TSTP Solution File: NUM156-1 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : NUM156-1 : TPTP v8.1.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n024.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 13:03:02 EDT 2022
% Result : Unsatisfiable 1.67s 1.28s
% Output : Proof 1.67s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.10 % Problem : NUM156-1 : TPTP v8.1.0. Bugfixed v2.1.0.
% 0.06/0.11 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.10/0.31 % Computer : n024.cluster.edu
% 0.10/0.31 % Model : x86_64 x86_64
% 0.10/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.31 % Memory : 8042.1875MB
% 0.10/0.31 % OS : Linux 3.10.0-693.el7.x86_64
% 0.10/0.31 % CPULimit : 300
% 0.10/0.31 % WCLimit : 300
% 0.10/0.31 % DateTime : Fri Sep 2 07:47:18 EDT 2022
% 0.10/0.31 % CPUTime :
% 0.10/0.32 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.10/0.32 Usage: tptp [options] [-file:]file
% 0.10/0.32 -h, -? prints this message.
% 0.10/0.32 -smt2 print SMT-LIB2 benchmark.
% 0.10/0.32 -m, -model generate model.
% 0.10/0.32 -p, -proof generate proof.
% 0.10/0.32 -c, -core generate unsat core of named formulas.
% 0.10/0.32 -st, -statistics display statistics.
% 0.10/0.32 -t:timeout set timeout (in second).
% 0.10/0.32 -smt2status display status in smt2 format instead of SZS.
% 0.10/0.32 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.10/0.32 -<param>:<value> configuration parameter and value.
% 0.10/0.32 -o:<output-file> file to place output in.
% 1.67/1.28 % SZS status Unsatisfiable
% 1.67/1.28 % SZS output start Proof
% 1.67/1.28 tff(member_type, type, (
% 1.67/1.28 member: ( $i * $i ) > $o)).
% 1.67/1.28 tff(complement_type, type, (
% 1.67/1.28 complement: $i > $i)).
% 1.67/1.28 tff(singleton_type, type, (
% 1.67/1.28 singleton: $i > $i)).
% 1.67/1.28 tff(null_class_type, type, (
% 1.67/1.28 null_class: $i)).
% 1.67/1.28 tff(intersection_type, type, (
% 1.67/1.28 intersection: ( $i * $i ) > $i)).
% 1.67/1.28 tff(image_type, type, (
% 1.67/1.28 image: ( $i * $i ) > $i)).
% 1.67/1.28 tff(ordinal_numbers_type, type, (
% 1.67/1.28 ordinal_numbers: $i)).
% 1.67/1.28 tff(successor_relation_type, type, (
% 1.67/1.28 successor_relation: $i)).
% 1.67/1.28 tff(universal_class_type, type, (
% 1.67/1.28 universal_class: $i)).
% 1.67/1.28 tff(subclass_type, type, (
% 1.67/1.28 subclass: ( $i * $i ) > $o)).
% 1.67/1.28 tff(omega_type, type, (
% 1.67/1.28 omega: $i)).
% 1.67/1.28 tff(inductive_type, type, (
% 1.67/1.28 inductive: $i > $o)).
% 1.67/1.28 tff(union_type, type, (
% 1.67/1.28 union: ( $i * $i ) > $i)).
% 1.67/1.28 tff(kind_1_ordinals_type, type, (
% 1.67/1.28 kind_1_ordinals: $i)).
% 1.67/1.28 tff(unordered_pair_type, type, (
% 1.67/1.28 unordered_pair: ( $i * $i ) > $i)).
% 1.67/1.28 tff(1,assumption,(~subclass(omega, universal_class)), introduced(assumption)).
% 1.67/1.28 tff(2,plain,
% 1.67/1.28 (^[X: $i] : refl(subclass(X, universal_class) <=> subclass(X, universal_class))),
% 1.67/1.28 inference(bind,[status(th)],[])).
% 1.67/1.28 tff(3,plain,
% 1.67/1.28 (![X: $i] : subclass(X, universal_class) <=> ![X: $i] : subclass(X, universal_class)),
% 1.67/1.28 inference(quant_intro,[status(thm)],[2])).
% 1.67/1.28 tff(4,plain,
% 1.67/1.28 (![X: $i] : subclass(X, universal_class) <=> ![X: $i] : subclass(X, universal_class)),
% 1.67/1.28 inference(rewrite,[status(thm)],[])).
% 1.67/1.28 tff(5,axiom,(![X: $i] : subclass(X, universal_class)), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','class_elements_are_sets')).
% 1.67/1.28 tff(6,plain,
% 1.67/1.28 (![X: $i] : subclass(X, universal_class)),
% 1.67/1.28 inference(modus_ponens,[status(thm)],[5, 4])).
% 1.67/1.28 tff(7,plain,(
% 1.67/1.28 ![X: $i] : subclass(X, universal_class)),
% 1.67/1.28 inference(skolemize,[status(sab)],[6])).
% 1.67/1.28 tff(8,plain,
% 1.67/1.28 (![X: $i] : subclass(X, universal_class)),
% 1.67/1.28 inference(modus_ponens,[status(thm)],[7, 3])).
% 1.67/1.28 tff(9,plain,
% 1.67/1.28 ((~![X: $i] : subclass(X, universal_class)) | subclass(omega, universal_class)),
% 1.67/1.28 inference(quant_inst,[status(thm)],[])).
% 1.67/1.28 tff(10,plain,
% 1.67/1.28 ($false),
% 1.67/1.28 inference(unit_resolution,[status(thm)],[9, 8, 1])).
% 1.67/1.28 tff(11,plain,(subclass(omega, universal_class)), inference(lemma,lemma(discharge,[]))).
% 1.67/1.28 tff(12,assumption,(~member(null_class, universal_class)), introduced(assumption)).
% 1.67/1.28 tff(13,plain,
% 1.67/1.28 (inductive(omega) <=> inductive(omega)),
% 1.67/1.28 inference(rewrite,[status(thm)],[])).
% 1.67/1.28 tff(14,axiom,(inductive(omega)), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','omega_is_inductive1')).
% 1.67/1.28 tff(15,plain,
% 1.67/1.28 (inductive(omega)),
% 1.67/1.28 inference(modus_ponens,[status(thm)],[14, 13])).
% 1.67/1.28 tff(16,plain,
% 1.67/1.28 (^[X: $i] : refl(((~inductive(X)) | member(null_class, X)) <=> ((~inductive(X)) | member(null_class, X)))),
% 1.67/1.28 inference(bind,[status(th)],[])).
% 1.67/1.28 tff(17,plain,
% 1.67/1.28 (![X: $i] : ((~inductive(X)) | member(null_class, X)) <=> ![X: $i] : ((~inductive(X)) | member(null_class, X))),
% 1.67/1.28 inference(quant_intro,[status(thm)],[16])).
% 1.67/1.28 tff(18,plain,
% 1.67/1.28 (![X: $i] : ((~inductive(X)) | member(null_class, X)) <=> ![X: $i] : ((~inductive(X)) | member(null_class, X))),
% 1.67/1.28 inference(rewrite,[status(thm)],[])).
% 1.67/1.28 tff(19,axiom,(![X: $i] : ((~inductive(X)) | member(null_class, X))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','inductive1')).
% 1.67/1.28 tff(20,plain,
% 1.67/1.28 (![X: $i] : ((~inductive(X)) | member(null_class, X))),
% 1.67/1.28 inference(modus_ponens,[status(thm)],[19, 18])).
% 1.67/1.28 tff(21,plain,(
% 1.67/1.28 ![X: $i] : ((~inductive(X)) | member(null_class, X))),
% 1.67/1.28 inference(skolemize,[status(sab)],[20])).
% 1.67/1.28 tff(22,plain,
% 1.67/1.28 (![X: $i] : ((~inductive(X)) | member(null_class, X))),
% 1.67/1.28 inference(modus_ponens,[status(thm)],[21, 17])).
% 1.67/1.28 tff(23,plain,
% 1.67/1.28 (((~![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))),
% 1.67/1.28 inference(rewrite,[status(thm)],[])).
% 1.67/1.28 tff(24,plain,
% 1.67/1.28 ((~![X: $i] : ((~inductive(X)) | member(null_class, X))) | ((~inductive(omega)) | member(null_class, omega))),
% 1.67/1.28 inference(quant_inst,[status(thm)],[])).
% 1.67/1.28 tff(25,plain,
% 1.67/1.28 ((~![X: $i] : ((~inductive(X)) | member(null_class, X))) | (~inductive(omega)) | member(null_class, omega)),
% 1.67/1.28 inference(modus_ponens,[status(thm)],[24, 23])).
% 1.67/1.28 tff(26,plain,
% 1.67/1.28 (member(null_class, omega)),
% 1.67/1.28 inference(unit_resolution,[status(thm)],[25, 22, 15])).
% 1.67/1.28 tff(27,plain,
% 1.67/1.28 (^[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))))),
% 1.67/1.28 inference(bind,[status(th)],[])).
% 1.67/1.28 tff(28,plain,
% 1.67/1.28 (![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)))),
% 1.67/1.28 inference(quant_intro,[status(thm)],[27])).
% 1.67/1.28 tff(29,plain,
% 1.67/1.28 (![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)))),
% 1.67/1.28 inference(rewrite,[status(thm)],[])).
% 1.67/1.28 tff(30,plain,
% 1.67/1.28 (^[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)))))),
% 1.67/1.28 inference(bind,[status(th)],[])).
% 1.67/1.28 tff(31,plain,
% 1.67/1.28 (![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)))),
% 1.67/1.28 inference(quant_intro,[status(thm)],[30])).
% 1.67/1.28 tff(32,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')).
% 1.67/1.28 tff(33,plain,
% 1.67/1.28 (![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 1.67/1.28 inference(modus_ponens,[status(thm)],[32, 31])).
% 1.67/1.28 tff(34,plain,
% 1.67/1.28 (![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 1.67/1.28 inference(modus_ponens,[status(thm)],[33, 29])).
% 1.67/1.28 tff(35,plain,(
% 1.67/1.28 ![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 1.67/1.28 inference(skolemize,[status(sab)],[34])).
% 1.67/1.28 tff(36,plain,
% 1.67/1.28 (![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 1.67/1.28 inference(modus_ponens,[status(thm)],[35, 28])).
% 1.67/1.28 tff(37,plain,
% 1.67/1.28 (((~![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)))),
% 1.67/1.28 inference(rewrite,[status(thm)],[])).
% 1.67/1.28 tff(38,plain,
% 1.67/1.28 ((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)))),
% 1.67/1.28 inference(rewrite,[status(thm)],[])).
% 1.67/1.28 tff(39,plain,
% 1.67/1.28 (((~![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))))),
% 1.67/1.28 inference(monotonicity,[status(thm)],[38])).
% 1.67/1.28 tff(40,plain,
% 1.67/1.28 (((~![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)))),
% 1.67/1.29 inference(transitivity,[status(thm)],[39, 37])).
% 1.67/1.29 tff(41,plain,
% 1.67/1.29 ((~![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)))),
% 1.67/1.29 inference(quant_inst,[status(thm)],[])).
% 1.67/1.29 tff(42,plain,
% 1.67/1.29 ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (~member(null_class, omega)) | member(null_class, universal_class) | (~subclass(omega, universal_class))),
% 1.67/1.29 inference(modus_ponens,[status(thm)],[41, 40])).
% 1.67/1.29 tff(43,plain,
% 1.67/1.29 ($false),
% 1.67/1.29 inference(unit_resolution,[status(thm)],[42, 36, 26, 12, 11])).
% 1.67/1.29 tff(44,plain,(member(null_class, universal_class)), inference(lemma,lemma(discharge,[]))).
% 1.67/1.29 tff(45,plain,
% 1.67/1.29 (^[Y: $i, X: $i] : refl((complement(intersection(complement(X), complement(Y))) = union(X, Y)) <=> (complement(intersection(complement(X), complement(Y))) = union(X, Y)))),
% 1.67/1.29 inference(bind,[status(th)],[])).
% 1.67/1.29 tff(46,plain,
% 1.67/1.29 (![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))),
% 1.67/1.29 inference(quant_intro,[status(thm)],[45])).
% 1.67/1.29 tff(47,plain,
% 1.67/1.29 (![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))),
% 1.67/1.29 inference(rewrite,[status(thm)],[])).
% 1.67/1.29 tff(48,axiom,(![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','union')).
% 1.67/1.29 tff(49,plain,
% 1.67/1.29 (![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))),
% 1.67/1.29 inference(modus_ponens,[status(thm)],[48, 47])).
% 1.67/1.29 tff(50,plain,(
% 1.67/1.29 ![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))),
% 1.67/1.29 inference(skolemize,[status(sab)],[49])).
% 1.67/1.29 tff(51,plain,
% 1.67/1.29 (![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))),
% 1.67/1.29 inference(modus_ponens,[status(thm)],[50, 46])).
% 1.67/1.29 tff(52,plain,
% 1.67/1.29 ((~![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)))),
% 1.67/1.29 inference(quant_inst,[status(thm)],[])).
% 1.67/1.29 tff(53,plain,
% 1.67/1.29 (complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) = union(singleton(null_class), image(successor_relation, ordinal_numbers))),
% 1.67/1.29 inference(unit_resolution,[status(thm)],[52, 51])).
% 1.67/1.29 tff(54,plain,
% 1.67/1.29 (member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))) <=> member(null_class, union(singleton(null_class), image(successor_relation, ordinal_numbers)))),
% 1.67/1.29 inference(monotonicity,[status(thm)],[53])).
% 1.67/1.29 tff(55,plain,
% 1.67/1.29 (member(null_class, union(singleton(null_class), image(successor_relation, ordinal_numbers))) <=> member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))))),
% 1.67/1.29 inference(symmetry,[status(thm)],[54])).
% 1.67/1.29 tff(56,plain,
% 1.67/1.29 ((~member(null_class, union(singleton(null_class), image(successor_relation, ordinal_numbers)))) <=> (~member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))))),
% 1.67/1.29 inference(monotonicity,[status(thm)],[55])).
% 1.67/1.29 tff(57,plain,
% 1.67/1.29 ((~member(null_class, kind_1_ordinals)) <=> (~member(null_class, union(singleton(null_class), image(successor_relation, ordinal_numbers))))),
% 1.67/1.29 inference(rewrite,[status(thm)],[])).
% 1.67/1.29 tff(58,plain,
% 1.67/1.29 ((~member(null_class, kind_1_ordinals)) <=> (~member(null_class, kind_1_ordinals))),
% 1.67/1.29 inference(rewrite,[status(thm)],[])).
% 1.67/1.29 tff(59,axiom,(~member(null_class, kind_1_ordinals)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_null_class_is_kind_1_1')).
% 1.67/1.29 tff(60,plain,
% 1.67/1.29 (~member(null_class, kind_1_ordinals)),
% 1.67/1.29 inference(modus_ponens,[status(thm)],[59, 58])).
% 1.67/1.29 tff(61,plain,
% 1.67/1.29 (~member(null_class, union(singleton(null_class), image(successor_relation, ordinal_numbers)))),
% 1.67/1.29 inference(modus_ponens,[status(thm)],[60, 57])).
% 1.67/1.29 tff(62,plain,
% 1.67/1.29 (~member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))))),
% 1.67/1.29 inference(modus_ponens,[status(thm)],[61, 56])).
% 1.67/1.29 tff(63,plain,
% 1.67/1.29 (^[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))))),
% 1.67/1.29 inference(bind,[status(th)],[])).
% 1.67/1.29 tff(64,plain,
% 1.67/1.29 (![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)))),
% 1.67/1.29 inference(quant_intro,[status(thm)],[63])).
% 1.67/1.29 tff(65,plain,
% 1.67/1.29 (![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)))),
% 1.67/1.29 inference(rewrite,[status(thm)],[])).
% 1.67/1.29 tff(66,plain,
% 1.67/1.29 (^[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))))),
% 1.67/1.29 inference(bind,[status(th)],[])).
% 1.67/1.29 tff(67,plain,
% 1.67/1.29 (![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)))),
% 1.67/1.29 inference(quant_intro,[status(thm)],[66])).
% 1.67/1.29 tff(68,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')).
% 1.67/1.29 tff(69,plain,
% 1.67/1.29 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 1.67/1.29 inference(modus_ponens,[status(thm)],[68, 67])).
% 1.67/1.29 tff(70,plain,
% 1.67/1.29 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 1.67/1.29 inference(modus_ponens,[status(thm)],[69, 65])).
% 1.67/1.29 tff(71,plain,(
% 1.67/1.29 ![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 1.67/1.29 inference(skolemize,[status(sab)],[70])).
% 1.67/1.29 tff(72,plain,
% 1.67/1.29 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 1.67/1.29 inference(modus_ponens,[status(thm)],[71, 64])).
% 1.67/1.29 tff(73,plain,
% 1.67/1.29 (((~![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))))))),
% 1.67/1.29 inference(rewrite,[status(thm)],[])).
% 1.67/1.29 tff(74,plain,
% 1.67/1.29 ((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))))))),
% 1.67/1.29 inference(rewrite,[status(thm)],[])).
% 1.67/1.29 tff(75,plain,
% 1.67/1.29 (((~![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)))))))),
% 1.67/1.29 inference(monotonicity,[status(thm)],[74])).
% 1.67/1.29 tff(76,plain,
% 1.67/1.29 (((~![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))))))),
% 1.67/1.29 inference(transitivity,[status(thm)],[75, 73])).
% 1.67/1.29 tff(77,plain,
% 1.67/1.29 ((~![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))))))),
% 1.67/1.29 inference(quant_inst,[status(thm)],[])).
% 1.67/1.29 tff(78,plain,
% 1.67/1.29 ((~![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)))))),
% 1.67/1.29 inference(modus_ponens,[status(thm)],[77, 76])).
% 1.67/1.29 tff(79,plain,
% 1.67/1.29 ((~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)))))),
% 1.67/1.29 inference(unit_resolution,[status(thm)],[78, 72])).
% 1.67/1.29 tff(80,plain,
% 1.67/1.29 ((~member(null_class, universal_class)) | member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))),
% 1.67/1.29 inference(unit_resolution,[status(thm)],[79, 62])).
% 1.67/1.29 tff(81,plain,
% 1.67/1.29 (member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))),
% 1.67/1.29 inference(unit_resolution,[status(thm)],[80, 44])).
% 1.67/1.29 tff(82,plain,
% 1.67/1.29 (^[Z: $i, Y: $i, X: $i] : refl(((~member(Z, intersection(X, Y))) | member(Z, X)) <=> ((~member(Z, intersection(X, Y))) | member(Z, X)))),
% 1.67/1.29 inference(bind,[status(th)],[])).
% 1.67/1.29 tff(83,plain,
% 1.67/1.29 (![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))),
% 1.67/1.29 inference(quant_intro,[status(thm)],[82])).
% 1.67/1.29 tff(84,plain,
% 1.67/1.29 (![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))),
% 1.67/1.29 inference(rewrite,[status(thm)],[])).
% 1.67/1.29 tff(85,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')).
% 1.67/1.29 tff(86,plain,
% 1.67/1.29 (![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))),
% 1.67/1.29 inference(modus_ponens,[status(thm)],[85, 84])).
% 1.67/1.29 tff(87,plain,(
% 1.67/1.29 ![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))),
% 1.67/1.29 inference(skolemize,[status(sab)],[86])).
% 1.67/1.29 tff(88,plain,
% 1.67/1.29 (![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))),
% 1.67/1.29 inference(modus_ponens,[status(thm)],[87, 83])).
% 1.67/1.29 tff(89,plain,
% 1.67/1.29 (((~![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))))),
% 1.67/1.29 inference(rewrite,[status(thm)],[])).
% 1.67/1.29 tff(90,plain,
% 1.67/1.29 ((~![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))))),
% 1.67/1.29 inference(quant_inst,[status(thm)],[])).
% 1.67/1.29 tff(91,plain,
% 1.67/1.29 ((~![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)))),
% 1.67/1.29 inference(modus_ponens,[status(thm)],[90, 89])).
% 1.67/1.29 tff(92,plain,
% 1.67/1.29 (member(null_class, complement(singleton(null_class)))),
% 1.67/1.29 inference(unit_resolution,[status(thm)],[91, 88, 81])).
% 1.67/1.29 tff(93,plain,
% 1.67/1.29 (^[X: $i] : refl((unordered_pair(X, X) = singleton(X)) <=> (unordered_pair(X, X) = singleton(X)))),
% 1.67/1.29 inference(bind,[status(th)],[])).
% 1.67/1.29 tff(94,plain,
% 1.67/1.29 (![X: $i] : (unordered_pair(X, X) = singleton(X)) <=> ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 1.67/1.29 inference(quant_intro,[status(thm)],[93])).
% 1.67/1.29 tff(95,plain,
% 1.67/1.29 (![X: $i] : (unordered_pair(X, X) = singleton(X)) <=> ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 1.67/1.29 inference(rewrite,[status(thm)],[])).
% 1.67/1.29 tff(96,axiom,(![X: $i] : (unordered_pair(X, X) = singleton(X))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','singleton_set')).
% 1.67/1.29 tff(97,plain,
% 1.67/1.29 (![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 1.67/1.29 inference(modus_ponens,[status(thm)],[96, 95])).
% 1.67/1.29 tff(98,plain,(
% 1.67/1.29 ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 1.67/1.29 inference(skolemize,[status(sab)],[97])).
% 1.67/1.29 tff(99,plain,
% 1.67/1.29 (![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 1.67/1.29 inference(modus_ponens,[status(thm)],[98, 94])).
% 1.67/1.29 tff(100,plain,
% 1.67/1.29 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(null_class, null_class) = singleton(null_class))),
% 1.67/1.29 inference(quant_inst,[status(thm)],[])).
% 1.67/1.29 tff(101,plain,
% 1.67/1.29 (unordered_pair(null_class, null_class) = singleton(null_class)),
% 1.67/1.29 inference(unit_resolution,[status(thm)],[100, 99])).
% 1.67/1.29 tff(102,plain,
% 1.67/1.29 (singleton(null_class) = unordered_pair(null_class, null_class)),
% 1.67/1.29 inference(symmetry,[status(thm)],[101])).
% 1.67/1.29 tff(103,plain,
% 1.67/1.29 (member(null_class, singleton(null_class)) <=> member(null_class, unordered_pair(null_class, null_class))),
% 1.67/1.29 inference(monotonicity,[status(thm)],[102])).
% 1.67/1.29 tff(104,plain,
% 1.67/1.29 (member(null_class, unordered_pair(null_class, null_class)) <=> member(null_class, singleton(null_class))),
% 1.67/1.29 inference(symmetry,[status(thm)],[103])).
% 1.67/1.29 tff(105,plain,
% 1.67/1.29 (^[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))))),
% 1.67/1.29 inference(bind,[status(th)],[])).
% 1.67/1.29 tff(106,plain,
% 1.67/1.29 (![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)))),
% 1.67/1.29 inference(quant_intro,[status(thm)],[105])).
% 1.67/1.29 tff(107,plain,
% 1.67/1.29 (![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)))),
% 1.67/1.29 inference(rewrite,[status(thm)],[])).
% 1.67/1.29 tff(108,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')).
% 1.67/1.29 tff(109,plain,
% 1.67/1.29 (![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 1.67/1.29 inference(modus_ponens,[status(thm)],[108, 107])).
% 1.67/1.29 tff(110,plain,(
% 1.67/1.29 ![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 1.67/1.29 inference(skolemize,[status(sab)],[109])).
% 1.67/1.29 tff(111,plain,
% 1.67/1.29 (![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 1.67/1.29 inference(modus_ponens,[status(thm)],[110, 106])).
% 1.67/1.29 tff(112,plain,
% 1.67/1.29 (((~![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)))),
% 1.67/1.29 inference(rewrite,[status(thm)],[])).
% 1.67/1.29 tff(113,plain,
% 1.67/1.29 ((~![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)))),
% 1.67/1.29 inference(quant_inst,[status(thm)],[])).
% 1.67/1.29 tff(114,plain,
% 1.67/1.29 ((~![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))),
% 1.67/1.29 inference(modus_ponens,[status(thm)],[113, 112])).
% 1.67/1.29 tff(115,plain,
% 1.67/1.29 ((~member(null_class, universal_class)) | member(null_class, unordered_pair(null_class, null_class))),
% 1.67/1.29 inference(unit_resolution,[status(thm)],[114, 111])).
% 1.67/1.29 tff(116,plain,
% 1.67/1.29 (member(null_class, unordered_pair(null_class, null_class))),
% 1.67/1.29 inference(unit_resolution,[status(thm)],[115, 44])).
% 1.67/1.29 tff(117,plain,
% 1.67/1.29 (member(null_class, singleton(null_class))),
% 1.67/1.29 inference(modus_ponens,[status(thm)],[116, 104])).
% 1.67/1.29 tff(118,plain,
% 1.67/1.29 (^[Z: $i, X: $i] : refl(((~member(Z, X)) | (~member(Z, complement(X)))) <=> ((~member(Z, X)) | (~member(Z, complement(X)))))),
% 1.67/1.29 inference(bind,[status(th)],[])).
% 1.67/1.29 tff(119,plain,
% 1.67/1.29 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X)))) <=> ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 1.67/1.29 inference(quant_intro,[status(thm)],[118])).
% 1.67/1.29 tff(120,plain,
% 1.67/1.29 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X)))) <=> ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 1.67/1.29 inference(rewrite,[status(thm)],[])).
% 1.67/1.29 tff(121,plain,
% 1.67/1.29 (^[Z: $i, X: $i] : rewrite(((~member(Z, complement(X))) | (~member(Z, X))) <=> ((~member(Z, X)) | (~member(Z, complement(X)))))),
% 1.67/1.29 inference(bind,[status(th)],[])).
% 1.67/1.29 tff(122,plain,
% 1.67/1.29 (![Z: $i, X: $i] : ((~member(Z, complement(X))) | (~member(Z, X))) <=> ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 1.67/1.29 inference(quant_intro,[status(thm)],[121])).
% 1.67/1.29 tff(123,axiom,(![Z: $i, X: $i] : ((~member(Z, complement(X))) | (~member(Z, X)))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','complement1')).
% 1.67/1.29 tff(124,plain,
% 1.67/1.29 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 1.67/1.29 inference(modus_ponens,[status(thm)],[123, 122])).
% 1.67/1.29 tff(125,plain,
% 1.67/1.29 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 1.67/1.29 inference(modus_ponens,[status(thm)],[124, 120])).
% 1.67/1.31 tff(126,plain,(
% 1.67/1.31 ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 1.67/1.31 inference(skolemize,[status(sab)],[125])).
% 1.67/1.31 tff(127,plain,
% 1.67/1.31 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 1.67/1.31 inference(modus_ponens,[status(thm)],[126, 119])).
% 1.67/1.31 tff(128,plain,
% 1.67/1.31 (((~![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)))))),
% 1.67/1.31 inference(rewrite,[status(thm)],[])).
% 1.67/1.31 tff(129,plain,
% 1.67/1.31 ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | ((~member(null_class, singleton(null_class))) | (~member(null_class, complement(singleton(null_class)))))),
% 1.67/1.31 inference(quant_inst,[status(thm)],[])).
% 1.67/1.31 tff(130,plain,
% 1.67/1.31 ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | (~member(null_class, singleton(null_class))) | (~member(null_class, complement(singleton(null_class))))),
% 1.67/1.31 inference(modus_ponens,[status(thm)],[129, 128])).
% 1.67/1.31 tff(131,plain,
% 1.67/1.31 ($false),
% 1.67/1.31 inference(unit_resolution,[status(thm)],[130, 127, 117, 92])).
% 1.67/1.31 % SZS output end Proof
%------------------------------------------------------------------------------