TSTP Solution File: NUM159-1 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : NUM159-1 : TPTP v8.1.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n016.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 : Sun Sep 18 13:03:03 EDT 2022
% Result : Unsatisfiable 104.91s 67.63s
% Output : Proof 104.91s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.11 % Problem : NUM159-1 : TPTP v8.1.0. Bugfixed v2.1.0.
% 0.06/0.12 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.33 % Computer : n016.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 Sep 2 08:09:05 EDT 2022
% 0.12/0.33 % CPUTime :
% 0.12/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.34 Usage: tptp [options] [-file:]file
% 0.12/0.34 -h, -? prints this message.
% 0.12/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.34 -m, -model generate model.
% 0.12/0.34 -p, -proof generate proof.
% 0.12/0.34 -c, -core generate unsat core of named formulas.
% 0.12/0.34 -st, -statistics display statistics.
% 0.12/0.34 -t:timeout set timeout (in second).
% 0.12/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.34 -<param>:<value> configuration parameter and value.
% 0.12/0.34 -o:<output-file> file to place output in.
% 104.91/67.63 % SZS status Unsatisfiable
% 104.91/67.63 % SZS output start Proof
% 104.91/67.63 tff(member_type, type, (
% 104.91/67.63 member: ( $i * $i ) > $o)).
% 104.91/67.63 tff(complement_type, type, (
% 104.91/67.63 complement: $i > $i)).
% 104.91/67.63 tff(image_type, type, (
% 104.91/67.63 image: ( $i * $i ) > $i)).
% 104.91/67.63 tff(ordinal_numbers_type, type, (
% 104.91/67.63 ordinal_numbers: $i)).
% 104.91/67.63 tff(successor_relation_type, type, (
% 104.91/67.63 successor_relation: $i)).
% 104.91/67.63 tff(not_subclass_element_type, type, (
% 104.91/67.63 not_subclass_element: ( $i * $i ) > $i)).
% 104.91/67.63 tff(union_type, type, (
% 104.91/67.63 union: ( $i * $i ) > $i)).
% 104.91/67.63 tff(singleton_type, type, (
% 104.91/67.63 singleton: $i > $i)).
% 104.91/67.63 tff(null_class_type, type, (
% 104.91/67.63 null_class: $i)).
% 104.91/67.63 tff(intersection_type, type, (
% 104.91/67.63 intersection: ( $i * $i ) > $i)).
% 104.91/67.63 tff(universal_class_type, type, (
% 104.91/67.63 universal_class: $i)).
% 104.91/67.63 tff(subclass_type, type, (
% 104.91/67.63 subclass: ( $i * $i ) > $o)).
% 104.91/67.63 tff(kind_1_ordinals_type, type, (
% 104.91/67.63 kind_1_ordinals: $i)).
% 104.91/67.63 tff(1,assumption,(~subclass(image(successor_relation, ordinal_numbers), universal_class)), introduced(assumption)).
% 104.91/67.63 tff(2,plain,
% 104.91/67.63 (^[X: $i] : refl(subclass(X, universal_class) <=> subclass(X, universal_class))),
% 104.91/67.63 inference(bind,[status(th)],[])).
% 104.91/67.63 tff(3,plain,
% 104.91/67.63 (![X: $i] : subclass(X, universal_class) <=> ![X: $i] : subclass(X, universal_class)),
% 104.91/67.63 inference(quant_intro,[status(thm)],[2])).
% 104.91/67.63 tff(4,plain,
% 104.91/67.63 (![X: $i] : subclass(X, universal_class) <=> ![X: $i] : subclass(X, universal_class)),
% 104.91/67.63 inference(rewrite,[status(thm)],[])).
% 104.91/67.63 tff(5,axiom,(![X: $i] : subclass(X, universal_class)), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','class_elements_are_sets')).
% 104.91/67.63 tff(6,plain,
% 104.91/67.63 (![X: $i] : subclass(X, universal_class)),
% 104.91/67.63 inference(modus_ponens,[status(thm)],[5, 4])).
% 104.91/67.63 tff(7,plain,(
% 104.91/67.63 ![X: $i] : subclass(X, universal_class)),
% 104.91/67.63 inference(skolemize,[status(sab)],[6])).
% 104.91/67.63 tff(8,plain,
% 104.91/67.63 (![X: $i] : subclass(X, universal_class)),
% 104.91/67.63 inference(modus_ponens,[status(thm)],[7, 3])).
% 104.91/67.63 tff(9,plain,
% 104.91/67.63 ((~![X: $i] : subclass(X, universal_class)) | subclass(image(successor_relation, ordinal_numbers), universal_class)),
% 104.91/67.63 inference(quant_inst,[status(thm)],[])).
% 104.91/67.63 tff(10,plain,
% 104.91/67.63 ($false),
% 104.91/67.63 inference(unit_resolution,[status(thm)],[9, 8, 1])).
% 104.91/67.63 tff(11,plain,(subclass(image(successor_relation, ordinal_numbers), universal_class)), inference(lemma,lemma(discharge,[]))).
% 104.91/67.63 tff(12,assumption,(~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), universal_class)), introduced(assumption)).
% 104.91/67.63 tff(13,plain,
% 104.91/67.63 ((~subclass(image(successor_relation, ordinal_numbers), kind_1_ordinals)) <=> (~subclass(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))))),
% 104.91/67.63 inference(rewrite,[status(thm)],[])).
% 104.91/67.63 tff(14,plain,
% 104.91/67.63 ((~subclass(image(successor_relation, ordinal_numbers), kind_1_ordinals)) <=> (~subclass(image(successor_relation, ordinal_numbers), kind_1_ordinals))),
% 104.91/67.63 inference(rewrite,[status(thm)],[])).
% 104.91/67.63 tff(15,axiom,(~subclass(image(successor_relation, ordinal_numbers), kind_1_ordinals)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_corollary_to_kind_1_ordinal_1')).
% 104.91/67.63 tff(16,plain,
% 104.91/67.63 (~subclass(image(successor_relation, ordinal_numbers), kind_1_ordinals)),
% 104.91/67.63 inference(modus_ponens,[status(thm)],[15, 14])).
% 104.91/67.63 tff(17,plain,
% 104.91/67.63 (~subclass(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers)))),
% 104.91/67.63 inference(modus_ponens,[status(thm)],[16, 13])).
% 104.91/67.63 tff(18,plain,
% 104.91/67.63 (^[Y: $i, X: $i] : refl((subclass(X, Y) | member(not_subclass_element(X, Y), X)) <=> (subclass(X, Y) | member(not_subclass_element(X, Y), X)))),
% 104.91/67.63 inference(bind,[status(th)],[])).
% 104.91/67.63 tff(19,plain,
% 104.91/67.63 (![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X)) <=> ![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))),
% 104.91/67.63 inference(quant_intro,[status(thm)],[18])).
% 104.91/67.63 tff(20,plain,
% 104.91/67.63 (![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X)) <=> ![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))),
% 104.91/67.63 inference(rewrite,[status(thm)],[])).
% 104.91/67.63 tff(21,plain,
% 104.91/67.63 (^[Y: $i, X: $i] : rewrite((member(not_subclass_element(X, Y), X) | subclass(X, Y)) <=> (subclass(X, Y) | member(not_subclass_element(X, Y), X)))),
% 104.91/67.63 inference(bind,[status(th)],[])).
% 104.91/67.63 tff(22,plain,
% 104.91/67.63 (![Y: $i, X: $i] : (member(not_subclass_element(X, Y), X) | subclass(X, Y)) <=> ![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))),
% 104.91/67.63 inference(quant_intro,[status(thm)],[21])).
% 104.91/67.63 tff(23,axiom,(![Y: $i, X: $i] : (member(not_subclass_element(X, Y), X) | subclass(X, Y))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','not_subclass_members1')).
% 104.91/67.63 tff(24,plain,
% 104.91/67.63 (![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))),
% 104.91/67.63 inference(modus_ponens,[status(thm)],[23, 22])).
% 104.91/67.63 tff(25,plain,
% 104.91/67.63 (![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))),
% 104.91/67.63 inference(modus_ponens,[status(thm)],[24, 20])).
% 104.91/67.63 tff(26,plain,(
% 104.91/67.63 ![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))),
% 104.91/67.63 inference(skolemize,[status(sab)],[25])).
% 104.91/67.63 tff(27,plain,
% 104.91/67.63 (![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))),
% 104.91/67.63 inference(modus_ponens,[status(thm)],[26, 19])).
% 104.91/67.63 tff(28,plain,
% 104.91/67.63 (((~![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))) | (subclass(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))) | member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), image(successor_relation, ordinal_numbers)))) <=> ((~![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))) | subclass(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))) | member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), image(successor_relation, ordinal_numbers)))),
% 104.91/67.63 inference(rewrite,[status(thm)],[])).
% 104.91/67.63 tff(29,plain,
% 104.91/67.63 ((~![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))) | (subclass(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))) | member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), image(successor_relation, ordinal_numbers)))),
% 104.91/67.63 inference(quant_inst,[status(thm)],[])).
% 104.91/67.63 tff(30,plain,
% 104.91/67.63 ((~![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))) | subclass(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))) | member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), image(successor_relation, ordinal_numbers))),
% 104.91/67.63 inference(modus_ponens,[status(thm)],[29, 28])).
% 104.91/67.63 tff(31,plain,
% 104.91/67.63 (member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), image(successor_relation, ordinal_numbers))),
% 104.91/67.63 inference(unit_resolution,[status(thm)],[30, 27, 17])).
% 104.91/67.63 tff(32,plain,
% 104.91/67.63 (^[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))))),
% 104.91/67.63 inference(bind,[status(th)],[])).
% 104.91/67.63 tff(33,plain,
% 104.91/67.63 (![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)))),
% 104.91/67.63 inference(quant_intro,[status(thm)],[32])).
% 104.91/67.63 tff(34,plain,
% 104.91/67.63 (![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)))),
% 104.91/67.63 inference(rewrite,[status(thm)],[])).
% 104.91/67.63 tff(35,plain,
% 104.91/67.63 (^[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)))))),
% 104.91/67.63 inference(bind,[status(th)],[])).
% 104.91/67.63 tff(36,plain,
% 104.91/67.63 (![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)))),
% 104.91/67.63 inference(quant_intro,[status(thm)],[35])).
% 104.91/67.63 tff(37,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')).
% 104.91/67.63 tff(38,plain,
% 104.91/67.63 (![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 104.91/67.63 inference(modus_ponens,[status(thm)],[37, 36])).
% 104.91/67.63 tff(39,plain,
% 104.91/67.63 (![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 104.91/67.63 inference(modus_ponens,[status(thm)],[38, 34])).
% 104.91/67.63 tff(40,plain,(
% 104.91/67.63 ![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 104.91/67.63 inference(skolemize,[status(sab)],[39])).
% 104.91/67.63 tff(41,plain,
% 104.91/67.63 (![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 104.91/67.63 inference(modus_ponens,[status(thm)],[40, 33])).
% 104.91/67.63 tff(42,plain,
% 104.91/67.63 (((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), universal_class) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), image(successor_relation, ordinal_numbers))) | (~subclass(image(successor_relation, ordinal_numbers), universal_class)))) <=> ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), universal_class) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), image(successor_relation, ordinal_numbers))) | (~subclass(image(successor_relation, ordinal_numbers), universal_class)))),
% 104.91/67.63 inference(rewrite,[status(thm)],[])).
% 104.91/67.63 tff(43,plain,
% 104.91/67.63 ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), universal_class) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), image(successor_relation, ordinal_numbers))) | (~subclass(image(successor_relation, ordinal_numbers), universal_class)))),
% 104.91/67.63 inference(quant_inst,[status(thm)],[])).
% 104.91/67.63 tff(44,plain,
% 104.91/67.63 ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), universal_class) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), image(successor_relation, ordinal_numbers))) | (~subclass(image(successor_relation, ordinal_numbers), universal_class))),
% 104.91/67.63 inference(modus_ponens,[status(thm)],[43, 42])).
% 104.91/67.63 tff(45,plain,
% 104.91/67.63 ($false),
% 104.91/67.63 inference(unit_resolution,[status(thm)],[44, 41, 31, 12, 11])).
% 104.91/67.63 tff(46,plain,(member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), universal_class)), inference(lemma,lemma(discharge,[]))).
% 104.91/67.63 tff(47,plain,
% 104.91/67.63 (^[Y: $i, X: $i] : refl((complement(intersection(complement(X), complement(Y))) = union(X, Y)) <=> (complement(intersection(complement(X), complement(Y))) = union(X, Y)))),
% 104.91/67.63 inference(bind,[status(th)],[])).
% 104.91/67.63 tff(48,plain,
% 104.91/67.63 (![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))),
% 104.91/67.63 inference(quant_intro,[status(thm)],[47])).
% 104.91/67.63 tff(49,plain,
% 104.91/67.63 (![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))),
% 104.91/67.63 inference(rewrite,[status(thm)],[])).
% 104.91/67.63 tff(50,axiom,(![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','union')).
% 104.91/67.63 tff(51,plain,
% 104.91/67.63 (![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))),
% 104.91/67.63 inference(modus_ponens,[status(thm)],[50, 49])).
% 104.91/67.63 tff(52,plain,(
% 104.91/67.63 ![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))),
% 104.91/67.63 inference(skolemize,[status(sab)],[51])).
% 104.91/67.63 tff(53,plain,
% 104.91/67.63 (![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))),
% 104.91/67.63 inference(modus_ponens,[status(thm)],[52, 48])).
% 104.91/67.63 tff(54,plain,
% 104.91/67.63 ((~![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)))),
% 104.91/67.63 inference(quant_inst,[status(thm)],[])).
% 104.91/67.63 tff(55,plain,
% 104.91/67.63 (complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) = union(singleton(null_class), image(successor_relation, ordinal_numbers))),
% 104.91/67.63 inference(unit_resolution,[status(thm)],[54, 53])).
% 104.91/67.63 tff(56,plain,
% 104.91/67.63 (member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))) <=> member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), union(singleton(null_class), image(successor_relation, ordinal_numbers)))),
% 104.91/67.63 inference(monotonicity,[status(thm)],[55])).
% 104.91/67.63 tff(57,plain,
% 104.91/67.63 (member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), union(singleton(null_class), image(successor_relation, ordinal_numbers))) <=> member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))))),
% 104.91/67.63 inference(symmetry,[status(thm)],[56])).
% 104.91/67.63 tff(58,plain,
% 104.91/67.63 ((~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), union(singleton(null_class), image(successor_relation, ordinal_numbers)))) <=> (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))))),
% 104.91/67.63 inference(monotonicity,[status(thm)],[57])).
% 104.91/67.63 tff(59,plain,
% 104.91/67.63 (^[Y: $i, X: $i] : refl(((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y)) <=> ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y)))),
% 104.91/67.63 inference(bind,[status(th)],[])).
% 104.91/67.63 tff(60,plain,
% 104.91/67.63 (![Y: $i, X: $i] : ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y)) <=> ![Y: $i, X: $i] : ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y))),
% 104.91/67.63 inference(quant_intro,[status(thm)],[59])).
% 104.91/67.63 tff(61,plain,
% 104.91/67.63 (![Y: $i, X: $i] : ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y)) <=> ![Y: $i, X: $i] : ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y))),
% 104.91/67.63 inference(rewrite,[status(thm)],[])).
% 104.91/67.63 tff(62,axiom,(![Y: $i, X: $i] : ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','not_subclass_members2')).
% 104.91/67.63 tff(63,plain,
% 104.91/67.63 (![Y: $i, X: $i] : ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y))),
% 104.91/67.63 inference(modus_ponens,[status(thm)],[62, 61])).
% 104.91/67.63 tff(64,plain,(
% 104.91/67.63 ![Y: $i, X: $i] : ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y))),
% 104.91/67.63 inference(skolemize,[status(sab)],[63])).
% 104.91/67.63 tff(65,plain,
% 104.91/67.63 (![Y: $i, X: $i] : ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y))),
% 104.91/67.63 inference(modus_ponens,[status(thm)],[64, 60])).
% 104.91/67.63 tff(66,plain,
% 104.91/67.63 (((~![Y: $i, X: $i] : ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y))) | ((~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), union(singleton(null_class), image(successor_relation, ordinal_numbers)))) | subclass(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))))) <=> ((~![Y: $i, X: $i] : ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y))) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), union(singleton(null_class), image(successor_relation, ordinal_numbers)))) | subclass(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))))),
% 104.91/67.63 inference(rewrite,[status(thm)],[])).
% 104.91/67.63 tff(67,plain,
% 104.91/67.63 ((~![Y: $i, X: $i] : ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y))) | ((~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), union(singleton(null_class), image(successor_relation, ordinal_numbers)))) | subclass(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))))),
% 104.91/67.63 inference(quant_inst,[status(thm)],[])).
% 104.91/67.63 tff(68,plain,
% 104.91/67.63 ((~![Y: $i, X: $i] : ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y))) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), union(singleton(null_class), image(successor_relation, ordinal_numbers)))) | subclass(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers)))),
% 104.91/67.63 inference(modus_ponens,[status(thm)],[67, 66])).
% 104.91/67.63 tff(69,plain,
% 104.91/67.63 (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), union(singleton(null_class), image(successor_relation, ordinal_numbers)))),
% 104.91/67.63 inference(unit_resolution,[status(thm)],[68, 65, 17])).
% 104.91/67.63 tff(70,plain,
% 104.91/67.63 (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))))),
% 104.91/67.63 inference(modus_ponens,[status(thm)],[69, 58])).
% 104.91/67.63 tff(71,plain,
% 104.91/67.63 (^[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))))),
% 104.91/67.63 inference(bind,[status(th)],[])).
% 104.91/67.63 tff(72,plain,
% 104.91/67.63 (![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)))),
% 104.91/67.63 inference(quant_intro,[status(thm)],[71])).
% 104.91/67.63 tff(73,plain,
% 104.91/67.63 (![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)))),
% 104.91/67.63 inference(rewrite,[status(thm)],[])).
% 104.91/67.63 tff(74,plain,
% 104.91/67.63 (^[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))))),
% 104.91/67.63 inference(bind,[status(th)],[])).
% 104.91/67.63 tff(75,plain,
% 104.91/67.63 (![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)))),
% 104.91/67.63 inference(quant_intro,[status(thm)],[74])).
% 104.91/67.63 tff(76,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')).
% 104.91/67.63 tff(77,plain,
% 104.91/67.63 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 104.91/67.63 inference(modus_ponens,[status(thm)],[76, 75])).
% 104.91/67.63 tff(78,plain,
% 104.91/67.63 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 104.91/67.63 inference(modus_ponens,[status(thm)],[77, 73])).
% 104.91/67.63 tff(79,plain,(
% 104.91/67.63 ![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 104.91/67.63 inference(skolemize,[status(sab)],[78])).
% 104.91/67.63 tff(80,plain,
% 104.91/67.63 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 104.91/67.63 inference(modus_ponens,[status(thm)],[79, 72])).
% 104.91/67.63 tff(81,plain,
% 104.91/67.63 (((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | (member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), universal_class)) | member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), 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(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), universal_class)) | member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))))),
% 104.91/67.63 inference(rewrite,[status(thm)],[])).
% 104.91/67.63 tff(82,plain,
% 104.91/67.63 ((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | (member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), universal_class)) | member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))))),
% 104.91/67.63 inference(quant_inst,[status(thm)],[])).
% 104.91/67.63 tff(83,plain,
% 104.91/67.63 ((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), universal_class)) | member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))))),
% 104.91/67.64 inference(modus_ponens,[status(thm)],[82, 81])).
% 104.91/67.64 tff(84,plain,
% 104.91/67.64 (member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), universal_class)) | member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))))),
% 104.91/67.64 inference(unit_resolution,[status(thm)],[83, 80])).
% 104.91/67.64 tff(85,plain,
% 104.91/67.64 (member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), universal_class))),
% 104.91/67.64 inference(unit_resolution,[status(thm)],[84, 70])).
% 104.91/67.64 tff(86,plain,
% 104.91/67.64 (member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))),
% 104.91/67.64 inference(unit_resolution,[status(thm)],[85, 46])).
% 104.91/67.64 tff(87,plain,
% 104.91/67.64 (^[Z: $i, Y: $i, X: $i] : refl(((~member(Z, intersection(X, Y))) | member(Z, Y)) <=> ((~member(Z, intersection(X, Y))) | member(Z, Y)))),
% 104.91/67.64 inference(bind,[status(th)],[])).
% 104.91/67.64 tff(88,plain,
% 104.91/67.64 (![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, Y)) <=> ![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, Y))),
% 104.91/67.64 inference(quant_intro,[status(thm)],[87])).
% 104.91/67.64 tff(89,plain,
% 104.91/67.64 (![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, Y)) <=> ![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, Y))),
% 104.91/67.64 inference(rewrite,[status(thm)],[])).
% 104.91/67.64 tff(90,axiom,(![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, Y))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','intersection2')).
% 104.91/67.64 tff(91,plain,
% 104.91/67.64 (![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, Y))),
% 104.91/67.64 inference(modus_ponens,[status(thm)],[90, 89])).
% 104.91/67.64 tff(92,plain,(
% 104.91/67.64 ![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, Y))),
% 104.91/67.64 inference(skolemize,[status(sab)],[91])).
% 104.91/67.64 tff(93,plain,
% 104.91/67.64 (![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, Y))),
% 104.91/67.64 inference(modus_ponens,[status(thm)],[92, 88])).
% 104.91/67.64 tff(94,plain,
% 104.91/67.64 (((~![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, Y))) | ((~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))) | member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), complement(image(successor_relation, ordinal_numbers))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, Y))) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))) | member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), complement(image(successor_relation, ordinal_numbers))))),
% 104.91/67.64 inference(rewrite,[status(thm)],[])).
% 104.91/67.64 tff(95,plain,
% 104.91/67.64 ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, Y))) | ((~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))) | member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), complement(image(successor_relation, ordinal_numbers))))),
% 104.91/67.64 inference(quant_inst,[status(thm)],[])).
% 104.91/67.64 tff(96,plain,
% 104.91/67.64 ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, Y))) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))) | member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), complement(image(successor_relation, ordinal_numbers)))),
% 104.91/67.64 inference(modus_ponens,[status(thm)],[95, 94])).
% 104.91/67.64 tff(97,plain,
% 104.91/67.64 (member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), complement(image(successor_relation, ordinal_numbers)))),
% 104.91/67.64 inference(unit_resolution,[status(thm)],[96, 93, 86])).
% 104.91/67.64 tff(98,plain,
% 104.91/67.64 (^[Z: $i, X: $i] : refl(((~member(Z, X)) | (~member(Z, complement(X)))) <=> ((~member(Z, X)) | (~member(Z, complement(X)))))),
% 104.91/67.64 inference(bind,[status(th)],[])).
% 104.91/67.64 tff(99,plain,
% 104.91/67.64 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X)))) <=> ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 104.91/67.64 inference(quant_intro,[status(thm)],[98])).
% 104.91/67.64 tff(100,plain,
% 104.91/67.64 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X)))) <=> ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 104.91/67.64 inference(rewrite,[status(thm)],[])).
% 104.91/67.64 tff(101,plain,
% 104.91/67.64 (^[Z: $i, X: $i] : rewrite(((~member(Z, complement(X))) | (~member(Z, X))) <=> ((~member(Z, X)) | (~member(Z, complement(X)))))),
% 104.91/67.64 inference(bind,[status(th)],[])).
% 104.91/67.64 tff(102,plain,
% 104.91/67.64 (![Z: $i, X: $i] : ((~member(Z, complement(X))) | (~member(Z, X))) <=> ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 104.91/67.64 inference(quant_intro,[status(thm)],[101])).
% 104.91/67.64 tff(103,axiom,(![Z: $i, X: $i] : ((~member(Z, complement(X))) | (~member(Z, X)))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','complement1')).
% 104.91/67.64 tff(104,plain,
% 104.91/67.64 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 104.91/67.64 inference(modus_ponens,[status(thm)],[103, 102])).
% 104.91/67.64 tff(105,plain,
% 104.91/67.64 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 104.91/67.64 inference(modus_ponens,[status(thm)],[104, 100])).
% 104.91/67.64 tff(106,plain,(
% 104.91/67.64 ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 104.91/67.64 inference(skolemize,[status(sab)],[105])).
% 104.91/67.64 tff(107,plain,
% 104.91/67.64 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 104.91/67.64 inference(modus_ponens,[status(thm)],[106, 99])).
% 104.91/67.64 tff(108,plain,
% 104.91/67.64 (((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | ((~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), complement(image(successor_relation, ordinal_numbers)))) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), image(successor_relation, ordinal_numbers))))) <=> ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), complement(image(successor_relation, ordinal_numbers)))) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), image(successor_relation, ordinal_numbers))))),
% 104.91/67.64 inference(rewrite,[status(thm)],[])).
% 104.91/67.64 tff(109,plain,
% 104.91/67.64 (((~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), image(successor_relation, ordinal_numbers))) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), complement(image(successor_relation, ordinal_numbers))))) <=> ((~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), complement(image(successor_relation, ordinal_numbers)))) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), image(successor_relation, ordinal_numbers))))),
% 104.91/67.64 inference(rewrite,[status(thm)],[])).
% 104.91/67.64 tff(110,plain,
% 104.91/67.64 (((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | ((~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), image(successor_relation, ordinal_numbers))) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), complement(image(successor_relation, ordinal_numbers)))))) <=> ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | ((~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), complement(image(successor_relation, ordinal_numbers)))) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), image(successor_relation, ordinal_numbers)))))),
% 104.91/67.64 inference(monotonicity,[status(thm)],[109])).
% 104.91/67.64 tff(111,plain,
% 104.91/67.64 (((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | ((~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), image(successor_relation, ordinal_numbers))) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), complement(image(successor_relation, ordinal_numbers)))))) <=> ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), complement(image(successor_relation, ordinal_numbers)))) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), image(successor_relation, ordinal_numbers))))),
% 104.91/67.64 inference(transitivity,[status(thm)],[110, 108])).
% 104.91/67.64 tff(112,plain,
% 104.91/67.64 ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | ((~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), image(successor_relation, ordinal_numbers))) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), complement(image(successor_relation, ordinal_numbers)))))),
% 104.91/67.65 inference(quant_inst,[status(thm)],[])).
% 104.91/67.65 tff(113,plain,
% 104.91/67.65 ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), complement(image(successor_relation, ordinal_numbers)))) | (~member(not_subclass_element(image(successor_relation, ordinal_numbers), union(singleton(null_class), image(successor_relation, ordinal_numbers))), image(successor_relation, ordinal_numbers)))),
% 104.91/67.65 inference(modus_ponens,[status(thm)],[112, 111])).
% 104.91/67.65 tff(114,plain,
% 104.91/67.65 ($false),
% 104.91/67.65 inference(unit_resolution,[status(thm)],[113, 107, 31, 97])).
% 104.91/67.65 % SZS output end Proof
%------------------------------------------------------------------------------