, complements_of_subsets(A!29, B!28)))) | (~element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29)))))),
% 0.21/0.60 inference(quant_inst,[status(thm)],[])).
% 0.21/0.60 tff(159,plain,
% 0.21/0.60 ((~![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) | (complements_of_subsets(A!29, B!28) = empty_set) | (subset_difference(A!29, cast_to_subset(A!29), union_of_subsets(A!29, complements_of_subsets(A!29, B!28))) = meet_of_subsets(A!29, complements_of_subsets(A!29, complements_of_subsets(A!29, B!28)))) | (~element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29))))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[158, 157])).
% 0.21/0.60 tff(160,plain,
% 0.21/0.60 (subset_difference(A!29, cast_to_subset(A!29), union_of_subsets(A!29, complements_of_subsets(A!29, B!28))) = meet_of_subsets(A!29, complements_of_subsets(A!29, complements_of_subsets(A!29, B!28)))),
% 0.21/0.60 inference(unit_resolution,[status(thm)],[159, 156, 146, 128])).
% 0.21/0.60 tff(161,plain,
% 0.21/0.60 (^[A: $i, B: $i] : refl(((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B))) <=> ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B))))),
% 0.21/0.60 inference(bind,[status(th)],[])).
% 0.21/0.60 tff(162,plain,
% 0.21/0.60 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))),
% 0.21/0.60 inference(quant_intro,[status(thm)],[161])).
% 0.21/0.60 tff(163,plain,
% 0.21/0.60 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))),
% 0.21/0.60 inference(rewrite,[status(thm)],[])).
% 0.21/0.60 tff(164,plain,
% 0.21/0.60 (^[A: $i, B: $i] : rewrite((element(B, powerset(powerset(A))) => (union_of_subsets(A, B) = union(B))) <=> ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B))))),
% 0.21/0.60 inference(bind,[status(th)],[])).
% 0.21/0.60 tff(165,plain,
% 0.21/0.60 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => (union_of_subsets(A, B) = union(B))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))),
% 0.21/0.60 inference(quant_intro,[status(thm)],[164])).
% 0.21/0.60 tff(166,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => (union_of_subsets(A, B) = union(B)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','redefinition_k5_setfam_1')).
% 0.21/0.60 tff(167,plain,
% 0.21/0.60 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[166, 165])).
% 0.21/0.60 tff(168,plain,
% 0.21/0.60 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[167, 163])).
% 0.21/0.60 tff(169,plain,(
% 0.21/0.60 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))),
% 0.21/0.60 inference(skolemize,[status(sab)],[168])).
% 0.21/0.60 tff(170,plain,
% 0.21/0.60 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[169, 162])).
% 0.21/0.60 tff(171,plain,
% 0.21/0.60 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))) | ((~element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29)))) | (union_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = union(complements_of_subsets(A!29, B!28))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))) | (~element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29)))) | (union_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = union(complements_of_subsets(A!29, B!28))))),
% 0.21/0.60 inference(rewrite,[status(thm)],[])).
% 0.21/0.60 tff(172,plain,
% 0.21/0.60 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))) | ((~element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29)))) | (union_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = union(complements_of_subsets(A!29, B!28))))),
% 0.21/0.60 inference(quant_inst,[status(thm)],[])).
% 0.21/0.60 tff(173,plain,
% 0.21/0.60 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))) | (~element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29)))) | (union_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = union(complements_of_subsets(A!29, B!28)))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[172, 171])).
% 0.21/0.60 tff(174,plain,
% 0.21/0.60 (union_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = union(complements_of_subsets(A!29, B!28))),
% 0.21/0.60 inference(unit_resolution,[status(thm)],[173, 170, 128])).
% 0.21/0.60 tff(175,plain,
% 0.21/0.60 (union(complements_of_subsets(A!29, B!28)) = union_of_subsets(A!29, complements_of_subsets(A!29, B!28))),
% 0.21/0.60 inference(symmetry,[status(thm)],[174])).
% 0.21/0.60 tff(176,plain,
% 0.21/0.60 (subset_difference(A!29, A!29, union(complements_of_subsets(A!29, B!28))) = subset_difference(A!29, cast_to_subset(A!29), union_of_subsets(A!29, complements_of_subsets(A!29, B!28)))),
% 0.21/0.60 inference(monotonicity,[status(thm)],[80, 175])).
% 0.21/0.60 tff(177,plain,
% 0.21/0.60 (element(union(complements_of_subsets(A!29, B!28)), powerset(A!29)) <=> element(union_of_subsets(A!29, complements_of_subsets(A!29, B!28)), powerset(A!29))),
% 0.21/0.60 inference(monotonicity,[status(thm)],[175])).
% 0.21/0.60 tff(178,plain,
% 0.21/0.60 (element(union_of_subsets(A!29, complements_of_subsets(A!29, B!28)), powerset(A!29)) <=> element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))),
% 0.21/0.60 inference(symmetry,[status(thm)],[177])).
% 0.21/0.60 tff(179,plain,
% 0.21/0.60 (^[A: $i, B: $i] : refl(((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A))) <=> ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A))))),
% 0.21/0.60 inference(bind,[status(th)],[])).
% 0.21/0.60 tff(180,plain,
% 0.21/0.60 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))),
% 0.21/0.60 inference(quant_intro,[status(thm)],[179])).
% 0.21/0.60 tff(181,plain,
% 0.21/0.60 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))),
% 0.21/0.60 inference(rewrite,[status(thm)],[])).
% 0.21/0.60 tff(182,plain,
% 0.21/0.60 (^[A: $i, B: $i] : rewrite((element(B, powerset(powerset(A))) => element(union_of_subsets(A, B), powerset(A))) <=> ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A))))),
% 0.21/0.60 inference(bind,[status(th)],[])).
% 0.21/0.60 tff(183,plain,
% 0.21/0.60 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => element(union_of_subsets(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))),
% 0.21/0.60 inference(quant_intro,[status(thm)],[182])).
% 0.21/0.60 tff(184,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => element(union_of_subsets(A, B), powerset(A)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','dt_k5_setfam_1')).
% 0.21/0.60 tff(185,plain,
% 0.21/0.60 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[184, 183])).
% 0.21/0.60 tff(186,plain,
% 0.21/0.60 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[185, 181])).
% 0.21/0.60 tff(187,plain,(
% 0.21/0.60 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))),
% 0.21/0.60 inference(skolemize,[status(sab)],[186])).
% 0.21/0.60 tff(188,plain,
% 0.21/0.60 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[187, 180])).
% 0.21/0.60 tff(189,plain,
% 0.21/0.60 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))) | ((~element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29)))) | element(union_of_subsets(A!29, complements_of_subsets(A!29, B!28)), powerset(A!29)))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))) | (~element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29)))) | element(union_of_subsets(A!29, complements_of_subsets(A!29, B!28)), powerset(A!29)))),
% 0.21/0.60 inference(rewrite,[status(thm)],[])).
% 0.21/0.60 tff(190,plain,
% 0.21/0.60 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))) | ((~element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29)))) | element(union_of_subsets(A!29, complements_of_subsets(A!29, B!28)), powerset(A!29)))),
% 0.21/0.60 inference(quant_inst,[status(thm)],[])).
% 0.21/0.60 tff(191,plain,
% 0.21/0.60 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))) | (~element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29)))) | element(union_of_subsets(A!29, complements_of_subsets(A!29, B!28)), powerset(A!29))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[190, 189])).
% 0.21/0.60 tff(192,plain,
% 0.21/0.60 (element(union_of_subsets(A!29, complements_of_subsets(A!29, B!28)), powerset(A!29))),
% 0.21/0.60 inference(unit_resolution,[status(thm)],[191, 188, 128])).
% 0.21/0.60 tff(193,plain,
% 0.21/0.60 (element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[192, 178])).
% 0.21/0.60 tff(194,plain,
% 0.21/0.60 (element(A!29, powerset(A!29)) <=> element(cast_to_subset(A!29), powerset(A!29))),
% 0.21/0.60 inference(monotonicity,[status(thm)],[80])).
% 0.21/0.60 tff(195,plain,
% 0.21/0.60 (element(cast_to_subset(A!29), powerset(A!29)) <=> element(A!29, powerset(A!29))),
% 0.21/0.60 inference(symmetry,[status(thm)],[194])).
% 0.21/0.60 tff(196,plain,
% 0.21/0.60 (element(A!29, powerset(A!29))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[9, 195])).
% 0.21/0.60 tff(197,plain,
% 0.21/0.60 (((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | ((~element(A!29, powerset(A!29))) | (subset_difference(A!29, A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))))) <=> ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | (~element(A!29, powerset(A!29))) | (subset_difference(A!29, A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))))),
% 0.21/0.60 inference(rewrite,[status(thm)],[])).
% 0.21/0.60 tff(198,plain,
% 0.21/0.60 (((subset_difference(A!29, A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (~element(A!29, powerset(A!29)))) <=> ((~element(A!29, powerset(A!29))) | (subset_difference(A!29, A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))))),
% 0.21/0.60 inference(rewrite,[status(thm)],[])).
% 0.21/0.60 tff(199,plain,
% 0.21/0.60 (((~![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!29, A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (~element(A!29, powerset(A!29))))) <=> ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | ((~element(A!29, powerset(A!29))) | (subset_difference(A!29, A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29)))))),
% 0.21/0.60 inference(monotonicity,[status(thm)],[198])).
% 0.21/0.60 tff(200,plain,
% 0.21/0.60 (((~![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!29, A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (~element(A!29, powerset(A!29))))) <=> ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | (~element(A!29, powerset(A!29))) | (subset_difference(A!29, A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))))),
% 0.21/0.60 inference(transitivity,[status(thm)],[199, 197])).
% 0.21/0.60 tff(201,plain,
% 0.21/0.60 ((~![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!29, A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (~element(A!29, powerset(A!29))))),
% 0.21/0.60 inference(quant_inst,[status(thm)],[])).
% 0.21/0.60 tff(202,plain,
% 0.21/0.60 ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | (~element(A!29, powerset(A!29))) | (subset_difference(A!29, A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29)))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[201, 200])).
% 0.21/0.60 tff(203,plain,
% 0.21/0.60 ((~element(A!29, powerset(A!29))) | (subset_difference(A!29, A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29)))),
% 0.21/0.61 inference(unit_resolution,[status(thm)],[202, 47])).
% 0.21/0.61 tff(204,plain,
% 0.21/0.61 (subset_difference(A!29, A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))),
% 0.21/0.61 inference(unit_resolution,[status(thm)],[203, 196, 193])).
% 0.21/0.61 tff(205,plain,
% 0.21/0.61 (set_difference(A!29, union(complements_of_subsets(A!29, B!28))) = subset_difference(A!29, A!29, union(complements_of_subsets(A!29, B!28)))),
% 0.21/0.61 inference(symmetry,[status(thm)],[204])).
% 0.21/0.61 tff(206,plain,
% 0.21/0.61 (((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | ((~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (subset_complement(A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (subset_complement(A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))))),
% 0.21/0.61 inference(rewrite,[status(thm)],[])).
% 0.21/0.61 tff(207,plain,
% 0.21/0.61 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | ((~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (subset_complement(A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))))),
% 0.21/0.61 inference(quant_inst,[status(thm)],[])).
% 0.21/0.61 tff(208,plain,
% 0.21/0.61 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (subset_complement(A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28))))),
% 0.21/0.61 inference(modus_ponens,[status(thm)],[207, 206])).
% 0.21/0.61 tff(209,plain,
% 0.21/0.61 ((~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (subset_complement(A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28))))),
% 0.21/0.61 inference(unit_resolution,[status(thm)],[208, 94])).
% 0.21/0.61 tff(210,plain,
% 0.21/0.61 (subset_complement(A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))),
% 0.21/0.61 inference(unit_resolution,[status(thm)],[209, 193])).
% 0.21/0.61 tff(211,plain,
% 0.21/0.61 (subset_complement(A!29, union(complements_of_subsets(A!29, B!28))) = set_meet(B!28)),
% 0.21/0.61 inference(transitivity,[status(thm)],[210, 205, 176, 160, 114, 69])).
% 0.21/0.61 tff(212,plain,
% 0.21/0.61 (subset_complement(A!29, subset_complement(A!29, union(complements_of_subsets(A!29, B!28)))) = subset_complement(A!29, set_meet(B!28))),
% 0.21/0.61 inference(monotonicity,[status(thm)],[211])).
% 0.21/0.61 tff(213,plain,
% 0.21/0.61 (^[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)))),
% 0.21/0.61 inference(bind,[status(th)],[])).
% 0.21/0.61 tff(214,plain,
% 0.21/0.61 (![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))),
% 0.21/0.61 inference(quant_intro,[status(thm)],[213])).
% 0.21/0.61 tff(215,plain,
% 0.21/0.61 (![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))),
% 0.21/0.61 inference(rewrite,[status(thm)],[])).
% 0.21/0.61 tff(216,plain,
% 0.21/0.61 (^[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)))),
% 0.21/0.61 inference(bind,[status(th)],[])).
% 0.21/0.61 tff(217,plain,
% 0.21/0.61 (![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))),
% 0.21/0.61 inference(quant_intro,[status(thm)],[216])).
% 0.21/0.61 tff(218,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')).
% 0.21/0.61 tff(219,plain,
% 0.21/0.61 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.21/0.61 inference(modus_ponens,[status(thm)],[218, 217])).
% 0.21/0.61 tff(220,plain,
% 0.21/0.61 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.21/0.61 inference(modus_ponens,[status(thm)],[219, 215])).
% 0.21/0.61 tff(221,plain,(
% 0.21/0.61 ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.21/0.61 inference(skolemize,[status(sab)],[220])).
% 0.21/0.61 tff(222,plain,
% 0.21/0.61 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.21/0.61 inference(modus_ponens,[status(thm)],[221, 214])).
% 0.21/0.61 tff(223,plain,
% 0.21/0.61 (((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))) | ((~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (subset_complement(A!29, subset_complement(A!29, union(complements_of_subsets(A!29, B!28)))) = union(complements_of_subsets(A!29, B!28))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (subset_complement(A!29, subset_complement(A!29, union(complements_of_subsets(A!29, B!28)))) = union(complements_of_subsets(A!29, B!28))))),
% 0.21/0.61 inference(rewrite,[status(thm)],[])).
% 0.21/0.61 tff(224,plain,
% 0.21/0.61 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))) | ((~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (subset_complement(A!29, subset_complement(A!29, union(complements_of_subsets(A!29, B!28)))) = union(complements_of_subsets(A!29, B!28))))),
% 0.21/0.61 inference(quant_inst,[status(thm)],[])).
% 0.21/0.61 tff(225,plain,
% 0.21/0.61 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (subset_complement(A!29, subset_complement(A!29, union(complements_of_subsets(A!29, B!28)))) = union(complements_of_subsets(A!29, B!28)))),
% 0.21/0.61 inference(modus_ponens,[status(thm)],[224, 223])).
% 0.21/0.61 tff(226,plain,
% 0.21/0.61 ((~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (subset_complement(A!29, subset_complement(A!29, union(complements_of_subsets(A!29, B!28)))) = union(complements_of_subsets(A!29, B!28)))),
% 0.21/0.61 inference(unit_resolution,[status(thm)],[225, 222])).
% 0.21/0.61 tff(227,plain,
% 0.21/0.61 (subset_complement(A!29, subset_complement(A!29, union(complements_of_subsets(A!29, B!28)))) = union(complements_of_subsets(A!29, B!28))),
% 0.21/0.61 inference(unit_resolution,[status(thm)],[226, 193])).
% 0.21/0.61 tff(228,plain,
% 0.21/0.61 (union(complements_of_subsets(A!29, B!28)) = subset_complement(A!29, subset_complement(A!29, union(complements_of_subsets(A!29, B!28))))),
% 0.21/0.61 inference(symmetry,[status(thm)],[227])).
% 0.21/0.61 tff(229,plain,
% 0.21/0.61 (union_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28))),
% 0.21/0.61 inference(transitivity,[status(thm)],[174, 228, 212, 99, 81, 55])).
% 0.21/0.61 tff(230,plain,
% 0.21/0.61 (~(union_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28)))),
% 0.21/0.61 inference(or_elim,[status(thm)],[19])).
% 0.21/0.61 tff(231,plain,
% 0.21/0.61 ($false),
% 0.21/0.61 inference(unit_resolution,[status(thm)],[230, 229])).
% 0.21/0.61 % SZS output end Proof
% 0.21/0.61 % E exiting
%------------------------------------------------------------------------------