TSTP Solution File: SET211-6 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET211-6 : TPTP v8.1.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n023.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 : Tue Sep 20 05:06:11 EDT 2022
% Result : Unsatisfiable 1.35s 1.16s
% Output : Proof 1.48s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SET211-6 : TPTP v8.1.0. Bugfixed v2.1.0.
% 0.00/0.12 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.11/0.33 % Computer : n023.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 : Sat Sep 3 03:46:18 EDT 2022
% 0.11/0.33 % CPUTime :
% 0.11/0.33 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.11/0.33 Usage: tptp [options] [-file:]file
% 0.11/0.33 -h, -? prints this message.
% 0.11/0.33 -smt2 print SMT-LIB2 benchmark.
% 0.11/0.33 -m, -model generate model.
% 0.11/0.33 -p, -proof generate proof.
% 0.11/0.33 -c, -core generate unsat core of named formulas.
% 0.11/0.33 -st, -statistics display statistics.
% 0.11/0.33 -t:timeout set timeout (in second).
% 0.11/0.33 -smt2status display status in smt2 format instead of SZS.
% 0.11/0.33 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.11/0.33 -<param>:<value> configuration parameter and value.
% 0.11/0.33 -o:<output-file> file to place output in.
% 1.35/1.16 % SZS status Unsatisfiable
% 1.35/1.16 % SZS output start Proof
% 1.35/1.16 tff(member_type, type, (
% 1.35/1.16 member: ( $i * $i ) > $o)).
% 1.35/1.16 tff(complement_type, type, (
% 1.35/1.16 complement: $i > $i)).
% 1.35/1.16 tff(y_type, type, (
% 1.35/1.16 y: $i)).
% 1.35/1.16 tff(first_type, type, (
% 1.35/1.16 first: $i > $i)).
% 1.35/1.16 tff(not_subclass_element_type, type, (
% 1.35/1.16 not_subclass_element: ( $i * $i ) > $i)).
% 1.35/1.16 tff(cross_product_type, type, (
% 1.35/1.16 cross_product: ( $i * $i ) > $i)).
% 1.35/1.16 tff(z_type, type, (
% 1.35/1.16 z: $i)).
% 1.35/1.16 tff(union_type, type, (
% 1.35/1.16 union: ( $i * $i ) > $i)).
% 1.35/1.16 tff(x_type, type, (
% 1.35/1.16 x: $i)).
% 1.35/1.16 tff(intersection_type, type, (
% 1.35/1.16 intersection: ( $i * $i ) > $i)).
% 1.35/1.16 tff(universal_class_type, type, (
% 1.35/1.16 universal_class: $i)).
% 1.35/1.16 tff(subclass_type, type, (
% 1.35/1.16 subclass: ( $i * $i ) > $o)).
% 1.35/1.16 tff(ordered_pair_type, type, (
% 1.35/1.16 ordered_pair: ( $i * $i ) > $i)).
% 1.35/1.16 tff(second_type, type, (
% 1.35/1.16 second: $i > $i)).
% 1.35/1.16 tff(1,assumption,(~subclass(y, universal_class)), introduced(assumption)).
% 1.35/1.16 tff(2,plain,
% 1.35/1.16 (^[X: $i] : refl(subclass(X, universal_class) <=> subclass(X, universal_class))),
% 1.35/1.16 inference(bind,[status(th)],[])).
% 1.35/1.16 tff(3,plain,
% 1.35/1.16 (![X: $i] : subclass(X, universal_class) <=> ![X: $i] : subclass(X, universal_class)),
% 1.35/1.16 inference(quant_intro,[status(thm)],[2])).
% 1.35/1.16 tff(4,plain,
% 1.35/1.16 (![X: $i] : subclass(X, universal_class) <=> ![X: $i] : subclass(X, universal_class)),
% 1.35/1.16 inference(rewrite,[status(thm)],[])).
% 1.35/1.16 tff(5,axiom,(![X: $i] : subclass(X, universal_class)), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','class_elements_are_sets')).
% 1.35/1.16 tff(6,plain,
% 1.35/1.16 (![X: $i] : subclass(X, universal_class)),
% 1.35/1.16 inference(modus_ponens,[status(thm)],[5, 4])).
% 1.35/1.16 tff(7,plain,(
% 1.35/1.16 ![X: $i] : subclass(X, universal_class)),
% 1.35/1.16 inference(skolemize,[status(sab)],[6])).
% 1.35/1.16 tff(8,plain,
% 1.35/1.16 (![X: $i] : subclass(X, universal_class)),
% 1.35/1.16 inference(modus_ponens,[status(thm)],[7, 3])).
% 1.35/1.16 tff(9,plain,
% 1.35/1.16 ((~![X: $i] : subclass(X, universal_class)) | subclass(y, universal_class)),
% 1.35/1.16 inference(quant_inst,[status(thm)],[])).
% 1.35/1.16 tff(10,plain,
% 1.35/1.16 ($false),
% 1.35/1.16 inference(unit_resolution,[status(thm)],[9, 8, 1])).
% 1.35/1.16 tff(11,plain,(subclass(y, universal_class)), inference(lemma,lemma(discharge,[]))).
% 1.35/1.16 tff(12,assumption,(~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), universal_class)), introduced(assumption)).
% 1.35/1.16 tff(13,plain,
% 1.35/1.16 ((~subclass(cross_product(y, z), cross_product(union(x, y), z))) <=> (~subclass(cross_product(y, z), cross_product(union(x, y), z)))),
% 1.35/1.16 inference(rewrite,[status(thm)],[])).
% 1.35/1.16 tff(14,axiom,(~subclass(cross_product(y, z), cross_product(union(x, y), z))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_corollary_2_to_X_product_monotonicity_1')).
% 1.35/1.16 tff(15,plain,
% 1.35/1.16 (~subclass(cross_product(y, z), cross_product(union(x, y), z))),
% 1.35/1.16 inference(modus_ponens,[status(thm)],[14, 13])).
% 1.35/1.16 tff(16,plain,
% 1.35/1.16 (^[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)))),
% 1.35/1.16 inference(bind,[status(th)],[])).
% 1.35/1.16 tff(17,plain,
% 1.35/1.16 (![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))),
% 1.35/1.16 inference(quant_intro,[status(thm)],[16])).
% 1.35/1.16 tff(18,plain,
% 1.35/1.16 (![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))),
% 1.35/1.16 inference(rewrite,[status(thm)],[])).
% 1.35/1.16 tff(19,plain,
% 1.35/1.16 (^[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)))),
% 1.35/1.16 inference(bind,[status(th)],[])).
% 1.35/1.16 tff(20,plain,
% 1.35/1.16 (![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))),
% 1.35/1.16 inference(quant_intro,[status(thm)],[19])).
% 1.35/1.16 tff(21,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')).
% 1.35/1.16 tff(22,plain,
% 1.35/1.16 (![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))),
% 1.35/1.16 inference(modus_ponens,[status(thm)],[21, 20])).
% 1.35/1.16 tff(23,plain,
% 1.35/1.16 (![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))),
% 1.35/1.16 inference(modus_ponens,[status(thm)],[22, 18])).
% 1.35/1.16 tff(24,plain,(
% 1.35/1.16 ![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))),
% 1.35/1.16 inference(skolemize,[status(sab)],[23])).
% 1.35/1.16 tff(25,plain,
% 1.35/1.16 (![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))),
% 1.35/1.16 inference(modus_ponens,[status(thm)],[24, 17])).
% 1.35/1.16 tff(26,plain,
% 1.35/1.16 (((~![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))) | (subclass(cross_product(y, z), cross_product(union(x, y), z)) | member(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)), cross_product(y, z)))) <=> ((~![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))) | subclass(cross_product(y, z), cross_product(union(x, y), z)) | member(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)), cross_product(y, z)))),
% 1.35/1.16 inference(rewrite,[status(thm)],[])).
% 1.35/1.16 tff(27,plain,
% 1.35/1.16 ((~![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))) | (subclass(cross_product(y, z), cross_product(union(x, y), z)) | member(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)), cross_product(y, z)))),
% 1.35/1.16 inference(quant_inst,[status(thm)],[])).
% 1.35/1.16 tff(28,plain,
% 1.35/1.16 ((~![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))) | subclass(cross_product(y, z), cross_product(union(x, y), z)) | member(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)), cross_product(y, z))),
% 1.35/1.16 inference(modus_ponens,[status(thm)],[27, 26])).
% 1.35/1.16 tff(29,plain,
% 1.35/1.16 (member(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)), cross_product(y, z))),
% 1.35/1.16 inference(unit_resolution,[status(thm)],[28, 25, 15])).
% 1.35/1.16 tff(30,plain,
% 1.35/1.16 (^[Z: $i, Y: $i, X: $i] : refl(((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z)) <=> ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z)))),
% 1.35/1.16 inference(bind,[status(th)],[])).
% 1.35/1.16 tff(31,plain,
% 1.35/1.16 (![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z)) <=> ![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))),
% 1.35/1.16 inference(quant_intro,[status(thm)],[30])).
% 1.35/1.16 tff(32,plain,
% 1.35/1.16 (![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z)) <=> ![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))),
% 1.35/1.16 inference(rewrite,[status(thm)],[])).
% 1.35/1.16 tff(33,axiom,(![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','cartesian_product4')).
% 1.35/1.16 tff(34,plain,
% 1.35/1.16 (![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))),
% 1.35/1.16 inference(modus_ponens,[status(thm)],[33, 32])).
% 1.35/1.16 tff(35,plain,(
% 1.35/1.16 ![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))),
% 1.35/1.16 inference(skolemize,[status(sab)],[34])).
% 1.35/1.16 tff(36,plain,
% 1.35/1.16 (![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))),
% 1.35/1.16 inference(modus_ponens,[status(thm)],[35, 31])).
% 1.35/1.16 tff(37,plain,
% 1.35/1.16 (((~![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))) | ((~member(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)), cross_product(y, z))) | (ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))) = not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))) | (~member(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)), cross_product(y, z))) | (ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))) = not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))))),
% 1.35/1.16 inference(rewrite,[status(thm)],[])).
% 1.35/1.16 tff(38,plain,
% 1.35/1.16 ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))) | ((~member(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)), cross_product(y, z))) | (ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))) = not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))))),
% 1.35/1.16 inference(quant_inst,[status(thm)],[])).
% 1.35/1.16 tff(39,plain,
% 1.35/1.16 ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))) | (~member(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)), cross_product(y, z))) | (ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))) = not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))),
% 1.35/1.16 inference(modus_ponens,[status(thm)],[38, 37])).
% 1.35/1.16 tff(40,plain,
% 1.35/1.16 (ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))) = not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))),
% 1.35/1.16 inference(unit_resolution,[status(thm)],[39, 36, 29])).
% 1.35/1.16 tff(41,plain,
% 1.35/1.16 (member(ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))), cross_product(y, z)) <=> member(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)), cross_product(y, z))),
% 1.35/1.16 inference(monotonicity,[status(thm)],[40])).
% 1.35/1.16 tff(42,plain,
% 1.35/1.16 (member(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)), cross_product(y, z)) <=> member(ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))), cross_product(y, z))),
% 1.35/1.16 inference(symmetry,[status(thm)],[41])).
% 1.35/1.16 tff(43,plain,
% 1.35/1.16 (member(ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))), cross_product(y, z))),
% 1.35/1.16 inference(modus_ponens,[status(thm)],[29, 42])).
% 1.35/1.16 tff(44,plain,
% 1.35/1.16 (^[V: $i, Y: $i, U: $i, X: $i] : refl(((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X)) <=> ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X)))),
% 1.35/1.16 inference(bind,[status(th)],[])).
% 1.35/1.16 tff(45,plain,
% 1.35/1.16 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X)) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))),
% 1.35/1.16 inference(quant_intro,[status(thm)],[44])).
% 1.35/1.16 tff(46,plain,
% 1.35/1.16 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X)) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))),
% 1.35/1.16 inference(rewrite,[status(thm)],[])).
% 1.35/1.16 tff(47,axiom,(![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','cartesian_product1')).
% 1.35/1.16 tff(48,plain,
% 1.35/1.16 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))),
% 1.35/1.16 inference(modus_ponens,[status(thm)],[47, 46])).
% 1.35/1.16 tff(49,plain,(
% 1.35/1.16 ![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))),
% 1.35/1.16 inference(skolemize,[status(sab)],[48])).
% 1.35/1.16 tff(50,plain,
% 1.35/1.16 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))),
% 1.35/1.16 inference(modus_ponens,[status(thm)],[49, 45])).
% 1.35/1.16 tff(51,plain,
% 1.35/1.16 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | ((~member(ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))), cross_product(y, z))) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), y))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | (~member(ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))), cross_product(y, z))) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), y))),
% 1.35/1.16 inference(rewrite,[status(thm)],[])).
% 1.35/1.16 tff(52,plain,
% 1.35/1.16 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | ((~member(ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))), cross_product(y, z))) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), y))),
% 1.35/1.16 inference(quant_inst,[status(thm)],[])).
% 1.35/1.16 tff(53,plain,
% 1.35/1.16 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | (~member(ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))), cross_product(y, z))) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), y)),
% 1.35/1.16 inference(modus_ponens,[status(thm)],[52, 51])).
% 1.35/1.16 tff(54,plain,
% 1.35/1.16 ((~member(ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))), cross_product(y, z))) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), y)),
% 1.35/1.16 inference(unit_resolution,[status(thm)],[53, 50])).
% 1.35/1.16 tff(55,plain,
% 1.35/1.16 (member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), y)),
% 1.35/1.16 inference(unit_resolution,[status(thm)],[54, 43])).
% 1.35/1.16 tff(56,plain,
% 1.35/1.16 (^[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))))),
% 1.35/1.16 inference(bind,[status(th)],[])).
% 1.35/1.16 tff(57,plain,
% 1.35/1.16 (![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)))),
% 1.35/1.16 inference(quant_intro,[status(thm)],[56])).
% 1.35/1.16 tff(58,plain,
% 1.35/1.16 (![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)))),
% 1.35/1.16 inference(rewrite,[status(thm)],[])).
% 1.35/1.16 tff(59,plain,
% 1.35/1.16 (^[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)))))),
% 1.35/1.16 inference(bind,[status(th)],[])).
% 1.35/1.16 tff(60,plain,
% 1.35/1.16 (![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)))),
% 1.35/1.16 inference(quant_intro,[status(thm)],[59])).
% 1.35/1.16 tff(61,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')).
% 1.35/1.16 tff(62,plain,
% 1.35/1.16 (![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 1.35/1.16 inference(modus_ponens,[status(thm)],[61, 60])).
% 1.35/1.16 tff(63,plain,
% 1.35/1.16 (![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 1.35/1.16 inference(modus_ponens,[status(thm)],[62, 58])).
% 1.35/1.16 tff(64,plain,(
% 1.35/1.16 ![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 1.35/1.16 inference(skolemize,[status(sab)],[63])).
% 1.35/1.16 tff(65,plain,
% 1.35/1.16 (![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 1.35/1.16 inference(modus_ponens,[status(thm)],[64, 57])).
% 1.35/1.16 tff(66,plain,
% 1.35/1.16 (((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), universal_class) | (~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), y)) | (~subclass(y, universal_class)))) <=> ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), universal_class) | (~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), y)) | (~subclass(y, universal_class)))),
% 1.35/1.16 inference(rewrite,[status(thm)],[])).
% 1.35/1.16 tff(67,plain,
% 1.35/1.16 ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), universal_class) | (~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), y)) | (~subclass(y, universal_class)))),
% 1.35/1.16 inference(quant_inst,[status(thm)],[])).
% 1.35/1.16 tff(68,plain,
% 1.35/1.16 ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), universal_class) | (~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), y)) | (~subclass(y, universal_class))),
% 1.35/1.16 inference(modus_ponens,[status(thm)],[67, 66])).
% 1.35/1.16 tff(69,plain,
% 1.35/1.16 ($false),
% 1.35/1.16 inference(unit_resolution,[status(thm)],[68, 65, 55, 12, 11])).
% 1.35/1.16 tff(70,plain,(member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), universal_class)), inference(lemma,lemma(discharge,[]))).
% 1.35/1.16 tff(71,plain,
% 1.35/1.16 (^[Y: $i, X: $i] : refl((complement(intersection(complement(X), complement(Y))) = union(X, Y)) <=> (complement(intersection(complement(X), complement(Y))) = union(X, Y)))),
% 1.35/1.16 inference(bind,[status(th)],[])).
% 1.35/1.16 tff(72,plain,
% 1.35/1.16 (![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))),
% 1.35/1.16 inference(quant_intro,[status(thm)],[71])).
% 1.35/1.16 tff(73,plain,
% 1.35/1.16 (![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))),
% 1.35/1.16 inference(rewrite,[status(thm)],[])).
% 1.35/1.16 tff(74,axiom,(![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','union')).
% 1.35/1.16 tff(75,plain,
% 1.35/1.16 (![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))),
% 1.35/1.16 inference(modus_ponens,[status(thm)],[74, 73])).
% 1.35/1.16 tff(76,plain,(
% 1.35/1.16 ![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))),
% 1.35/1.16 inference(skolemize,[status(sab)],[75])).
% 1.35/1.16 tff(77,plain,
% 1.35/1.16 (![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))),
% 1.35/1.16 inference(modus_ponens,[status(thm)],[76, 72])).
% 1.35/1.16 tff(78,plain,
% 1.35/1.16 ((~![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))) | (complement(intersection(complement(x), complement(y))) = union(x, y))),
% 1.35/1.16 inference(quant_inst,[status(thm)],[])).
% 1.35/1.16 tff(79,plain,
% 1.35/1.16 (complement(intersection(complement(x), complement(y))) = union(x, y)),
% 1.35/1.17 inference(unit_resolution,[status(thm)],[78, 77])).
% 1.35/1.17 tff(80,plain,
% 1.35/1.17 (member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), complement(intersection(complement(x), complement(y)))) <=> member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), union(x, y))),
% 1.35/1.17 inference(monotonicity,[status(thm)],[79])).
% 1.35/1.17 tff(81,plain,
% 1.35/1.17 (member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), union(x, y)) <=> member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), complement(intersection(complement(x), complement(y))))),
% 1.35/1.17 inference(symmetry,[status(thm)],[80])).
% 1.35/1.17 tff(82,plain,
% 1.35/1.17 ((~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), union(x, y))) <=> (~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), complement(intersection(complement(x), complement(y)))))),
% 1.35/1.17 inference(monotonicity,[status(thm)],[81])).
% 1.35/1.17 tff(83,plain,
% 1.35/1.17 (member(ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))), cross_product(union(x, y), z)) <=> member(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)), cross_product(union(x, y), z))),
% 1.35/1.17 inference(monotonicity,[status(thm)],[40])).
% 1.35/1.17 tff(84,plain,
% 1.35/1.17 (member(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)), cross_product(union(x, y), z)) <=> member(ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))), cross_product(union(x, y), z))),
% 1.35/1.17 inference(symmetry,[status(thm)],[83])).
% 1.35/1.17 tff(85,plain,
% 1.35/1.17 ((~member(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)), cross_product(union(x, y), z))) <=> (~member(ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))), cross_product(union(x, y), z)))),
% 1.35/1.17 inference(monotonicity,[status(thm)],[84])).
% 1.35/1.17 tff(86,plain,
% 1.35/1.17 (^[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)))),
% 1.35/1.17 inference(bind,[status(th)],[])).
% 1.35/1.17 tff(87,plain,
% 1.35/1.17 (![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))),
% 1.35/1.17 inference(quant_intro,[status(thm)],[86])).
% 1.35/1.17 tff(88,plain,
% 1.35/1.17 (![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))),
% 1.35/1.17 inference(rewrite,[status(thm)],[])).
% 1.35/1.17 tff(89,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')).
% 1.35/1.17 tff(90,plain,
% 1.35/1.17 (![Y: $i, X: $i] : ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y))),
% 1.35/1.17 inference(modus_ponens,[status(thm)],[89, 88])).
% 1.35/1.17 tff(91,plain,(
% 1.35/1.17 ![Y: $i, X: $i] : ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y))),
% 1.35/1.17 inference(skolemize,[status(sab)],[90])).
% 1.35/1.17 tff(92,plain,
% 1.35/1.17 (![Y: $i, X: $i] : ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y))),
% 1.35/1.17 inference(modus_ponens,[status(thm)],[91, 87])).
% 1.35/1.17 tff(93,plain,
% 1.35/1.17 (((~![Y: $i, X: $i] : ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y))) | ((~member(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)), cross_product(union(x, y), z))) | subclass(cross_product(y, z), cross_product(union(x, y), z)))) <=> ((~![Y: $i, X: $i] : ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y))) | (~member(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)), cross_product(union(x, y), z))) | subclass(cross_product(y, z), cross_product(union(x, y), z)))),
% 1.35/1.17 inference(rewrite,[status(thm)],[])).
% 1.35/1.17 tff(94,plain,
% 1.35/1.17 ((~![Y: $i, X: $i] : ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y))) | ((~member(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)), cross_product(union(x, y), z))) | subclass(cross_product(y, z), cross_product(union(x, y), z)))),
% 1.35/1.17 inference(quant_inst,[status(thm)],[])).
% 1.35/1.17 tff(95,plain,
% 1.35/1.17 ((~![Y: $i, X: $i] : ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y))) | (~member(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)), cross_product(union(x, y), z))) | subclass(cross_product(y, z), cross_product(union(x, y), z))),
% 1.35/1.17 inference(modus_ponens,[status(thm)],[94, 93])).
% 1.35/1.17 tff(96,plain,
% 1.35/1.17 (~member(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)), cross_product(union(x, y), z))),
% 1.35/1.17 inference(unit_resolution,[status(thm)],[95, 92, 15])).
% 1.35/1.17 tff(97,plain,
% 1.35/1.17 (~member(ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))), cross_product(union(x, y), z))),
% 1.35/1.17 inference(modus_ponens,[status(thm)],[96, 85])).
% 1.35/1.17 tff(98,plain,
% 1.35/1.17 (^[V: $i, Y: $i, U: $i, X: $i] : refl(((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y)) <=> ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y)))),
% 1.35/1.17 inference(bind,[status(th)],[])).
% 1.35/1.17 tff(99,plain,
% 1.35/1.17 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y)) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))),
% 1.35/1.17 inference(quant_intro,[status(thm)],[98])).
% 1.35/1.17 tff(100,plain,
% 1.35/1.17 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y)) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))),
% 1.35/1.17 inference(rewrite,[status(thm)],[])).
% 1.35/1.17 tff(101,axiom,(![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','cartesian_product2')).
% 1.35/1.17 tff(102,plain,
% 1.35/1.17 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))),
% 1.35/1.17 inference(modus_ponens,[status(thm)],[101, 100])).
% 1.35/1.17 tff(103,plain,(
% 1.35/1.17 ![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))),
% 1.35/1.17 inference(skolemize,[status(sab)],[102])).
% 1.35/1.17 tff(104,plain,
% 1.35/1.17 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))),
% 1.35/1.17 inference(modus_ponens,[status(thm)],[103, 99])).
% 1.35/1.17 tff(105,plain,
% 1.35/1.17 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))) | ((~member(ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))), cross_product(y, z))) | member(second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), z))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))) | (~member(ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))), cross_product(y, z))) | member(second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), z))),
% 1.35/1.17 inference(rewrite,[status(thm)],[])).
% 1.35/1.17 tff(106,plain,
% 1.35/1.17 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))) | ((~member(ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))), cross_product(y, z))) | member(second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), z))),
% 1.35/1.17 inference(quant_inst,[status(thm)],[])).
% 1.35/1.17 tff(107,plain,
% 1.35/1.17 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))) | (~member(ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))), cross_product(y, z))) | member(second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), z)),
% 1.35/1.17 inference(modus_ponens,[status(thm)],[106, 105])).
% 1.35/1.17 tff(108,plain,
% 1.35/1.17 ((~member(ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))), cross_product(y, z))) | member(second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), z)),
% 1.35/1.17 inference(unit_resolution,[status(thm)],[107, 104])).
% 1.35/1.17 tff(109,plain,
% 1.35/1.17 (member(second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), z)),
% 1.35/1.17 inference(unit_resolution,[status(thm)],[108, 43])).
% 1.35/1.17 tff(110,plain,
% 1.35/1.17 (^[V: $i, Y: $i, U: $i, X: $i] : refl(((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y))) <=> ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y))))),
% 1.35/1.17 inference(bind,[status(th)],[])).
% 1.35/1.17 tff(111,plain,
% 1.35/1.17 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y))) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))),
% 1.35/1.17 inference(quant_intro,[status(thm)],[110])).
% 1.35/1.17 tff(112,plain,
% 1.35/1.17 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y))) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))),
% 1.35/1.17 inference(rewrite,[status(thm)],[])).
% 1.35/1.17 tff(113,plain,
% 1.35/1.17 (^[V: $i, Y: $i, U: $i, X: $i] : rewrite((((~member(U, X)) | (~member(V, Y))) | member(ordered_pair(U, V), cross_product(X, Y))) <=> ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y))))),
% 1.35/1.17 inference(bind,[status(th)],[])).
% 1.35/1.17 tff(114,plain,
% 1.35/1.17 (![V: $i, Y: $i, U: $i, X: $i] : (((~member(U, X)) | (~member(V, Y))) | member(ordered_pair(U, V), cross_product(X, Y))) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))),
% 1.35/1.17 inference(quant_intro,[status(thm)],[113])).
% 1.35/1.17 tff(115,axiom,(![V: $i, Y: $i, U: $i, X: $i] : (((~member(U, X)) | (~member(V, Y))) | member(ordered_pair(U, V), cross_product(X, Y)))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','cartesian_product3')).
% 1.35/1.17 tff(116,plain,
% 1.35/1.17 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))),
% 1.35/1.17 inference(modus_ponens,[status(thm)],[115, 114])).
% 1.35/1.17 tff(117,plain,
% 1.35/1.17 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))),
% 1.35/1.17 inference(modus_ponens,[status(thm)],[116, 112])).
% 1.35/1.17 tff(118,plain,(
% 1.35/1.17 ![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))),
% 1.35/1.17 inference(skolemize,[status(sab)],[117])).
% 1.35/1.17 tff(119,plain,
% 1.35/1.17 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))),
% 1.35/1.17 inference(modus_ponens,[status(thm)],[118, 111])).
% 1.35/1.17 tff(120,plain,
% 1.35/1.17 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | ((~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), union(x, y))) | member(ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))), cross_product(union(x, y), z)) | (~member(second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), z)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | (~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), union(x, y))) | member(ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))), cross_product(union(x, y), z)) | (~member(second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), z)))),
% 1.48/1.17 inference(rewrite,[status(thm)],[])).
% 1.48/1.17 tff(121,plain,
% 1.48/1.17 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | ((~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), union(x, y))) | member(ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))), cross_product(union(x, y), z)) | (~member(second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), z)))),
% 1.48/1.17 inference(quant_inst,[status(thm)],[])).
% 1.48/1.17 tff(122,plain,
% 1.48/1.17 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | (~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), union(x, y))) | member(ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))), cross_product(union(x, y), z)) | (~member(second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), z))),
% 1.48/1.17 inference(modus_ponens,[status(thm)],[121, 120])).
% 1.48/1.17 tff(123,plain,
% 1.48/1.17 ((~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), union(x, y))) | member(ordered_pair(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z)))), cross_product(union(x, y), z)) | (~member(second(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), z))),
% 1.48/1.17 inference(unit_resolution,[status(thm)],[122, 119])).
% 1.48/1.17 tff(124,plain,
% 1.48/1.17 (~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), union(x, y))),
% 1.48/1.17 inference(unit_resolution,[status(thm)],[123, 109, 97])).
% 1.48/1.17 tff(125,plain,
% 1.48/1.17 (~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), complement(intersection(complement(x), complement(y))))),
% 1.48/1.17 inference(modus_ponens,[status(thm)],[124, 82])).
% 1.48/1.17 tff(126,plain,
% 1.48/1.17 (^[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))))),
% 1.48/1.17 inference(bind,[status(th)],[])).
% 1.48/1.17 tff(127,plain,
% 1.48/1.17 (![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)))),
% 1.48/1.17 inference(quant_intro,[status(thm)],[126])).
% 1.48/1.17 tff(128,plain,
% 1.48/1.17 (![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)))),
% 1.48/1.17 inference(rewrite,[status(thm)],[])).
% 1.48/1.17 tff(129,plain,
% 1.48/1.17 (^[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))))),
% 1.48/1.17 inference(bind,[status(th)],[])).
% 1.48/1.17 tff(130,plain,
% 1.48/1.17 (![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)))),
% 1.48/1.17 inference(quant_intro,[status(thm)],[129])).
% 1.48/1.17 tff(131,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')).
% 1.48/1.17 tff(132,plain,
% 1.48/1.17 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 1.48/1.17 inference(modus_ponens,[status(thm)],[131, 130])).
% 1.48/1.17 tff(133,plain,
% 1.48/1.17 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 1.48/1.17 inference(modus_ponens,[status(thm)],[132, 128])).
% 1.48/1.17 tff(134,plain,(
% 1.48/1.17 ![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 1.48/1.17 inference(skolemize,[status(sab)],[133])).
% 1.48/1.17 tff(135,plain,
% 1.48/1.17 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 1.48/1.17 inference(modus_ponens,[status(thm)],[134, 127])).
% 1.48/1.17 tff(136,plain,
% 1.48/1.17 (((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | ((~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), universal_class)) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), intersection(complement(x), complement(y))) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), complement(intersection(complement(x), complement(y)))))) <=> ((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | (~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), universal_class)) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), intersection(complement(x), complement(y))) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), complement(intersection(complement(x), complement(y)))))),
% 1.48/1.17 inference(rewrite,[status(thm)],[])).
% 1.48/1.17 tff(137,plain,
% 1.48/1.17 ((member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), intersection(complement(x), complement(y))) | (~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), universal_class)) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), complement(intersection(complement(x), complement(y))))) <=> ((~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), universal_class)) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), intersection(complement(x), complement(y))) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), complement(intersection(complement(x), complement(y)))))),
% 1.48/1.17 inference(rewrite,[status(thm)],[])).
% 1.48/1.17 tff(138,plain,
% 1.48/1.17 (((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | (member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), intersection(complement(x), complement(y))) | (~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), universal_class)) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), complement(intersection(complement(x), complement(y)))))) <=> ((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | ((~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), universal_class)) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), intersection(complement(x), complement(y))) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), complement(intersection(complement(x), complement(y))))))),
% 1.48/1.17 inference(monotonicity,[status(thm)],[137])).
% 1.48/1.17 tff(139,plain,
% 1.48/1.17 (((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | (member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), intersection(complement(x), complement(y))) | (~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), universal_class)) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), complement(intersection(complement(x), complement(y)))))) <=> ((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | (~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), universal_class)) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), intersection(complement(x), complement(y))) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), complement(intersection(complement(x), complement(y)))))),
% 1.48/1.17 inference(transitivity,[status(thm)],[138, 136])).
% 1.48/1.17 tff(140,plain,
% 1.48/1.17 ((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | (member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), intersection(complement(x), complement(y))) | (~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), universal_class)) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), complement(intersection(complement(x), complement(y)))))),
% 1.48/1.17 inference(quant_inst,[status(thm)],[])).
% 1.48/1.17 tff(141,plain,
% 1.48/1.17 ((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | (~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), universal_class)) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), intersection(complement(x), complement(y))) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), complement(intersection(complement(x), complement(y))))),
% 1.48/1.17 inference(modus_ponens,[status(thm)],[140, 139])).
% 1.48/1.17 tff(142,plain,
% 1.48/1.17 ((~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), universal_class)) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), intersection(complement(x), complement(y))) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), complement(intersection(complement(x), complement(y))))),
% 1.48/1.17 inference(unit_resolution,[status(thm)],[141, 135])).
% 1.48/1.17 tff(143,plain,
% 1.48/1.17 ((~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), universal_class)) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), intersection(complement(x), complement(y)))),
% 1.48/1.17 inference(unit_resolution,[status(thm)],[142, 125])).
% 1.48/1.17 tff(144,plain,
% 1.48/1.17 (member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), intersection(complement(x), complement(y)))),
% 1.48/1.17 inference(unit_resolution,[status(thm)],[143, 70])).
% 1.48/1.17 tff(145,plain,
% 1.48/1.17 (^[Z: $i, Y: $i, X: $i] : refl(((~member(Z, intersection(X, Y))) | member(Z, Y)) <=> ((~member(Z, intersection(X, Y))) | member(Z, Y)))),
% 1.48/1.17 inference(bind,[status(th)],[])).
% 1.48/1.17 tff(146,plain,
% 1.48/1.17 (![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))),
% 1.48/1.17 inference(quant_intro,[status(thm)],[145])).
% 1.48/1.17 tff(147,plain,
% 1.48/1.17 (![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))),
% 1.48/1.17 inference(rewrite,[status(thm)],[])).
% 1.48/1.17 tff(148,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')).
% 1.48/1.17 tff(149,plain,
% 1.48/1.17 (![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, Y))),
% 1.48/1.17 inference(modus_ponens,[status(thm)],[148, 147])).
% 1.48/1.17 tff(150,plain,(
% 1.48/1.17 ![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, Y))),
% 1.48/1.17 inference(skolemize,[status(sab)],[149])).
% 1.48/1.17 tff(151,plain,
% 1.48/1.17 (![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, Y))),
% 1.48/1.17 inference(modus_ponens,[status(thm)],[150, 146])).
% 1.48/1.17 tff(152,plain,
% 1.48/1.17 (((~![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, Y))) | ((~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), intersection(complement(x), complement(y)))) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), complement(y)))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, Y))) | (~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), intersection(complement(x), complement(y)))) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), complement(y)))),
% 1.48/1.18 inference(rewrite,[status(thm)],[])).
% 1.48/1.18 tff(153,plain,
% 1.48/1.18 ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, Y))) | ((~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), intersection(complement(x), complement(y)))) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), complement(y)))),
% 1.48/1.18 inference(quant_inst,[status(thm)],[])).
% 1.48/1.18 tff(154,plain,
% 1.48/1.18 ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, Y))) | (~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), intersection(complement(x), complement(y)))) | member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), complement(y))),
% 1.48/1.18 inference(modus_ponens,[status(thm)],[153, 152])).
% 1.48/1.18 tff(155,plain,
% 1.48/1.18 (member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), complement(y))),
% 1.48/1.18 inference(unit_resolution,[status(thm)],[154, 151, 144])).
% 1.48/1.18 tff(156,plain,
% 1.48/1.18 (^[Z: $i, X: $i] : refl(((~member(Z, X)) | (~member(Z, complement(X)))) <=> ((~member(Z, X)) | (~member(Z, complement(X)))))),
% 1.48/1.18 inference(bind,[status(th)],[])).
% 1.48/1.18 tff(157,plain,
% 1.48/1.18 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X)))) <=> ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 1.48/1.18 inference(quant_intro,[status(thm)],[156])).
% 1.48/1.18 tff(158,plain,
% 1.48/1.18 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X)))) <=> ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 1.48/1.18 inference(rewrite,[status(thm)],[])).
% 1.48/1.18 tff(159,plain,
% 1.48/1.18 (^[Z: $i, X: $i] : rewrite(((~member(Z, complement(X))) | (~member(Z, X))) <=> ((~member(Z, X)) | (~member(Z, complement(X)))))),
% 1.48/1.18 inference(bind,[status(th)],[])).
% 1.48/1.18 tff(160,plain,
% 1.48/1.18 (![Z: $i, X: $i] : ((~member(Z, complement(X))) | (~member(Z, X))) <=> ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 1.48/1.18 inference(quant_intro,[status(thm)],[159])).
% 1.48/1.18 tff(161,axiom,(![Z: $i, X: $i] : ((~member(Z, complement(X))) | (~member(Z, X)))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','complement1')).
% 1.48/1.18 tff(162,plain,
% 1.48/1.18 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 1.48/1.18 inference(modus_ponens,[status(thm)],[161, 160])).
% 1.48/1.18 tff(163,plain,
% 1.48/1.18 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 1.48/1.18 inference(modus_ponens,[status(thm)],[162, 158])).
% 1.48/1.18 tff(164,plain,(
% 1.48/1.18 ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 1.48/1.18 inference(skolemize,[status(sab)],[163])).
% 1.48/1.18 tff(165,plain,
% 1.48/1.18 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 1.48/1.18 inference(modus_ponens,[status(thm)],[164, 157])).
% 1.48/1.18 tff(166,plain,
% 1.48/1.18 (((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | ((~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), y)) | (~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), complement(y))))) <=> ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | (~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), y)) | (~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), complement(y))))),
% 1.48/1.18 inference(rewrite,[status(thm)],[])).
% 1.48/1.18 tff(167,plain,
% 1.48/1.18 ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | ((~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), y)) | (~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), complement(y))))),
% 1.48/1.18 inference(quant_inst,[status(thm)],[])).
% 1.48/1.18 tff(168,plain,
% 1.48/1.18 ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | (~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), y)) | (~member(first(not_subclass_element(cross_product(y, z), cross_product(union(x, y), z))), complement(y)))),
% 1.48/1.18 inference(modus_ponens,[status(thm)],[167, 166])).
% 1.48/1.18 tff(169,plain,
% 1.48/1.18 ($false),
% 1.48/1.18 inference(unit_resolution,[status(thm)],[168, 165, 55, 155])).
% 1.48/1.18 % SZS output end Proof
%------------------------------------------------------------------------------