%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : NUM156-1 : 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 05:18:50 AM UTC 2025
% Result : Unsatisfiable 147.54s 147.73s
% Output : Proof 148.16s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : NUM156-1 : TPTP v9.0.0. Bugfixed v2.1.0.
% 0.11/0.12 % Command : run_E %s %d THM
% 0.11/0.33 % Computer : n007.cluster.edu
% 0.11/0.33 % Model : x86_64 x86_64
% 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.33 % Memory : 8042.1875MB
% 0.11/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.33 % CPULimit : 300
% 0.11/0.33 % WCLimit : 300
% 0.11/0.33 % DateTime : Fri Jun 20 00:22:35 EDT 2025
% 0.11/0.33 % CPUTime :
% 147.54/147.73 % SZS status Unsatisfiable
% 147.54/147.73 % SZS output start Proof
% 147.54/147.73 tff(subclass_type, type, (
% 147.54/147.73 subclass: ( $i * $i ) > $o)).
% 147.54/147.73 tff(universal_class_type, type, (
% 147.54/147.73 universal_class: $i)).
% 147.54/147.73 tff(omega_type, type, (
% 147.54/147.73 omega: $i)).
% 147.54/147.73 tff(member_type, type, (
% 147.54/147.73 member: ( $i * $i ) > $o)).
% 147.54/147.73 tff(null_class_type, type, (
% 147.54/147.73 null_class: $i)).
% 147.54/147.73 tff(complement_type, type, (
% 147.54/147.73 complement: $i > $i)).
% 147.54/147.73 tff(singleton_type, type, (
% 147.54/147.73 singleton: $i > $i)).
% 147.54/147.73 tff(intersection_type, type, (
% 147.54/147.73 intersection: ( $i * $i ) > $i)).
% 147.54/147.73 tff(image_type, type, (
% 147.54/147.73 image: ( $i * $i ) > $i)).
% 147.54/147.73 tff(ordinal_numbers_type, type, (
% 147.54/147.73 ordinal_numbers: $i)).
% 147.54/147.73 tff(successor_relation_type, type, (
% 147.54/147.73 successor_relation: $i)).
% 147.54/147.73 tff(kind_1_ordinals_type, type, (
% 147.54/147.73 kind_1_ordinals: $i)).
% 147.54/147.73 tff(union_type, type, (
% 147.54/147.73 union: ( $i * $i ) > $i)).
% 147.54/147.73 tff(unordered_pair_type, type, (
% 147.54/147.73 unordered_pair: ( $i * $i ) > $i)).
% 147.54/147.73 tff(inductive_type, type, (
% 147.54/147.73 inductive: $i > $o)).
% 147.54/147.73 tff(1,assumption,(member(null_class, universal_class)), introduced(assumption)).
% 147.54/147.73 tff(2,plain,
% 147.54/147.73 ((union(singleton(null_class), image(successor_relation, ordinal_numbers)) = kind_1_ordinals) <=> (union(singleton(null_class), image(successor_relation, ordinal_numbers)) = kind_1_ordinals)),
% 147.54/147.73 inference(rewrite,[status(thm)],[])).
% 147.54/147.73 tff(3,axiom,(union(singleton(null_class), image(successor_relation, ordinal_numbers)) = kind_1_ordinals), file('/export/starexec/sandbox/benchmark/Axioms/NUM004-0.ax','kind_1_ordinals')).
% 147.54/147.73 tff(4,plain,
% 147.54/147.73 (union(singleton(null_class), image(successor_relation, ordinal_numbers)) = kind_1_ordinals),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[3, 2])).
% 147.54/147.73 tff(5,plain,
% 147.54/147.73 (^[Y: $i, X: $i] : refl((complement(intersection(complement(X), complement(Y))) = union(X, Y)) <=> (complement(intersection(complement(X), complement(Y))) = union(X, Y)))),
% 147.54/147.73 inference(bind,[status(th)],[])).
% 147.54/147.73 tff(6,plain,
% 147.54/147.73 (![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))),
% 147.54/147.73 inference(quant_intro,[status(thm)],[5])).
% 147.54/147.73 tff(7,plain,
% 147.54/147.73 (![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))),
% 147.54/147.73 inference(rewrite,[status(thm)],[])).
% 147.54/147.73 tff(8,axiom,(![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','union')).
% 147.54/147.73 tff(9,plain,
% 147.54/147.73 (![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[8, 7])).
% 147.54/147.73 tff(10,plain,(
% 147.54/147.73 ![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))),
% 147.54/147.73 inference(skolemize,[status(sab)],[9])).
% 147.54/147.73 tff(11,plain,
% 147.54/147.73 (![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[10, 6])).
% 147.54/147.73 tff(12,plain,
% 147.54/147.73 ((~![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)))),
% 147.54/147.73 inference(quant_inst,[status(thm)],[])).
% 147.54/147.73 tff(13,plain,
% 147.54/147.73 (complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) = union(singleton(null_class), image(successor_relation, ordinal_numbers))),
% 147.54/147.73 inference(unit_resolution,[status(thm)],[12, 11])).
% 147.54/147.73 tff(14,plain,
% 147.54/147.73 (complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) = kind_1_ordinals),
% 147.54/147.73 inference(transitivity,[status(thm)],[13, 4])).
% 147.54/147.73 tff(15,plain,
% 147.54/147.73 (member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))) <=> member(null_class, kind_1_ordinals)),
% 147.54/147.73 inference(monotonicity,[status(thm)],[14])).
% 147.54/147.73 tff(16,plain,
% 147.54/147.73 (member(null_class, kind_1_ordinals) <=> member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))))),
% 147.54/147.73 inference(symmetry,[status(thm)],[15])).
% 147.54/147.73 tff(17,plain,
% 147.54/147.73 ((~member(null_class, kind_1_ordinals)) <=> (~member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))))),
% 147.54/147.73 inference(monotonicity,[status(thm)],[16])).
% 147.54/147.73 tff(18,plain,
% 147.54/147.73 ((~member(null_class, kind_1_ordinals)) <=> (~member(null_class, kind_1_ordinals))),
% 147.54/147.73 inference(rewrite,[status(thm)],[])).
% 147.54/147.73 tff(19,axiom,(~member(null_class, kind_1_ordinals)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_null_class_is_kind_1_1')).
% 147.54/147.73 tff(20,plain,
% 147.54/147.73 (~member(null_class, kind_1_ordinals)),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[19, 18])).
% 147.54/147.73 tff(21,plain,
% 147.54/147.73 (~member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))))),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[20, 17])).
% 147.54/147.73 tff(22,plain,
% 147.54/147.73 (^[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))))),
% 147.54/147.73 inference(bind,[status(th)],[])).
% 147.54/147.73 tff(23,plain,
% 147.54/147.73 (![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)))),
% 147.54/147.73 inference(quant_intro,[status(thm)],[22])).
% 147.54/147.73 tff(24,plain,
% 147.54/147.73 (![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)))),
% 147.54/147.73 inference(rewrite,[status(thm)],[])).
% 147.54/147.73 tff(25,plain,
% 147.54/147.73 (^[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))))),
% 147.54/147.73 inference(bind,[status(th)],[])).
% 147.54/147.73 tff(26,plain,
% 147.54/147.73 (![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)))),
% 147.54/147.73 inference(quant_intro,[status(thm)],[25])).
% 147.54/147.73 tff(27,axiom,(![Z: $i, X: $i] : (((~member(Z, universal_class)) | member(Z, complement(X))) | member(Z, X))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','complement2')).
% 147.54/147.73 tff(28,plain,
% 147.54/147.73 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[27, 26])).
% 147.54/147.73 tff(29,plain,
% 147.54/147.73 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[28, 24])).
% 147.54/147.73 tff(30,plain,(
% 147.54/147.73 ![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 147.54/147.73 inference(skolemize,[status(sab)],[29])).
% 147.54/147.73 tff(31,plain,
% 147.54/147.73 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[30, 23])).
% 147.54/147.73 tff(32,plain,
% 147.54/147.73 (((~![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))))))),
% 147.54/147.73 inference(rewrite,[status(thm)],[])).
% 147.54/147.73 tff(33,plain,
% 147.54/147.73 ((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))))))),
% 147.54/147.73 inference(rewrite,[status(thm)],[])).
% 147.54/147.73 tff(34,plain,
% 147.54/147.73 (((~![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)))))))),
% 147.54/147.73 inference(monotonicity,[status(thm)],[33])).
% 147.54/147.73 tff(35,plain,
% 147.54/147.73 (((~![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))))))),
% 147.54/147.73 inference(transitivity,[status(thm)],[34, 32])).
% 147.54/147.73 tff(36,plain,
% 147.54/147.73 ((~![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))))))),
% 147.54/147.73 inference(quant_inst,[status(thm)],[])).
% 147.54/147.73 tff(37,plain,
% 147.54/147.73 ((~![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)))))),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[36, 35])).
% 147.54/147.73 tff(38,plain,
% 147.54/147.73 ((~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)))))),
% 147.54/147.73 inference(unit_resolution,[status(thm)],[37, 31])).
% 147.54/147.73 tff(39,plain,
% 147.54/147.73 ((~member(null_class, universal_class)) | member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))),
% 147.54/147.73 inference(unit_resolution,[status(thm)],[38, 21])).
% 147.54/147.73 tff(40,plain,
% 147.54/147.73 (member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))),
% 147.54/147.73 inference(unit_resolution,[status(thm)],[39, 1])).
% 147.54/147.73 tff(41,plain,
% 147.54/147.73 (^[Z: $i, Y: $i, X: $i] : refl(((~member(Z, intersection(X, Y))) | member(Z, X)) <=> ((~member(Z, intersection(X, Y))) | member(Z, X)))),
% 147.54/147.73 inference(bind,[status(th)],[])).
% 147.54/147.73 tff(42,plain,
% 147.54/147.73 (![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))),
% 147.54/147.73 inference(quant_intro,[status(thm)],[41])).
% 147.54/147.73 tff(43,plain,
% 147.54/147.73 (![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))),
% 147.54/147.73 inference(rewrite,[status(thm)],[])).
% 147.54/147.73 tff(44,axiom,(![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','intersection1')).
% 147.54/147.73 tff(45,plain,
% 147.54/147.73 (![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[44, 43])).
% 147.54/147.73 tff(46,plain,(
% 147.54/147.73 ![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))),
% 147.54/147.73 inference(skolemize,[status(sab)],[45])).
% 147.54/147.73 tff(47,plain,
% 147.54/147.73 (![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[46, 42])).
% 147.54/147.73 tff(48,plain,
% 147.54/147.73 (((~![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))))),
% 147.54/147.73 inference(rewrite,[status(thm)],[])).
% 147.54/147.73 tff(49,plain,
% 147.54/147.73 ((~![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))))),
% 147.54/147.73 inference(quant_inst,[status(thm)],[])).
% 147.54/147.73 tff(50,plain,
% 147.54/147.73 ((~![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)))),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[49, 48])).
% 147.54/147.73 tff(51,plain,
% 147.54/147.73 (member(null_class, complement(singleton(null_class)))),
% 147.54/147.73 inference(unit_resolution,[status(thm)],[50, 47, 40])).
% 147.54/147.73 tff(52,plain,
% 147.54/147.73 (^[X: $i] : refl((unordered_pair(X, X) = singleton(X)) <=> (unordered_pair(X, X) = singleton(X)))),
% 147.54/147.73 inference(bind,[status(th)],[])).
% 147.54/147.73 tff(53,plain,
% 147.54/147.73 (![X: $i] : (unordered_pair(X, X) = singleton(X)) <=> ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 147.54/147.73 inference(quant_intro,[status(thm)],[52])).
% 147.54/147.73 tff(54,plain,
% 147.54/147.73 (![X: $i] : (unordered_pair(X, X) = singleton(X)) <=> ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 147.54/147.73 inference(rewrite,[status(thm)],[])).
% 147.54/147.73 tff(55,axiom,(![X: $i] : (unordered_pair(X, X) = singleton(X))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','singleton_set')).
% 147.54/147.73 tff(56,plain,
% 147.54/147.73 (![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[55, 54])).
% 147.54/147.73 tff(57,plain,(
% 147.54/147.73 ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 147.54/147.73 inference(skolemize,[status(sab)],[56])).
% 147.54/147.73 tff(58,plain,
% 147.54/147.73 (![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[57, 53])).
% 147.54/147.73 tff(59,plain,
% 147.54/147.73 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(null_class, null_class) = singleton(null_class))),
% 147.54/147.73 inference(quant_inst,[status(thm)],[])).
% 147.54/147.73 tff(60,plain,
% 147.54/147.73 (unordered_pair(null_class, null_class) = singleton(null_class)),
% 147.54/147.73 inference(unit_resolution,[status(thm)],[59, 58])).
% 147.54/147.73 tff(61,plain,
% 147.54/147.73 (singleton(null_class) = unordered_pair(null_class, null_class)),
% 147.54/147.73 inference(symmetry,[status(thm)],[60])).
% 147.54/147.73 tff(62,plain,
% 147.54/147.73 (member(null_class, singleton(null_class)) <=> member(null_class, unordered_pair(null_class, null_class))),
% 147.54/147.73 inference(monotonicity,[status(thm)],[61])).
% 147.54/147.73 tff(63,plain,
% 147.54/147.73 (member(null_class, unordered_pair(null_class, null_class)) <=> member(null_class, singleton(null_class))),
% 147.54/147.73 inference(symmetry,[status(thm)],[62])).
% 147.54/147.73 tff(64,plain,
% 147.54/147.73 (^[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))))),
% 147.54/147.73 inference(bind,[status(th)],[])).
% 147.54/147.73 tff(65,plain,
% 147.54/147.73 (![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)))),
% 147.54/147.73 inference(quant_intro,[status(thm)],[64])).
% 147.54/147.73 tff(66,plain,
% 147.54/147.73 (![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)))),
% 147.54/147.73 inference(rewrite,[status(thm)],[])).
% 147.54/147.73 tff(67,axiom,(![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','unordered_pair2')).
% 147.54/147.73 tff(68,plain,
% 147.54/147.73 (![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[67, 66])).
% 147.54/147.73 tff(69,plain,(
% 147.54/147.73 ![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 147.54/147.73 inference(skolemize,[status(sab)],[68])).
% 147.54/147.73 tff(70,plain,
% 147.54/147.73 (![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[69, 65])).
% 147.54/147.73 tff(71,plain,
% 147.54/147.73 (((~![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)))),
% 147.54/147.73 inference(rewrite,[status(thm)],[])).
% 147.54/147.73 tff(72,plain,
% 147.54/147.73 ((~![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)))),
% 147.54/147.73 inference(quant_inst,[status(thm)],[])).
% 147.54/147.73 tff(73,plain,
% 147.54/147.73 ((~![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))),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[72, 71])).
% 147.54/147.73 tff(74,plain,
% 147.54/147.73 ((~member(null_class, universal_class)) | member(null_class, unordered_pair(null_class, null_class))),
% 147.54/147.73 inference(unit_resolution,[status(thm)],[73, 70])).
% 147.54/147.73 tff(75,plain,
% 147.54/147.73 (member(null_class, unordered_pair(null_class, null_class))),
% 147.54/147.73 inference(unit_resolution,[status(thm)],[74, 1])).
% 147.54/147.73 tff(76,plain,
% 147.54/147.73 (member(null_class, singleton(null_class))),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[75, 63])).
% 147.54/147.73 tff(77,plain,
% 147.54/147.73 (^[Z: $i, X: $i] : refl(((~member(Z, X)) | (~member(Z, complement(X)))) <=> ((~member(Z, X)) | (~member(Z, complement(X)))))),
% 147.54/147.73 inference(bind,[status(th)],[])).
% 147.54/147.73 tff(78,plain,
% 147.54/147.73 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X)))) <=> ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 147.54/147.73 inference(quant_intro,[status(thm)],[77])).
% 147.54/147.73 tff(79,plain,
% 147.54/147.73 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X)))) <=> ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 147.54/147.73 inference(rewrite,[status(thm)],[])).
% 147.54/147.73 tff(80,plain,
% 147.54/147.73 (^[Z: $i, X: $i] : rewrite(((~member(Z, complement(X))) | (~member(Z, X))) <=> ((~member(Z, X)) | (~member(Z, complement(X)))))),
% 147.54/147.73 inference(bind,[status(th)],[])).
% 147.54/147.73 tff(81,plain,
% 147.54/147.73 (![Z: $i, X: $i] : ((~member(Z, complement(X))) | (~member(Z, X))) <=> ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 147.54/147.73 inference(quant_intro,[status(thm)],[80])).
% 147.54/147.73 tff(82,axiom,(![Z: $i, X: $i] : ((~member(Z, complement(X))) | (~member(Z, X)))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','complement1')).
% 147.54/147.73 tff(83,plain,
% 147.54/147.73 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[82, 81])).
% 147.54/147.73 tff(84,plain,
% 147.54/147.73 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[83, 79])).
% 147.54/147.73 tff(85,plain,(
% 147.54/147.73 ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 147.54/147.73 inference(skolemize,[status(sab)],[84])).
% 147.54/147.73 tff(86,plain,
% 147.54/147.73 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[85, 78])).
% 147.54/147.73 tff(87,plain,
% 147.54/147.73 (((~![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)))))),
% 147.54/147.73 inference(rewrite,[status(thm)],[])).
% 147.54/147.73 tff(88,plain,
% 147.54/147.73 ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | ((~member(null_class, singleton(null_class))) | (~member(null_class, complement(singleton(null_class)))))),
% 147.54/147.73 inference(quant_inst,[status(thm)],[])).
% 147.54/147.73 tff(89,plain,
% 147.54/147.73 ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | (~member(null_class, singleton(null_class))) | (~member(null_class, complement(singleton(null_class))))),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[88, 87])).
% 147.54/147.73 tff(90,plain,
% 147.54/147.73 ($false),
% 147.54/147.73 inference(unit_resolution,[status(thm)],[89, 86, 76, 51])).
% 147.54/147.73 tff(91,plain,(~member(null_class, universal_class)), inference(lemma,lemma(discharge,[]))).
% 147.54/147.73 tff(92,plain,
% 147.54/147.73 (inductive(omega) <=> inductive(omega)),
% 147.54/147.73 inference(rewrite,[status(thm)],[])).
% 147.54/147.73 tff(93,axiom,(inductive(omega)), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','omega_is_inductive1')).
% 147.54/147.73 tff(94,plain,
% 147.54/147.73 (inductive(omega)),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[93, 92])).
% 147.54/147.73 tff(95,plain,
% 147.54/147.73 (^[X: $i] : refl(((~inductive(X)) | member(null_class, X)) <=> ((~inductive(X)) | member(null_class, X)))),
% 147.54/147.73 inference(bind,[status(th)],[])).
% 147.54/147.73 tff(96,plain,
% 147.54/147.73 (![X: $i] : ((~inductive(X)) | member(null_class, X)) <=> ![X: $i] : ((~inductive(X)) | member(null_class, X))),
% 147.54/147.73 inference(quant_intro,[status(thm)],[95])).
% 147.54/147.73 tff(97,plain,
% 147.54/147.73 (![X: $i] : ((~inductive(X)) | member(null_class, X)) <=> ![X: $i] : ((~inductive(X)) | member(null_class, X))),
% 147.54/147.73 inference(rewrite,[status(thm)],[])).
% 147.54/147.73 tff(98,axiom,(![X: $i] : ((~inductive(X)) | member(null_class, X))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','inductive1')).
% 147.54/147.73 tff(99,plain,
% 147.54/147.73 (![X: $i] : ((~inductive(X)) | member(null_class, X))),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[98, 97])).
% 147.54/147.73 tff(100,plain,(
% 147.54/147.73 ![X: $i] : ((~inductive(X)) | member(null_class, X))),
% 147.54/147.73 inference(skolemize,[status(sab)],[99])).
% 147.54/147.73 tff(101,plain,
% 147.54/147.73 (![X: $i] : ((~inductive(X)) | member(null_class, X))),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[100, 96])).
% 147.54/147.73 tff(102,plain,
% 147.54/147.73 (((~![X: $i] : ((~inductive(X)) | member(null_class, X))) | ((~inductive(omega)) | member(null_class, omega))) <=> ((~![X: $i] : ((~inductive(X)) | member(null_class, X))) | (~inductive(omega)) | member(null_class, omega))),
% 147.54/147.73 inference(rewrite,[status(thm)],[])).
% 147.54/147.73 tff(103,plain,
% 147.54/147.73 ((~![X: $i] : ((~inductive(X)) | member(null_class, X))) | ((~inductive(omega)) | member(null_class, omega))),
% 147.54/147.73 inference(quant_inst,[status(thm)],[])).
% 147.54/147.73 tff(104,plain,
% 147.54/147.73 ((~![X: $i] : ((~inductive(X)) | member(null_class, X))) | (~inductive(omega)) | member(null_class, omega)),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[103, 102])).
% 147.54/147.73 tff(105,plain,
% 147.54/147.73 (member(null_class, omega)),
% 147.54/147.73 inference(unit_resolution,[status(thm)],[104, 101, 94])).
% 147.54/147.73 tff(106,plain,
% 147.54/147.73 (^[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))))),
% 147.54/147.73 inference(bind,[status(th)],[])).
% 147.54/147.73 tff(107,plain,
% 147.54/147.73 (![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)))),
% 147.54/147.73 inference(quant_intro,[status(thm)],[106])).
% 147.54/147.73 tff(108,plain,
% 147.54/147.73 (![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)))),
% 147.54/147.73 inference(rewrite,[status(thm)],[])).
% 147.54/147.73 tff(109,plain,
% 147.54/147.73 (^[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)))))),
% 147.54/147.73 inference(bind,[status(th)],[])).
% 147.54/147.73 tff(110,plain,
% 147.54/147.73 (![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)))),
% 147.54/147.73 inference(quant_intro,[status(thm)],[109])).
% 147.54/147.73 tff(111,axiom,(![Y: $i, U: $i, X: $i] : (((~subclass(X, Y)) | (~member(U, X))) | member(U, Y))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','subclass_members')).
% 147.54/147.73 tff(112,plain,
% 147.54/147.73 (![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[111, 110])).
% 147.54/147.73 tff(113,plain,
% 147.54/147.73 (![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[112, 108])).
% 147.54/147.73 tff(114,plain,(
% 147.54/147.73 ![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 147.54/147.73 inference(skolemize,[status(sab)],[113])).
% 147.54/147.73 tff(115,plain,
% 147.54/147.73 (![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 147.54/147.73 inference(modus_ponens,[status(thm)],[114, 107])).
% 147.54/147.73 tff(116,plain,
% 147.54/147.73 (((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | ((~member(null_class, omega)) | member(null_class, universal_class) | (~subclass(omega, universal_class)))) <=> ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (~member(null_class, omega)) | member(null_class, universal_class) | (~subclass(omega, universal_class)))),
% 147.54/147.73 inference(rewrite,[status(thm)],[])).
% 147.54/147.73 tff(117,plain,
% 147.54/147.73 ((member(null_class, universal_class) | (~member(null_class, omega)) | (~subclass(omega, universal_class))) <=> ((~member(null_class, omega)) | member(null_class, universal_class) | (~subclass(omega, universal_class)))),
% 147.54/147.73 inference(rewrite,[status(thm)],[])).
% 147.54/147.73 tff(118,plain,
% 147.54/147.73 (((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (member(null_class, universal_class) | (~member(null_class, omega)) | (~subclass(omega, universal_class)))) <=> ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | ((~member(null_class, omega)) | member(null_class, universal_class) | (~subclass(omega, universal_class))))),
% 147.54/147.73 inference(monotonicity,[status(thm)],[117])).
% 147.54/147.73 tff(119,plain,
% 147.54/147.73 (((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (member(null_class, universal_class) | (~member(null_class, omega)) | (~subclass(omega, universal_class)))) <=> ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (~member(null_class, omega)) | member(null_class, universal_class) | (~subclass(omega, universal_class)))),
% 147.54/147.73 inference(transitivity,[status(thm)],[118, 116])).
% 147.54/147.73 tff(120,plain,
% 147.54/147.73 ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (member(null_class, universal_class) | (~member(null_class, omega)) | (~subclass(omega, universal_class)))),
% 147.54/147.73 inference(quant_inst,[status(thm)],[])).
% 147.54/147.73 tff(121,plain,
% 147.54/147.73 ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (~member(null_class, omega)) | member(null_class, universal_class) | (~subclass(omega, universal_class))),
% 148.16/148.33 inference(modus_ponens,[status(thm)],[120, 119])).
% 148.16/148.33 tff(122,plain,
% 148.16/148.33 (~subclass(omega, universal_class)),
% 148.16/148.33 inference(unit_resolution,[status(thm)],[121, 115, 105, 91])).
% 148.16/148.33 tff(123,plain,
% 148.16/148.33 (^[X: $i] : refl(subclass(X, universal_class) <=> subclass(X, universal_class))),
% 148.16/148.33 inference(bind,[status(th)],[])).
% 148.16/148.33 tff(124,plain,
% 148.16/148.33 (![X: $i] : subclass(X, universal_class) <=> ![X: $i] : subclass(X, universal_class)),
% 148.16/148.33 inference(quant_intro,[status(thm)],[123])).
% 148.16/148.33 tff(125,plain,
% 148.16/148.33 (![X: $i] : subclass(X, universal_class) <=> ![X: $i] : subclass(X, universal_class)),
% 148.16/148.33 inference(rewrite,[status(thm)],[])).
% 148.16/148.33 tff(126,axiom,(![X: $i] : subclass(X, universal_class)), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','class_elements_are_sets')).
% 148.16/148.33 tff(127,plain,
% 148.16/148.33 (![X: $i] : subclass(X, universal_class)),
% 148.16/148.33 inference(modus_ponens,[status(thm)],[126, 125])).
% 148.16/148.33 tff(128,plain,(
% 148.16/148.33 ![X: $i] : subclass(X, universal_class)),
% 148.16/148.33 inference(skolemize,[status(sab)],[127])).
% 148.16/148.33 tff(129,plain,
% 148.16/148.33 (![X: $i] : subclass(X, universal_class)),
% 148.16/148.33 inference(modus_ponens,[status(thm)],[128, 124])).
% 148.16/148.33 tff(130,plain,
% 148.16/148.33 ((~![X: $i] : subclass(X, universal_class)) | subclass(omega, universal_class)),
% 148.16/148.33 inference(quant_inst,[status(thm)],[])).
% 148.16/148.33 tff(131,plain,
% 148.16/148.33 ($false),
% 148.16/148.33 inference(unit_resolution,[status(thm)],[130, 129, 122])).
% 148.16/148.33 % SZS output end Proof
% 148.16/148.33 % E exiting
%------------------------------------------------------------------------------