TSTP Solution File: SET021-6 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET021-6 : TPTP v8.1.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n022.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 05:04:54 EDT 2022
% Result : Unsatisfiable 0.46s 0.52s
% Output : Proof 0.48s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : SET021-6 : TPTP v8.1.0. Bugfixed v2.1.0.
% 0.08/0.10 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.09/0.30 % Computer : n022.cluster.edu
% 0.09/0.30 % Model : x86_64 x86_64
% 0.09/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.30 % Memory : 8042.1875MB
% 0.09/0.30 % OS : Linux 3.10.0-693.el7.x86_64
% 0.09/0.30 % CPULimit : 300
% 0.09/0.30 % WCLimit : 300
% 0.09/0.30 % DateTime : Sat Sep 3 01:19:09 EDT 2022
% 0.09/0.30 % CPUTime :
% 0.09/0.30 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.09/0.30 Usage: tptp [options] [-file:]file
% 0.09/0.30 -h, -? prints this message.
% 0.09/0.30 -smt2 print SMT-LIB2 benchmark.
% 0.09/0.30 -m, -model generate model.
% 0.09/0.30 -p, -proof generate proof.
% 0.09/0.30 -c, -core generate unsat core of named formulas.
% 0.09/0.30 -st, -statistics display statistics.
% 0.09/0.30 -t:timeout set timeout (in second).
% 0.09/0.30 -smt2status display status in smt2 format instead of SZS.
% 0.09/0.30 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.09/0.30 -<param>:<value> configuration parameter and value.
% 0.09/0.30 -o:<output-file> file to place output in.
% 0.46/0.52 % SZS status Unsatisfiable
% 0.46/0.52 % SZS output start Proof
% 0.46/0.52 tff(member_type, type, (
% 0.46/0.52 member: ( $i * $i ) > $o)).
% 0.46/0.52 tff(unordered_pair_type, type, (
% 0.46/0.52 unordered_pair: ( $i * $i ) > $i)).
% 0.46/0.52 tff(v_type, type, (
% 0.46/0.52 v: $i)).
% 0.46/0.52 tff(second_type, type, (
% 0.46/0.52 second: $i > $i)).
% 0.46/0.52 tff(ordered_pair_type, type, (
% 0.46/0.52 ordered_pair: ( $i * $i ) > $i)).
% 0.46/0.52 tff(u_type, type, (
% 0.46/0.52 u: $i)).
% 0.46/0.52 tff(singleton_type, type, (
% 0.46/0.52 singleton: $i > $i)).
% 0.46/0.52 tff(first_type, type, (
% 0.46/0.52 first: $i > $i)).
% 0.46/0.52 tff(cross_product_type, type, (
% 0.46/0.52 cross_product: ( $i * $i ) > $i)).
% 0.46/0.52 tff(universal_class_type, type, (
% 0.46/0.52 universal_class: $i)).
% 0.46/0.52 tff(1,plain,
% 0.46/0.52 (^[X: $i] : refl((unordered_pair(X, X) = singleton(X)) <=> (unordered_pair(X, X) = singleton(X)))),
% 0.46/0.52 inference(bind,[status(th)],[])).
% 0.46/0.52 tff(2,plain,
% 0.46/0.52 (![X: $i] : (unordered_pair(X, X) = singleton(X)) <=> ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.46/0.52 inference(quant_intro,[status(thm)],[1])).
% 0.46/0.52 tff(3,plain,
% 0.46/0.52 (![X: $i] : (unordered_pair(X, X) = singleton(X)) <=> ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.46/0.52 inference(rewrite,[status(thm)],[])).
% 0.46/0.52 tff(4,axiom,(![X: $i] : (unordered_pair(X, X) = singleton(X))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','singleton_set')).
% 0.46/0.52 tff(5,plain,
% 0.46/0.52 (![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.46/0.52 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.46/0.52 tff(6,plain,(
% 0.46/0.52 ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.46/0.52 inference(skolemize,[status(sab)],[5])).
% 0.46/0.52 tff(7,plain,
% 0.46/0.52 (![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.46/0.52 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.46/0.52 tff(8,plain,
% 0.46/0.52 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v))) = singleton(second(ordered_pair(u, v))))),
% 0.46/0.52 inference(quant_inst,[status(thm)],[])).
% 0.46/0.52 tff(9,plain,
% 0.46/0.52 (unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v))) = singleton(second(ordered_pair(u, v)))),
% 0.46/0.52 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.46/0.52 tff(10,plain,
% 0.46/0.52 (singleton(second(ordered_pair(u, v))) = unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v)))),
% 0.46/0.52 inference(symmetry,[status(thm)],[9])).
% 0.46/0.52 tff(11,plain,
% 0.46/0.52 ((singleton(second(ordered_pair(u, v))) = singleton(v)) <=> (singleton(v) = singleton(second(ordered_pair(u, v))))),
% 0.46/0.52 inference(commutativity,[status(thm)],[])).
% 0.46/0.52 tff(12,plain,
% 0.46/0.52 ((singleton(v) = singleton(second(ordered_pair(u, v)))) <=> (singleton(second(ordered_pair(u, v))) = singleton(v))),
% 0.46/0.52 inference(symmetry,[status(thm)],[11])).
% 0.46/0.52 tff(13,plain,
% 0.46/0.52 ((~(singleton(v) = singleton(second(ordered_pair(u, v))))) <=> (~(singleton(second(ordered_pair(u, v))) = singleton(v)))),
% 0.46/0.52 inference(monotonicity,[status(thm)],[12])).
% 0.46/0.52 tff(14,assumption,(~(singleton(v) = singleton(second(ordered_pair(u, v))))), introduced(assumption)).
% 0.46/0.52 tff(15,plain,
% 0.46/0.52 (~(singleton(second(ordered_pair(u, v))) = singleton(v))),
% 0.46/0.52 inference(modus_ponens,[status(thm)],[14, 13])).
% 0.46/0.52 tff(16,plain,
% 0.46/0.52 ((unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = unordered_pair(u, singleton(v))) <=> (unordered_pair(u, singleton(v)) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))),
% 0.46/0.52 inference(commutativity,[status(thm)],[])).
% 0.46/0.52 tff(17,plain,
% 0.46/0.52 ((singleton(u) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) <=> (unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = singleton(u))),
% 0.46/0.52 inference(commutativity,[status(thm)],[])).
% 0.46/0.52 tff(18,assumption,(~(singleton(u) = singleton(first(ordered_pair(u, v))))), introduced(assumption)).
% 0.46/0.52 tff(19,plain,
% 0.46/0.52 (^[Y: $i, X: $i] : refl((unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y)) <=> (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y)))),
% 0.46/0.52 inference(bind,[status(th)],[])).
% 0.46/0.52 tff(20,plain,
% 0.46/0.52 (![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y)) <=> ![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))),
% 0.46/0.52 inference(quant_intro,[status(thm)],[19])).
% 0.46/0.52 tff(21,plain,
% 0.46/0.52 (![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y)) <=> ![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))),
% 0.46/0.52 inference(rewrite,[status(thm)],[])).
% 0.46/0.52 tff(22,axiom,(![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','ordered_pair')).
% 0.46/0.52 tff(23,plain,
% 0.46/0.52 (![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))),
% 0.46/0.52 inference(modus_ponens,[status(thm)],[22, 21])).
% 0.46/0.52 tff(24,plain,(
% 0.46/0.52 ![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))),
% 0.46/0.52 inference(skolemize,[status(sab)],[23])).
% 0.46/0.52 tff(25,plain,
% 0.46/0.52 (![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))),
% 0.46/0.52 inference(modus_ponens,[status(thm)],[24, 20])).
% 0.46/0.52 tff(26,plain,
% 0.46/0.52 ((~![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))) | (unordered_pair(singleton(u), unordered_pair(u, singleton(v))) = ordered_pair(u, v))),
% 0.46/0.52 inference(quant_inst,[status(thm)],[])).
% 0.46/0.52 tff(27,plain,
% 0.46/0.52 (unordered_pair(singleton(u), unordered_pair(u, singleton(v))) = ordered_pair(u, v)),
% 0.46/0.52 inference(unit_resolution,[status(thm)],[26, 25])).
% 0.46/0.52 tff(28,plain,
% 0.46/0.52 (ordered_pair(u, v) = unordered_pair(singleton(u), unordered_pair(u, singleton(v)))),
% 0.46/0.52 inference(symmetry,[status(thm)],[27])).
% 0.46/0.52 tff(29,plain,
% 0.46/0.52 (member(ordered_pair(u, v), cross_product(universal_class, universal_class)) <=> member(ordered_pair(u, v), cross_product(universal_class, universal_class))),
% 0.46/0.52 inference(rewrite,[status(thm)],[])).
% 0.46/0.52 tff(30,axiom,(member(ordered_pair(u, v), cross_product(universal_class, universal_class))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_unique_1st_and_2nd_in_pair_of_sets2_1')).
% 0.46/0.52 tff(31,plain,
% 0.46/0.52 (member(ordered_pair(u, v), cross_product(universal_class, universal_class))),
% 0.46/0.52 inference(modus_ponens,[status(thm)],[30, 29])).
% 0.46/0.52 tff(32,plain,
% 0.46/0.52 (^[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)))),
% 0.46/0.52 inference(bind,[status(th)],[])).
% 0.46/0.52 tff(33,plain,
% 0.46/0.52 (![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))),
% 0.46/0.52 inference(quant_intro,[status(thm)],[32])).
% 0.46/0.52 tff(34,plain,
% 0.46/0.52 (![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))),
% 0.46/0.52 inference(rewrite,[status(thm)],[])).
% 0.46/0.52 tff(35,axiom,(![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','cartesian_product4')).
% 0.46/0.52 tff(36,plain,
% 0.46/0.52 (![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))),
% 0.46/0.52 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.46/0.52 tff(37,plain,(
% 0.46/0.52 ![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))),
% 0.46/0.52 inference(skolemize,[status(sab)],[36])).
% 0.46/0.52 tff(38,plain,
% 0.46/0.52 (![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))),
% 0.46/0.52 inference(modus_ponens,[status(thm)],[37, 33])).
% 0.46/0.52 tff(39,plain,
% 0.46/0.52 (((~![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))) | ((~member(ordered_pair(u, v), cross_product(universal_class, universal_class))) | (ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))) = ordered_pair(u, v)))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))) | (~member(ordered_pair(u, v), cross_product(universal_class, universal_class))) | (ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))) = ordered_pair(u, v)))),
% 0.46/0.52 inference(rewrite,[status(thm)],[])).
% 0.46/0.52 tff(40,plain,
% 0.46/0.52 ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))) | ((~member(ordered_pair(u, v), cross_product(universal_class, universal_class))) | (ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))) = ordered_pair(u, v)))),
% 0.46/0.52 inference(quant_inst,[status(thm)],[])).
% 0.46/0.52 tff(41,plain,
% 0.46/0.52 ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))) | (~member(ordered_pair(u, v), cross_product(universal_class, universal_class))) | (ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))) = ordered_pair(u, v))),
% 0.46/0.52 inference(modus_ponens,[status(thm)],[40, 39])).
% 0.46/0.52 tff(42,plain,
% 0.46/0.52 (ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))) = ordered_pair(u, v)),
% 0.46/0.52 inference(unit_resolution,[status(thm)],[41, 38, 31])).
% 0.46/0.52 tff(43,plain,
% 0.46/0.52 ((~![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))) | (unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) = ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))))),
% 0.46/0.52 inference(quant_inst,[status(thm)],[])).
% 0.46/0.52 tff(44,plain,
% 0.46/0.52 (unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) = ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v)))),
% 0.46/0.52 inference(unit_resolution,[status(thm)],[43, 25])).
% 0.46/0.52 tff(45,plain,
% 0.46/0.52 (unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) = unordered_pair(singleton(u), unordered_pair(u, singleton(v)))),
% 0.46/0.52 inference(transitivity,[status(thm)],[44, 42, 28])).
% 0.46/0.52 tff(46,plain,
% 0.46/0.52 (member(singleton(u), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))) <=> member(singleton(u), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))),
% 0.46/0.52 inference(monotonicity,[status(thm)],[45])).
% 0.46/0.52 tff(47,plain,
% 0.46/0.52 (member(singleton(u), unordered_pair(singleton(u), unordered_pair(u, singleton(v)))) <=> member(singleton(u), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))),
% 0.46/0.52 inference(symmetry,[status(thm)],[46])).
% 0.46/0.52 tff(48,plain,
% 0.46/0.52 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(u, u) = singleton(u))),
% 0.46/0.52 inference(quant_inst,[status(thm)],[])).
% 0.46/0.52 tff(49,plain,
% 0.46/0.52 (unordered_pair(u, u) = singleton(u)),
% 0.46/0.52 inference(unit_resolution,[status(thm)],[48, 7])).
% 0.46/0.52 tff(50,plain,
% 0.46/0.52 (member(unordered_pair(u, u), universal_class) <=> member(singleton(u), universal_class)),
% 0.46/0.52 inference(monotonicity,[status(thm)],[49])).
% 0.46/0.52 tff(51,plain,
% 0.46/0.52 (^[Y: $i, X: $i] : refl(member(unordered_pair(X, Y), universal_class) <=> member(unordered_pair(X, Y), universal_class))),
% 0.46/0.52 inference(bind,[status(th)],[])).
% 0.46/0.52 tff(52,plain,
% 0.46/0.52 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class) <=> ![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.46/0.52 inference(quant_intro,[status(thm)],[51])).
% 0.46/0.52 tff(53,plain,
% 0.46/0.52 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class) <=> ![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.46/0.52 inference(rewrite,[status(thm)],[])).
% 0.46/0.52 tff(54,axiom,(![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','unordered_pairs_in_universal')).
% 0.46/0.52 tff(55,plain,
% 0.46/0.52 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.46/0.52 inference(modus_ponens,[status(thm)],[54, 53])).
% 0.46/0.52 tff(56,plain,(
% 0.46/0.52 ![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.46/0.52 inference(skolemize,[status(sab)],[55])).
% 0.46/0.52 tff(57,plain,
% 0.46/0.52 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.46/0.52 inference(modus_ponens,[status(thm)],[56, 52])).
% 0.46/0.52 tff(58,plain,
% 0.46/0.52 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(u, u), universal_class)),
% 0.46/0.52 inference(quant_inst,[status(thm)],[])).
% 0.46/0.52 tff(59,plain,
% 0.46/0.52 (member(unordered_pair(u, u), universal_class)),
% 0.46/0.52 inference(unit_resolution,[status(thm)],[58, 57])).
% 0.46/0.52 tff(60,plain,
% 0.46/0.52 (member(singleton(u), universal_class)),
% 0.46/0.52 inference(modus_ponens,[status(thm)],[59, 50])).
% 0.46/0.52 tff(61,plain,
% 0.46/0.52 (^[Y: $i, X: $i] : refl(((~member(X, universal_class)) | member(X, unordered_pair(X, Y))) <=> ((~member(X, universal_class)) | member(X, unordered_pair(X, Y))))),
% 0.46/0.52 inference(bind,[status(th)],[])).
% 0.46/0.52 tff(62,plain,
% 0.46/0.52 (![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y))) <=> ![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 0.46/0.52 inference(quant_intro,[status(thm)],[61])).
% 0.46/0.52 tff(63,plain,
% 0.46/0.52 (![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y))) <=> ![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 0.46/0.52 inference(rewrite,[status(thm)],[])).
% 0.46/0.52 tff(64,axiom,(![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','unordered_pair2')).
% 0.46/0.52 tff(65,plain,
% 0.46/0.52 (![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 0.46/0.52 inference(modus_ponens,[status(thm)],[64, 63])).
% 0.46/0.52 tff(66,plain,(
% 0.46/0.52 ![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 0.46/0.52 inference(skolemize,[status(sab)],[65])).
% 0.46/0.52 tff(67,plain,
% 0.46/0.52 (![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 0.46/0.52 inference(modus_ponens,[status(thm)],[66, 62])).
% 0.46/0.52 tff(68,plain,
% 0.46/0.52 (((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(singleton(u), universal_class)) | member(singleton(u), unordered_pair(singleton(u), unordered_pair(u, singleton(v)))))) <=> ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(singleton(u), universal_class)) | member(singleton(u), unordered_pair(singleton(u), unordered_pair(u, singleton(v)))))),
% 0.46/0.52 inference(rewrite,[status(thm)],[])).
% 0.46/0.52 tff(69,plain,
% 0.46/0.52 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(singleton(u), universal_class)) | member(singleton(u), unordered_pair(singleton(u), unordered_pair(u, singleton(v)))))),
% 0.46/0.52 inference(quant_inst,[status(thm)],[])).
% 0.46/0.52 tff(70,plain,
% 0.46/0.52 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(singleton(u), universal_class)) | member(singleton(u), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))),
% 0.46/0.52 inference(modus_ponens,[status(thm)],[69, 68])).
% 0.46/0.52 tff(71,plain,
% 0.46/0.52 ((~member(singleton(u), universal_class)) | member(singleton(u), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))),
% 0.46/0.52 inference(unit_resolution,[status(thm)],[70, 67])).
% 0.46/0.52 tff(72,plain,
% 0.46/0.52 (member(singleton(u), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))),
% 0.46/0.52 inference(unit_resolution,[status(thm)],[71, 60])).
% 0.46/0.52 tff(73,plain,
% 0.46/0.52 (member(singleton(u), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))),
% 0.46/0.52 inference(modus_ponens,[status(thm)],[72, 47])).
% 0.46/0.52 tff(74,plain,
% 0.46/0.52 (^[Y: $i, U: $i, X: $i] : refl(((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y)))) <=> ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y)))))),
% 0.46/0.52 inference(bind,[status(th)],[])).
% 0.46/0.52 tff(75,plain,
% 0.46/0.52 (![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y)))) <=> ![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 0.46/0.52 inference(quant_intro,[status(thm)],[74])).
% 0.46/0.52 tff(76,plain,
% 0.46/0.52 (![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y)))) <=> ![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 0.46/0.53 inference(rewrite,[status(thm)],[])).
% 0.46/0.53 tff(77,plain,
% 0.46/0.53 (^[Y: $i, U: $i, X: $i] : rewrite((((~member(U, unordered_pair(X, Y))) | (U = X)) | (U = Y)) <=> ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y)))))),
% 0.46/0.53 inference(bind,[status(th)],[])).
% 0.46/0.53 tff(78,plain,
% 0.46/0.53 (![Y: $i, U: $i, X: $i] : (((~member(U, unordered_pair(X, Y))) | (U = X)) | (U = Y)) <=> ![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 0.46/0.53 inference(quant_intro,[status(thm)],[77])).
% 0.46/0.53 tff(79,axiom,(![Y: $i, U: $i, X: $i] : (((~member(U, unordered_pair(X, Y))) | (U = X)) | (U = Y))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','unordered_pair_member')).
% 0.46/0.53 tff(80,plain,
% 0.46/0.53 (![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 0.46/0.53 inference(modus_ponens,[status(thm)],[79, 78])).
% 0.46/0.53 tff(81,plain,
% 0.46/0.53 (![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 0.46/0.53 inference(modus_ponens,[status(thm)],[80, 76])).
% 0.46/0.53 tff(82,plain,(
% 0.46/0.53 ![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 0.46/0.53 inference(skolemize,[status(sab)],[81])).
% 0.46/0.53 tff(83,plain,
% 0.46/0.53 (![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 0.46/0.53 inference(modus_ponens,[status(thm)],[82, 75])).
% 0.46/0.53 tff(84,plain,
% 0.46/0.53 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(u) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) | (singleton(u) = singleton(first(ordered_pair(u, v)))) | (~member(singleton(u), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(u) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) | (singleton(u) = singleton(first(ordered_pair(u, v)))) | (~member(singleton(u), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))))),
% 0.46/0.53 inference(rewrite,[status(thm)],[])).
% 0.46/0.53 tff(85,plain,
% 0.46/0.53 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(u) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) | (singleton(u) = singleton(first(ordered_pair(u, v)))) | (~member(singleton(u), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))))),
% 0.46/0.53 inference(quant_inst,[status(thm)],[])).
% 0.46/0.53 tff(86,plain,
% 0.46/0.53 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(u) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) | (singleton(u) = singleton(first(ordered_pair(u, v)))) | (~member(singleton(u), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))))),
% 0.46/0.53 inference(modus_ponens,[status(thm)],[85, 84])).
% 0.46/0.53 tff(87,plain,
% 0.46/0.53 ((singleton(u) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) | (singleton(u) = singleton(first(ordered_pair(u, v)))) | (~member(singleton(u), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))))),
% 0.46/0.53 inference(unit_resolution,[status(thm)],[86, 83])).
% 0.46/0.53 tff(88,plain,
% 0.46/0.53 ((singleton(u) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) | (singleton(u) = singleton(first(ordered_pair(u, v))))),
% 0.46/0.53 inference(unit_resolution,[status(thm)],[87, 73])).
% 0.46/0.53 tff(89,plain,
% 0.46/0.53 (singleton(u) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))),
% 0.46/0.53 inference(unit_resolution,[status(thm)],[88, 18])).
% 0.46/0.53 tff(90,plain,
% 0.48/0.53 (unordered_pair(u, u) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))),
% 0.48/0.53 inference(transitivity,[status(thm)],[49, 89])).
% 0.48/0.53 tff(91,plain,
% 0.48/0.53 (member(first(ordered_pair(u, v)), unordered_pair(u, u)) <=> member(first(ordered_pair(u, v)), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))),
% 0.48/0.53 inference(monotonicity,[status(thm)],[90])).
% 0.48/0.53 tff(92,plain,
% 0.48/0.53 (member(first(ordered_pair(u, v)), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) <=> member(first(ordered_pair(u, v)), unordered_pair(u, u))),
% 0.48/0.53 inference(symmetry,[status(thm)],[91])).
% 0.48/0.53 tff(93,plain,
% 0.48/0.53 (member(ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), cross_product(universal_class, universal_class)) <=> member(ordered_pair(u, v), cross_product(universal_class, universal_class))),
% 0.48/0.53 inference(monotonicity,[status(thm)],[42])).
% 0.48/0.53 tff(94,plain,
% 0.48/0.53 (member(ordered_pair(u, v), cross_product(universal_class, universal_class)) <=> member(ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), cross_product(universal_class, universal_class))),
% 0.48/0.53 inference(symmetry,[status(thm)],[93])).
% 0.48/0.53 tff(95,plain,
% 0.48/0.53 (member(ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), cross_product(universal_class, universal_class))),
% 0.48/0.53 inference(modus_ponens,[status(thm)],[31, 94])).
% 0.48/0.53 tff(96,plain,
% 0.48/0.53 (^[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)))),
% 0.48/0.53 inference(bind,[status(th)],[])).
% 0.48/0.53 tff(97,plain,
% 0.48/0.53 (![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))),
% 0.48/0.53 inference(quant_intro,[status(thm)],[96])).
% 0.48/0.53 tff(98,plain,
% 0.48/0.53 (![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))),
% 0.48/0.53 inference(rewrite,[status(thm)],[])).
% 0.48/0.53 tff(99,axiom,(![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','cartesian_product1')).
% 0.48/0.53 tff(100,plain,
% 0.48/0.53 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))),
% 0.48/0.53 inference(modus_ponens,[status(thm)],[99, 98])).
% 0.48/0.53 tff(101,plain,(
% 0.48/0.53 ![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))),
% 0.48/0.53 inference(skolemize,[status(sab)],[100])).
% 0.48/0.53 tff(102,plain,
% 0.48/0.53 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))),
% 0.48/0.53 inference(modus_ponens,[status(thm)],[101, 97])).
% 0.48/0.53 tff(103,plain,
% 0.48/0.53 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | ((~member(ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), cross_product(universal_class, universal_class))) | member(first(ordered_pair(u, v)), universal_class))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | (~member(ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), cross_product(universal_class, universal_class))) | member(first(ordered_pair(u, v)), universal_class))),
% 0.48/0.53 inference(rewrite,[status(thm)],[])).
% 0.48/0.53 tff(104,plain,
% 0.48/0.53 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | ((~member(ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), cross_product(universal_class, universal_class))) | member(first(ordered_pair(u, v)), universal_class))),
% 0.48/0.53 inference(quant_inst,[status(thm)],[])).
% 0.48/0.53 tff(105,plain,
% 0.48/0.53 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | (~member(ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), cross_product(universal_class, universal_class))) | member(first(ordered_pair(u, v)), universal_class)),
% 0.48/0.53 inference(modus_ponens,[status(thm)],[104, 103])).
% 0.48/0.53 tff(106,plain,
% 0.48/0.53 ((~member(ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), cross_product(universal_class, universal_class))) | member(first(ordered_pair(u, v)), universal_class)),
% 0.48/0.53 inference(unit_resolution,[status(thm)],[105, 102])).
% 0.48/0.53 tff(107,plain,
% 0.48/0.53 (member(first(ordered_pair(u, v)), universal_class)),
% 0.48/0.53 inference(unit_resolution,[status(thm)],[106, 95])).
% 0.48/0.53 tff(108,plain,
% 0.48/0.53 (((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(first(ordered_pair(u, v)), universal_class)) | member(first(ordered_pair(u, v)), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))) <=> ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(first(ordered_pair(u, v)), universal_class)) | member(first(ordered_pair(u, v)), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))),
% 0.48/0.53 inference(rewrite,[status(thm)],[])).
% 0.48/0.53 tff(109,plain,
% 0.48/0.53 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(first(ordered_pair(u, v)), universal_class)) | member(first(ordered_pair(u, v)), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))),
% 0.48/0.53 inference(quant_inst,[status(thm)],[])).
% 0.48/0.53 tff(110,plain,
% 0.48/0.53 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(first(ordered_pair(u, v)), universal_class)) | member(first(ordered_pair(u, v)), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))),
% 0.48/0.53 inference(modus_ponens,[status(thm)],[109, 108])).
% 0.48/0.53 tff(111,plain,
% 0.48/0.53 ((~member(first(ordered_pair(u, v)), universal_class)) | member(first(ordered_pair(u, v)), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))),
% 0.48/0.53 inference(unit_resolution,[status(thm)],[110, 67])).
% 0.48/0.53 tff(112,plain,
% 0.48/0.53 (member(first(ordered_pair(u, v)), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))),
% 0.48/0.53 inference(unit_resolution,[status(thm)],[111, 107])).
% 0.48/0.53 tff(113,plain,
% 0.48/0.53 (member(first(ordered_pair(u, v)), unordered_pair(u, u))),
% 0.48/0.53 inference(modus_ponens,[status(thm)],[112, 92])).
% 0.48/0.53 tff(114,plain,
% 0.48/0.53 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((~member(first(ordered_pair(u, v)), unordered_pair(u, u))) | (first(ordered_pair(u, v)) = u))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(first(ordered_pair(u, v)), unordered_pair(u, u))) | (first(ordered_pair(u, v)) = u))),
% 0.48/0.53 inference(rewrite,[status(thm)],[])).
% 0.48/0.53 tff(115,plain,
% 0.48/0.53 (((first(ordered_pair(u, v)) = u) | (first(ordered_pair(u, v)) = u) | (~member(first(ordered_pair(u, v)), unordered_pair(u, u)))) <=> ((~member(first(ordered_pair(u, v)), unordered_pair(u, u))) | (first(ordered_pair(u, v)) = u))),
% 0.48/0.53 inference(rewrite,[status(thm)],[])).
% 0.48/0.53 tff(116,plain,
% 0.48/0.53 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((first(ordered_pair(u, v)) = u) | (first(ordered_pair(u, v)) = u) | (~member(first(ordered_pair(u, v)), unordered_pair(u, u))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((~member(first(ordered_pair(u, v)), unordered_pair(u, u))) | (first(ordered_pair(u, v)) = u)))),
% 0.48/0.53 inference(monotonicity,[status(thm)],[115])).
% 0.48/0.53 tff(117,plain,
% 0.48/0.53 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((first(ordered_pair(u, v)) = u) | (first(ordered_pair(u, v)) = u) | (~member(first(ordered_pair(u, v)), unordered_pair(u, u))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(first(ordered_pair(u, v)), unordered_pair(u, u))) | (first(ordered_pair(u, v)) = u))),
% 0.48/0.53 inference(transitivity,[status(thm)],[116, 114])).
% 0.48/0.53 tff(118,plain,
% 0.48/0.53 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((first(ordered_pair(u, v)) = u) | (first(ordered_pair(u, v)) = u) | (~member(first(ordered_pair(u, v)), unordered_pair(u, u))))),
% 0.48/0.53 inference(quant_inst,[status(thm)],[])).
% 0.48/0.53 tff(119,plain,
% 0.48/0.53 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(first(ordered_pair(u, v)), unordered_pair(u, u))) | (first(ordered_pair(u, v)) = u)),
% 0.48/0.53 inference(modus_ponens,[status(thm)],[118, 117])).
% 0.48/0.53 tff(120,plain,
% 0.48/0.53 ((~member(first(ordered_pair(u, v)), unordered_pair(u, u))) | (first(ordered_pair(u, v)) = u)),
% 0.48/0.53 inference(unit_resolution,[status(thm)],[119, 83])).
% 0.48/0.53 tff(121,plain,
% 0.48/0.53 (first(ordered_pair(u, v)) = u),
% 0.48/0.53 inference(unit_resolution,[status(thm)],[120, 113])).
% 0.48/0.53 tff(122,plain,
% 0.48/0.53 (singleton(first(ordered_pair(u, v))) = singleton(u)),
% 0.48/0.53 inference(monotonicity,[status(thm)],[121])).
% 0.48/0.53 tff(123,plain,
% 0.48/0.53 (singleton(u) = singleton(first(ordered_pair(u, v)))),
% 0.48/0.53 inference(symmetry,[status(thm)],[122])).
% 0.48/0.53 tff(124,plain,
% 0.48/0.53 ($false),
% 0.48/0.53 inference(unit_resolution,[status(thm)],[18, 123])).
% 0.48/0.53 tff(125,plain,(singleton(u) = singleton(first(ordered_pair(u, v)))), inference(lemma,lemma(discharge,[]))).
% 0.48/0.53 tff(126,plain,
% 0.48/0.53 (singleton(first(ordered_pair(u, v))) = singleton(u)),
% 0.48/0.53 inference(symmetry,[status(thm)],[125])).
% 0.48/0.53 tff(127,assumption,(~(unordered_pair(u, singleton(v)) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))), introduced(assumption)).
% 0.48/0.53 tff(128,plain,
% 0.48/0.53 (member(unordered_pair(u, singleton(v)), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))) <=> member(unordered_pair(u, singleton(v)), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))),
% 0.48/0.53 inference(monotonicity,[status(thm)],[45])).
% 0.48/0.53 tff(129,plain,
% 0.48/0.53 (member(unordered_pair(u, singleton(v)), unordered_pair(singleton(u), unordered_pair(u, singleton(v)))) <=> member(unordered_pair(u, singleton(v)), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))),
% 0.48/0.53 inference(symmetry,[status(thm)],[128])).
% 0.48/0.53 tff(130,plain,
% 0.48/0.53 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(u, singleton(v)), universal_class)),
% 0.48/0.53 inference(quant_inst,[status(thm)],[])).
% 0.48/0.53 tff(131,plain,
% 0.48/0.53 (member(unordered_pair(u, singleton(v)), universal_class)),
% 0.48/0.53 inference(unit_resolution,[status(thm)],[130, 57])).
% 0.48/0.53 tff(132,plain,
% 0.48/0.53 (^[Y: $i, X: $i] : refl(((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y))) <=> ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y))))),
% 0.48/0.53 inference(bind,[status(th)],[])).
% 0.48/0.53 tff(133,plain,
% 0.48/0.53 (![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y))) <=> ![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))),
% 0.48/0.53 inference(quant_intro,[status(thm)],[132])).
% 0.48/0.53 tff(134,plain,
% 0.48/0.53 (![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y))) <=> ![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))),
% 0.48/0.53 inference(rewrite,[status(thm)],[])).
% 0.48/0.53 tff(135,axiom,(![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','unordered_pair3')).
% 0.48/0.53 tff(136,plain,
% 0.48/0.53 (![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))),
% 0.48/0.53 inference(modus_ponens,[status(thm)],[135, 134])).
% 0.48/0.53 tff(137,plain,(
% 0.48/0.53 ![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))),
% 0.48/0.53 inference(skolemize,[status(sab)],[136])).
% 0.48/0.53 tff(138,plain,
% 0.48/0.53 (![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))),
% 0.48/0.53 inference(modus_ponens,[status(thm)],[137, 133])).
% 0.48/0.53 tff(139,plain,
% 0.48/0.53 (((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | ((~member(unordered_pair(u, singleton(v)), universal_class)) | member(unordered_pair(u, singleton(v)), unordered_pair(singleton(u), unordered_pair(u, singleton(v)))))) <=> ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | (~member(unordered_pair(u, singleton(v)), universal_class)) | member(unordered_pair(u, singleton(v)), unordered_pair(singleton(u), unordered_pair(u, singleton(v)))))),
% 0.48/0.54 inference(rewrite,[status(thm)],[])).
% 0.48/0.54 tff(140,plain,
% 0.48/0.54 ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | ((~member(unordered_pair(u, singleton(v)), universal_class)) | member(unordered_pair(u, singleton(v)), unordered_pair(singleton(u), unordered_pair(u, singleton(v)))))),
% 0.48/0.54 inference(quant_inst,[status(thm)],[])).
% 0.48/0.54 tff(141,plain,
% 0.48/0.54 ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | (~member(unordered_pair(u, singleton(v)), universal_class)) | member(unordered_pair(u, singleton(v)), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))),
% 0.48/0.54 inference(modus_ponens,[status(thm)],[140, 139])).
% 0.48/0.54 tff(142,plain,
% 0.48/0.54 ((~member(unordered_pair(u, singleton(v)), universal_class)) | member(unordered_pair(u, singleton(v)), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))),
% 0.48/0.54 inference(unit_resolution,[status(thm)],[141, 138])).
% 0.48/0.54 tff(143,plain,
% 0.48/0.54 (member(unordered_pair(u, singleton(v)), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))),
% 0.48/0.54 inference(unit_resolution,[status(thm)],[142, 131])).
% 0.48/0.54 tff(144,plain,
% 0.48/0.54 (member(unordered_pair(u, singleton(v)), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))),
% 0.48/0.54 inference(modus_ponens,[status(thm)],[143, 129])).
% 0.48/0.54 tff(145,plain,
% 0.48/0.54 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((unordered_pair(u, singleton(v)) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) | (unordered_pair(u, singleton(v)) = singleton(first(ordered_pair(u, v)))) | (~member(unordered_pair(u, singleton(v)), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (unordered_pair(u, singleton(v)) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) | (unordered_pair(u, singleton(v)) = singleton(first(ordered_pair(u, v)))) | (~member(unordered_pair(u, singleton(v)), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))))),
% 0.48/0.54 inference(rewrite,[status(thm)],[])).
% 0.48/0.54 tff(146,plain,
% 0.48/0.54 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((unordered_pair(u, singleton(v)) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) | (unordered_pair(u, singleton(v)) = singleton(first(ordered_pair(u, v)))) | (~member(unordered_pair(u, singleton(v)), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))))),
% 0.48/0.54 inference(quant_inst,[status(thm)],[])).
% 0.48/0.54 tff(147,plain,
% 0.48/0.54 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (unordered_pair(u, singleton(v)) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) | (unordered_pair(u, singleton(v)) = singleton(first(ordered_pair(u, v)))) | (~member(unordered_pair(u, singleton(v)), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))))),
% 0.48/0.54 inference(modus_ponens,[status(thm)],[146, 145])).
% 0.48/0.54 tff(148,plain,
% 0.48/0.54 ((unordered_pair(u, singleton(v)) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) | (unordered_pair(u, singleton(v)) = singleton(first(ordered_pair(u, v)))) | (~member(unordered_pair(u, singleton(v)), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))))),
% 0.48/0.54 inference(unit_resolution,[status(thm)],[147, 83])).
% 0.48/0.54 tff(149,plain,
% 0.48/0.54 ((unordered_pair(u, singleton(v)) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) | (unordered_pair(u, singleton(v)) = singleton(first(ordered_pair(u, v))))),
% 0.48/0.54 inference(unit_resolution,[status(thm)],[148, 144])).
% 0.48/0.54 tff(150,plain,
% 0.48/0.54 (unordered_pair(u, singleton(v)) = singleton(first(ordered_pair(u, v)))),
% 0.48/0.54 inference(unit_resolution,[status(thm)],[149, 127])).
% 0.48/0.54 tff(151,plain,
% 0.48/0.54 (unordered_pair(u, singleton(v)) = singleton(u)),
% 0.48/0.54 inference(transitivity,[status(thm)],[150, 126])).
% 0.48/0.54 tff(152,plain,
% 0.48/0.54 ((unordered_pair(u, singleton(v)) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) <=> (singleton(u) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))),
% 0.48/0.54 inference(monotonicity,[status(thm)],[151])).
% 0.48/0.54 tff(153,plain,
% 0.48/0.54 ((unordered_pair(u, singleton(v)) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) <=> (unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = singleton(u))),
% 0.48/0.54 inference(transitivity,[status(thm)],[152, 17])).
% 0.48/0.54 tff(154,plain,
% 0.48/0.54 ((~(unordered_pair(u, singleton(v)) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))) <=> (~(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = singleton(u)))),
% 0.48/0.54 inference(monotonicity,[status(thm)],[153])).
% 0.48/0.54 tff(155,plain,
% 0.48/0.54 (~(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = singleton(u))),
% 0.48/0.54 inference(modus_ponens,[status(thm)],[127, 154])).
% 0.48/0.54 tff(156,plain,
% 0.48/0.54 (ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))) = unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))),
% 0.48/0.54 inference(symmetry,[status(thm)],[44])).
% 0.48/0.54 tff(157,plain,
% 0.48/0.54 (ordered_pair(u, v) = ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v)))),
% 0.48/0.54 inference(symmetry,[status(thm)],[42])).
% 0.48/0.54 tff(158,plain,
% 0.48/0.54 (unordered_pair(singleton(u), unordered_pair(u, singleton(v))) = unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))),
% 0.48/0.54 inference(transitivity,[status(thm)],[27, 157, 156])).
% 0.48/0.54 tff(159,plain,
% 0.48/0.54 (member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(u), unordered_pair(u, singleton(v)))) <=> member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))),
% 0.48/0.54 inference(monotonicity,[status(thm)],[158])).
% 0.48/0.54 tff(160,plain,
% 0.48/0.54 (member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))) <=> member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))),
% 0.48/0.54 inference(symmetry,[status(thm)],[159])).
% 0.48/0.54 tff(161,plain,
% 0.48/0.54 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), universal_class)),
% 0.48/0.54 inference(quant_inst,[status(thm)],[])).
% 0.48/0.54 tff(162,plain,
% 0.48/0.54 (member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), universal_class)),
% 0.48/0.54 inference(unit_resolution,[status(thm)],[161, 57])).
% 0.48/0.54 tff(163,plain,
% 0.48/0.54 (((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | ((~member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), universal_class)) | member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))))) <=> ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | (~member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), universal_class)) | member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))))),
% 0.48/0.54 inference(rewrite,[status(thm)],[])).
% 0.48/0.54 tff(164,plain,
% 0.48/0.54 ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | ((~member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), universal_class)) | member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))))),
% 0.48/0.54 inference(quant_inst,[status(thm)],[])).
% 0.48/0.54 tff(165,plain,
% 0.48/0.54 ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | (~member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), universal_class)) | member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))),
% 0.48/0.54 inference(modus_ponens,[status(thm)],[164, 163])).
% 0.48/0.54 tff(166,plain,
% 0.48/0.54 ((~member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), universal_class)) | member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))),
% 0.48/0.54 inference(unit_resolution,[status(thm)],[165, 138])).
% 0.48/0.54 tff(167,plain,
% 0.48/0.54 (member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))),
% 0.48/0.54 inference(unit_resolution,[status(thm)],[166, 162])).
% 0.48/0.54 tff(168,plain,
% 0.48/0.54 (member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))),
% 0.48/0.54 inference(modus_ponens,[status(thm)],[167, 160])).
% 0.48/0.54 tff(169,plain,
% 0.48/0.54 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = unordered_pair(u, singleton(v))) | (unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = singleton(u)) | (~member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = unordered_pair(u, singleton(v))) | (unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = singleton(u)) | (~member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))))),
% 0.48/0.54 inference(rewrite,[status(thm)],[])).
% 0.48/0.54 tff(170,plain,
% 0.48/0.54 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = unordered_pair(u, singleton(v))) | (unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = singleton(u)) | (~member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))))),
% 0.48/0.54 inference(quant_inst,[status(thm)],[])).
% 0.48/0.55 tff(171,plain,
% 0.48/0.55 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = unordered_pair(u, singleton(v))) | (unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = singleton(u)) | (~member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(u), unordered_pair(u, singleton(v)))))),
% 0.48/0.55 inference(modus_ponens,[status(thm)],[170, 169])).
% 0.48/0.55 tff(172,plain,
% 0.48/0.55 ((unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = unordered_pair(u, singleton(v))) | (unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = singleton(u)) | (~member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(u), unordered_pair(u, singleton(v)))))),
% 0.48/0.55 inference(unit_resolution,[status(thm)],[171, 83])).
% 0.48/0.55 tff(173,plain,
% 0.48/0.55 ((unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = unordered_pair(u, singleton(v))) | (unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = singleton(u))),
% 0.48/0.55 inference(unit_resolution,[status(thm)],[172, 168])).
% 0.48/0.55 tff(174,plain,
% 0.48/0.55 (unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = unordered_pair(u, singleton(v))),
% 0.48/0.55 inference(unit_resolution,[status(thm)],[173, 155])).
% 0.48/0.55 tff(175,plain,
% 0.48/0.55 (unordered_pair(u, singleton(v)) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))),
% 0.48/0.55 inference(modus_ponens,[status(thm)],[174, 16])).
% 0.48/0.55 tff(176,plain,
% 0.48/0.55 ($false),
% 0.48/0.55 inference(unit_resolution,[status(thm)],[127, 175])).
% 0.48/0.55 tff(177,plain,(unordered_pair(u, singleton(v)) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))), inference(lemma,lemma(discharge,[]))).
% 0.48/0.55 tff(178,plain,
% 0.48/0.55 (unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = unordered_pair(u, singleton(v))),
% 0.48/0.55 inference(symmetry,[status(thm)],[177])).
% 0.48/0.55 tff(179,plain,
% 0.48/0.55 (member(singleton(v), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) <=> member(singleton(v), unordered_pair(u, singleton(v)))),
% 0.48/0.55 inference(monotonicity,[status(thm)],[178])).
% 0.48/0.55 tff(180,plain,
% 0.48/0.55 (member(singleton(v), unordered_pair(u, singleton(v))) <=> member(singleton(v), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))),
% 0.48/0.55 inference(symmetry,[status(thm)],[179])).
% 0.48/0.55 tff(181,plain,
% 0.48/0.55 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(v, v) = singleton(v))),
% 0.48/0.55 inference(quant_inst,[status(thm)],[])).
% 0.48/0.55 tff(182,plain,
% 0.48/0.55 (unordered_pair(v, v) = singleton(v)),
% 0.48/0.55 inference(unit_resolution,[status(thm)],[181, 7])).
% 0.48/0.55 tff(183,plain,
% 0.48/0.55 (member(unordered_pair(v, v), universal_class) <=> member(singleton(v), universal_class)),
% 0.48/0.55 inference(monotonicity,[status(thm)],[182])).
% 0.48/0.55 tff(184,plain,
% 0.48/0.55 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(v, v), universal_class)),
% 0.48/0.55 inference(quant_inst,[status(thm)],[])).
% 0.48/0.55 tff(185,plain,
% 0.48/0.55 (member(unordered_pair(v, v), universal_class)),
% 0.48/0.55 inference(unit_resolution,[status(thm)],[184, 57])).
% 0.48/0.55 tff(186,plain,
% 0.48/0.55 (member(singleton(v), universal_class)),
% 0.48/0.55 inference(modus_ponens,[status(thm)],[185, 183])).
% 0.48/0.55 tff(187,plain,
% 0.48/0.55 (((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | ((~member(singleton(v), universal_class)) | member(singleton(v), unordered_pair(u, singleton(v))))) <=> ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | (~member(singleton(v), universal_class)) | member(singleton(v), unordered_pair(u, singleton(v))))),
% 0.48/0.55 inference(rewrite,[status(thm)],[])).
% 0.48/0.55 tff(188,plain,
% 0.48/0.55 ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | ((~member(singleton(v), universal_class)) | member(singleton(v), unordered_pair(u, singleton(v))))),
% 0.48/0.55 inference(quant_inst,[status(thm)],[])).
% 0.48/0.55 tff(189,plain,
% 0.48/0.55 ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | (~member(singleton(v), universal_class)) | member(singleton(v), unordered_pair(u, singleton(v)))),
% 0.48/0.55 inference(modus_ponens,[status(thm)],[188, 187])).
% 0.48/0.55 tff(190,plain,
% 0.48/0.55 ((~member(singleton(v), universal_class)) | member(singleton(v), unordered_pair(u, singleton(v)))),
% 0.48/0.55 inference(unit_resolution,[status(thm)],[189, 138])).
% 0.48/0.55 tff(191,plain,
% 0.48/0.55 (member(singleton(v), unordered_pair(u, singleton(v)))),
% 0.48/0.55 inference(unit_resolution,[status(thm)],[190, 186])).
% 0.48/0.55 tff(192,plain,
% 0.48/0.55 (member(singleton(v), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))),
% 0.48/0.55 inference(modus_ponens,[status(thm)],[191, 180])).
% 0.48/0.55 tff(193,plain,
% 0.48/0.55 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((~member(singleton(v), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))) | (singleton(v) = first(ordered_pair(u, v))) | (singleton(v) = singleton(second(ordered_pair(u, v)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(singleton(v), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))) | (singleton(v) = first(ordered_pair(u, v))) | (singleton(v) = singleton(second(ordered_pair(u, v)))))),
% 0.48/0.55 inference(rewrite,[status(thm)],[])).
% 0.48/0.55 tff(194,plain,
% 0.48/0.55 (((singleton(v) = singleton(second(ordered_pair(u, v)))) | (singleton(v) = first(ordered_pair(u, v))) | (~member(singleton(v), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))) <=> ((~member(singleton(v), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))) | (singleton(v) = first(ordered_pair(u, v))) | (singleton(v) = singleton(second(ordered_pair(u, v)))))),
% 0.48/0.55 inference(rewrite,[status(thm)],[])).
% 0.48/0.55 tff(195,plain,
% 0.48/0.55 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(v) = singleton(second(ordered_pair(u, v)))) | (singleton(v) = first(ordered_pair(u, v))) | (~member(singleton(v), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((~member(singleton(v), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))) | (singleton(v) = first(ordered_pair(u, v))) | (singleton(v) = singleton(second(ordered_pair(u, v))))))),
% 0.48/0.55 inference(monotonicity,[status(thm)],[194])).
% 0.48/0.55 tff(196,plain,
% 0.48/0.55 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(v) = singleton(second(ordered_pair(u, v)))) | (singleton(v) = first(ordered_pair(u, v))) | (~member(singleton(v), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(singleton(v), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))) | (singleton(v) = first(ordered_pair(u, v))) | (singleton(v) = singleton(second(ordered_pair(u, v)))))),
% 0.48/0.55 inference(transitivity,[status(thm)],[195, 193])).
% 0.48/0.55 tff(197,plain,
% 0.48/0.55 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(v) = singleton(second(ordered_pair(u, v)))) | (singleton(v) = first(ordered_pair(u, v))) | (~member(singleton(v), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))))),
% 0.48/0.55 inference(quant_inst,[status(thm)],[])).
% 0.48/0.55 tff(198,plain,
% 0.48/0.55 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(singleton(v), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))) | (singleton(v) = first(ordered_pair(u, v))) | (singleton(v) = singleton(second(ordered_pair(u, v))))),
% 0.48/0.55 inference(modus_ponens,[status(thm)],[197, 196])).
% 0.48/0.55 tff(199,plain,
% 0.48/0.55 ((~member(singleton(v), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))) | (singleton(v) = first(ordered_pair(u, v))) | (singleton(v) = singleton(second(ordered_pair(u, v))))),
% 0.48/0.55 inference(unit_resolution,[status(thm)],[198, 83])).
% 0.48/0.55 tff(200,plain,
% 0.48/0.55 ((singleton(v) = first(ordered_pair(u, v))) | (singleton(v) = singleton(second(ordered_pair(u, v))))),
% 0.48/0.55 inference(unit_resolution,[status(thm)],[199, 192])).
% 0.48/0.55 tff(201,plain,
% 0.48/0.55 (singleton(v) = first(ordered_pair(u, v))),
% 0.48/0.55 inference(unit_resolution,[status(thm)],[200, 14])).
% 0.48/0.55 tff(202,plain,
% 0.48/0.55 (first(ordered_pair(u, v)) = singleton(v)),
% 0.48/0.55 inference(symmetry,[status(thm)],[201])).
% 0.48/0.55 tff(203,plain,
% 0.48/0.55 ((singleton(second(ordered_pair(u, v))) = first(ordered_pair(u, v))) <=> (singleton(second(ordered_pair(u, v))) = singleton(v))),
% 0.48/0.55 inference(monotonicity,[status(thm)],[202])).
% 0.48/0.55 tff(204,plain,
% 0.48/0.55 ((singleton(second(ordered_pair(u, v))) = first(ordered_pair(u, v))) <=> (singleton(v) = singleton(second(ordered_pair(u, v))))),
% 0.48/0.55 inference(transitivity,[status(thm)],[203, 11])).
% 0.48/0.55 tff(205,plain,
% 0.48/0.55 ((singleton(v) = singleton(second(ordered_pair(u, v)))) <=> (singleton(second(ordered_pair(u, v))) = first(ordered_pair(u, v)))),
% 0.48/0.55 inference(symmetry,[status(thm)],[204])).
% 0.48/0.55 tff(206,plain,
% 0.48/0.55 ((~(singleton(v) = singleton(second(ordered_pair(u, v))))) <=> (~(singleton(second(ordered_pair(u, v))) = first(ordered_pair(u, v))))),
% 0.48/0.55 inference(monotonicity,[status(thm)],[205])).
% 0.48/0.55 tff(207,plain,
% 0.48/0.55 (~(singleton(second(ordered_pair(u, v))) = first(ordered_pair(u, v)))),
% 0.48/0.55 inference(modus_ponens,[status(thm)],[14, 206])).
% 0.48/0.55 tff(208,plain,
% 0.48/0.55 (singleton(u) = unordered_pair(u, u)),
% 0.48/0.55 inference(symmetry,[status(thm)],[49])).
% 0.48/0.55 tff(209,plain,
% 0.48/0.55 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))) = singleton(first(ordered_pair(u, v))))),
% 0.48/0.55 inference(quant_inst,[status(thm)],[])).
% 0.48/0.55 tff(210,plain,
% 0.48/0.55 (unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))) = singleton(first(ordered_pair(u, v)))),
% 0.48/0.55 inference(unit_resolution,[status(thm)],[209, 7])).
% 0.48/0.55 tff(211,plain,
% 0.48/0.55 (unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))) = unordered_pair(u, u)),
% 0.48/0.55 inference(transitivity,[status(thm)],[210, 126, 208])).
% 0.48/0.55 tff(212,plain,
% 0.48/0.55 (member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v)))) <=> member(u, unordered_pair(u, u))),
% 0.48/0.55 inference(monotonicity,[status(thm)],[211])).
% 0.48/0.55 tff(213,plain,
% 0.48/0.55 (member(u, unordered_pair(u, u)) <=> member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))))),
% 0.48/0.55 inference(symmetry,[status(thm)],[212])).
% 0.48/0.55 tff(214,plain,
% 0.48/0.55 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | ((~member(ordered_pair(u, v), cross_product(universal_class, universal_class))) | member(u, universal_class))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | (~member(ordered_pair(u, v), cross_product(universal_class, universal_class))) | member(u, universal_class))),
% 0.48/0.55 inference(rewrite,[status(thm)],[])).
% 0.48/0.55 tff(215,plain,
% 0.48/0.55 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | ((~member(ordered_pair(u, v), cross_product(universal_class, universal_class))) | member(u, universal_class))),
% 0.48/0.55 inference(quant_inst,[status(thm)],[])).
% 0.48/0.55 tff(216,plain,
% 0.48/0.55 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | (~member(ordered_pair(u, v), cross_product(universal_class, universal_class))) | member(u, universal_class)),
% 0.48/0.55 inference(modus_ponens,[status(thm)],[215, 214])).
% 0.48/0.55 tff(217,plain,
% 0.48/0.55 (member(u, universal_class)),
% 0.48/0.55 inference(unit_resolution,[status(thm)],[216, 102, 31])).
% 0.48/0.55 tff(218,plain,
% 0.48/0.55 (((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(u, universal_class)) | member(u, unordered_pair(u, u)))) <=> ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(u, universal_class)) | member(u, unordered_pair(u, u)))),
% 0.48/0.56 inference(rewrite,[status(thm)],[])).
% 0.48/0.56 tff(219,plain,
% 0.48/0.56 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(u, universal_class)) | member(u, unordered_pair(u, u)))),
% 0.48/0.56 inference(quant_inst,[status(thm)],[])).
% 0.48/0.56 tff(220,plain,
% 0.48/0.56 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(u, universal_class)) | member(u, unordered_pair(u, u))),
% 0.48/0.56 inference(modus_ponens,[status(thm)],[219, 218])).
% 0.48/0.56 tff(221,plain,
% 0.48/0.56 (member(u, unordered_pair(u, u))),
% 0.48/0.56 inference(unit_resolution,[status(thm)],[220, 67, 217])).
% 0.48/0.56 tff(222,plain,
% 0.48/0.56 (member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))))),
% 0.48/0.56 inference(modus_ponens,[status(thm)],[221, 213])).
% 0.48/0.56 tff(223,plain,
% 0.48/0.56 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((~member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))))) | (u = first(ordered_pair(u, v))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))))) | (u = first(ordered_pair(u, v))))),
% 0.48/0.56 inference(rewrite,[status(thm)],[])).
% 0.48/0.56 tff(224,plain,
% 0.48/0.56 (((u = first(ordered_pair(u, v))) | (u = first(ordered_pair(u, v))) | (~member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v)))))) <=> ((~member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))))) | (u = first(ordered_pair(u, v))))),
% 0.48/0.56 inference(rewrite,[status(thm)],[])).
% 0.48/0.56 tff(225,plain,
% 0.48/0.56 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((u = first(ordered_pair(u, v))) | (u = first(ordered_pair(u, v))) | (~member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((~member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))))) | (u = first(ordered_pair(u, v)))))),
% 0.48/0.56 inference(monotonicity,[status(thm)],[224])).
% 0.48/0.56 tff(226,plain,
% 0.48/0.56 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((u = first(ordered_pair(u, v))) | (u = first(ordered_pair(u, v))) | (~member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))))) | (u = first(ordered_pair(u, v))))),
% 0.48/0.56 inference(transitivity,[status(thm)],[225, 223])).
% 0.48/0.56 tff(227,plain,
% 0.48/0.56 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((u = first(ordered_pair(u, v))) | (u = first(ordered_pair(u, v))) | (~member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))))))),
% 0.48/0.56 inference(quant_inst,[status(thm)],[])).
% 0.48/0.56 tff(228,plain,
% 0.48/0.56 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))))) | (u = first(ordered_pair(u, v)))),
% 0.48/0.56 inference(modus_ponens,[status(thm)],[227, 226])).
% 0.48/0.56 tff(229,plain,
% 0.48/0.56 ((~member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))))) | (u = first(ordered_pair(u, v)))),
% 0.48/0.56 inference(unit_resolution,[status(thm)],[228, 83])).
% 0.48/0.56 tff(230,plain,
% 0.48/0.56 (u = first(ordered_pair(u, v))),
% 0.48/0.56 inference(unit_resolution,[status(thm)],[229, 222])).
% 0.48/0.56 tff(231,plain,
% 0.48/0.56 (first(ordered_pair(u, v)) = u),
% 0.48/0.56 inference(symmetry,[status(thm)],[230])).
% 0.48/0.56 tff(232,plain,
% 0.48/0.56 (unordered_pair(first(ordered_pair(u, v)), singleton(v)) = unordered_pair(u, singleton(v))),
% 0.48/0.56 inference(monotonicity,[status(thm)],[231])).
% 0.48/0.56 tff(233,plain,
% 0.48/0.56 (unordered_pair(first(ordered_pair(u, v)), singleton(v)) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))),
% 0.48/0.56 inference(transitivity,[status(thm)],[232, 177])).
% 0.48/0.56 tff(234,plain,
% 0.48/0.56 (member(singleton(second(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(v))) <=> member(singleton(second(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))),
% 0.48/0.56 inference(monotonicity,[status(thm)],[233])).
% 0.48/0.56 tff(235,plain,
% 0.48/0.56 (member(singleton(second(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) <=> member(singleton(second(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(v)))),
% 0.48/0.56 inference(symmetry,[status(thm)],[234])).
% 0.48/0.56 tff(236,plain,
% 0.48/0.56 (member(unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v))), universal_class) <=> member(singleton(second(ordered_pair(u, v))), universal_class)),
% 0.48/0.56 inference(monotonicity,[status(thm)],[9])).
% 0.48/0.56 tff(237,plain,
% 0.48/0.56 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v))), universal_class)),
% 0.48/0.56 inference(quant_inst,[status(thm)],[])).
% 0.48/0.56 tff(238,plain,
% 0.48/0.56 (member(unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v))), universal_class)),
% 0.48/0.56 inference(unit_resolution,[status(thm)],[237, 57])).
% 0.48/0.56 tff(239,plain,
% 0.48/0.56 (member(singleton(second(ordered_pair(u, v))), universal_class)),
% 0.48/0.56 inference(modus_ponens,[status(thm)],[238, 236])).
% 0.48/0.56 tff(240,plain,
% 0.48/0.56 (((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | ((~member(singleton(second(ordered_pair(u, v))), universal_class)) | member(singleton(second(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))) <=> ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | (~member(singleton(second(ordered_pair(u, v))), universal_class)) | member(singleton(second(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))),
% 0.48/0.56 inference(rewrite,[status(thm)],[])).
% 0.48/0.56 tff(241,plain,
% 0.48/0.56 ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | ((~member(singleton(second(ordered_pair(u, v))), universal_class)) | member(singleton(second(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))),
% 0.48/0.56 inference(quant_inst,[status(thm)],[])).
% 0.48/0.56 tff(242,plain,
% 0.48/0.56 ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | (~member(singleton(second(ordered_pair(u, v))), universal_class)) | member(singleton(second(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))),
% 0.48/0.56 inference(modus_ponens,[status(thm)],[241, 240])).
% 0.48/0.56 tff(243,plain,
% 0.48/0.56 ((~member(singleton(second(ordered_pair(u, v))), universal_class)) | member(singleton(second(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))),
% 0.48/0.56 inference(unit_resolution,[status(thm)],[242, 138])).
% 0.48/0.56 tff(244,plain,
% 0.48/0.56 (member(singleton(second(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))),
% 0.48/0.56 inference(unit_resolution,[status(thm)],[243, 239])).
% 0.48/0.56 tff(245,plain,
% 0.48/0.56 (member(singleton(second(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(v)))),
% 0.48/0.56 inference(modus_ponens,[status(thm)],[244, 235])).
% 0.48/0.56 tff(246,plain,
% 0.48/0.56 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(second(ordered_pair(u, v))) = singleton(v)) | (singleton(second(ordered_pair(u, v))) = first(ordered_pair(u, v))) | (~member(singleton(second(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(v)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(second(ordered_pair(u, v))) = singleton(v)) | (singleton(second(ordered_pair(u, v))) = first(ordered_pair(u, v))) | (~member(singleton(second(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(v)))))),
% 0.48/0.56 inference(rewrite,[status(thm)],[])).
% 0.48/0.56 tff(247,plain,
% 0.48/0.56 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(second(ordered_pair(u, v))) = singleton(v)) | (singleton(second(ordered_pair(u, v))) = first(ordered_pair(u, v))) | (~member(singleton(second(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(v)))))),
% 0.48/0.56 inference(quant_inst,[status(thm)],[])).
% 0.48/0.56 tff(248,plain,
% 0.48/0.56 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(second(ordered_pair(u, v))) = singleton(v)) | (singleton(second(ordered_pair(u, v))) = first(ordered_pair(u, v))) | (~member(singleton(second(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(v))))),
% 0.48/0.56 inference(modus_ponens,[status(thm)],[247, 246])).
% 0.48/0.56 tff(249,plain,
% 0.48/0.56 ((singleton(second(ordered_pair(u, v))) = singleton(v)) | (singleton(second(ordered_pair(u, v))) = first(ordered_pair(u, v))) | (~member(singleton(second(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(v))))),
% 0.48/0.56 inference(unit_resolution,[status(thm)],[248, 83])).
% 0.48/0.56 tff(250,plain,
% 0.48/0.56 ((singleton(second(ordered_pair(u, v))) = singleton(v)) | (singleton(second(ordered_pair(u, v))) = first(ordered_pair(u, v)))),
% 0.48/0.56 inference(unit_resolution,[status(thm)],[249, 245])).
% 0.48/0.56 tff(251,plain,
% 0.48/0.56 ($false),
% 0.48/0.56 inference(unit_resolution,[status(thm)],[250, 207, 15])).
% 0.48/0.56 tff(252,plain,(singleton(v) = singleton(second(ordered_pair(u, v)))), inference(lemma,lemma(discharge,[]))).
% 0.48/0.56 tff(253,plain,
% 0.48/0.56 (unordered_pair(v, v) = unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v)))),
% 0.48/0.56 inference(transitivity,[status(thm)],[182, 252, 10])).
% 0.48/0.56 tff(254,plain,
% 0.48/0.56 (member(second(ordered_pair(u, v)), unordered_pair(v, v)) <=> member(second(ordered_pair(u, v)), unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v))))),
% 0.48/0.56 inference(monotonicity,[status(thm)],[253])).
% 0.48/0.56 tff(255,plain,
% 0.48/0.56 (member(second(ordered_pair(u, v)), unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v)))) <=> member(second(ordered_pair(u, v)), unordered_pair(v, v))),
% 0.48/0.56 inference(symmetry,[status(thm)],[254])).
% 0.48/0.56 tff(256,plain,
% 0.48/0.56 (^[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)))),
% 0.48/0.56 inference(bind,[status(th)],[])).
% 0.48/0.56 tff(257,plain,
% 0.48/0.56 (![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))),
% 0.48/0.56 inference(quant_intro,[status(thm)],[256])).
% 0.48/0.56 tff(258,plain,
% 0.48/0.56 (![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))),
% 0.48/0.56 inference(rewrite,[status(thm)],[])).
% 0.48/0.56 tff(259,axiom,(![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','cartesian_product2')).
% 0.48/0.56 tff(260,plain,
% 0.48/0.56 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))),
% 0.48/0.56 inference(modus_ponens,[status(thm)],[259, 258])).
% 0.48/0.56 tff(261,plain,(
% 0.48/0.56 ![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))),
% 0.48/0.56 inference(skolemize,[status(sab)],[260])).
% 0.48/0.56 tff(262,plain,
% 0.48/0.56 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))),
% 0.48/0.56 inference(modus_ponens,[status(thm)],[261, 257])).
% 0.48/0.56 tff(263,plain,
% 0.48/0.56 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))) | ((~member(ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), cross_product(universal_class, universal_class))) | member(second(ordered_pair(u, v)), universal_class))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))) | (~member(ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), cross_product(universal_class, universal_class))) | member(second(ordered_pair(u, v)), universal_class))),
% 0.48/0.56 inference(rewrite,[status(thm)],[])).
% 0.48/0.56 tff(264,plain,
% 0.48/0.56 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))) | ((~member(ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), cross_product(universal_class, universal_class))) | member(second(ordered_pair(u, v)), universal_class))),
% 0.48/0.56 inference(quant_inst,[status(thm)],[])).
% 0.48/0.56 tff(265,plain,
% 0.48/0.56 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))) | (~member(ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), cross_product(universal_class, universal_class))) | member(second(ordered_pair(u, v)), universal_class)),
% 0.48/0.56 inference(modus_ponens,[status(thm)],[264, 263])).
% 0.48/0.56 tff(266,plain,
% 0.48/0.56 ((~member(ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), cross_product(universal_class, universal_class))) | member(second(ordered_pair(u, v)), universal_class)),
% 0.48/0.56 inference(unit_resolution,[status(thm)],[265, 262])).
% 0.48/0.56 tff(267,plain,
% 0.48/0.56 (member(second(ordered_pair(u, v)), universal_class)),
% 0.48/0.56 inference(unit_resolution,[status(thm)],[266, 95])).
% 0.48/0.56 tff(268,plain,
% 0.48/0.56 (((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(second(ordered_pair(u, v)), universal_class)) | member(second(ordered_pair(u, v)), unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v)))))) <=> ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(second(ordered_pair(u, v)), universal_class)) | member(second(ordered_pair(u, v)), unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v)))))),
% 0.48/0.56 inference(rewrite,[status(thm)],[])).
% 0.48/0.56 tff(269,plain,
% 0.48/0.56 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(second(ordered_pair(u, v)), universal_class)) | member(second(ordered_pair(u, v)), unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v)))))),
% 0.48/0.56 inference(quant_inst,[status(thm)],[])).
% 0.48/0.56 tff(270,plain,
% 0.48/0.56 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(second(ordered_pair(u, v)), universal_class)) | member(second(ordered_pair(u, v)), unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v))))),
% 0.48/0.56 inference(modus_ponens,[status(thm)],[269, 268])).
% 0.48/0.56 tff(271,plain,
% 0.48/0.56 (member(second(ordered_pair(u, v)), unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v))))),
% 0.48/0.56 inference(unit_resolution,[status(thm)],[270, 67, 267])).
% 0.48/0.56 tff(272,plain,
% 0.48/0.56 (member(second(ordered_pair(u, v)), unordered_pair(v, v))),
% 0.48/0.56 inference(modus_ponens,[status(thm)],[271, 255])).
% 0.48/0.56 tff(273,plain,
% 0.48/0.56 ((~(second(ordered_pair(u, v)) = v)) <=> (~(second(ordered_pair(u, v)) = v))),
% 0.48/0.56 inference(rewrite,[status(thm)],[])).
% 0.48/0.56 tff(274,axiom,(~(second(ordered_pair(u, v)) = v)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_unique_1st_and_2nd_in_pair_of_sets2_2')).
% 0.48/0.56 tff(275,plain,
% 0.48/0.56 (~(second(ordered_pair(u, v)) = v)),
% 0.48/0.56 inference(modus_ponens,[status(thm)],[274, 273])).
% 0.48/0.56 tff(276,plain,
% 0.48/0.56 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((second(ordered_pair(u, v)) = v) | (~member(second(ordered_pair(u, v)), unordered_pair(v, v))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (second(ordered_pair(u, v)) = v) | (~member(second(ordered_pair(u, v)), unordered_pair(v, v))))),
% 0.48/0.56 inference(rewrite,[status(thm)],[])).
% 0.48/0.56 tff(277,plain,
% 0.48/0.56 (((second(ordered_pair(u, v)) = v) | (second(ordered_pair(u, v)) = v) | (~member(second(ordered_pair(u, v)), unordered_pair(v, v)))) <=> ((second(ordered_pair(u, v)) = v) | (~member(second(ordered_pair(u, v)), unordered_pair(v, v))))),
% 0.48/0.56 inference(rewrite,[status(thm)],[])).
% 0.48/0.56 tff(278,plain,
% 0.48/0.56 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((second(ordered_pair(u, v)) = v) | (second(ordered_pair(u, v)) = v) | (~member(second(ordered_pair(u, v)), unordered_pair(v, v))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((second(ordered_pair(u, v)) = v) | (~member(second(ordered_pair(u, v)), unordered_pair(v, v)))))),
% 0.48/0.57 inference(monotonicity,[status(thm)],[277])).
% 0.48/0.57 tff(279,plain,
% 0.48/0.57 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((second(ordered_pair(u, v)) = v) | (second(ordered_pair(u, v)) = v) | (~member(second(ordered_pair(u, v)), unordered_pair(v, v))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (second(ordered_pair(u, v)) = v) | (~member(second(ordered_pair(u, v)), unordered_pair(v, v))))),
% 0.48/0.57 inference(transitivity,[status(thm)],[278, 276])).
% 0.48/0.57 tff(280,plain,
% 0.48/0.57 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((second(ordered_pair(u, v)) = v) | (second(ordered_pair(u, v)) = v) | (~member(second(ordered_pair(u, v)), unordered_pair(v, v))))),
% 0.48/0.57 inference(quant_inst,[status(thm)],[])).
% 0.48/0.57 tff(281,plain,
% 0.48/0.57 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (second(ordered_pair(u, v)) = v) | (~member(second(ordered_pair(u, v)), unordered_pair(v, v)))),
% 0.48/0.57 inference(modus_ponens,[status(thm)],[280, 279])).
% 0.48/0.57 tff(282,plain,
% 0.48/0.57 (~member(second(ordered_pair(u, v)), unordered_pair(v, v))),
% 0.48/0.57 inference(unit_resolution,[status(thm)],[281, 83, 275])).
% 0.48/0.57 tff(283,plain,
% 0.48/0.57 ($false),
% 0.48/0.57 inference(unit_resolution,[status(thm)],[282, 272])).
% 0.48/0.57 % SZS output end Proof
%------------------------------------------------------------------------------