TSTP Solution File: SET202-6 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET202-6 : TPTP v8.1.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n005.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 05:06:09 EDT 2022
% Result : Unsatisfiable 0.58s 0.61s
% Output : Proof 0.60s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : SET202-6 : TPTP v8.1.0. Bugfixed v2.1.0.
% 0.07/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.34 % Computer : n005.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Sat Sep 3 03:22:33 EDT 2022
% 0.13/0.35 % CPUTime :
% 0.13/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.13/0.35 Usage: tptp [options] [-file:]file
% 0.13/0.35 -h, -? prints this message.
% 0.13/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.13/0.35 -m, -model generate model.
% 0.13/0.35 -p, -proof generate proof.
% 0.13/0.35 -c, -core generate unsat core of named formulas.
% 0.13/0.35 -st, -statistics display statistics.
% 0.13/0.35 -t:timeout set timeout (in second).
% 0.13/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.13/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.13/0.35 -<param>:<value> configuration parameter and value.
% 0.13/0.35 -o:<output-file> file to place output in.
% 0.58/0.61 % SZS status Unsatisfiable
% 0.58/0.61 % SZS output start Proof
% 0.58/0.61 tff(subclass_type, type, (
% 0.58/0.61 subclass: ( $i * $i ) > $o)).
% 0.58/0.61 tff(universal_class_type, type, (
% 0.58/0.61 universal_class: $i)).
% 0.58/0.61 tff(x_type, type, (
% 0.58/0.61 x: $i)).
% 0.58/0.61 tff(member_type, type, (
% 0.58/0.61 member: ( $i * $i ) > $o)).
% 0.58/0.61 tff(first_type, type, (
% 0.58/0.61 first: $i > $i)).
% 0.58/0.61 tff(not_subclass_element_type, type, (
% 0.58/0.61 not_subclass_element: ( $i * $i ) > $i)).
% 0.58/0.61 tff(cross_product_type, type, (
% 0.58/0.61 cross_product: ( $i * $i ) > $i)).
% 0.58/0.61 tff(y_type, type, (
% 0.58/0.61 y: $i)).
% 0.58/0.61 tff(second_type, type, (
% 0.58/0.61 second: $i > $i)).
% 0.58/0.61 tff(ordered_pair_type, type, (
% 0.58/0.61 ordered_pair: ( $i * $i ) > $i)).
% 0.58/0.61 tff(1,assumption,(~subclass(y, universal_class)), introduced(assumption)).
% 0.58/0.61 tff(2,plain,
% 0.58/0.61 (^[X: $i] : refl(subclass(X, universal_class) <=> subclass(X, universal_class))),
% 0.58/0.61 inference(bind,[status(th)],[])).
% 0.58/0.61 tff(3,plain,
% 0.58/0.61 (![X: $i] : subclass(X, universal_class) <=> ![X: $i] : subclass(X, universal_class)),
% 0.58/0.61 inference(quant_intro,[status(thm)],[2])).
% 0.58/0.61 tff(4,plain,
% 0.58/0.61 (![X: $i] : subclass(X, universal_class) <=> ![X: $i] : subclass(X, universal_class)),
% 0.58/0.61 inference(rewrite,[status(thm)],[])).
% 0.58/0.61 tff(5,axiom,(![X: $i] : subclass(X, universal_class)), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','class_elements_are_sets')).
% 0.58/0.61 tff(6,plain,
% 0.58/0.61 (![X: $i] : subclass(X, universal_class)),
% 0.58/0.61 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.58/0.61 tff(7,plain,(
% 0.58/0.61 ![X: $i] : subclass(X, universal_class)),
% 0.58/0.61 inference(skolemize,[status(sab)],[6])).
% 0.58/0.61 tff(8,plain,
% 0.58/0.61 (![X: $i] : subclass(X, universal_class)),
% 0.58/0.61 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.58/0.61 tff(9,plain,
% 0.58/0.61 ((~![X: $i] : subclass(X, universal_class)) | subclass(y, universal_class)),
% 0.58/0.61 inference(quant_inst,[status(thm)],[])).
% 0.58/0.61 tff(10,plain,
% 0.58/0.61 ($false),
% 0.58/0.61 inference(unit_resolution,[status(thm)],[9, 8, 1])).
% 0.58/0.61 tff(11,plain,(subclass(y, universal_class)), inference(lemma,lemma(discharge,[]))).
% 0.58/0.61 tff(12,assumption,(~member(second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), universal_class)), introduced(assumption)).
% 0.58/0.61 tff(13,plain,
% 0.58/0.61 ((~subclass(cross_product(x, y), cross_product(universal_class, universal_class))) <=> (~subclass(cross_product(x, y), cross_product(universal_class, universal_class)))),
% 0.58/0.61 inference(rewrite,[status(thm)],[])).
% 0.58/0.61 tff(14,axiom,(~subclass(cross_product(x, y), cross_product(universal_class, universal_class))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_cross_product_property1_1')).
% 0.58/0.61 tff(15,plain,
% 0.58/0.61 (~subclass(cross_product(x, y), cross_product(universal_class, universal_class))),
% 0.58/0.61 inference(modus_ponens,[status(thm)],[14, 13])).
% 0.58/0.61 tff(16,plain,
% 0.58/0.61 (^[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)))),
% 0.58/0.61 inference(bind,[status(th)],[])).
% 0.58/0.61 tff(17,plain,
% 0.58/0.61 (![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))),
% 0.58/0.61 inference(quant_intro,[status(thm)],[16])).
% 0.58/0.61 tff(18,plain,
% 0.58/0.61 (![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))),
% 0.58/0.61 inference(rewrite,[status(thm)],[])).
% 0.58/0.61 tff(19,plain,
% 0.58/0.61 (^[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)))),
% 0.58/0.61 inference(bind,[status(th)],[])).
% 0.58/0.61 tff(20,plain,
% 0.58/0.61 (![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))),
% 0.58/0.61 inference(quant_intro,[status(thm)],[19])).
% 0.58/0.61 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')).
% 0.58/0.61 tff(22,plain,
% 0.58/0.61 (![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))),
% 0.58/0.61 inference(modus_ponens,[status(thm)],[21, 20])).
% 0.58/0.61 tff(23,plain,
% 0.58/0.61 (![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[22, 18])).
% 0.58/0.62 tff(24,plain,(
% 0.58/0.62 ![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))),
% 0.58/0.62 inference(skolemize,[status(sab)],[23])).
% 0.58/0.62 tff(25,plain,
% 0.58/0.62 (![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[24, 17])).
% 0.58/0.62 tff(26,plain,
% 0.58/0.62 (((~![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))) | (subclass(cross_product(x, y), cross_product(universal_class, universal_class)) | member(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)), cross_product(x, y)))) <=> ((~![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))) | subclass(cross_product(x, y), cross_product(universal_class, universal_class)) | member(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)), cross_product(x, y)))),
% 0.58/0.62 inference(rewrite,[status(thm)],[])).
% 0.58/0.62 tff(27,plain,
% 0.58/0.62 ((~![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))) | (subclass(cross_product(x, y), cross_product(universal_class, universal_class)) | member(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)), cross_product(x, y)))),
% 0.58/0.62 inference(quant_inst,[status(thm)],[])).
% 0.58/0.62 tff(28,plain,
% 0.58/0.62 ((~![Y: $i, X: $i] : (subclass(X, Y) | member(not_subclass_element(X, Y), X))) | subclass(cross_product(x, y), cross_product(universal_class, universal_class)) | member(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)), cross_product(x, y))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[27, 26])).
% 0.58/0.62 tff(29,plain,
% 0.58/0.62 (member(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)), cross_product(x, y))),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[28, 25, 15])).
% 0.58/0.62 tff(30,plain,
% 0.58/0.62 (^[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)))),
% 0.58/0.62 inference(bind,[status(th)],[])).
% 0.58/0.62 tff(31,plain,
% 0.58/0.62 (![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))),
% 0.58/0.62 inference(quant_intro,[status(thm)],[30])).
% 0.58/0.62 tff(32,plain,
% 0.58/0.62 (![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))),
% 0.58/0.62 inference(rewrite,[status(thm)],[])).
% 0.58/0.62 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')).
% 0.58/0.62 tff(34,plain,
% 0.58/0.62 (![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[33, 32])).
% 0.58/0.62 tff(35,plain,(
% 0.58/0.62 ![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))),
% 0.58/0.62 inference(skolemize,[status(sab)],[34])).
% 0.58/0.62 tff(36,plain,
% 0.58/0.62 (![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[35, 31])).
% 0.58/0.62 tff(37,plain,
% 0.58/0.62 (((~![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(x, y), cross_product(universal_class, universal_class)), cross_product(x, y))) | (ordered_pair(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))) = not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))))) <=> ((~![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(x, y), cross_product(universal_class, universal_class)), cross_product(x, y))) | (ordered_pair(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))) = not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))))),
% 0.58/0.62 inference(rewrite,[status(thm)],[])).
% 0.58/0.62 tff(38,plain,
% 0.58/0.62 ((~![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(x, y), cross_product(universal_class, universal_class)), cross_product(x, y))) | (ordered_pair(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))) = not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))))),
% 0.58/0.62 inference(quant_inst,[status(thm)],[])).
% 0.58/0.62 tff(39,plain,
% 0.58/0.62 ((~![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(x, y), cross_product(universal_class, universal_class)), cross_product(x, y))) | (ordered_pair(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))) = not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[38, 37])).
% 0.58/0.62 tff(40,plain,
% 0.58/0.62 (ordered_pair(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))) = not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[39, 36, 29])).
% 0.58/0.62 tff(41,plain,
% 0.58/0.62 (member(ordered_pair(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))), cross_product(x, y)) <=> member(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)), cross_product(x, y))),
% 0.58/0.62 inference(monotonicity,[status(thm)],[40])).
% 0.58/0.62 tff(42,plain,
% 0.58/0.62 (member(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)), cross_product(x, y)) <=> member(ordered_pair(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))), cross_product(x, y))),
% 0.58/0.62 inference(symmetry,[status(thm)],[41])).
% 0.58/0.62 tff(43,plain,
% 0.58/0.62 (member(ordered_pair(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))), cross_product(x, y))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[29, 42])).
% 0.58/0.62 tff(44,plain,
% 0.58/0.62 (^[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)))),
% 0.58/0.62 inference(bind,[status(th)],[])).
% 0.58/0.62 tff(45,plain,
% 0.58/0.62 (![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))),
% 0.58/0.62 inference(quant_intro,[status(thm)],[44])).
% 0.58/0.62 tff(46,plain,
% 0.58/0.62 (![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))),
% 0.58/0.62 inference(rewrite,[status(thm)],[])).
% 0.58/0.62 tff(47,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')).
% 0.58/0.62 tff(48,plain,
% 0.58/0.62 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[47, 46])).
% 0.58/0.62 tff(49,plain,(
% 0.58/0.62 ![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))),
% 0.58/0.62 inference(skolemize,[status(sab)],[48])).
% 0.58/0.62 tff(50,plain,
% 0.58/0.62 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[49, 45])).
% 0.58/0.62 tff(51,plain,
% 0.58/0.62 (((~![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(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))), cross_product(x, y))) | member(second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), y))) <=> ((~![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(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))), cross_product(x, y))) | member(second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), y))),
% 0.58/0.62 inference(rewrite,[status(thm)],[])).
% 0.58/0.62 tff(52,plain,
% 0.58/0.62 ((~![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(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))), cross_product(x, y))) | member(second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), y))),
% 0.58/0.62 inference(quant_inst,[status(thm)],[])).
% 0.58/0.62 tff(53,plain,
% 0.58/0.62 ((~![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(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))), cross_product(x, y))) | member(second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), y)),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[52, 51])).
% 0.58/0.62 tff(54,plain,
% 0.58/0.62 ((~member(ordered_pair(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))), cross_product(x, y))) | member(second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), y)),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[53, 50])).
% 0.58/0.62 tff(55,plain,
% 0.58/0.62 (member(second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), y)),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[54, 43])).
% 0.58/0.62 tff(56,plain,
% 0.58/0.62 (^[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))))),
% 0.58/0.62 inference(bind,[status(th)],[])).
% 0.58/0.62 tff(57,plain,
% 0.58/0.62 (![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)))),
% 0.58/0.62 inference(quant_intro,[status(thm)],[56])).
% 0.58/0.62 tff(58,plain,
% 0.58/0.62 (![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)))),
% 0.58/0.62 inference(rewrite,[status(thm)],[])).
% 0.58/0.62 tff(59,plain,
% 0.58/0.62 (^[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)))))),
% 0.58/0.62 inference(bind,[status(th)],[])).
% 0.58/0.62 tff(60,plain,
% 0.58/0.62 (![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)))),
% 0.58/0.62 inference(quant_intro,[status(thm)],[59])).
% 0.58/0.62 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')).
% 0.58/0.62 tff(62,plain,
% 0.58/0.62 (![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[61, 60])).
% 0.58/0.62 tff(63,plain,
% 0.58/0.62 (![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[62, 58])).
% 0.58/0.62 tff(64,plain,(
% 0.58/0.62 ![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 0.58/0.62 inference(skolemize,[status(sab)],[63])).
% 0.58/0.62 tff(65,plain,
% 0.58/0.62 (![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[64, 57])).
% 0.58/0.62 tff(66,plain,
% 0.58/0.62 (((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (member(second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), universal_class) | (~member(second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), y)) | (~subclass(y, universal_class)))) <=> ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | member(second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), universal_class) | (~member(second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), y)) | (~subclass(y, universal_class)))),
% 0.58/0.62 inference(rewrite,[status(thm)],[])).
% 0.58/0.62 tff(67,plain,
% 0.58/0.62 ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (member(second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), universal_class) | (~member(second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), y)) | (~subclass(y, universal_class)))),
% 0.58/0.62 inference(quant_inst,[status(thm)],[])).
% 0.58/0.62 tff(68,plain,
% 0.58/0.62 ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | member(second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), universal_class) | (~member(second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), y)) | (~subclass(y, universal_class))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[67, 66])).
% 0.58/0.62 tff(69,plain,
% 0.58/0.62 ($false),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[68, 65, 55, 12, 11])).
% 0.58/0.62 tff(70,plain,(member(second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), universal_class)), inference(lemma,lemma(discharge,[]))).
% 0.58/0.62 tff(71,plain,
% 0.58/0.62 (member(ordered_pair(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))), cross_product(universal_class, universal_class)) <=> member(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)), cross_product(universal_class, universal_class))),
% 0.58/0.62 inference(monotonicity,[status(thm)],[40])).
% 0.58/0.62 tff(72,plain,
% 0.58/0.62 (member(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)), cross_product(universal_class, universal_class)) <=> member(ordered_pair(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))), cross_product(universal_class, universal_class))),
% 0.58/0.62 inference(symmetry,[status(thm)],[71])).
% 0.58/0.62 tff(73,plain,
% 0.58/0.62 ((~member(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)), cross_product(universal_class, universal_class))) <=> (~member(ordered_pair(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))), cross_product(universal_class, universal_class)))),
% 0.58/0.62 inference(monotonicity,[status(thm)],[72])).
% 0.58/0.62 tff(74,plain,
% 0.58/0.62 (^[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)))),
% 0.58/0.62 inference(bind,[status(th)],[])).
% 0.58/0.62 tff(75,plain,
% 0.58/0.62 (![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))),
% 0.58/0.62 inference(quant_intro,[status(thm)],[74])).
% 0.58/0.62 tff(76,plain,
% 0.58/0.62 (![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))),
% 0.58/0.62 inference(rewrite,[status(thm)],[])).
% 0.58/0.62 tff(77,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')).
% 0.58/0.62 tff(78,plain,
% 0.58/0.62 (![Y: $i, X: $i] : ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[77, 76])).
% 0.58/0.62 tff(79,plain,(
% 0.58/0.62 ![Y: $i, X: $i] : ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y))),
% 0.58/0.62 inference(skolemize,[status(sab)],[78])).
% 0.58/0.62 tff(80,plain,
% 0.58/0.62 (![Y: $i, X: $i] : ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[79, 75])).
% 0.58/0.62 tff(81,plain,
% 0.58/0.62 (((~![Y: $i, X: $i] : ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y))) | ((~member(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)), cross_product(universal_class, universal_class))) | subclass(cross_product(x, y), cross_product(universal_class, universal_class)))) <=> ((~![Y: $i, X: $i] : ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y))) | (~member(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)), cross_product(universal_class, universal_class))) | subclass(cross_product(x, y), cross_product(universal_class, universal_class)))),
% 0.58/0.62 inference(rewrite,[status(thm)],[])).
% 0.58/0.62 tff(82,plain,
% 0.58/0.62 ((~![Y: $i, X: $i] : ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y))) | ((~member(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)), cross_product(universal_class, universal_class))) | subclass(cross_product(x, y), cross_product(universal_class, universal_class)))),
% 0.58/0.62 inference(quant_inst,[status(thm)],[])).
% 0.58/0.62 tff(83,plain,
% 0.58/0.62 ((~![Y: $i, X: $i] : ((~member(not_subclass_element(X, Y), Y)) | subclass(X, Y))) | (~member(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)), cross_product(universal_class, universal_class))) | subclass(cross_product(x, y), cross_product(universal_class, universal_class))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[82, 81])).
% 0.58/0.62 tff(84,plain,
% 0.58/0.62 (~member(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)), cross_product(universal_class, universal_class))),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[83, 80, 15])).
% 0.58/0.62 tff(85,plain,
% 0.58/0.62 (~member(ordered_pair(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))), cross_product(universal_class, universal_class))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[84, 73])).
% 0.58/0.62 tff(86,plain,
% 0.58/0.62 (^[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))))),
% 0.58/0.62 inference(bind,[status(th)],[])).
% 0.58/0.62 tff(87,plain,
% 0.58/0.62 (![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)))),
% 0.58/0.62 inference(quant_intro,[status(thm)],[86])).
% 0.58/0.62 tff(88,plain,
% 0.58/0.62 (![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)))),
% 0.58/0.62 inference(rewrite,[status(thm)],[])).
% 0.58/0.62 tff(89,plain,
% 0.58/0.62 (^[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))))),
% 0.58/0.62 inference(bind,[status(th)],[])).
% 0.58/0.62 tff(90,plain,
% 0.58/0.62 (![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)))),
% 0.58/0.62 inference(quant_intro,[status(thm)],[89])).
% 0.58/0.62 tff(91,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')).
% 0.58/0.62 tff(92,plain,
% 0.58/0.62 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[91, 90])).
% 0.58/0.62 tff(93,plain,
% 0.58/0.62 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[92, 88])).
% 0.58/0.62 tff(94,plain,(
% 0.58/0.62 ![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))),
% 0.58/0.62 inference(skolemize,[status(sab)],[93])).
% 0.58/0.62 tff(95,plain,
% 0.58/0.62 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[94, 87])).
% 0.58/0.62 tff(96,plain,
% 0.58/0.62 (((~![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(x, y), cross_product(universal_class, universal_class))), universal_class)) | member(ordered_pair(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))), cross_product(universal_class, universal_class)) | (~member(second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), universal_class)))) <=> ((~![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(x, y), cross_product(universal_class, universal_class))), universal_class)) | member(ordered_pair(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))), cross_product(universal_class, universal_class)) | (~member(second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), universal_class)))),
% 0.58/0.62 inference(rewrite,[status(thm)],[])).
% 0.58/0.62 tff(97,plain,
% 0.58/0.62 ((~![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(x, y), cross_product(universal_class, universal_class))), universal_class)) | member(ordered_pair(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))), cross_product(universal_class, universal_class)) | (~member(second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), universal_class)))),
% 0.58/0.62 inference(quant_inst,[status(thm)],[])).
% 0.58/0.62 tff(98,plain,
% 0.58/0.62 ((~![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(x, y), cross_product(universal_class, universal_class))), universal_class)) | member(ordered_pair(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))), cross_product(universal_class, universal_class)) | (~member(second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), universal_class))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[97, 96])).
% 0.58/0.62 tff(99,plain,
% 0.58/0.62 ((~member(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), universal_class)) | member(ordered_pair(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))), cross_product(universal_class, universal_class)) | (~member(second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), universal_class))),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[98, 95])).
% 0.58/0.62 tff(100,plain,
% 0.58/0.62 ((~member(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), universal_class)) | (~member(second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), universal_class))),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[99, 85])).
% 0.58/0.62 tff(101,plain,
% 0.58/0.62 (~member(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), universal_class)),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[100, 70])).
% 0.58/0.62 tff(102,plain,
% 0.58/0.62 (^[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)))),
% 0.58/0.62 inference(bind,[status(th)],[])).
% 0.58/0.62 tff(103,plain,
% 0.58/0.62 (![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))),
% 0.58/0.62 inference(quant_intro,[status(thm)],[102])).
% 0.58/0.62 tff(104,plain,
% 0.58/0.62 (![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))),
% 0.58/0.62 inference(rewrite,[status(thm)],[])).
% 0.58/0.62 tff(105,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')).
% 0.58/0.62 tff(106,plain,
% 0.58/0.62 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[105, 104])).
% 0.58/0.62 tff(107,plain,(
% 0.58/0.62 ![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))),
% 0.58/0.62 inference(skolemize,[status(sab)],[106])).
% 0.58/0.62 tff(108,plain,
% 0.58/0.62 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[107, 103])).
% 0.58/0.62 tff(109,plain,
% 0.58/0.62 (((~![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(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))), cross_product(x, y))) | member(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), x))) <=> ((~![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(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))), cross_product(x, y))) | member(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), x))),
% 0.58/0.63 inference(rewrite,[status(thm)],[])).
% 0.58/0.63 tff(110,plain,
% 0.58/0.63 ((~![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(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))), cross_product(x, y))) | member(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), x))),
% 0.58/0.63 inference(quant_inst,[status(thm)],[])).
% 0.58/0.63 tff(111,plain,
% 0.58/0.63 ((~![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(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))), cross_product(x, y))) | member(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), x)),
% 0.58/0.63 inference(modus_ponens,[status(thm)],[110, 109])).
% 0.58/0.63 tff(112,plain,
% 0.58/0.63 ((~member(ordered_pair(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), second(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class)))), cross_product(x, y))) | member(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), x)),
% 0.58/0.63 inference(unit_resolution,[status(thm)],[111, 108])).
% 0.58/0.63 tff(113,plain,
% 0.58/0.63 (member(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), x)),
% 0.58/0.63 inference(unit_resolution,[status(thm)],[112, 43])).
% 0.58/0.63 tff(114,plain,
% 0.58/0.63 (((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (member(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), universal_class) | (~member(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), x)) | (~subclass(x, universal_class)))) <=> ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | member(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), universal_class) | (~member(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), x)) | (~subclass(x, universal_class)))),
% 0.58/0.63 inference(rewrite,[status(thm)],[])).
% 0.58/0.63 tff(115,plain,
% 0.58/0.63 ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (member(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), universal_class) | (~member(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), x)) | (~subclass(x, universal_class)))),
% 0.58/0.63 inference(quant_inst,[status(thm)],[])).
% 0.58/0.63 tff(116,plain,
% 0.58/0.63 ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | member(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), universal_class) | (~member(first(not_subclass_element(cross_product(x, y), cross_product(universal_class, universal_class))), x)) | (~subclass(x, universal_class))),
% 0.58/0.63 inference(modus_ponens,[status(thm)],[115, 114])).
% 0.58/0.63 tff(117,plain,
% 0.58/0.63 (~subclass(x, universal_class)),
% 0.60/0.63 inference(unit_resolution,[status(thm)],[116, 65, 113, 101])).
% 0.60/0.63 tff(118,plain,
% 0.60/0.63 ((~![X: $i] : subclass(X, universal_class)) | subclass(x, universal_class)),
% 0.60/0.63 inference(quant_inst,[status(thm)],[])).
% 0.60/0.63 tff(119,plain,
% 0.60/0.63 ($false),
% 0.60/0.63 inference(unit_resolution,[status(thm)],[118, 8, 117])).
% 0.60/0.63 % SZS output end Proof
%------------------------------------------------------------------------------