%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : NUM181-1 : TPTP v9.0.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n007.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 05:18:52 AM UTC 2025
% Result : Unsatisfiable 161.60s 161.81s
% Output : Proof 162.29s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : NUM181-1 : TPTP v9.0.0. Bugfixed v2.1.0.
% 0.03/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n007.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Jun 20 00:26:20 EDT 2025
% 0.12/0.33 % CPUTime :
% 161.60/161.81 % SZS status Unsatisfiable
% 161.60/161.81 % SZS output start Proof
% 161.60/161.81 tff(subclass_type, type, (
% 161.60/161.81 subclass: ( $i * $i ) > $o)).
% 161.60/161.81 tff(universal_class_type, type, (
% 161.60/161.81 universal_class: $i)).
% 161.60/161.81 tff(limit_ordinals_type, type, (
% 161.60/161.81 limit_ordinals: $i)).
% 161.60/161.81 tff(member_type, type, (
% 161.60/161.81 member: ( $i * $i ) > $o)).
% 161.60/161.81 tff(null_class_type, type, (
% 161.60/161.81 null_class: $i)).
% 161.60/161.81 tff(singleton_type, type, (
% 161.60/161.81 singleton: $i > $i)).
% 161.60/161.81 tff(unordered_pair_type, type, (
% 161.60/161.81 unordered_pair: ( $i * $i ) > $i)).
% 161.60/161.81 tff(complement_type, type, (
% 161.60/161.81 complement: $i > $i)).
% 161.60/161.81 tff(intersection_type, type, (
% 161.60/161.81 intersection: ( $i * $i ) > $i)).
% 161.60/161.81 tff(image_type, type, (
% 161.60/161.81 image: ( $i * $i ) > $i)).
% 161.60/161.81 tff(ordinal_numbers_type, type, (
% 161.60/161.81 ordinal_numbers: $i)).
% 161.60/161.81 tff(successor_relation_type, type, (
% 161.60/161.81 successor_relation: $i)).
% 161.60/161.81 tff(kind_1_ordinals_type, type, (
% 161.60/161.81 kind_1_ordinals: $i)).
% 161.60/161.81 tff(union_type, type, (
% 161.60/161.81 union: ( $i * $i ) > $i)).
% 161.60/161.81 tff(1,plain,
% 161.60/161.81 (^[X: $i] : refl((unordered_pair(X, X) = singleton(X)) <=> (unordered_pair(X, X) = singleton(X)))),
% 161.60/161.81 inference(bind,[status(th)],[])).
% 161.60/161.81 tff(2,plain,
% 161.60/161.81 (![X: $i] : (unordered_pair(X, X) = singleton(X)) <=> ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 161.60/161.81 inference(quant_intro,[status(thm)],[1])).
% 161.60/161.81 tff(3,plain,
% 161.60/161.81 (![X: $i] : (unordered_pair(X, X) = singleton(X)) <=> ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 161.60/161.81 inference(rewrite,[status(thm)],[])).
% 161.60/161.81 tff(4,axiom,(![X: $i] : (unordered_pair(X, X) = singleton(X))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','singleton_set')).
% 161.60/161.81 tff(5,plain,
% 161.60/161.81 (![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 161.60/161.81 inference(modus_ponens,[status(thm)],[4, 3])).
% 161.60/161.81 tff(6,plain,(
% 161.60/161.81 ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 161.60/161.81 inference(skolemize,[status(sab)],[5])).
% 161.60/161.81 tff(7,plain,
% 161.60/161.81 (![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 161.60/161.81 inference(modus_ponens,[status(thm)],[6, 2])).
% 161.60/161.81 tff(8,plain,
% 161.60/161.81 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(null_class, null_class) = singleton(null_class))),
% 161.60/161.81 inference(quant_inst,[status(thm)],[])).
% 161.60/161.81 tff(9,plain,
% 161.60/161.81 (unordered_pair(null_class, null_class) = singleton(null_class)),
% 161.60/161.81 inference(unit_resolution,[status(thm)],[8, 7])).
% 161.60/161.81 tff(10,plain,
% 161.60/161.81 (singleton(null_class) = unordered_pair(null_class, null_class)),
% 161.60/161.81 inference(symmetry,[status(thm)],[9])).
% 161.60/161.81 tff(11,plain,
% 161.60/161.81 (member(null_class, singleton(null_class)) <=> member(null_class, unordered_pair(null_class, null_class))),
% 161.60/161.81 inference(monotonicity,[status(thm)],[10])).
% 161.60/161.81 tff(12,plain,
% 161.60/161.81 (member(null_class, unordered_pair(null_class, null_class)) <=> member(null_class, singleton(null_class))),
% 161.60/161.81 inference(symmetry,[status(thm)],[11])).
% 161.60/161.81 tff(13,assumption,(member(null_class, universal_class)), introduced(assumption)).
% 161.60/161.81 tff(14,plain,
% 161.60/161.81 (^[Y: $i, X: $i] : refl(((~member(X, universal_class)) | member(X, unordered_pair(X, Y))) <=> ((~member(X, universal_class)) | member(X, unordered_pair(X, Y))))),
% 161.60/161.81 inference(bind,[status(th)],[])).
% 161.60/161.81 tff(15,plain,
% 161.60/161.81 (![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y))) <=> ![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 161.60/161.81 inference(quant_intro,[status(thm)],[14])).
% 161.60/161.81 tff(16,plain,
% 161.60/161.81 (![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y))) <=> ![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 161.60/161.81 inference(rewrite,[status(thm)],[])).
% 161.60/161.81 tff(17,axiom,(![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','unordered_pair2')).
% 161.60/161.81 tff(18,plain,
% 161.60/161.81 (![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 161.60/161.81 inference(modus_ponens,[status(thm)],[17, 16])).
% 161.60/161.81 tff(19,plain,(
% 161.60/161.81 ![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 161.60/161.81 inference(skolemize,[status(sab)],[18])).
% 161.60/161.81 tff(20,plain,
% 161.60/161.81 (![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 161.60/161.81 inference(modus_ponens,[status(thm)],[19, 15])).
% 161.60/161.81 tff(21,plain,
% 161.60/161.81 (((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(null_class, universal_class)) | member(null_class, unordered_pair(null_class, null_class)))) <=> ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(null_class, universal_class)) | member(null_class, unordered_pair(null_class, null_class)))),
% 161.60/161.81 inference(rewrite,[status(thm)],[])).
% 161.60/161.81 tff(22,plain,
% 161.60/161.81 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(null_class, universal_class)) | member(null_class, unordered_pair(null_class, null_class)))),
% 161.60/161.81 inference(quant_inst,[status(thm)],[])).
% 161.60/161.81 tff(23,plain,
% 161.60/161.81 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(null_class, universal_class)) | member(null_class, unordered_pair(null_class, null_class))),
% 161.60/161.81 inference(modus_ponens,[status(thm)],[22, 21])).
% 161.60/161.81 tff(24,plain,
% 161.60/161.81 ((~member(null_class, universal_class)) | member(null_class, unordered_pair(null_class, null_class))),
% 161.60/161.81 inference(unit_resolution,[status(thm)],[23, 20])).
% 161.60/161.81 tff(25,plain,
% 161.60/161.81 (member(null_class, unordered_pair(null_class, null_class))),
% 161.60/161.81 inference(unit_resolution,[status(thm)],[24, 13])).
% 161.60/161.81 tff(26,plain,
% 161.60/161.81 (member(null_class, singleton(null_class))),
% 161.60/161.81 inference(modus_ponens,[status(thm)],[25, 12])).
% 161.60/161.81 tff(27,plain,
% 161.60/161.81 ((union(singleton(null_class), image(successor_relation, ordinal_numbers)) = kind_1_ordinals) <=> (union(singleton(null_class), image(successor_relation, ordinal_numbers)) = kind_1_ordinals)),
% 161.60/161.81 inference(rewrite,[status(thm)],[])).
% 161.60/161.81 tff(28,axiom,(union(singleton(null_class), image(successor_relation, ordinal_numbers)) = kind_1_ordinals), file('/export/starexec/sandbox2/benchmark/Axioms/NUM004-0.ax','kind_1_ordinals')).
% 161.60/161.81 tff(29,plain,
% 161.60/161.81 (union(singleton(null_class), image(successor_relation, ordinal_numbers)) = kind_1_ordinals),
% 161.60/161.81 inference(modus_ponens,[status(thm)],[28, 27])).
% 161.60/161.81 tff(30,plain,
% 161.60/161.81 (^[Y: $i, X: $i] : refl((complement(intersection(complement(X), complement(Y))) = union(X, Y)) <=> (complement(intersection(complement(X), complement(Y))) = union(X, Y)))),
% 161.60/161.81 inference(bind,[status(th)],[])).
% 161.60/161.81 tff(31,plain,
% 161.60/161.81 (![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y)) <=> ![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))),
% 161.60/161.81 inference(quant_intro,[status(thm)],[30])).
% 161.60/161.81 tff(32,plain,
% 161.60/161.81 (![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y)) <=> ![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))),
% 161.60/161.81 inference(rewrite,[status(thm)],[])).
% 161.60/161.81 tff(33,axiom,(![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','union')).
% 161.60/161.81 tff(34,plain,
% 161.60/161.81 (![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))),
% 161.60/161.81 inference(modus_ponens,[status(thm)],[33, 32])).
% 161.60/161.81 tff(35,plain,(
% 161.60/161.81 ![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))),
% 161.60/161.81 inference(skolemize,[status(sab)],[34])).
% 161.60/161.81 tff(36,plain,
% 161.60/161.81 (![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))),
% 161.60/161.81 inference(modus_ponens,[status(thm)],[35, 31])).
% 161.60/161.81 tff(37,plain,
% 161.60/161.81 ((~![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))) | (complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) = union(singleton(null_class), image(successor_relation, ordinal_numbers)))),
% 161.60/161.81 inference(quant_inst,[status(thm)],[])).
% 161.60/161.81 tff(38,plain,
% 161.60/161.81 (complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) = union(singleton(null_class), image(successor_relation, ordinal_numbers))),
% 161.60/161.81 inference(unit_resolution,[status(thm)],[37, 36])).
% 161.60/161.81 tff(39,plain,
% 161.60/161.81 (complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) = kind_1_ordinals),
% 161.60/161.81 inference(transitivity,[status(thm)],[38, 29])).
% 161.60/161.81 tff(40,plain,
% 161.60/161.81 (member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))) <=> member(null_class, kind_1_ordinals)),
% 161.60/161.81 inference(monotonicity,[status(thm)],[39])).
% 161.60/161.81 tff(41,plain,
% 161.60/161.81 (member(null_class, kind_1_ordinals) <=> member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))))),
% 161.60/161.81 inference(symmetry,[status(thm)],[40])).
% 161.60/161.81 tff(42,plain,
% 161.60/161.81 ((~member(null_class, kind_1_ordinals)) <=> (~member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))))),
% 161.60/161.81 inference(monotonicity,[status(thm)],[41])).
% 161.60/161.81 tff(43,plain,
% 161.60/161.81 ((intersection(complement(kind_1_ordinals), ordinal_numbers) = limit_ordinals) <=> (intersection(complement(kind_1_ordinals), ordinal_numbers) = limit_ordinals)),
% 161.60/161.81 inference(rewrite,[status(thm)],[])).
% 161.60/161.81 tff(44,axiom,(intersection(complement(kind_1_ordinals), ordinal_numbers) = limit_ordinals), file('/export/starexec/sandbox2/benchmark/Axioms/NUM004-0.ax','limit_ordinals')).
% 161.60/161.81 tff(45,plain,
% 161.60/161.81 (intersection(complement(kind_1_ordinals), ordinal_numbers) = limit_ordinals),
% 161.60/161.81 inference(modus_ponens,[status(thm)],[44, 43])).
% 161.60/161.81 tff(46,plain,
% 161.60/161.81 (member(null_class, intersection(complement(kind_1_ordinals), ordinal_numbers)) <=> member(null_class, limit_ordinals)),
% 161.60/161.81 inference(monotonicity,[status(thm)],[45])).
% 161.60/161.81 tff(47,plain,
% 161.60/161.81 (member(null_class, limit_ordinals) <=> member(null_class, intersection(complement(kind_1_ordinals), ordinal_numbers))),
% 161.60/161.81 inference(symmetry,[status(thm)],[46])).
% 161.60/161.81 tff(48,plain,
% 161.60/161.81 (member(null_class, limit_ordinals) <=> member(null_class, limit_ordinals)),
% 161.60/161.81 inference(rewrite,[status(thm)],[])).
% 161.60/161.81 tff(49,axiom,(member(null_class, limit_ordinals)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_null_class_is_not_a_limit_ordinal_1')).
% 161.60/161.81 tff(50,plain,
% 161.60/161.81 (member(null_class, limit_ordinals)),
% 161.60/161.81 inference(modus_ponens,[status(thm)],[49, 48])).
% 161.60/161.81 tff(51,plain,
% 161.60/161.81 (member(null_class, intersection(complement(kind_1_ordinals), ordinal_numbers))),
% 161.60/161.81 inference(modus_ponens,[status(thm)],[50, 47])).
% 161.60/161.81 tff(52,plain,
% 161.60/161.81 (^[Z: $i, Y: $i, X: $i] : refl(((~member(Z, intersection(X, Y))) | member(Z, X)) <=> ((~member(Z, intersection(X, Y))) | member(Z, X)))),
% 161.60/161.81 inference(bind,[status(th)],[])).
% 161.60/161.81 tff(53,plain,
% 161.60/161.81 (![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X)) <=> ![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))),
% 161.60/161.81 inference(quant_intro,[status(thm)],[52])).
% 161.60/161.81 tff(54,plain,
% 161.60/161.81 (![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X)) <=> ![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))),
% 161.60/161.81 inference(rewrite,[status(thm)],[])).
% 161.60/161.81 tff(55,axiom,(![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','intersection1')).
% 161.60/161.81 tff(56,plain,
% 161.60/161.81 (![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))),
% 161.60/161.81 inference(modus_ponens,[status(thm)],[55, 54])).
% 161.60/161.81 tff(57,plain,(
% 161.60/161.81 ![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))),
% 161.60/161.81 inference(skolemize,[status(sab)],[56])).
% 161.60/161.81 tff(58,plain,
% 161.60/161.81 (![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))),
% 161.60/161.81 inference(modus_ponens,[status(thm)],[57, 53])).
% 161.60/161.81 tff(59,plain,
% 161.60/161.81 (((~![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))) | ((~member(null_class, intersection(complement(kind_1_ordinals), ordinal_numbers))) | member(null_class, complement(kind_1_ordinals)))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))) | (~member(null_class, intersection(complement(kind_1_ordinals), ordinal_numbers))) | member(null_class, complement(kind_1_ordinals)))),
% 161.60/161.81 inference(rewrite,[status(thm)],[])).
% 161.60/161.81 tff(60,plain,
% 161.60/161.81 ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))) | ((~member(null_class, intersection(complement(kind_1_ordinals), ordinal_numbers))) | member(null_class, complement(kind_1_ordinals)))),
% 161.60/161.82 inference(quant_inst,[status(thm)],[])).
% 161.60/161.82 tff(61,plain,
% 161.60/161.82 ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))) | (~member(null_class, intersection(complement(kind_1_ordinals), ordinal_numbers))) | member(null_class, complement(kind_1_ordinals))),
% 161.60/161.82 inference(modus_ponens,[status(thm)],[60, 59])).
% 161.60/161.82 tff(62,plain,
% 161.60/161.82 ((~member(null_class, intersection(complement(kind_1_ordinals), ordinal_numbers))) | member(null_class, complement(kind_1_ordinals))),
% 161.60/161.82 inference(unit_resolution,[status(thm)],[61, 58])).
% 161.60/161.82 tff(63,plain,
% 161.60/161.82 (member(null_class, complement(kind_1_ordinals))),
% 161.60/161.82 inference(unit_resolution,[status(thm)],[62, 51])).
% 161.60/161.82 tff(64,plain,
% 161.60/161.82 (^[Z: $i, X: $i] : refl(((~member(Z, X)) | (~member(Z, complement(X)))) <=> ((~member(Z, X)) | (~member(Z, complement(X)))))),
% 161.60/161.82 inference(bind,[status(th)],[])).
% 161.60/161.82 tff(65,plain,
% 161.60/161.82 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X)))) <=> ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 161.60/161.82 inference(quant_intro,[status(thm)],[64])).
% 161.60/161.82 tff(66,plain,
% 161.60/161.82 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X)))) <=> ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 161.60/161.82 inference(rewrite,[status(thm)],[])).
% 161.60/161.82 tff(67,plain,
% 161.60/161.82 (^[Z: $i, X: $i] : rewrite(((~member(Z, complement(X))) | (~member(Z, X))) <=> ((~member(Z, X)) | (~member(Z, complement(X)))))),
% 161.60/161.82 inference(bind,[status(th)],[])).
% 161.60/161.82 tff(68,plain,
% 161.60/161.82 (![Z: $i, X: $i] : ((~member(Z, complement(X))) | (~member(Z, X))) <=> ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 161.60/161.82 inference(quant_intro,[status(thm)],[67])).
% 161.60/161.82 tff(69,axiom,(![Z: $i, X: $i] : ((~member(Z, complement(X))) | (~member(Z, X)))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','complement1')).
% 161.60/161.82 tff(70,plain,
% 161.60/161.82 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 161.60/161.82 inference(modus_ponens,[status(thm)],[69, 68])).
% 161.60/161.82 tff(71,plain,
% 161.60/161.82 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 161.60/161.82 inference(modus_ponens,[status(thm)],[70, 66])).
% 161.60/161.82 tff(72,plain,(
% 161.60/161.82 ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 161.60/161.82 inference(skolemize,[status(sab)],[71])).
% 161.60/161.82 tff(73,plain,
% 161.60/161.82 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 161.60/161.82 inference(modus_ponens,[status(thm)],[72, 65])).
% 161.60/161.82 tff(74,plain,
% 161.60/161.82 (((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | ((~member(null_class, kind_1_ordinals)) | (~member(null_class, complement(kind_1_ordinals))))) <=> ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | (~member(null_class, kind_1_ordinals)) | (~member(null_class, complement(kind_1_ordinals))))),
% 161.60/161.82 inference(rewrite,[status(thm)],[])).
% 161.60/161.82 tff(75,plain,
% 161.60/161.82 ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | ((~member(null_class, kind_1_ordinals)) | (~member(null_class, complement(kind_1_ordinals))))),
% 161.60/161.82 inference(quant_inst,[status(thm)],[])).
% 161.60/161.82 tff(76,plain,
% 161.60/161.82 ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | (~member(null_class, kind_1_ordinals)) | (~member(null_class, complement(kind_1_ordinals)))),
% 161.60/161.82 inference(modus_ponens,[status(thm)],[75, 74])).
% 161.60/161.82 tff(77,plain,
% 161.60/161.82 (~member(null_class, kind_1_ordinals)),
% 161.60/161.82 inference(unit_resolution,[status(thm)],[76, 73, 63])).
% 161.60/161.82 tff(78,plain,
% 161.60/161.82 (~member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))))),
% 161.60/161.82 inference(modus_ponens,[status(thm)],[77, 42])).
% 161.60/161.82 tff(79,plain,
% 161.60/161.82 (^[Z: $i, X: $i] : refl((member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X))) <=> (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X))))),
% 161.60/161.82 inference(bind,[status(th)],[])).
% 161.60/161.82 tff(80,plain,
% 161.60/161.82 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X))) <=> ![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 161.60/161.82 inference(quant_intro,[status(thm)],[79])).
% 161.60/161.82 tff(81,plain,
% 161.60/161.82 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X))) <=> ![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 161.60/161.82 inference(rewrite,[status(thm)],[])).
% 161.60/161.82 tff(82,plain,
% 161.60/161.82 (^[Z: $i, X: $i] : rewrite((((~member(Z, universal_class)) | member(Z, complement(X))) | member(Z, X)) <=> (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X))))),
% 161.60/161.82 inference(bind,[status(th)],[])).
% 161.60/161.82 tff(83,plain,
% 161.60/161.82 (![Z: $i, X: $i] : (((~member(Z, universal_class)) | member(Z, complement(X))) | member(Z, X)) <=> ![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 161.60/161.82 inference(quant_intro,[status(thm)],[82])).
% 161.60/161.82 tff(84,axiom,(![Z: $i, X: $i] : (((~member(Z, universal_class)) | member(Z, complement(X))) | member(Z, X))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','complement2')).
% 161.60/161.82 tff(85,plain,
% 161.60/161.82 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 161.60/161.82 inference(modus_ponens,[status(thm)],[84, 83])).
% 161.60/161.82 tff(86,plain,
% 161.60/161.82 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 161.60/161.82 inference(modus_ponens,[status(thm)],[85, 81])).
% 161.60/161.82 tff(87,plain,(
% 161.60/161.82 ![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 161.60/161.82 inference(skolemize,[status(sab)],[86])).
% 161.60/161.82 tff(88,plain,
% 161.60/161.82 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 161.60/161.82 inference(modus_ponens,[status(thm)],[87, 80])).
% 161.60/161.82 tff(89,plain,
% 161.60/161.82 (((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | ((~member(null_class, universal_class)) | member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))))) <=> ((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | (~member(null_class, universal_class)) | member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))))),
% 161.60/161.82 inference(rewrite,[status(thm)],[])).
% 161.60/161.82 tff(90,plain,
% 161.60/161.82 ((member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | (~member(null_class, universal_class)) | member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))))) <=> ((~member(null_class, universal_class)) | member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))))),
% 161.60/161.82 inference(rewrite,[status(thm)],[])).
% 161.60/161.82 tff(91,plain,
% 161.60/161.82 (((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | (member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | (~member(null_class, universal_class)) | member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))))) <=> ((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | ((~member(null_class, universal_class)) | member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))))))),
% 161.60/161.82 inference(monotonicity,[status(thm)],[90])).
% 161.60/161.82 tff(92,plain,
% 161.60/161.82 (((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | (member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | (~member(null_class, universal_class)) | member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))))) <=> ((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | (~member(null_class, universal_class)) | member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))))),
% 161.60/161.82 inference(transitivity,[status(thm)],[91, 89])).
% 161.60/161.82 tff(93,plain,
% 161.60/161.82 ((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | (member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | (~member(null_class, universal_class)) | member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))))),
% 161.60/161.82 inference(quant_inst,[status(thm)],[])).
% 161.60/161.82 tff(94,plain,
% 161.60/161.82 ((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | (~member(null_class, universal_class)) | member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))))),
% 161.60/161.82 inference(modus_ponens,[status(thm)],[93, 92])).
% 161.60/161.82 tff(95,plain,
% 161.60/161.82 ((~member(null_class, universal_class)) | member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))))),
% 161.60/161.82 inference(unit_resolution,[status(thm)],[94, 88])).
% 161.60/161.82 tff(96,plain,
% 161.60/161.82 ((~member(null_class, universal_class)) | member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))),
% 161.60/161.82 inference(unit_resolution,[status(thm)],[95, 78])).
% 161.60/161.82 tff(97,plain,
% 161.60/161.82 (member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))),
% 161.60/161.82 inference(unit_resolution,[status(thm)],[96, 13])).
% 161.60/161.82 tff(98,plain,
% 161.60/161.82 (((~![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))) | ((~member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))) | member(null_class, complement(singleton(null_class))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))) | (~member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))) | member(null_class, complement(singleton(null_class))))),
% 161.60/161.82 inference(rewrite,[status(thm)],[])).
% 161.60/161.82 tff(99,plain,
% 161.60/161.82 ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))) | ((~member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))) | member(null_class, complement(singleton(null_class))))),
% 161.60/161.82 inference(quant_inst,[status(thm)],[])).
% 161.60/161.82 tff(100,plain,
% 161.60/161.82 ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))) | (~member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))) | member(null_class, complement(singleton(null_class)))),
% 161.60/161.82 inference(modus_ponens,[status(thm)],[99, 98])).
% 161.60/161.82 tff(101,plain,
% 161.60/161.82 (member(null_class, complement(singleton(null_class)))),
% 161.60/161.82 inference(unit_resolution,[status(thm)],[100, 58, 97])).
% 161.60/161.82 tff(102,plain,
% 161.60/161.82 (((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | ((~member(null_class, singleton(null_class))) | (~member(null_class, complement(singleton(null_class)))))) <=> ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | (~member(null_class, singleton(null_class))) | (~member(null_class, complement(singleton(null_class)))))),
% 161.60/161.82 inference(rewrite,[status(thm)],[])).
% 161.60/161.82 tff(103,plain,
% 161.60/161.82 ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | ((~member(null_class, singleton(null_class))) | (~member(null_class, complement(singleton(null_class)))))),
% 161.60/161.82 inference(quant_inst,[status(thm)],[])).
% 161.60/161.82 tff(104,plain,
% 161.60/161.82 ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | (~member(null_class, singleton(null_class))) | (~member(null_class, complement(singleton(null_class))))),
% 161.60/161.82 inference(modus_ponens,[status(thm)],[103, 102])).
% 161.60/161.82 tff(105,plain,
% 161.60/161.82 ($false),
% 161.60/161.82 inference(unit_resolution,[status(thm)],[104, 73, 101, 26])).
% 161.60/161.82 tff(106,plain,(~member(null_class, universal_class)), inference(lemma,lemma(discharge,[]))).
% 161.60/161.82 tff(107,plain,
% 161.60/161.82 (^[Y: $i, U: $i, X: $i] : refl((member(U, Y) | (~member(U, X)) | (~subclass(X, Y))) <=> (member(U, Y) | (~member(U, X)) | (~subclass(X, Y))))),
% 161.60/161.82 inference(bind,[status(th)],[])).
% 161.60/161.82 tff(108,plain,
% 161.60/161.82 (![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y))) <=> ![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 161.60/161.82 inference(quant_intro,[status(thm)],[107])).
% 161.60/161.82 tff(109,plain,
% 161.60/161.82 (![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y))) <=> ![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 161.60/161.82 inference(rewrite,[status(thm)],[])).
% 161.60/161.82 tff(110,plain,
% 161.60/161.82 (^[Y: $i, U: $i, X: $i] : trans(monotonicity(rewrite(((~subclass(X, Y)) | (~member(U, X))) <=> ((~member(U, X)) | (~subclass(X, Y)))), ((((~subclass(X, Y)) | (~member(U, X))) | member(U, Y)) <=> (((~member(U, X)) | (~subclass(X, Y))) | member(U, Y)))), rewrite((((~member(U, X)) | (~subclass(X, Y))) | member(U, Y)) <=> (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))), ((((~subclass(X, Y)) | (~member(U, X))) | member(U, Y)) <=> (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))))),
% 161.60/161.82 inference(bind,[status(th)],[])).
% 161.60/161.82 tff(111,plain,
% 161.60/161.82 (![Y: $i, U: $i, X: $i] : (((~subclass(X, Y)) | (~member(U, X))) | member(U, Y)) <=> ![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 161.60/161.82 inference(quant_intro,[status(thm)],[110])).
% 161.60/161.82 tff(112,axiom,(![Y: $i, U: $i, X: $i] : (((~subclass(X, Y)) | (~member(U, X))) | member(U, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','subclass_members')).
% 161.60/161.82 tff(113,plain,
% 161.60/161.82 (![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 161.60/161.82 inference(modus_ponens,[status(thm)],[112, 111])).
% 161.60/161.82 tff(114,plain,
% 161.60/161.82 (![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 161.60/161.82 inference(modus_ponens,[status(thm)],[113, 109])).
% 161.60/161.82 tff(115,plain,(
% 161.60/161.82 ![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 161.60/161.82 inference(skolemize,[status(sab)],[114])).
% 161.60/161.82 tff(116,plain,
% 161.60/161.82 (![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 161.60/161.82 inference(modus_ponens,[status(thm)],[115, 108])).
% 161.60/161.82 tff(117,plain,
% 161.60/161.82 (((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | ((~member(null_class, limit_ordinals)) | member(null_class, universal_class) | (~subclass(limit_ordinals, universal_class)))) <=> ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (~member(null_class, limit_ordinals)) | member(null_class, universal_class) | (~subclass(limit_ordinals, universal_class)))),
% 161.60/161.82 inference(rewrite,[status(thm)],[])).
% 161.60/161.82 tff(118,plain,
% 161.60/161.82 ((member(null_class, universal_class) | (~member(null_class, limit_ordinals)) | (~subclass(limit_ordinals, universal_class))) <=> ((~member(null_class, limit_ordinals)) | member(null_class, universal_class) | (~subclass(limit_ordinals, universal_class)))),
% 162.29/162.47 inference(rewrite,[status(thm)],[])).
% 162.29/162.47 tff(119,plain,
% 162.29/162.47 (((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (member(null_class, universal_class) | (~member(null_class, limit_ordinals)) | (~subclass(limit_ordinals, universal_class)))) <=> ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | ((~member(null_class, limit_ordinals)) | member(null_class, universal_class) | (~subclass(limit_ordinals, universal_class))))),
% 162.29/162.47 inference(monotonicity,[status(thm)],[118])).
% 162.29/162.47 tff(120,plain,
% 162.29/162.47 (((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (member(null_class, universal_class) | (~member(null_class, limit_ordinals)) | (~subclass(limit_ordinals, universal_class)))) <=> ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (~member(null_class, limit_ordinals)) | member(null_class, universal_class) | (~subclass(limit_ordinals, universal_class)))),
% 162.29/162.47 inference(transitivity,[status(thm)],[119, 117])).
% 162.29/162.47 tff(121,plain,
% 162.29/162.47 ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (member(null_class, universal_class) | (~member(null_class, limit_ordinals)) | (~subclass(limit_ordinals, universal_class)))),
% 162.29/162.47 inference(quant_inst,[status(thm)],[])).
% 162.29/162.47 tff(122,plain,
% 162.29/162.47 ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (~member(null_class, limit_ordinals)) | member(null_class, universal_class) | (~subclass(limit_ordinals, universal_class))),
% 162.29/162.47 inference(modus_ponens,[status(thm)],[121, 120])).
% 162.29/162.47 tff(123,plain,
% 162.29/162.47 (~subclass(limit_ordinals, universal_class)),
% 162.29/162.47 inference(unit_resolution,[status(thm)],[122, 116, 50, 106])).
% 162.29/162.47 tff(124,plain,
% 162.29/162.47 (^[X: $i] : refl(subclass(X, universal_class) <=> subclass(X, universal_class))),
% 162.29/162.47 inference(bind,[status(th)],[])).
% 162.29/162.47 tff(125,plain,
% 162.29/162.47 (![X: $i] : subclass(X, universal_class) <=> ![X: $i] : subclass(X, universal_class)),
% 162.29/162.47 inference(quant_intro,[status(thm)],[124])).
% 162.29/162.47 tff(126,plain,
% 162.29/162.47 (![X: $i] : subclass(X, universal_class) <=> ![X: $i] : subclass(X, universal_class)),
% 162.29/162.47 inference(rewrite,[status(thm)],[])).
% 162.29/162.47 tff(127,axiom,(![X: $i] : subclass(X, universal_class)), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','class_elements_are_sets')).
% 162.29/162.47 tff(128,plain,
% 162.29/162.47 (![X: $i] : subclass(X, universal_class)),
% 162.29/162.47 inference(modus_ponens,[status(thm)],[127, 126])).
% 162.29/162.47 tff(129,plain,(
% 162.29/162.47 ![X: $i] : subclass(X, universal_class)),
% 162.29/162.47 inference(skolemize,[status(sab)],[128])).
% 162.29/162.47 tff(130,plain,
% 162.29/162.47 (![X: $i] : subclass(X, universal_class)),
% 162.29/162.47 inference(modus_ponens,[status(thm)],[129, 125])).
% 162.29/162.47 tff(131,plain,
% 162.29/162.47 ((~![X: $i] : subclass(X, universal_class)) | subclass(limit_ordinals, universal_class)),
% 162.29/162.47 inference(quant_inst,[status(thm)],[])).
% 162.29/162.47 tff(132,plain,
% 162.29/162.47 ($false),
% 162.29/162.47 inference(unit_resolution,[status(thm)],[131, 130, 123])).
% 162.29/162.47 % SZS output end Proof
% 162.29/162.47 % E exiting
%------------------------------------------------------------------------------