TSTP Solution File: SET018-6 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET018-6 : TPTP v8.1.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n024.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 20 05:04:51 EDT 2022
% Result : Unsatisfiable 0.21s 0.57s
% Output : Proof 0.59s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.12 % Problem : SET018-6 : TPTP v8.1.0. Bugfixed v2.1.0.
% 0.10/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.14/0.35 % Computer : n024.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Sat Sep 3 01:15:18 EDT 2022
% 0.14/0.35 % CPUTime :
% 0.14/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.14/0.35 Usage: tptp [options] [-file:]file
% 0.14/0.35 -h, -? prints this message.
% 0.14/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.14/0.35 -m, -model generate model.
% 0.14/0.35 -p, -proof generate proof.
% 0.14/0.35 -c, -core generate unsat core of named formulas.
% 0.14/0.35 -st, -statistics display statistics.
% 0.14/0.35 -t:timeout set timeout (in second).
% 0.14/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.14/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.14/0.35 -<param>:<value> configuration parameter and value.
% 0.14/0.35 -o:<output-file> file to place output in.
% 0.21/0.57 % SZS status Unsatisfiable
% 0.21/0.57 % SZS output start Proof
% 0.21/0.57 tff(singleton_type, type, (
% 0.21/0.57 singleton: $i > $i)).
% 0.21/0.57 tff(y_type, type, (
% 0.21/0.57 y: $i)).
% 0.21/0.57 tff(unordered_pair_type, type, (
% 0.21/0.57 unordered_pair: ( $i * $i ) > $i)).
% 0.21/0.57 tff(x_type, type, (
% 0.21/0.57 x: $i)).
% 0.21/0.57 tff(w_type, type, (
% 0.21/0.57 w: $i)).
% 0.21/0.57 tff(z_type, type, (
% 0.21/0.57 z: $i)).
% 0.21/0.57 tff(member_type, type, (
% 0.21/0.57 member: ( $i * $i ) > $o)).
% 0.21/0.57 tff(universal_class_type, type, (
% 0.21/0.57 universal_class: $i)).
% 0.21/0.57 tff(apply_type, type, (
% 0.21/0.57 apply: ( $i * $i ) > $i)).
% 0.21/0.57 tff(choice_type, type, (
% 0.21/0.57 choice: $i)).
% 0.21/0.57 tff(null_class_type, type, (
% 0.21/0.57 null_class: $i)).
% 0.21/0.57 tff(ordered_pair_type, type, (
% 0.21/0.57 ordered_pair: ( $i * $i ) > $i)).
% 0.21/0.57 tff(image_type, type, (
% 0.21/0.57 image: ( $i * $i ) > $i)).
% 0.21/0.57 tff(sum_class_type, type, (
% 0.21/0.57 sum_class: $i > $i)).
% 0.21/0.57 tff(function_type, type, (
% 0.21/0.57 function: $i > $o)).
% 0.21/0.57 tff(1,plain,
% 0.21/0.57 (^[X: $i] : refl((unordered_pair(X, X) = singleton(X)) <=> (unordered_pair(X, X) = singleton(X)))),
% 0.21/0.57 inference(bind,[status(th)],[])).
% 0.21/0.57 tff(2,plain,
% 0.21/0.57 (![X: $i] : (unordered_pair(X, X) = singleton(X)) <=> ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.21/0.57 inference(quant_intro,[status(thm)],[1])).
% 0.21/0.57 tff(3,plain,
% 0.21/0.57 (![X: $i] : (unordered_pair(X, X) = singleton(X)) <=> ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.21/0.57 inference(rewrite,[status(thm)],[])).
% 0.21/0.57 tff(4,axiom,(![X: $i] : (unordered_pair(X, X) = singleton(X))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','singleton_set')).
% 0.21/0.57 tff(5,plain,
% 0.21/0.57 (![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.21/0.57 tff(6,plain,(
% 0.21/0.57 ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.21/0.57 inference(skolemize,[status(sab)],[5])).
% 0.21/0.57 tff(7,plain,
% 0.21/0.57 (![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.21/0.57 tff(8,plain,
% 0.21/0.57 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(y, y) = singleton(y))),
% 0.21/0.57 inference(quant_inst,[status(thm)],[])).
% 0.21/0.57 tff(9,plain,
% 0.21/0.57 (unordered_pair(y, y) = singleton(y)),
% 0.21/0.57 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.21/0.57 tff(10,plain,
% 0.21/0.57 ((singleton(x) = singleton(z)) <=> (singleton(z) = singleton(x))),
% 0.21/0.57 inference(commutativity,[status(thm)],[])).
% 0.21/0.57 tff(11,plain,
% 0.21/0.57 ((singleton(z) = singleton(x)) <=> (singleton(x) = singleton(z))),
% 0.21/0.57 inference(symmetry,[status(thm)],[10])).
% 0.21/0.57 tff(12,plain,
% 0.21/0.57 ((~(singleton(z) = singleton(x))) <=> (~(singleton(x) = singleton(z)))),
% 0.21/0.57 inference(monotonicity,[status(thm)],[11])).
% 0.21/0.57 tff(13,plain,
% 0.21/0.57 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(x, x) = singleton(x))),
% 0.21/0.57 inference(quant_inst,[status(thm)],[])).
% 0.21/0.57 tff(14,plain,
% 0.21/0.57 (unordered_pair(x, x) = singleton(x)),
% 0.21/0.57 inference(unit_resolution,[status(thm)],[13, 7])).
% 0.21/0.57 tff(15,plain,
% 0.21/0.57 (singleton(x) = unordered_pair(x, x)),
% 0.21/0.57 inference(symmetry,[status(thm)],[14])).
% 0.21/0.57 tff(16,assumption,(singleton(z) = singleton(x)), introduced(assumption)).
% 0.21/0.57 tff(17,plain,
% 0.21/0.57 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(z, z) = singleton(z))),
% 0.21/0.57 inference(quant_inst,[status(thm)],[])).
% 0.21/0.57 tff(18,plain,
% 0.21/0.57 (unordered_pair(z, z) = singleton(z)),
% 0.21/0.57 inference(unit_resolution,[status(thm)],[17, 7])).
% 0.21/0.57 tff(19,plain,
% 0.21/0.57 (unordered_pair(z, z) = unordered_pair(x, x)),
% 0.21/0.57 inference(transitivity,[status(thm)],[18, 16, 15])).
% 0.21/0.57 tff(20,plain,
% 0.21/0.57 (member(x, unordered_pair(z, z)) <=> member(x, unordered_pair(x, x))),
% 0.21/0.57 inference(monotonicity,[status(thm)],[19])).
% 0.21/0.57 tff(21,plain,
% 0.21/0.57 (member(x, unordered_pair(x, x)) <=> member(x, unordered_pair(z, z))),
% 0.21/0.57 inference(symmetry,[status(thm)],[20])).
% 0.21/0.57 tff(22,plain,
% 0.21/0.57 (member(x, universal_class) <=> member(x, universal_class)),
% 0.21/0.57 inference(rewrite,[status(thm)],[])).
% 0.21/0.57 tff(23,axiom,(member(x, universal_class)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_ordered_pair_determines_components2_2')).
% 0.21/0.57 tff(24,plain,
% 0.21/0.57 (member(x, universal_class)),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[23, 22])).
% 0.21/0.57 tff(25,plain,
% 0.21/0.57 (^[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))))),
% 0.21/0.57 inference(bind,[status(th)],[])).
% 0.21/0.57 tff(26,plain,
% 0.21/0.57 (![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)))),
% 0.21/0.57 inference(quant_intro,[status(thm)],[25])).
% 0.21/0.57 tff(27,plain,
% 0.21/0.57 (![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)))),
% 0.21/0.57 inference(rewrite,[status(thm)],[])).
% 0.21/0.57 tff(28,axiom,(![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','unordered_pair2')).
% 0.21/0.57 tff(29,plain,
% 0.21/0.57 (![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[28, 27])).
% 0.21/0.57 tff(30,plain,(
% 0.21/0.57 ![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 0.21/0.57 inference(skolemize,[status(sab)],[29])).
% 0.21/0.57 tff(31,plain,
% 0.21/0.57 (![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[30, 26])).
% 0.21/0.57 tff(32,plain,
% 0.21/0.57 (((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(x, universal_class)) | member(x, unordered_pair(x, x)))) <=> ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(x, universal_class)) | member(x, unordered_pair(x, x)))),
% 0.21/0.57 inference(rewrite,[status(thm)],[])).
% 0.21/0.57 tff(33,plain,
% 0.21/0.57 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(x, universal_class)) | member(x, unordered_pair(x, x)))),
% 0.21/0.57 inference(quant_inst,[status(thm)],[])).
% 0.21/0.57 tff(34,plain,
% 0.21/0.57 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(x, universal_class)) | member(x, unordered_pair(x, x))),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[33, 32])).
% 0.21/0.57 tff(35,plain,
% 0.21/0.57 (member(x, unordered_pair(x, x))),
% 0.21/0.57 inference(unit_resolution,[status(thm)],[34, 31, 24])).
% 0.21/0.57 tff(36,plain,
% 0.21/0.57 (member(x, unordered_pair(z, z))),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[35, 21])).
% 0.21/0.57 tff(37,plain,
% 0.21/0.57 ((~(x = z)) <=> (~(x = z))),
% 0.21/0.57 inference(rewrite,[status(thm)],[])).
% 0.21/0.57 tff(38,axiom,(~(x = z)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_ordered_pair_determines_components2_3')).
% 0.21/0.57 tff(39,plain,
% 0.21/0.57 (~(x = z)),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[38, 37])).
% 0.21/0.57 tff(40,plain,
% 0.21/0.57 (^[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)))))),
% 0.21/0.57 inference(bind,[status(th)],[])).
% 0.21/0.57 tff(41,plain,
% 0.21/0.57 (![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))))),
% 0.21/0.57 inference(quant_intro,[status(thm)],[40])).
% 0.21/0.57 tff(42,plain,
% 0.21/0.57 (![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))))),
% 0.21/0.57 inference(rewrite,[status(thm)],[])).
% 0.21/0.57 tff(43,plain,
% 0.21/0.57 (^[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)))))),
% 0.21/0.57 inference(bind,[status(th)],[])).
% 0.21/0.57 tff(44,plain,
% 0.21/0.57 (![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))))),
% 0.21/0.57 inference(quant_intro,[status(thm)],[43])).
% 0.21/0.57 tff(45,axiom,(![Y: $i, U: $i, X: $i] : (((~member(U, unordered_pair(X, Y))) | (U = X)) | (U = Y))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','unordered_pair_member')).
% 0.21/0.57 tff(46,plain,
% 0.21/0.57 (![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[45, 44])).
% 0.21/0.57 tff(47,plain,
% 0.21/0.57 (![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[46, 42])).
% 0.21/0.57 tff(48,plain,(
% 0.21/0.57 ![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 0.21/0.57 inference(skolemize,[status(sab)],[47])).
% 0.21/0.57 tff(49,plain,
% 0.21/0.57 (![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[48, 41])).
% 0.21/0.57 tff(50,plain,
% 0.21/0.57 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((x = z) | (~member(x, unordered_pair(z, z))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (x = z) | (~member(x, unordered_pair(z, z))))),
% 0.21/0.57 inference(rewrite,[status(thm)],[])).
% 0.21/0.57 tff(51,plain,
% 0.21/0.57 (((x = z) | (x = z) | (~member(x, unordered_pair(z, z)))) <=> ((x = z) | (~member(x, unordered_pair(z, z))))),
% 0.21/0.57 inference(rewrite,[status(thm)],[])).
% 0.21/0.57 tff(52,plain,
% 0.21/0.57 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((x = z) | (x = z) | (~member(x, unordered_pair(z, z))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((x = z) | (~member(x, unordered_pair(z, z)))))),
% 0.21/0.57 inference(monotonicity,[status(thm)],[51])).
% 0.21/0.57 tff(53,plain,
% 0.21/0.57 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((x = z) | (x = z) | (~member(x, unordered_pair(z, z))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (x = z) | (~member(x, unordered_pair(z, z))))),
% 0.21/0.57 inference(transitivity,[status(thm)],[52, 50])).
% 0.21/0.57 tff(54,plain,
% 0.21/0.57 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((x = z) | (x = z) | (~member(x, unordered_pair(z, z))))),
% 0.21/0.57 inference(quant_inst,[status(thm)],[])).
% 0.21/0.57 tff(55,plain,
% 0.21/0.57 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (x = z) | (~member(x, unordered_pair(z, z)))),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[54, 53])).
% 0.21/0.57 tff(56,plain,
% 0.21/0.57 (~member(x, unordered_pair(z, z))),
% 0.21/0.57 inference(unit_resolution,[status(thm)],[55, 49, 39])).
% 0.21/0.57 tff(57,plain,
% 0.21/0.57 ($false),
% 0.21/0.57 inference(unit_resolution,[status(thm)],[56, 36])).
% 0.21/0.57 tff(58,plain,(~(singleton(z) = singleton(x))), inference(lemma,lemma(discharge,[]))).
% 0.21/0.57 tff(59,plain,
% 0.21/0.57 (~(singleton(x) = singleton(z))),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[58, 12])).
% 0.21/0.57 tff(60,assumption,(unordered_pair(y, singleton(z)) = null_class), introduced(assumption)).
% 0.21/0.57 tff(61,plain,
% 0.21/0.57 (null_class = unordered_pair(y, singleton(z))),
% 0.21/0.57 inference(symmetry,[status(thm)],[60])).
% 0.21/0.57 tff(62,assumption,(~(singleton(y) = null_class)), introduced(assumption)).
% 0.21/0.57 tff(63,plain,
% 0.21/0.57 ((null_class = singleton(y)) <=> (singleton(y) = null_class)),
% 0.21/0.57 inference(commutativity,[status(thm)],[])).
% 0.21/0.57 tff(64,plain,
% 0.21/0.57 ((w = apply(choice, singleton(y))) <=> (apply(choice, singleton(y)) = w)),
% 0.21/0.57 inference(commutativity,[status(thm)],[])).
% 0.21/0.57 tff(65,assumption,(~(singleton(w) = singleton(y))), introduced(assumption)).
% 0.21/0.57 tff(66,plain,
% 0.21/0.57 (^[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)))),
% 0.21/0.57 inference(bind,[status(th)],[])).
% 0.21/0.57 tff(67,plain,
% 0.21/0.57 (![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))),
% 0.21/0.57 inference(quant_intro,[status(thm)],[66])).
% 0.21/0.57 tff(68,plain,
% 0.21/0.57 (![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))),
% 0.21/0.57 inference(rewrite,[status(thm)],[])).
% 0.21/0.57 tff(69,axiom,(![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','ordered_pair')).
% 0.21/0.57 tff(70,plain,
% 0.21/0.57 (![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[69, 68])).
% 0.21/0.57 tff(71,plain,(
% 0.21/0.57 ![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))),
% 0.21/0.57 inference(skolemize,[status(sab)],[70])).
% 0.21/0.57 tff(72,plain,
% 0.21/0.57 (![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[71, 67])).
% 0.21/0.57 tff(73,plain,
% 0.21/0.57 ((~![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))) | (unordered_pair(singleton(w), unordered_pair(w, singleton(x))) = ordered_pair(w, x))),
% 0.21/0.57 inference(quant_inst,[status(thm)],[])).
% 0.21/0.57 tff(74,plain,
% 0.21/0.57 (unordered_pair(singleton(w), unordered_pair(w, singleton(x))) = ordered_pair(w, x)),
% 0.21/0.57 inference(unit_resolution,[status(thm)],[73, 72])).
% 0.21/0.57 tff(75,plain,
% 0.21/0.57 (ordered_pair(w, x) = unordered_pair(singleton(w), unordered_pair(w, singleton(x)))),
% 0.21/0.57 inference(symmetry,[status(thm)],[74])).
% 0.21/0.57 tff(76,plain,
% 0.21/0.57 ((ordered_pair(w, x) = ordered_pair(y, z)) <=> (ordered_pair(w, x) = ordered_pair(y, z))),
% 0.21/0.57 inference(rewrite,[status(thm)],[])).
% 0.21/0.57 tff(77,axiom,(ordered_pair(w, x) = ordered_pair(y, z)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_ordered_pair_determines_components2_1')).
% 0.21/0.57 tff(78,plain,
% 0.21/0.57 (ordered_pair(w, x) = ordered_pair(y, z)),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[77, 76])).
% 0.21/0.57 tff(79,plain,
% 0.21/0.57 (ordered_pair(y, z) = ordered_pair(w, x)),
% 0.21/0.57 inference(symmetry,[status(thm)],[78])).
% 0.21/0.57 tff(80,plain,
% 0.21/0.57 ((~![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))) | (unordered_pair(singleton(y), unordered_pair(y, singleton(z))) = ordered_pair(y, z))),
% 0.21/0.57 inference(quant_inst,[status(thm)],[])).
% 0.21/0.57 tff(81,plain,
% 0.21/0.57 (unordered_pair(singleton(y), unordered_pair(y, singleton(z))) = ordered_pair(y, z)),
% 0.21/0.57 inference(unit_resolution,[status(thm)],[80, 72])).
% 0.21/0.57 tff(82,plain,
% 0.21/0.57 (unordered_pair(singleton(y), unordered_pair(y, singleton(z))) = unordered_pair(singleton(w), unordered_pair(w, singleton(x)))),
% 0.21/0.57 inference(transitivity,[status(thm)],[81, 79, 75])).
% 0.21/0.57 tff(83,plain,
% 0.21/0.57 (member(singleton(w), unordered_pair(singleton(y), unordered_pair(y, singleton(z)))) <=> member(singleton(w), unordered_pair(singleton(w), unordered_pair(w, singleton(x))))),
% 0.21/0.57 inference(monotonicity,[status(thm)],[82])).
% 0.21/0.57 tff(84,plain,
% 0.21/0.57 (member(singleton(w), unordered_pair(singleton(w), unordered_pair(w, singleton(x)))) <=> member(singleton(w), unordered_pair(singleton(y), unordered_pair(y, singleton(z))))),
% 0.21/0.57 inference(symmetry,[status(thm)],[83])).
% 0.21/0.57 tff(85,plain,
% 0.21/0.57 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(w, w) = singleton(w))),
% 0.21/0.57 inference(quant_inst,[status(thm)],[])).
% 0.21/0.57 tff(86,plain,
% 0.21/0.57 (unordered_pair(w, w) = singleton(w)),
% 0.21/0.57 inference(unit_resolution,[status(thm)],[85, 7])).
% 0.21/0.57 tff(87,plain,
% 0.21/0.57 (member(unordered_pair(w, w), universal_class) <=> member(singleton(w), universal_class)),
% 0.21/0.57 inference(monotonicity,[status(thm)],[86])).
% 0.21/0.57 tff(88,plain,
% 0.21/0.57 (^[Y: $i, X: $i] : refl(member(unordered_pair(X, Y), universal_class) <=> member(unordered_pair(X, Y), universal_class))),
% 0.21/0.57 inference(bind,[status(th)],[])).
% 0.21/0.57 tff(89,plain,
% 0.21/0.57 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class) <=> ![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.21/0.57 inference(quant_intro,[status(thm)],[88])).
% 0.21/0.57 tff(90,plain,
% 0.21/0.57 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class) <=> ![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.21/0.57 inference(rewrite,[status(thm)],[])).
% 0.21/0.57 tff(91,axiom,(![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','unordered_pairs_in_universal')).
% 0.21/0.57 tff(92,plain,
% 0.21/0.57 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[91, 90])).
% 0.21/0.57 tff(93,plain,(
% 0.21/0.57 ![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.21/0.57 inference(skolemize,[status(sab)],[92])).
% 0.21/0.57 tff(94,plain,
% 0.21/0.57 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[93, 89])).
% 0.21/0.57 tff(95,plain,
% 0.21/0.57 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(w, w), universal_class)),
% 0.21/0.57 inference(quant_inst,[status(thm)],[])).
% 0.21/0.57 tff(96,plain,
% 0.21/0.57 (member(unordered_pair(w, w), universal_class)),
% 0.21/0.57 inference(unit_resolution,[status(thm)],[95, 94])).
% 0.21/0.57 tff(97,plain,
% 0.21/0.57 (member(singleton(w), universal_class)),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[96, 87])).
% 0.21/0.57 tff(98,plain,
% 0.21/0.57 (((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(singleton(w), universal_class)) | member(singleton(w), unordered_pair(singleton(w), unordered_pair(w, singleton(x)))))) <=> ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(singleton(w), universal_class)) | member(singleton(w), unordered_pair(singleton(w), unordered_pair(w, singleton(x)))))),
% 0.21/0.57 inference(rewrite,[status(thm)],[])).
% 0.21/0.57 tff(99,plain,
% 0.21/0.57 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(singleton(w), universal_class)) | member(singleton(w), unordered_pair(singleton(w), unordered_pair(w, singleton(x)))))),
% 0.21/0.57 inference(quant_inst,[status(thm)],[])).
% 0.21/0.57 tff(100,plain,
% 0.21/0.57 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(singleton(w), universal_class)) | member(singleton(w), unordered_pair(singleton(w), unordered_pair(w, singleton(x))))),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[99, 98])).
% 0.21/0.57 tff(101,plain,
% 0.21/0.57 ((~member(singleton(w), universal_class)) | member(singleton(w), unordered_pair(singleton(w), unordered_pair(w, singleton(x))))),
% 0.21/0.57 inference(unit_resolution,[status(thm)],[100, 31])).
% 0.21/0.57 tff(102,plain,
% 0.21/0.57 (member(singleton(w), unordered_pair(singleton(w), unordered_pair(w, singleton(x))))),
% 0.21/0.57 inference(unit_resolution,[status(thm)],[101, 97])).
% 0.21/0.57 tff(103,plain,
% 0.21/0.57 (member(singleton(w), unordered_pair(singleton(y), unordered_pair(y, singleton(z))))),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[102, 84])).
% 0.21/0.57 tff(104,plain,
% 0.21/0.57 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(w) = unordered_pair(y, singleton(z))) | (singleton(w) = singleton(y)) | (~member(singleton(w), unordered_pair(singleton(y), unordered_pair(y, singleton(z))))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(w) = unordered_pair(y, singleton(z))) | (singleton(w) = singleton(y)) | (~member(singleton(w), unordered_pair(singleton(y), unordered_pair(y, singleton(z))))))),
% 0.21/0.57 inference(rewrite,[status(thm)],[])).
% 0.21/0.57 tff(105,plain,
% 0.21/0.57 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(w) = unordered_pair(y, singleton(z))) | (singleton(w) = singleton(y)) | (~member(singleton(w), unordered_pair(singleton(y), unordered_pair(y, singleton(z))))))),
% 0.21/0.57 inference(quant_inst,[status(thm)],[])).
% 0.21/0.57 tff(106,plain,
% 0.21/0.57 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(w) = unordered_pair(y, singleton(z))) | (singleton(w) = singleton(y)) | (~member(singleton(w), unordered_pair(singleton(y), unordered_pair(y, singleton(z)))))),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[105, 104])).
% 0.21/0.57 tff(107,plain,
% 0.21/0.57 ((singleton(w) = unordered_pair(y, singleton(z))) | (singleton(w) = singleton(y)) | (~member(singleton(w), unordered_pair(singleton(y), unordered_pair(y, singleton(z)))))),
% 0.21/0.57 inference(unit_resolution,[status(thm)],[106, 49])).
% 0.21/0.57 tff(108,plain,
% 0.21/0.57 ((singleton(w) = unordered_pair(y, singleton(z))) | (singleton(w) = singleton(y))),
% 0.21/0.57 inference(unit_resolution,[status(thm)],[107, 103])).
% 0.21/0.57 tff(109,plain,
% 0.21/0.57 (singleton(w) = unordered_pair(y, singleton(z))),
% 0.21/0.57 inference(unit_resolution,[status(thm)],[108, 65])).
% 0.21/0.57 tff(110,plain,
% 0.21/0.57 (unordered_pair(w, w) = unordered_pair(y, singleton(z))),
% 0.21/0.57 inference(transitivity,[status(thm)],[86, 109])).
% 0.21/0.57 tff(111,plain,
% 0.21/0.57 (member(singleton(z), unordered_pair(w, w)) <=> member(singleton(z), unordered_pair(y, singleton(z)))),
% 0.21/0.57 inference(monotonicity,[status(thm)],[110])).
% 0.21/0.57 tff(112,plain,
% 0.21/0.57 (member(singleton(z), unordered_pair(y, singleton(z))) <=> member(singleton(z), unordered_pair(w, w))),
% 0.21/0.57 inference(symmetry,[status(thm)],[111])).
% 0.21/0.57 tff(113,plain,
% 0.21/0.57 (member(unordered_pair(z, z), universal_class) <=> member(singleton(z), universal_class)),
% 0.21/0.57 inference(monotonicity,[status(thm)],[18])).
% 0.21/0.57 tff(114,plain,
% 0.21/0.57 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(z, z), universal_class)),
% 0.21/0.57 inference(quant_inst,[status(thm)],[])).
% 0.21/0.57 tff(115,plain,
% 0.21/0.57 (member(unordered_pair(z, z), universal_class)),
% 0.21/0.57 inference(unit_resolution,[status(thm)],[114, 94])).
% 0.21/0.57 tff(116,plain,
% 0.21/0.57 (member(singleton(z), universal_class)),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[115, 113])).
% 0.21/0.57 tff(117,plain,
% 0.21/0.57 (^[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))))),
% 0.21/0.57 inference(bind,[status(th)],[])).
% 0.21/0.57 tff(118,plain,
% 0.21/0.57 (![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)))),
% 0.21/0.57 inference(quant_intro,[status(thm)],[117])).
% 0.21/0.57 tff(119,plain,
% 0.21/0.57 (![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)))),
% 0.21/0.57 inference(rewrite,[status(thm)],[])).
% 0.21/0.57 tff(120,axiom,(![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','unordered_pair3')).
% 0.21/0.57 tff(121,plain,
% 0.21/0.57 (![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[120, 119])).
% 0.21/0.57 tff(122,plain,(
% 0.21/0.57 ![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))),
% 0.21/0.57 inference(skolemize,[status(sab)],[121])).
% 0.21/0.57 tff(123,plain,
% 0.21/0.57 (![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[122, 118])).
% 0.21/0.57 tff(124,plain,
% 0.21/0.57 (((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | ((~member(singleton(z), universal_class)) | member(singleton(z), unordered_pair(y, singleton(z))))) <=> ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | (~member(singleton(z), universal_class)) | member(singleton(z), unordered_pair(y, singleton(z))))),
% 0.21/0.57 inference(rewrite,[status(thm)],[])).
% 0.21/0.57 tff(125,plain,
% 0.21/0.57 ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | ((~member(singleton(z), universal_class)) | member(singleton(z), unordered_pair(y, singleton(z))))),
% 0.21/0.57 inference(quant_inst,[status(thm)],[])).
% 0.21/0.57 tff(126,plain,
% 0.21/0.57 ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | (~member(singleton(z), universal_class)) | member(singleton(z), unordered_pair(y, singleton(z)))),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[125, 124])).
% 0.21/0.57 tff(127,plain,
% 0.21/0.57 ((~member(singleton(z), universal_class)) | member(singleton(z), unordered_pair(y, singleton(z)))),
% 0.21/0.57 inference(unit_resolution,[status(thm)],[126, 123])).
% 0.21/0.57 tff(128,plain,
% 0.21/0.57 (member(singleton(z), unordered_pair(y, singleton(z)))),
% 0.21/0.57 inference(unit_resolution,[status(thm)],[127, 116])).
% 0.21/0.57 tff(129,plain,
% 0.21/0.57 (member(singleton(z), unordered_pair(w, w))),
% 0.21/0.57 inference(modus_ponens,[status(thm)],[128, 112])).
% 0.21/0.57 tff(130,plain,
% 0.21/0.57 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = w) | (~member(singleton(z), unordered_pair(w, w))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(z) = w) | (~member(singleton(z), unordered_pair(w, w))))),
% 0.21/0.57 inference(rewrite,[status(thm)],[])).
% 0.21/0.57 tff(131,plain,
% 0.21/0.57 (((singleton(z) = w) | (singleton(z) = w) | (~member(singleton(z), unordered_pair(w, w)))) <=> ((singleton(z) = w) | (~member(singleton(z), unordered_pair(w, w))))),
% 0.21/0.57 inference(rewrite,[status(thm)],[])).
% 0.21/0.57 tff(132,plain,
% 0.21/0.57 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = w) | (singleton(z) = w) | (~member(singleton(z), unordered_pair(w, w))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = w) | (~member(singleton(z), unordered_pair(w, w)))))),
% 0.21/0.57 inference(monotonicity,[status(thm)],[131])).
% 0.21/0.57 tff(133,plain,
% 0.21/0.57 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = w) | (singleton(z) = w) | (~member(singleton(z), unordered_pair(w, w))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(z) = w) | (~member(singleton(z), unordered_pair(w, w))))),
% 0.21/0.57 inference(transitivity,[status(thm)],[132, 130])).
% 0.21/0.57 tff(134,plain,
% 0.21/0.57 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = w) | (singleton(z) = w) | (~member(singleton(z), unordered_pair(w, w))))),
% 0.21/0.58 inference(quant_inst,[status(thm)],[])).
% 0.21/0.58 tff(135,plain,
% 0.21/0.58 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(z) = w) | (~member(singleton(z), unordered_pair(w, w)))),
% 0.21/0.58 inference(modus_ponens,[status(thm)],[134, 133])).
% 0.21/0.58 tff(136,plain,
% 0.21/0.58 ((singleton(z) = w) | (~member(singleton(z), unordered_pair(w, w)))),
% 0.21/0.58 inference(unit_resolution,[status(thm)],[135, 49])).
% 0.21/0.58 tff(137,plain,
% 0.21/0.58 (singleton(z) = w),
% 0.21/0.58 inference(unit_resolution,[status(thm)],[136, 129])).
% 0.21/0.58 tff(138,plain,
% 0.21/0.58 ((singleton(z) = apply(choice, singleton(y))) <=> (w = apply(choice, singleton(y)))),
% 0.21/0.58 inference(monotonicity,[status(thm)],[137])).
% 0.21/0.58 tff(139,plain,
% 0.21/0.58 ((singleton(z) = apply(choice, singleton(y))) <=> (apply(choice, singleton(y)) = w)),
% 0.21/0.58 inference(transitivity,[status(thm)],[138, 64])).
% 0.21/0.58 tff(140,plain,
% 0.21/0.58 ((apply(choice, singleton(y)) = w) <=> (singleton(z) = apply(choice, singleton(y)))),
% 0.21/0.58 inference(symmetry,[status(thm)],[139])).
% 0.21/0.58 tff(141,plain,
% 0.21/0.58 ((~(apply(choice, singleton(y)) = w)) <=> (~(singleton(z) = apply(choice, singleton(y))))),
% 0.21/0.58 inference(monotonicity,[status(thm)],[140])).
% 0.21/0.58 tff(142,plain,
% 0.21/0.58 (member(apply(choice, singleton(y)), unordered_pair(y, y)) <=> member(apply(choice, singleton(y)), singleton(y))),
% 0.21/0.58 inference(monotonicity,[status(thm)],[9])).
% 0.21/0.58 tff(143,plain,
% 0.21/0.58 (member(apply(choice, singleton(y)), singleton(y)) <=> member(apply(choice, singleton(y)), unordered_pair(y, y))),
% 0.21/0.58 inference(symmetry,[status(thm)],[142])).
% 0.21/0.58 tff(144,plain,
% 0.21/0.58 ((singleton(y) = singleton(w)) <=> (singleton(w) = singleton(y))),
% 0.21/0.58 inference(commutativity,[status(thm)],[])).
% 0.21/0.58 tff(145,plain,
% 0.21/0.58 (unordered_pair(y, singleton(z)) = singleton(w)),
% 0.21/0.58 inference(symmetry,[status(thm)],[109])).
% 0.21/0.58 tff(146,assumption,(unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x))), introduced(assumption)).
% 0.21/0.58 tff(147,plain,
% 0.21/0.58 (singleton(w) = unordered_pair(w, singleton(x))),
% 0.21/0.58 inference(transitivity,[status(thm)],[109, 146])).
% 0.21/0.58 tff(148,plain,
% 0.21/0.58 ((singleton(w) = singleton(y)) <=> (unordered_pair(w, singleton(x)) = singleton(y))),
% 0.21/0.58 inference(monotonicity,[status(thm)],[147])).
% 0.21/0.58 tff(149,plain,
% 0.21/0.58 ((unordered_pair(w, singleton(x)) = singleton(y)) <=> (singleton(w) = singleton(y))),
% 0.21/0.58 inference(symmetry,[status(thm)],[148])).
% 0.21/0.58 tff(150,plain,
% 0.21/0.58 ((singleton(y) = unordered_pair(w, singleton(x))) <=> (unordered_pair(w, singleton(x)) = singleton(y))),
% 0.21/0.58 inference(commutativity,[status(thm)],[])).
% 0.21/0.58 tff(151,assumption,(singleton(y) = singleton(w)), introduced(assumption)).
% 0.21/0.58 tff(152,plain,
% 0.21/0.58 (singleton(w) = singleton(y)),
% 0.21/0.58 inference(modus_ponens,[status(thm)],[151, 144])).
% 0.21/0.58 tff(153,plain,
% 0.21/0.58 ($false),
% 0.21/0.58 inference(unit_resolution,[status(thm)],[65, 152])).
% 0.21/0.58 tff(154,plain,((~(singleton(y) = singleton(w))) | (singleton(w) = singleton(y))), inference(lemma,lemma(discharge,[]))).
% 0.21/0.58 tff(155,plain,
% 0.21/0.58 (~(singleton(y) = singleton(w))),
% 0.21/0.58 inference(unit_resolution,[status(thm)],[154, 65])).
% 0.21/0.58 tff(156,plain,
% 0.21/0.58 (ordered_pair(y, z) = unordered_pair(singleton(y), unordered_pair(y, singleton(z)))),
% 0.21/0.58 inference(symmetry,[status(thm)],[81])).
% 0.21/0.58 tff(157,plain,
% 0.21/0.58 (unordered_pair(singleton(w), unordered_pair(w, singleton(x))) = unordered_pair(singleton(y), unordered_pair(y, singleton(z)))),
% 0.21/0.58 inference(transitivity,[status(thm)],[74, 78, 156])).
% 0.21/0.58 tff(158,plain,
% 0.21/0.58 (member(singleton(y), unordered_pair(singleton(w), unordered_pair(w, singleton(x)))) <=> member(singleton(y), unordered_pair(singleton(y), unordered_pair(y, singleton(z))))),
% 0.21/0.58 inference(monotonicity,[status(thm)],[157])).
% 0.21/0.58 tff(159,plain,
% 0.21/0.58 (member(singleton(y), unordered_pair(singleton(y), unordered_pair(y, singleton(z)))) <=> member(singleton(y), unordered_pair(singleton(w), unordered_pair(w, singleton(x))))),
% 0.21/0.58 inference(symmetry,[status(thm)],[158])).
% 0.21/0.58 tff(160,plain,
% 0.21/0.58 (member(unordered_pair(y, y), universal_class) <=> member(singleton(y), universal_class)),
% 0.21/0.58 inference(monotonicity,[status(thm)],[9])).
% 0.21/0.58 tff(161,plain,
% 0.21/0.58 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(y, y), universal_class)),
% 0.21/0.58 inference(quant_inst,[status(thm)],[])).
% 0.21/0.58 tff(162,plain,
% 0.21/0.58 (member(unordered_pair(y, y), universal_class)),
% 0.21/0.58 inference(unit_resolution,[status(thm)],[161, 94])).
% 0.21/0.58 tff(163,plain,
% 0.21/0.58 (member(singleton(y), universal_class)),
% 0.21/0.58 inference(modus_ponens,[status(thm)],[162, 160])).
% 0.21/0.58 tff(164,plain,
% 0.21/0.58 (((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(singleton(y), universal_class)) | member(singleton(y), unordered_pair(singleton(y), unordered_pair(y, singleton(z)))))) <=> ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(singleton(y), universal_class)) | member(singleton(y), unordered_pair(singleton(y), unordered_pair(y, singleton(z)))))),
% 0.21/0.58 inference(rewrite,[status(thm)],[])).
% 0.21/0.58 tff(165,plain,
% 0.21/0.58 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(singleton(y), universal_class)) | member(singleton(y), unordered_pair(singleton(y), unordered_pair(y, singleton(z)))))),
% 0.21/0.58 inference(quant_inst,[status(thm)],[])).
% 0.21/0.58 tff(166,plain,
% 0.21/0.58 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(singleton(y), universal_class)) | member(singleton(y), unordered_pair(singleton(y), unordered_pair(y, singleton(z))))),
% 0.21/0.58 inference(modus_ponens,[status(thm)],[165, 164])).
% 0.21/0.58 tff(167,plain,
% 0.21/0.58 ((~member(singleton(y), universal_class)) | member(singleton(y), unordered_pair(singleton(y), unordered_pair(y, singleton(z))))),
% 0.21/0.58 inference(unit_resolution,[status(thm)],[166, 31])).
% 0.21/0.58 tff(168,plain,
% 0.21/0.58 (member(singleton(y), unordered_pair(singleton(y), unordered_pair(y, singleton(z))))),
% 0.21/0.58 inference(unit_resolution,[status(thm)],[167, 163])).
% 0.21/0.58 tff(169,plain,
% 0.21/0.58 (member(singleton(y), unordered_pair(singleton(w), unordered_pair(w, singleton(x))))),
% 0.21/0.58 inference(modus_ponens,[status(thm)],[168, 159])).
% 0.21/0.58 tff(170,plain,
% 0.21/0.58 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(y) = unordered_pair(w, singleton(x))) | (singleton(y) = singleton(w)) | (~member(singleton(y), unordered_pair(singleton(w), unordered_pair(w, singleton(x))))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(y) = unordered_pair(w, singleton(x))) | (singleton(y) = singleton(w)) | (~member(singleton(y), unordered_pair(singleton(w), unordered_pair(w, singleton(x))))))),
% 0.21/0.58 inference(rewrite,[status(thm)],[])).
% 0.21/0.58 tff(171,plain,
% 0.21/0.58 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(y) = unordered_pair(w, singleton(x))) | (singleton(y) = singleton(w)) | (~member(singleton(y), unordered_pair(singleton(w), unordered_pair(w, singleton(x))))))),
% 0.21/0.58 inference(quant_inst,[status(thm)],[])).
% 0.21/0.58 tff(172,plain,
% 0.21/0.58 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(y) = unordered_pair(w, singleton(x))) | (singleton(y) = singleton(w)) | (~member(singleton(y), unordered_pair(singleton(w), unordered_pair(w, singleton(x)))))),
% 0.21/0.58 inference(modus_ponens,[status(thm)],[171, 170])).
% 0.21/0.58 tff(173,plain,
% 0.21/0.58 ((singleton(y) = unordered_pair(w, singleton(x))) | (singleton(y) = singleton(w)) | (~member(singleton(y), unordered_pair(singleton(w), unordered_pair(w, singleton(x)))))),
% 0.21/0.58 inference(unit_resolution,[status(thm)],[172, 49])).
% 0.21/0.58 tff(174,plain,
% 0.21/0.58 ((singleton(y) = unordered_pair(w, singleton(x))) | (singleton(y) = singleton(w))),
% 0.21/0.58 inference(unit_resolution,[status(thm)],[173, 169])).
% 0.21/0.58 tff(175,plain,
% 0.21/0.58 (singleton(y) = unordered_pair(w, singleton(x))),
% 0.21/0.58 inference(unit_resolution,[status(thm)],[174, 155])).
% 0.21/0.58 tff(176,plain,
% 0.21/0.58 (unordered_pair(w, singleton(x)) = singleton(y)),
% 0.21/0.58 inference(modus_ponens,[status(thm)],[175, 150])).
% 0.21/0.58 tff(177,plain,
% 0.21/0.58 (singleton(w) = singleton(y)),
% 0.21/0.58 inference(modus_ponens,[status(thm)],[176, 149])).
% 0.21/0.58 tff(178,plain,
% 0.21/0.58 ($false),
% 0.21/0.58 inference(unit_resolution,[status(thm)],[65, 177])).
% 0.21/0.58 tff(179,plain,((singleton(w) = singleton(y)) | (~(unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x))))), inference(lemma,lemma(discharge,[]))).
% 0.21/0.58 tff(180,plain,
% 0.21/0.58 (~(unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x)))),
% 0.21/0.58 inference(unit_resolution,[status(thm)],[179, 65])).
% 0.21/0.58 tff(181,plain,
% 0.21/0.58 ((unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x))) <=> (unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z)))),
% 0.21/0.58 inference(commutativity,[status(thm)],[])).
% 0.21/0.58 tff(182,plain,
% 0.21/0.58 ((unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))) <=> (unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x)))),
% 0.21/0.58 inference(symmetry,[status(thm)],[181])).
% 0.21/0.58 tff(183,assumption,(~(unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x)))), introduced(assumption)).
% 0.21/0.58 tff(184,plain,
% 0.21/0.58 (member(unordered_pair(y, singleton(z)), unordered_pair(singleton(w), unordered_pair(w, singleton(x)))) <=> member(unordered_pair(y, singleton(z)), unordered_pair(singleton(y), unordered_pair(y, singleton(z))))),
% 0.21/0.58 inference(monotonicity,[status(thm)],[157])).
% 0.21/0.58 tff(185,plain,
% 0.21/0.58 (member(unordered_pair(y, singleton(z)), unordered_pair(singleton(y), unordered_pair(y, singleton(z)))) <=> member(unordered_pair(y, singleton(z)), unordered_pair(singleton(w), unordered_pair(w, singleton(x))))),
% 0.21/0.58 inference(symmetry,[status(thm)],[184])).
% 0.21/0.58 tff(186,plain,
% 0.21/0.58 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(y, singleton(z)), universal_class)),
% 0.21/0.58 inference(quant_inst,[status(thm)],[])).
% 0.21/0.58 tff(187,plain,
% 0.21/0.58 (member(unordered_pair(y, singleton(z)), universal_class)),
% 0.21/0.58 inference(unit_resolution,[status(thm)],[186, 94])).
% 0.21/0.58 tff(188,plain,
% 0.21/0.58 (((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | ((~member(unordered_pair(y, singleton(z)), universal_class)) | member(unordered_pair(y, singleton(z)), unordered_pair(singleton(y), unordered_pair(y, singleton(z)))))) <=> ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | (~member(unordered_pair(y, singleton(z)), universal_class)) | member(unordered_pair(y, singleton(z)), unordered_pair(singleton(y), unordered_pair(y, singleton(z)))))),
% 0.21/0.58 inference(rewrite,[status(thm)],[])).
% 0.21/0.58 tff(189,plain,
% 0.21/0.58 ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | ((~member(unordered_pair(y, singleton(z)), universal_class)) | member(unordered_pair(y, singleton(z)), unordered_pair(singleton(y), unordered_pair(y, singleton(z)))))),
% 0.21/0.58 inference(quant_inst,[status(thm)],[])).
% 0.21/0.58 tff(190,plain,
% 0.21/0.58 ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | (~member(unordered_pair(y, singleton(z)), universal_class)) | member(unordered_pair(y, singleton(z)), unordered_pair(singleton(y), unordered_pair(y, singleton(z))))),
% 0.21/0.58 inference(modus_ponens,[status(thm)],[189, 188])).
% 0.21/0.58 tff(191,plain,
% 0.21/0.58 ((~member(unordered_pair(y, singleton(z)), universal_class)) | member(unordered_pair(y, singleton(z)), unordered_pair(singleton(y), unordered_pair(y, singleton(z))))),
% 0.21/0.58 inference(unit_resolution,[status(thm)],[190, 123])).
% 0.21/0.58 tff(192,plain,
% 0.21/0.58 (member(unordered_pair(y, singleton(z)), unordered_pair(singleton(y), unordered_pair(y, singleton(z))))),
% 0.21/0.58 inference(unit_resolution,[status(thm)],[191, 187])).
% 0.21/0.58 tff(193,plain,
% 0.21/0.58 (member(unordered_pair(y, singleton(z)), unordered_pair(singleton(w), unordered_pair(w, singleton(x))))),
% 0.21/0.58 inference(modus_ponens,[status(thm)],[192, 185])).
% 0.21/0.58 tff(194,plain,
% 0.21/0.58 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x))) | (unordered_pair(y, singleton(z)) = singleton(w)) | (~member(unordered_pair(y, singleton(z)), unordered_pair(singleton(w), unordered_pair(w, singleton(x))))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x))) | (unordered_pair(y, singleton(z)) = singleton(w)) | (~member(unordered_pair(y, singleton(z)), unordered_pair(singleton(w), unordered_pair(w, singleton(x))))))),
% 0.21/0.58 inference(rewrite,[status(thm)],[])).
% 0.21/0.58 tff(195,plain,
% 0.21/0.58 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x))) | (unordered_pair(y, singleton(z)) = singleton(w)) | (~member(unordered_pair(y, singleton(z)), unordered_pair(singleton(w), unordered_pair(w, singleton(x))))))),
% 0.21/0.58 inference(quant_inst,[status(thm)],[])).
% 0.21/0.58 tff(196,plain,
% 0.21/0.58 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x))) | (unordered_pair(y, singleton(z)) = singleton(w)) | (~member(unordered_pair(y, singleton(z)), unordered_pair(singleton(w), unordered_pair(w, singleton(x)))))),
% 0.21/0.58 inference(modus_ponens,[status(thm)],[195, 194])).
% 0.21/0.58 tff(197,plain,
% 0.21/0.58 ((unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x))) | (unordered_pair(y, singleton(z)) = singleton(w)) | (~member(unordered_pair(y, singleton(z)), unordered_pair(singleton(w), unordered_pair(w, singleton(x)))))),
% 0.21/0.58 inference(unit_resolution,[status(thm)],[196, 49])).
% 0.21/0.58 tff(198,plain,
% 0.21/0.58 ((unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x))) | (unordered_pair(y, singleton(z)) = singleton(w))),
% 0.21/0.58 inference(unit_resolution,[status(thm)],[197, 193])).
% 0.21/0.58 tff(199,plain,
% 0.21/0.58 (unordered_pair(y, singleton(z)) = singleton(w)),
% 0.21/0.58 inference(unit_resolution,[status(thm)],[198, 183])).
% 0.21/0.58 tff(200,plain,
% 0.21/0.58 (singleton(w) = unordered_pair(y, singleton(z))),
% 0.21/0.58 inference(symmetry,[status(thm)],[199])).
% 0.21/0.58 tff(201,plain,
% 0.21/0.58 ((unordered_pair(w, singleton(x)) = singleton(w)) <=> (unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z)))),
% 0.21/0.58 inference(monotonicity,[status(thm)],[200])).
% 0.21/0.58 tff(202,plain,
% 0.21/0.58 ((unordered_pair(w, singleton(x)) = singleton(w)) <=> (unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x)))),
% 0.21/0.58 inference(transitivity,[status(thm)],[201, 182])).
% 0.21/0.58 tff(203,plain,
% 0.21/0.58 (unordered_pair(w, w) = unordered_pair(y, singleton(z))),
% 0.21/0.58 inference(transitivity,[status(thm)],[86, 200])).
% 0.21/0.58 tff(204,plain,
% 0.21/0.58 (member(singleton(z), unordered_pair(w, w)) <=> member(singleton(z), unordered_pair(y, singleton(z)))),
% 0.21/0.58 inference(monotonicity,[status(thm)],[203])).
% 0.21/0.58 tff(205,plain,
% 0.21/0.58 (member(singleton(z), unordered_pair(y, singleton(z))) <=> member(singleton(z), unordered_pair(w, w))),
% 0.21/0.58 inference(symmetry,[status(thm)],[204])).
% 0.21/0.58 tff(206,plain,
% 0.21/0.58 (member(singleton(z), unordered_pair(w, w))),
% 0.21/0.58 inference(modus_ponens,[status(thm)],[128, 205])).
% 0.21/0.58 tff(207,plain,
% 0.21/0.58 (singleton(z) = w),
% 0.21/0.58 inference(unit_resolution,[status(thm)],[136, 206])).
% 0.21/0.58 tff(208,plain,
% 0.21/0.58 (w = singleton(z)),
% 0.21/0.58 inference(symmetry,[status(thm)],[207])).
% 0.21/0.58 tff(209,plain,
% 0.21/0.58 (member(w, universal_class) <=> member(singleton(z), universal_class)),
% 0.21/0.58 inference(monotonicity,[status(thm)],[208])).
% 0.21/0.58 tff(210,plain,
% 0.21/0.58 (member(singleton(z), universal_class) <=> member(w, universal_class)),
% 0.21/0.58 inference(symmetry,[status(thm)],[209])).
% 0.21/0.58 tff(211,plain,
% 0.21/0.58 (member(w, universal_class)),
% 0.21/0.58 inference(modus_ponens,[status(thm)],[116, 210])).
% 0.21/0.58 tff(212,plain,
% 0.21/0.58 (((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(w, universal_class)) | member(w, unordered_pair(w, singleton(x))))) <=> ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(w, universal_class)) | member(w, unordered_pair(w, singleton(x))))),
% 0.21/0.58 inference(rewrite,[status(thm)],[])).
% 0.21/0.58 tff(213,plain,
% 0.21/0.58 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(w, universal_class)) | member(w, unordered_pair(w, singleton(x))))),
% 0.21/0.58 inference(quant_inst,[status(thm)],[])).
% 0.21/0.58 tff(214,plain,
% 0.21/0.58 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(w, universal_class)) | member(w, unordered_pair(w, singleton(x)))),
% 0.21/0.58 inference(modus_ponens,[status(thm)],[213, 212])).
% 0.21/0.58 tff(215,plain,
% 0.21/0.58 ((~member(w, universal_class)) | member(w, unordered_pair(w, singleton(x)))),
% 0.21/0.58 inference(unit_resolution,[status(thm)],[214, 31])).
% 0.21/0.58 tff(216,plain,
% 0.21/0.58 (member(w, unordered_pair(w, singleton(x)))),
% 0.21/0.58 inference(unit_resolution,[status(thm)],[215, 211])).
% 0.21/0.58 tff(217,plain,
% 0.21/0.58 (member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(w, w)) <=> member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, singleton(z)))),
% 0.21/0.58 inference(monotonicity,[status(thm)],[203])).
% 0.21/0.58 tff(218,plain,
% 0.21/0.58 (member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, singleton(z))) <=> member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(w, w))),
% 0.21/0.58 inference(symmetry,[status(thm)],[217])).
% 0.21/0.58 tff(219,assumption,(member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, singleton(z)))), introduced(assumption)).
% 0.21/0.58 tff(220,plain,
% 0.21/0.58 (member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(w, w))),
% 0.21/0.58 inference(modus_ponens,[status(thm)],[219, 218])).
% 0.21/0.58 tff(221,plain,
% 0.21/0.58 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, unordered_pair(y, singleton(z))) = w) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(w, w))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, unordered_pair(y, singleton(z))) = w) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(w, w))))),
% 0.21/0.58 inference(rewrite,[status(thm)],[])).
% 0.21/0.58 tff(222,plain,
% 0.21/0.58 (((apply(choice, unordered_pair(y, singleton(z))) = w) | (apply(choice, unordered_pair(y, singleton(z))) = w) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(w, w)))) <=> ((apply(choice, unordered_pair(y, singleton(z))) = w) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(w, w))))),
% 0.21/0.58 inference(rewrite,[status(thm)],[])).
% 0.21/0.58 tff(223,plain,
% 0.21/0.58 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, unordered_pair(y, singleton(z))) = w) | (apply(choice, unordered_pair(y, singleton(z))) = w) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(w, w))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, unordered_pair(y, singleton(z))) = w) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(w, w)))))),
% 0.21/0.58 inference(monotonicity,[status(thm)],[222])).
% 0.21/0.58 tff(224,plain,
% 0.21/0.58 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, unordered_pair(y, singleton(z))) = w) | (apply(choice, unordered_pair(y, singleton(z))) = w) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(w, w))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, unordered_pair(y, singleton(z))) = w) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(w, w))))),
% 0.21/0.58 inference(transitivity,[status(thm)],[223, 221])).
% 0.21/0.58 tff(225,plain,
% 0.21/0.58 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, unordered_pair(y, singleton(z))) = w) | (apply(choice, unordered_pair(y, singleton(z))) = w) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(w, w))))),
% 0.21/0.58 inference(quant_inst,[status(thm)],[])).
% 0.21/0.58 tff(226,plain,
% 0.21/0.58 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, unordered_pair(y, singleton(z))) = w) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(w, w)))),
% 0.21/0.58 inference(modus_ponens,[status(thm)],[225, 224])).
% 0.21/0.58 tff(227,plain,
% 0.21/0.58 ((apply(choice, unordered_pair(y, singleton(z))) = w) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(w, w)))),
% 0.21/0.58 inference(unit_resolution,[status(thm)],[226, 49])).
% 0.21/0.58 tff(228,plain,
% 0.21/0.58 (apply(choice, unordered_pair(y, singleton(z))) = w),
% 0.21/0.58 inference(unit_resolution,[status(thm)],[227, 220])).
% 0.21/0.58 tff(229,assumption,(apply(choice, unordered_pair(y, singleton(z))) = w), introduced(assumption)).
% 0.21/0.58 tff(230,plain,
% 0.21/0.58 (unordered_pair(y, y) = unordered_pair(w, singleton(x))),
% 0.21/0.58 inference(transitivity,[status(thm)],[9, 175])).
% 0.21/0.58 tff(231,plain,
% 0.21/0.58 (member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, y)) <=> member(w, unordered_pair(w, singleton(x)))),
% 0.21/0.58 inference(monotonicity,[status(thm)],[229, 230])).
% 0.21/0.58 tff(232,plain,
% 0.21/0.58 (member(w, unordered_pair(w, singleton(x))) <=> member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, y))),
% 0.21/0.58 inference(symmetry,[status(thm)],[231])).
% 0.21/0.58 tff(233,assumption,(member(w, unordered_pair(w, singleton(x)))), introduced(assumption)).
% 0.21/0.58 tff(234,plain,
% 0.21/0.58 (member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, y))),
% 0.21/0.58 inference(modus_ponens,[status(thm)],[233, 232])).
% 0.21/0.58 tff(235,plain,
% 0.21/0.58 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, unordered_pair(y, singleton(z))) = y) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, y))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, unordered_pair(y, singleton(z))) = y) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, y))))),
% 0.21/0.58 inference(rewrite,[status(thm)],[])).
% 0.21/0.58 tff(236,plain,
% 0.21/0.58 (((apply(choice, unordered_pair(y, singleton(z))) = y) | (apply(choice, unordered_pair(y, singleton(z))) = y) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, y)))) <=> ((apply(choice, unordered_pair(y, singleton(z))) = y) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, y))))),
% 0.21/0.58 inference(rewrite,[status(thm)],[])).
% 0.21/0.58 tff(237,plain,
% 0.21/0.58 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, unordered_pair(y, singleton(z))) = y) | (apply(choice, unordered_pair(y, singleton(z))) = y) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, y))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, unordered_pair(y, singleton(z))) = y) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, y)))))),
% 0.21/0.58 inference(monotonicity,[status(thm)],[236])).
% 0.21/0.58 tff(238,plain,
% 0.21/0.58 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, unordered_pair(y, singleton(z))) = y) | (apply(choice, unordered_pair(y, singleton(z))) = y) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, y))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, unordered_pair(y, singleton(z))) = y) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, y))))),
% 0.21/0.58 inference(transitivity,[status(thm)],[237, 235])).
% 0.21/0.59 tff(239,plain,
% 0.21/0.59 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, unordered_pair(y, singleton(z))) = y) | (apply(choice, unordered_pair(y, singleton(z))) = y) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, y))))),
% 0.21/0.59 inference(quant_inst,[status(thm)],[])).
% 0.21/0.59 tff(240,plain,
% 0.21/0.59 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, unordered_pair(y, singleton(z))) = y) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, y)))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[239, 238])).
% 0.21/0.59 tff(241,plain,
% 0.21/0.59 ((apply(choice, unordered_pair(y, singleton(z))) = y) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, y)))),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[240, 49])).
% 0.21/0.59 tff(242,plain,
% 0.21/0.59 (apply(choice, unordered_pair(y, singleton(z))) = y),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[241, 234])).
% 0.21/0.59 tff(243,plain,
% 0.21/0.59 (y = apply(choice, unordered_pair(y, singleton(z)))),
% 0.21/0.59 inference(symmetry,[status(thm)],[242])).
% 0.21/0.59 tff(244,plain,
% 0.21/0.59 (y = w),
% 0.21/0.59 inference(transitivity,[status(thm)],[243, 229])).
% 0.21/0.59 tff(245,plain,
% 0.21/0.59 (singleton(y) = singleton(w)),
% 0.21/0.59 inference(monotonicity,[status(thm)],[244])).
% 0.21/0.59 tff(246,plain,
% 0.21/0.59 (singleton(w) = singleton(y)),
% 0.21/0.59 inference(symmetry,[status(thm)],[245])).
% 0.21/0.59 tff(247,plain,
% 0.21/0.59 ($false),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[65, 246])).
% 0.21/0.59 tff(248,plain,((singleton(w) = singleton(y)) | (~(apply(choice, unordered_pair(y, singleton(z))) = w)) | (~member(w, unordered_pair(w, singleton(x))))), inference(lemma,lemma(discharge,[]))).
% 0.21/0.59 tff(249,plain,
% 0.21/0.59 (singleton(w) = singleton(y)),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[248, 228, 216])).
% 0.21/0.59 tff(250,plain,
% 0.21/0.59 (singleton(y) = singleton(w)),
% 0.21/0.59 inference(symmetry,[status(thm)],[249])).
% 0.21/0.59 tff(251,plain,
% 0.21/0.59 (singleton(y) = unordered_pair(y, singleton(z))),
% 0.21/0.59 inference(transitivity,[status(thm)],[250, 200])).
% 0.21/0.59 tff(252,plain,
% 0.21/0.59 (unordered_pair(singleton(y), unordered_pair(y, singleton(z))) = unordered_pair(unordered_pair(y, singleton(z)), unordered_pair(y, singleton(z)))),
% 0.21/0.59 inference(monotonicity,[status(thm)],[251])).
% 0.21/0.59 tff(253,plain,
% 0.21/0.59 (unordered_pair(unordered_pair(y, singleton(z)), unordered_pair(y, singleton(z))) = unordered_pair(singleton(y), unordered_pair(y, singleton(z)))),
% 0.21/0.59 inference(symmetry,[status(thm)],[252])).
% 0.21/0.59 tff(254,plain,
% 0.21/0.59 (unordered_pair(singleton(w), singleton(w)) = unordered_pair(unordered_pair(y, singleton(z)), unordered_pair(y, singleton(z)))),
% 0.21/0.59 inference(monotonicity,[status(thm)],[200, 200])).
% 0.21/0.59 tff(255,plain,
% 0.21/0.59 (unordered_pair(singleton(w), singleton(w)) = unordered_pair(singleton(w), unordered_pair(w, singleton(x)))),
% 0.21/0.59 inference(transitivity,[status(thm)],[254, 253, 81, 79, 75])).
% 0.21/0.59 tff(256,plain,
% 0.21/0.59 (member(unordered_pair(w, singleton(x)), unordered_pair(singleton(w), singleton(w))) <=> member(unordered_pair(w, singleton(x)), unordered_pair(singleton(w), unordered_pair(w, singleton(x))))),
% 0.21/0.59 inference(monotonicity,[status(thm)],[255])).
% 0.21/0.59 tff(257,plain,
% 0.21/0.59 (member(unordered_pair(w, singleton(x)), unordered_pair(singleton(w), unordered_pair(w, singleton(x)))) <=> member(unordered_pair(w, singleton(x)), unordered_pair(singleton(w), singleton(w)))),
% 0.21/0.59 inference(symmetry,[status(thm)],[256])).
% 0.21/0.59 tff(258,plain,
% 0.21/0.59 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(w, singleton(x)), universal_class)),
% 0.21/0.59 inference(quant_inst,[status(thm)],[])).
% 0.21/0.59 tff(259,plain,
% 0.21/0.59 (member(unordered_pair(w, singleton(x)), universal_class)),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[258, 94])).
% 0.21/0.59 tff(260,plain,
% 0.21/0.59 (((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | ((~member(unordered_pair(w, singleton(x)), universal_class)) | member(unordered_pair(w, singleton(x)), unordered_pair(singleton(w), unordered_pair(w, singleton(x)))))) <=> ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | (~member(unordered_pair(w, singleton(x)), universal_class)) | member(unordered_pair(w, singleton(x)), unordered_pair(singleton(w), unordered_pair(w, singleton(x)))))),
% 0.21/0.59 inference(rewrite,[status(thm)],[])).
% 0.21/0.59 tff(261,plain,
% 0.21/0.59 ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | ((~member(unordered_pair(w, singleton(x)), universal_class)) | member(unordered_pair(w, singleton(x)), unordered_pair(singleton(w), unordered_pair(w, singleton(x)))))),
% 0.21/0.59 inference(quant_inst,[status(thm)],[])).
% 0.21/0.59 tff(262,plain,
% 0.21/0.59 ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | (~member(unordered_pair(w, singleton(x)), universal_class)) | member(unordered_pair(w, singleton(x)), unordered_pair(singleton(w), unordered_pair(w, singleton(x))))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[261, 260])).
% 0.21/0.59 tff(263,plain,
% 0.21/0.59 ((~member(unordered_pair(w, singleton(x)), universal_class)) | member(unordered_pair(w, singleton(x)), unordered_pair(singleton(w), unordered_pair(w, singleton(x))))),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[262, 123])).
% 0.21/0.59 tff(264,plain,
% 0.21/0.59 (member(unordered_pair(w, singleton(x)), unordered_pair(singleton(w), unordered_pair(w, singleton(x))))),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[263, 259])).
% 0.21/0.59 tff(265,plain,
% 0.21/0.59 (member(unordered_pair(w, singleton(x)), unordered_pair(singleton(w), singleton(w)))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[264, 257])).
% 0.21/0.59 tff(266,plain,
% 0.21/0.59 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((~member(unordered_pair(w, singleton(x)), unordered_pair(singleton(w), singleton(w)))) | (unordered_pair(w, singleton(x)) = singleton(w)))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(unordered_pair(w, singleton(x)), unordered_pair(singleton(w), singleton(w)))) | (unordered_pair(w, singleton(x)) = singleton(w)))),
% 0.21/0.59 inference(rewrite,[status(thm)],[])).
% 0.21/0.59 tff(267,plain,
% 0.21/0.59 (((unordered_pair(w, singleton(x)) = singleton(w)) | (unordered_pair(w, singleton(x)) = singleton(w)) | (~member(unordered_pair(w, singleton(x)), unordered_pair(singleton(w), singleton(w))))) <=> ((~member(unordered_pair(w, singleton(x)), unordered_pair(singleton(w), singleton(w)))) | (unordered_pair(w, singleton(x)) = singleton(w)))),
% 0.21/0.59 inference(rewrite,[status(thm)],[])).
% 0.21/0.59 tff(268,plain,
% 0.21/0.59 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((unordered_pair(w, singleton(x)) = singleton(w)) | (unordered_pair(w, singleton(x)) = singleton(w)) | (~member(unordered_pair(w, singleton(x)), unordered_pair(singleton(w), singleton(w)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((~member(unordered_pair(w, singleton(x)), unordered_pair(singleton(w), singleton(w)))) | (unordered_pair(w, singleton(x)) = singleton(w))))),
% 0.21/0.59 inference(monotonicity,[status(thm)],[267])).
% 0.21/0.59 tff(269,plain,
% 0.21/0.59 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((unordered_pair(w, singleton(x)) = singleton(w)) | (unordered_pair(w, singleton(x)) = singleton(w)) | (~member(unordered_pair(w, singleton(x)), unordered_pair(singleton(w), singleton(w)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(unordered_pair(w, singleton(x)), unordered_pair(singleton(w), singleton(w)))) | (unordered_pair(w, singleton(x)) = singleton(w)))),
% 0.21/0.59 inference(transitivity,[status(thm)],[268, 266])).
% 0.21/0.59 tff(270,plain,
% 0.21/0.59 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((unordered_pair(w, singleton(x)) = singleton(w)) | (unordered_pair(w, singleton(x)) = singleton(w)) | (~member(unordered_pair(w, singleton(x)), unordered_pair(singleton(w), singleton(w)))))),
% 0.21/0.59 inference(quant_inst,[status(thm)],[])).
% 0.21/0.59 tff(271,plain,
% 0.21/0.59 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(unordered_pair(w, singleton(x)), unordered_pair(singleton(w), singleton(w)))) | (unordered_pair(w, singleton(x)) = singleton(w))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[270, 269])).
% 0.21/0.59 tff(272,plain,
% 0.21/0.59 ((~member(unordered_pair(w, singleton(x)), unordered_pair(singleton(w), singleton(w)))) | (unordered_pair(w, singleton(x)) = singleton(w))),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[271, 49])).
% 0.21/0.59 tff(273,plain,
% 0.21/0.59 (unordered_pair(w, singleton(x)) = singleton(w)),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[272, 265])).
% 0.21/0.59 tff(274,plain,
% 0.21/0.59 (unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[273, 202])).
% 0.21/0.59 tff(275,plain,
% 0.21/0.59 ($false),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[183, 274])).
% 0.21/0.59 tff(276,plain,((unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x))) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, singleton(z))))), inference(lemma,lemma(discharge,[]))).
% 0.21/0.59 tff(277,plain,
% 0.21/0.59 (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, singleton(z)))),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[276, 180])).
% 0.21/0.59 tff(278,plain,
% 0.21/0.59 (^[Y: $i] : refl(((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y)) <=> ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y)))),
% 0.21/0.59 inference(bind,[status(th)],[])).
% 0.21/0.59 tff(279,plain,
% 0.21/0.59 (![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y)) <=> ![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))),
% 0.21/0.59 inference(quant_intro,[status(thm)],[278])).
% 0.21/0.59 tff(280,plain,
% 0.21/0.59 (![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y)) <=> ![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))),
% 0.21/0.59 inference(rewrite,[status(thm)],[])).
% 0.21/0.59 tff(281,plain,
% 0.21/0.59 (^[Y: $i] : rewrite((((~member(Y, universal_class)) | (Y = null_class)) | member(apply(choice, Y), Y)) <=> ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y)))),
% 0.21/0.59 inference(bind,[status(th)],[])).
% 0.21/0.59 tff(282,plain,
% 0.21/0.59 (![Y: $i] : (((~member(Y, universal_class)) | (Y = null_class)) | member(apply(choice, Y), Y)) <=> ![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))),
% 0.21/0.59 inference(quant_intro,[status(thm)],[281])).
% 0.21/0.59 tff(283,axiom,(![Y: $i] : (((~member(Y, universal_class)) | (Y = null_class)) | member(apply(choice, Y), Y))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','choice2')).
% 0.21/0.59 tff(284,plain,
% 0.21/0.59 (![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[283, 282])).
% 0.21/0.59 tff(285,plain,
% 0.21/0.59 (![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[284, 280])).
% 0.21/0.59 tff(286,plain,(
% 0.21/0.59 ![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))),
% 0.21/0.59 inference(skolemize,[status(sab)],[285])).
% 0.21/0.59 tff(287,plain,
% 0.21/0.59 (![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[286, 279])).
% 0.21/0.59 tff(288,plain,
% 0.21/0.59 (((~![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))) | ((~member(unordered_pair(y, singleton(z)), universal_class)) | (unordered_pair(y, singleton(z)) = null_class) | member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, singleton(z))))) <=> ((~![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))) | (~member(unordered_pair(y, singleton(z)), universal_class)) | (unordered_pair(y, singleton(z)) = null_class) | member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, singleton(z))))),
% 0.21/0.59 inference(rewrite,[status(thm)],[])).
% 0.21/0.59 tff(289,plain,
% 0.21/0.59 ((~![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))) | ((~member(unordered_pair(y, singleton(z)), universal_class)) | (unordered_pair(y, singleton(z)) = null_class) | member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, singleton(z))))),
% 0.21/0.59 inference(quant_inst,[status(thm)],[])).
% 0.21/0.59 tff(290,plain,
% 0.21/0.59 ((~![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))) | (~member(unordered_pair(y, singleton(z)), universal_class)) | (unordered_pair(y, singleton(z)) = null_class) | member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, singleton(z)))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[289, 288])).
% 0.21/0.59 tff(291,plain,
% 0.21/0.59 ((unordered_pair(y, singleton(z)) = null_class) | member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, singleton(z)))),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[290, 287, 187])).
% 0.21/0.59 tff(292,plain,
% 0.21/0.59 (unordered_pair(y, singleton(z)) = null_class),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[291, 277])).
% 0.21/0.59 tff(293,plain,
% 0.21/0.59 (null_class = unordered_pair(y, singleton(z))),
% 0.21/0.59 inference(symmetry,[status(thm)],[292])).
% 0.21/0.59 tff(294,plain,
% 0.21/0.59 (null_class = singleton(w)),
% 0.21/0.59 inference(transitivity,[status(thm)],[293, 145])).
% 0.21/0.59 tff(295,plain,
% 0.21/0.59 ((singleton(y) = null_class) <=> (singleton(y) = singleton(w))),
% 0.21/0.59 inference(monotonicity,[status(thm)],[294])).
% 0.21/0.59 tff(296,plain,
% 0.21/0.59 ((singleton(y) = null_class) <=> (singleton(w) = singleton(y))),
% 0.21/0.59 inference(transitivity,[status(thm)],[295, 144])).
% 0.21/0.59 tff(297,plain,
% 0.21/0.59 ((singleton(w) = singleton(y)) <=> (singleton(y) = null_class)),
% 0.21/0.59 inference(symmetry,[status(thm)],[296])).
% 0.21/0.59 tff(298,plain,
% 0.21/0.59 ((~(singleton(w) = singleton(y))) <=> (~(singleton(y) = null_class))),
% 0.21/0.59 inference(monotonicity,[status(thm)],[297])).
% 0.21/0.59 tff(299,plain,
% 0.21/0.59 (~(singleton(y) = null_class)),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[65, 298])).
% 0.21/0.59 tff(300,plain,
% 0.21/0.59 (((~![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))) | ((~member(singleton(y), universal_class)) | (singleton(y) = null_class) | member(apply(choice, singleton(y)), singleton(y)))) <=> ((~![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))) | (~member(singleton(y), universal_class)) | (singleton(y) = null_class) | member(apply(choice, singleton(y)), singleton(y)))),
% 0.21/0.59 inference(rewrite,[status(thm)],[])).
% 0.21/0.59 tff(301,plain,
% 0.21/0.59 ((~![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))) | ((~member(singleton(y), universal_class)) | (singleton(y) = null_class) | member(apply(choice, singleton(y)), singleton(y)))),
% 0.21/0.59 inference(quant_inst,[status(thm)],[])).
% 0.21/0.59 tff(302,plain,
% 0.21/0.59 ((~![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))) | (~member(singleton(y), universal_class)) | (singleton(y) = null_class) | member(apply(choice, singleton(y)), singleton(y))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[301, 300])).
% 0.21/0.59 tff(303,plain,
% 0.21/0.59 ((singleton(y) = null_class) | member(apply(choice, singleton(y)), singleton(y))),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[302, 287, 163])).
% 0.21/0.59 tff(304,plain,
% 0.21/0.59 (member(apply(choice, singleton(y)), singleton(y))),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[303, 299])).
% 0.21/0.59 tff(305,plain,
% 0.21/0.59 (member(apply(choice, singleton(y)), unordered_pair(y, y))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[304, 143])).
% 0.21/0.59 tff(306,plain,
% 0.21/0.59 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, singleton(y)) = y) | (~member(apply(choice, singleton(y)), unordered_pair(y, y))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, singleton(y)) = y) | (~member(apply(choice, singleton(y)), unordered_pair(y, y))))),
% 0.21/0.59 inference(rewrite,[status(thm)],[])).
% 0.21/0.59 tff(307,plain,
% 0.21/0.59 (((apply(choice, singleton(y)) = y) | (apply(choice, singleton(y)) = y) | (~member(apply(choice, singleton(y)), unordered_pair(y, y)))) <=> ((apply(choice, singleton(y)) = y) | (~member(apply(choice, singleton(y)), unordered_pair(y, y))))),
% 0.21/0.59 inference(rewrite,[status(thm)],[])).
% 0.21/0.59 tff(308,plain,
% 0.21/0.59 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, singleton(y)) = y) | (apply(choice, singleton(y)) = y) | (~member(apply(choice, singleton(y)), unordered_pair(y, y))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, singleton(y)) = y) | (~member(apply(choice, singleton(y)), unordered_pair(y, y)))))),
% 0.21/0.59 inference(monotonicity,[status(thm)],[307])).
% 0.21/0.59 tff(309,plain,
% 0.21/0.59 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, singleton(y)) = y) | (apply(choice, singleton(y)) = y) | (~member(apply(choice, singleton(y)), unordered_pair(y, y))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, singleton(y)) = y) | (~member(apply(choice, singleton(y)), unordered_pair(y, y))))),
% 0.21/0.59 inference(transitivity,[status(thm)],[308, 306])).
% 0.21/0.59 tff(310,plain,
% 0.21/0.59 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, singleton(y)) = y) | (apply(choice, singleton(y)) = y) | (~member(apply(choice, singleton(y)), unordered_pair(y, y))))),
% 0.21/0.59 inference(quant_inst,[status(thm)],[])).
% 0.21/0.59 tff(311,plain,
% 0.21/0.59 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, singleton(y)) = y) | (~member(apply(choice, singleton(y)), unordered_pair(y, y)))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[310, 309])).
% 0.21/0.59 tff(312,plain,
% 0.21/0.59 ((apply(choice, singleton(y)) = y) | (~member(apply(choice, singleton(y)), unordered_pair(y, y)))),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[311, 49])).
% 0.21/0.59 tff(313,plain,
% 0.21/0.59 (apply(choice, singleton(y)) = y),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[312, 305])).
% 0.21/0.59 tff(314,plain,
% 0.21/0.59 (y = apply(choice, singleton(y))),
% 0.21/0.59 inference(symmetry,[status(thm)],[313])).
% 0.21/0.59 tff(315,plain,
% 0.21/0.59 (member(singleton(x), unordered_pair(y, y)) <=> member(singleton(x), unordered_pair(w, singleton(x)))),
% 0.21/0.59 inference(monotonicity,[status(thm)],[230])).
% 0.21/0.59 tff(316,plain,
% 0.21/0.59 (member(singleton(x), unordered_pair(w, singleton(x))) <=> member(singleton(x), unordered_pair(y, y))),
% 0.21/0.59 inference(symmetry,[status(thm)],[315])).
% 0.21/0.59 tff(317,plain,
% 0.21/0.59 (member(unordered_pair(x, x), universal_class) <=> member(singleton(x), universal_class)),
% 0.21/0.59 inference(monotonicity,[status(thm)],[14])).
% 0.21/0.59 tff(318,plain,
% 0.21/0.59 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(x, x), universal_class)),
% 0.21/0.59 inference(quant_inst,[status(thm)],[])).
% 0.21/0.59 tff(319,plain,
% 0.21/0.59 (member(unordered_pair(x, x), universal_class)),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[318, 94])).
% 0.21/0.59 tff(320,plain,
% 0.21/0.59 (member(singleton(x), universal_class)),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[319, 317])).
% 0.21/0.59 tff(321,plain,
% 0.21/0.59 (((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | ((~member(singleton(x), universal_class)) | member(singleton(x), unordered_pair(w, singleton(x))))) <=> ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | (~member(singleton(x), universal_class)) | member(singleton(x), unordered_pair(w, singleton(x))))),
% 0.21/0.59 inference(rewrite,[status(thm)],[])).
% 0.21/0.59 tff(322,plain,
% 0.21/0.59 ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | ((~member(singleton(x), universal_class)) | member(singleton(x), unordered_pair(w, singleton(x))))),
% 0.21/0.59 inference(quant_inst,[status(thm)],[])).
% 0.21/0.59 tff(323,plain,
% 0.21/0.59 ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | (~member(singleton(x), universal_class)) | member(singleton(x), unordered_pair(w, singleton(x)))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[322, 321])).
% 0.21/0.59 tff(324,plain,
% 0.21/0.59 ((~member(singleton(x), universal_class)) | member(singleton(x), unordered_pair(w, singleton(x)))),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[323, 123])).
% 0.21/0.59 tff(325,plain,
% 0.21/0.59 (member(singleton(x), unordered_pair(w, singleton(x)))),
% 0.21/0.59 inference(unit_resolution,[status(thm)],[324, 320])).
% 0.21/0.59 tff(326,plain,
% 0.21/0.59 (member(singleton(x), unordered_pair(y, y))),
% 0.21/0.59 inference(modus_ponens,[status(thm)],[325, 316])).
% 0.21/0.59 tff(327,plain,
% 0.21/0.60 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(x) = y) | (~member(singleton(x), unordered_pair(y, y))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(x) = y) | (~member(singleton(x), unordered_pair(y, y))))),
% 0.21/0.60 inference(rewrite,[status(thm)],[])).
% 0.21/0.60 tff(328,plain,
% 0.21/0.60 (((singleton(x) = y) | (singleton(x) = y) | (~member(singleton(x), unordered_pair(y, y)))) <=> ((singleton(x) = y) | (~member(singleton(x), unordered_pair(y, y))))),
% 0.21/0.60 inference(rewrite,[status(thm)],[])).
% 0.21/0.60 tff(329,plain,
% 0.21/0.60 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(x) = y) | (singleton(x) = y) | (~member(singleton(x), unordered_pair(y, y))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(x) = y) | (~member(singleton(x), unordered_pair(y, y)))))),
% 0.21/0.60 inference(monotonicity,[status(thm)],[328])).
% 0.21/0.60 tff(330,plain,
% 0.21/0.60 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(x) = y) | (singleton(x) = y) | (~member(singleton(x), unordered_pair(y, y))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(x) = y) | (~member(singleton(x), unordered_pair(y, y))))),
% 0.21/0.60 inference(transitivity,[status(thm)],[329, 327])).
% 0.21/0.60 tff(331,plain,
% 0.21/0.60 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(x) = y) | (singleton(x) = y) | (~member(singleton(x), unordered_pair(y, y))))),
% 0.21/0.60 inference(quant_inst,[status(thm)],[])).
% 0.21/0.60 tff(332,plain,
% 0.21/0.60 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(x) = y) | (~member(singleton(x), unordered_pair(y, y)))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[331, 330])).
% 0.21/0.60 tff(333,plain,
% 0.21/0.60 ((singleton(x) = y) | (~member(singleton(x), unordered_pair(y, y)))),
% 0.21/0.60 inference(unit_resolution,[status(thm)],[332, 49])).
% 0.21/0.60 tff(334,plain,
% 0.21/0.60 (singleton(x) = y),
% 0.21/0.60 inference(unit_resolution,[status(thm)],[333, 326])).
% 0.21/0.60 tff(335,plain,
% 0.21/0.60 (singleton(x) = apply(choice, singleton(y))),
% 0.21/0.60 inference(transitivity,[status(thm)],[334, 314])).
% 0.21/0.60 tff(336,plain,
% 0.21/0.60 ((singleton(z) = singleton(x)) <=> (w = apply(choice, singleton(y)))),
% 0.21/0.60 inference(monotonicity,[status(thm)],[137, 335])).
% 0.21/0.60 tff(337,plain,
% 0.21/0.60 ((singleton(z) = singleton(x)) <=> (apply(choice, singleton(y)) = w)),
% 0.21/0.60 inference(transitivity,[status(thm)],[336, 64])).
% 0.21/0.60 tff(338,plain,
% 0.21/0.60 ((~(singleton(z) = singleton(x))) <=> (~(apply(choice, singleton(y)) = w))),
% 0.21/0.60 inference(monotonicity,[status(thm)],[337])).
% 0.21/0.60 tff(339,plain,
% 0.21/0.60 (~(apply(choice, singleton(y)) = w)),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[58, 338])).
% 0.21/0.60 tff(340,plain,
% 0.21/0.60 (~(singleton(z) = apply(choice, singleton(y)))),
% 0.21/0.60 inference(modus_ponens,[status(thm)],[339, 141])).
% 0.21/0.60 tff(341,plain,
% 0.21/0.60 (unordered_pair(apply(choice, singleton(y)), apply(choice, singleton(y))) = unordered_pair(y, y)),
% 0.21/0.60 inference(monotonicity,[status(thm)],[313, 313])).
% 0.21/0.60 tff(342,plain,
% 0.21/0.60 (unordered_pair(apply(choice, singleton(y)), apply(choice, singleton(y))) = unordered_pair(w, singleton(x))),
% 0.21/0.60 inference(transitivity,[status(thm)],[341, 9, 175])).
% 0.21/0.60 tff(343,plain,
% 0.21/0.60 (member(singleton(z), unordered_pair(apply(choice, singleton(y)), apply(choice, singleton(y)))) <=> member(w, unordered_pair(w, singleton(x)))),
% 0.21/0.60 inference(monotonicity,[status(thm)],[137, 342])).
% 0.21/0.60 tff(344,plain,
% 0.21/0.60 (member(w, unordered_pair(w, singleton(x))) <=> member(singleton(z), unordered_pair(apply(choice, singleton(y)), apply(choice, singleton(y))))),
% 0.21/0.60 inference(symmetry,[status(thm)],[343])).
% 0.21/0.60 tff(345,plain,
% 0.21/0.60 (w = singleton(z)),
% 0.21/0.60 inference(symmetry,[status(thm)],[137])).
% 0.21/0.60 tff(346,plain,
% 0.21/0.60 (member(w, universal_class) <=> member(singleton(z), universal_class)),
% 0.21/0.60 inference(monotonicity,[status(thm)],[345])).
% 0.21/0.60 tff(347,plain,
% 0.21/0.60 (member(singleton(z), universal_class) <=> member(w, universal_class)),
% 0.21/0.60 inference(symmetry,[status(thm)],[346])).
% 0.21/0.60 tff(348,plain,
% 0.55/0.60 (member(w, universal_class)),
% 0.55/0.60 inference(modus_ponens,[status(thm)],[116, 347])).
% 0.55/0.60 tff(349,plain,
% 0.55/0.60 (member(w, unordered_pair(w, singleton(x)))),
% 0.55/0.60 inference(unit_resolution,[status(thm)],[215, 348])).
% 0.55/0.60 tff(350,plain,
% 0.55/0.60 (member(singleton(z), unordered_pair(apply(choice, singleton(y)), apply(choice, singleton(y))))),
% 0.55/0.60 inference(modus_ponens,[status(thm)],[349, 344])).
% 0.55/0.60 tff(351,plain,
% 0.55/0.60 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = apply(choice, singleton(y))) | (~member(singleton(z), unordered_pair(apply(choice, singleton(y)), apply(choice, singleton(y))))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(z) = apply(choice, singleton(y))) | (~member(singleton(z), unordered_pair(apply(choice, singleton(y)), apply(choice, singleton(y))))))),
% 0.55/0.60 inference(rewrite,[status(thm)],[])).
% 0.55/0.60 tff(352,plain,
% 0.55/0.60 (((singleton(z) = apply(choice, singleton(y))) | (singleton(z) = apply(choice, singleton(y))) | (~member(singleton(z), unordered_pair(apply(choice, singleton(y)), apply(choice, singleton(y)))))) <=> ((singleton(z) = apply(choice, singleton(y))) | (~member(singleton(z), unordered_pair(apply(choice, singleton(y)), apply(choice, singleton(y))))))),
% 0.55/0.60 inference(rewrite,[status(thm)],[])).
% 0.55/0.60 tff(353,plain,
% 0.55/0.60 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = apply(choice, singleton(y))) | (singleton(z) = apply(choice, singleton(y))) | (~member(singleton(z), unordered_pair(apply(choice, singleton(y)), apply(choice, singleton(y))))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = apply(choice, singleton(y))) | (~member(singleton(z), unordered_pair(apply(choice, singleton(y)), apply(choice, singleton(y)))))))),
% 0.55/0.60 inference(monotonicity,[status(thm)],[352])).
% 0.55/0.60 tff(354,plain,
% 0.55/0.60 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = apply(choice, singleton(y))) | (singleton(z) = apply(choice, singleton(y))) | (~member(singleton(z), unordered_pair(apply(choice, singleton(y)), apply(choice, singleton(y))))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(z) = apply(choice, singleton(y))) | (~member(singleton(z), unordered_pair(apply(choice, singleton(y)), apply(choice, singleton(y))))))),
% 0.55/0.60 inference(transitivity,[status(thm)],[353, 351])).
% 0.55/0.60 tff(355,plain,
% 0.55/0.60 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = apply(choice, singleton(y))) | (singleton(z) = apply(choice, singleton(y))) | (~member(singleton(z), unordered_pair(apply(choice, singleton(y)), apply(choice, singleton(y))))))),
% 0.55/0.60 inference(quant_inst,[status(thm)],[])).
% 0.55/0.60 tff(356,plain,
% 0.55/0.60 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(z) = apply(choice, singleton(y))) | (~member(singleton(z), unordered_pair(apply(choice, singleton(y)), apply(choice, singleton(y)))))),
% 0.55/0.60 inference(modus_ponens,[status(thm)],[355, 354])).
% 0.55/0.60 tff(357,plain,
% 0.55/0.60 ((singleton(z) = apply(choice, singleton(y))) | (~member(singleton(z), unordered_pair(apply(choice, singleton(y)), apply(choice, singleton(y)))))),
% 0.55/0.60 inference(unit_resolution,[status(thm)],[356, 49])).
% 0.55/0.60 tff(358,plain,
% 0.55/0.60 ($false),
% 0.55/0.60 inference(unit_resolution,[status(thm)],[357, 350, 340])).
% 0.55/0.60 tff(359,plain,(singleton(w) = singleton(y)), inference(lemma,lemma(discharge,[]))).
% 0.55/0.60 tff(360,plain,
% 0.55/0.60 ((unordered_pair(y, singleton(z)) = singleton(w)) <=> (null_class = singleton(y))),
% 0.55/0.60 inference(monotonicity,[status(thm)],[60, 359])).
% 0.55/0.60 tff(361,plain,
% 0.55/0.60 ((unordered_pair(y, singleton(z)) = singleton(w)) <=> (singleton(y) = null_class)),
% 0.55/0.60 inference(transitivity,[status(thm)],[360, 63])).
% 0.55/0.60 tff(362,plain,
% 0.55/0.60 ((singleton(y) = null_class) <=> (unordered_pair(y, singleton(z)) = singleton(w))),
% 0.55/0.60 inference(symmetry,[status(thm)],[361])).
% 0.55/0.60 tff(363,plain,
% 0.55/0.60 ((~(singleton(y) = null_class)) <=> (~(unordered_pair(y, singleton(z)) = singleton(w)))),
% 0.55/0.60 inference(monotonicity,[status(thm)],[362])).
% 0.55/0.60 tff(364,plain,
% 0.55/0.60 (~(unordered_pair(y, singleton(z)) = singleton(w))),
% 0.55/0.60 inference(modus_ponens,[status(thm)],[62, 363])).
% 0.55/0.60 tff(365,plain,
% 0.55/0.60 (unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x))),
% 0.55/0.60 inference(unit_resolution,[status(thm)],[198, 364])).
% 0.55/0.60 tff(366,plain,
% 0.55/0.60 (unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))),
% 0.55/0.60 inference(symmetry,[status(thm)],[365])).
% 0.55/0.60 tff(367,plain,
% 0.55/0.60 (member(apply(choice, singleton(y)), singleton(y))),
% 0.55/0.60 inference(unit_resolution,[status(thm)],[303, 62])).
% 0.55/0.60 tff(368,plain,
% 0.55/0.60 (member(apply(choice, singleton(y)), unordered_pair(y, y))),
% 0.55/0.60 inference(modus_ponens,[status(thm)],[367, 143])).
% 0.55/0.60 tff(369,plain,
% 0.55/0.60 (apply(choice, singleton(y)) = y),
% 0.55/0.60 inference(unit_resolution,[status(thm)],[312, 368])).
% 0.55/0.60 tff(370,plain,
% 0.55/0.60 (unordered_pair(apply(choice, singleton(y)), singleton(z)) = unordered_pair(y, singleton(z))),
% 0.55/0.60 inference(monotonicity,[status(thm)],[369])).
% 0.55/0.60 tff(371,plain,
% 0.55/0.60 (unordered_pair(apply(choice, singleton(y)), singleton(z)) = unordered_pair(w, singleton(x))),
% 0.55/0.60 inference(transitivity,[status(thm)],[370, 365])).
% 0.55/0.60 tff(372,plain,
% 0.55/0.60 (member(singleton(x), unordered_pair(apply(choice, singleton(y)), singleton(z))) <=> member(singleton(x), unordered_pair(w, singleton(x)))),
% 0.55/0.60 inference(monotonicity,[status(thm)],[371])).
% 0.55/0.60 tff(373,plain,
% 0.55/0.60 (member(singleton(x), unordered_pair(w, singleton(x))) <=> member(singleton(x), unordered_pair(apply(choice, singleton(y)), singleton(z)))),
% 0.55/0.60 inference(symmetry,[status(thm)],[372])).
% 0.55/0.60 tff(374,plain,
% 0.55/0.60 (member(singleton(x), unordered_pair(apply(choice, singleton(y)), singleton(z)))),
% 0.55/0.60 inference(modus_ponens,[status(thm)],[325, 373])).
% 0.55/0.60 tff(375,plain,
% 0.55/0.60 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(x) = singleton(z)) | (singleton(x) = apply(choice, singleton(y))) | (~member(singleton(x), unordered_pair(apply(choice, singleton(y)), singleton(z)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(x) = singleton(z)) | (singleton(x) = apply(choice, singleton(y))) | (~member(singleton(x), unordered_pair(apply(choice, singleton(y)), singleton(z)))))),
% 0.55/0.60 inference(rewrite,[status(thm)],[])).
% 0.55/0.60 tff(376,plain,
% 0.55/0.60 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(x) = singleton(z)) | (singleton(x) = apply(choice, singleton(y))) | (~member(singleton(x), unordered_pair(apply(choice, singleton(y)), singleton(z)))))),
% 0.55/0.60 inference(quant_inst,[status(thm)],[])).
% 0.55/0.60 tff(377,plain,
% 0.55/0.60 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(x) = singleton(z)) | (singleton(x) = apply(choice, singleton(y))) | (~member(singleton(x), unordered_pair(apply(choice, singleton(y)), singleton(z))))),
% 0.55/0.60 inference(modus_ponens,[status(thm)],[376, 375])).
% 0.55/0.60 tff(378,plain,
% 0.55/0.60 ((singleton(x) = singleton(z)) | (singleton(x) = apply(choice, singleton(y))) | (~member(singleton(x), unordered_pair(apply(choice, singleton(y)), singleton(z))))),
% 0.55/0.60 inference(unit_resolution,[status(thm)],[377, 49])).
% 0.55/0.60 tff(379,plain,
% 0.55/0.60 (singleton(x) = apply(choice, singleton(y))),
% 0.55/0.60 inference(unit_resolution,[status(thm)],[378, 59, 374])).
% 0.55/0.60 tff(380,plain,
% 0.55/0.60 (singleton(x) = y),
% 0.55/0.60 inference(transitivity,[status(thm)],[379, 369])).
% 0.55/0.60 tff(381,plain,
% 0.55/0.60 (singleton(w) = unordered_pair(w, w)),
% 0.55/0.60 inference(symmetry,[status(thm)],[86])).
% 0.55/0.60 tff(382,plain,
% 0.55/0.60 (singleton(y) = singleton(w)),
% 0.55/0.60 inference(symmetry,[status(thm)],[359])).
% 0.55/0.60 tff(383,plain,
% 0.55/0.60 (unordered_pair(y, y) = unordered_pair(w, w)),
% 0.55/0.60 inference(transitivity,[status(thm)],[9, 382, 381])).
% 0.55/0.60 tff(384,plain,
% 0.55/0.60 (member(w, unordered_pair(y, y)) <=> member(w, unordered_pair(w, w))),
% 0.55/0.60 inference(monotonicity,[status(thm)],[383])).
% 0.55/0.60 tff(385,plain,
% 0.55/0.60 (member(w, unordered_pair(w, w)) <=> member(w, unordered_pair(y, y))),
% 0.55/0.60 inference(symmetry,[status(thm)],[384])).
% 0.55/0.60 tff(386,assumption,(~member(w, universal_class)), introduced(assumption)).
% 0.55/0.60 tff(387,plain,
% 0.55/0.60 (singleton(y) = unordered_pair(y, y)),
% 0.55/0.60 inference(symmetry,[status(thm)],[9])).
% 0.55/0.60 tff(388,plain,
% 0.55/0.60 (unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x))),
% 0.55/0.60 inference(unit_resolution,[status(thm)],[276, 219])).
% 0.55/0.60 tff(389,plain,
% 0.55/0.60 (singleton(w) = singleton(y)),
% 0.55/0.60 inference(unit_resolution,[status(thm)],[179, 388])).
% 0.55/0.60 tff(390,plain,
% 0.55/0.60 (unordered_pair(w, w) = unordered_pair(y, y)),
% 0.55/0.60 inference(transitivity,[status(thm)],[86, 389, 387])).
% 0.55/0.60 tff(391,plain,
% 0.55/0.60 (member(y, unordered_pair(w, w)) <=> member(y, unordered_pair(y, y))),
% 0.55/0.60 inference(monotonicity,[status(thm)],[390])).
% 0.55/0.60 tff(392,plain,
% 0.55/0.60 (member(y, unordered_pair(y, y)) <=> member(y, unordered_pair(w, w))),
% 0.55/0.60 inference(symmetry,[status(thm)],[391])).
% 0.55/0.60 tff(393,plain,
% 0.55/0.60 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(singleton(y), singleton(y)) = singleton(singleton(y)))),
% 0.55/0.60 inference(quant_inst,[status(thm)],[])).
% 0.55/0.60 tff(394,plain,
% 0.55/0.60 (unordered_pair(singleton(y), singleton(y)) = singleton(singleton(y))),
% 0.55/0.60 inference(unit_resolution,[status(thm)],[393, 7])).
% 0.55/0.60 tff(395,plain,
% 0.55/0.60 (member(unordered_pair(singleton(y), singleton(y)), universal_class) <=> member(singleton(singleton(y)), universal_class)),
% 0.55/0.60 inference(monotonicity,[status(thm)],[394])).
% 0.55/0.60 tff(396,plain,
% 0.55/0.60 (member(singleton(singleton(y)), universal_class) <=> member(unordered_pair(singleton(y), singleton(y)), universal_class)),
% 0.55/0.60 inference(symmetry,[status(thm)],[395])).
% 0.55/0.60 tff(397,plain,
% 0.55/0.60 ((~member(singleton(singleton(y)), universal_class)) <=> (~member(unordered_pair(singleton(y), singleton(y)), universal_class))),
% 0.55/0.60 inference(monotonicity,[status(thm)],[396])).
% 0.55/0.60 tff(398,plain,
% 0.55/0.60 (^[Y: $i, Xf: $i] : refl((sum_class(image(Xf, singleton(Y))) = apply(Xf, Y)) <=> (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y)))),
% 0.55/0.60 inference(bind,[status(th)],[])).
% 0.55/0.60 tff(399,plain,
% 0.55/0.60 (![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y)) <=> ![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))),
% 0.55/0.60 inference(quant_intro,[status(thm)],[398])).
% 0.55/0.60 tff(400,plain,
% 0.55/0.60 (![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y)) <=> ![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))),
% 0.55/0.60 inference(rewrite,[status(thm)],[])).
% 0.55/0.60 tff(401,axiom,(![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','apply')).
% 0.55/0.60 tff(402,plain,
% 0.55/0.60 (![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))),
% 0.55/0.60 inference(modus_ponens,[status(thm)],[401, 400])).
% 0.55/0.60 tff(403,plain,(
% 0.55/0.60 ![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))),
% 0.55/0.60 inference(skolemize,[status(sab)],[402])).
% 0.55/0.60 tff(404,plain,
% 0.55/0.60 (![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))),
% 0.55/0.60 inference(modus_ponens,[status(thm)],[403, 399])).
% 0.55/0.60 tff(405,plain,
% 0.55/0.60 ((~![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))) | (sum_class(image(choice, singleton(singleton(y)))) = apply(choice, singleton(y)))),
% 0.55/0.60 inference(quant_inst,[status(thm)],[])).
% 0.55/0.60 tff(406,plain,
% 0.55/0.60 (sum_class(image(choice, singleton(singleton(y)))) = apply(choice, singleton(y))),
% 0.55/0.60 inference(unit_resolution,[status(thm)],[405, 404])).
% 0.55/0.60 tff(407,plain,
% 0.55/0.60 (apply(choice, singleton(y)) = sum_class(image(choice, singleton(singleton(y))))),
% 0.55/0.60 inference(symmetry,[status(thm)],[406])).
% 0.55/0.60 tff(408,assumption,(singleton(w) = singleton(y)), introduced(assumption)).
% 0.55/0.60 tff(409,plain,
% 0.55/0.60 (unordered_pair(w, w) = singleton(y)),
% 0.55/0.60 inference(transitivity,[status(thm)],[86, 408])).
% 0.55/0.60 tff(410,plain,
% 0.55/0.60 (member(apply(choice, singleton(y)), unordered_pair(w, w)) <=> member(apply(choice, singleton(y)), singleton(y))),
% 0.55/0.60 inference(monotonicity,[status(thm)],[409])).
% 0.55/0.60 tff(411,plain,
% 0.55/0.60 (member(apply(choice, singleton(y)), singleton(y)) <=> member(apply(choice, singleton(y)), unordered_pair(w, w))),
% 0.55/0.60 inference(symmetry,[status(thm)],[410])).
% 0.55/0.60 tff(412,assumption,(member(apply(choice, singleton(y)), singleton(y))), introduced(assumption)).
% 0.55/0.60 tff(413,plain,
% 0.55/0.60 (member(apply(choice, singleton(y)), unordered_pair(w, w))),
% 0.55/0.60 inference(modus_ponens,[status(thm)],[412, 411])).
% 0.55/0.60 tff(414,plain,
% 0.55/0.60 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, singleton(y)) = w) | (~member(apply(choice, singleton(y)), unordered_pair(w, w))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, singleton(y)) = w) | (~member(apply(choice, singleton(y)), unordered_pair(w, w))))),
% 0.55/0.60 inference(rewrite,[status(thm)],[])).
% 0.55/0.60 tff(415,plain,
% 0.55/0.60 (((apply(choice, singleton(y)) = w) | (apply(choice, singleton(y)) = w) | (~member(apply(choice, singleton(y)), unordered_pair(w, w)))) <=> ((apply(choice, singleton(y)) = w) | (~member(apply(choice, singleton(y)), unordered_pair(w, w))))),
% 0.55/0.60 inference(rewrite,[status(thm)],[])).
% 0.55/0.60 tff(416,plain,
% 0.55/0.60 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, singleton(y)) = w) | (apply(choice, singleton(y)) = w) | (~member(apply(choice, singleton(y)), unordered_pair(w, w))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, singleton(y)) = w) | (~member(apply(choice, singleton(y)), unordered_pair(w, w)))))),
% 0.55/0.60 inference(monotonicity,[status(thm)],[415])).
% 0.55/0.60 tff(417,plain,
% 0.55/0.60 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, singleton(y)) = w) | (apply(choice, singleton(y)) = w) | (~member(apply(choice, singleton(y)), unordered_pair(w, w))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, singleton(y)) = w) | (~member(apply(choice, singleton(y)), unordered_pair(w, w))))),
% 0.55/0.60 inference(transitivity,[status(thm)],[416, 414])).
% 0.55/0.60 tff(418,plain,
% 0.55/0.60 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, singleton(y)) = w) | (apply(choice, singleton(y)) = w) | (~member(apply(choice, singleton(y)), unordered_pair(w, w))))),
% 0.55/0.60 inference(quant_inst,[status(thm)],[])).
% 0.55/0.60 tff(419,plain,
% 0.55/0.60 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, singleton(y)) = w) | (~member(apply(choice, singleton(y)), unordered_pair(w, w)))),
% 0.55/0.60 inference(modus_ponens,[status(thm)],[418, 417])).
% 0.55/0.60 tff(420,plain,
% 0.55/0.60 ((apply(choice, singleton(y)) = w) | (~member(apply(choice, singleton(y)), unordered_pair(w, w)))),
% 0.55/0.60 inference(unit_resolution,[status(thm)],[419, 49])).
% 0.55/0.60 tff(421,plain,
% 0.55/0.60 (apply(choice, singleton(y)) = w),
% 0.55/0.60 inference(unit_resolution,[status(thm)],[420, 413])).
% 0.55/0.60 tff(422,plain,
% 0.55/0.60 (w = apply(choice, singleton(y))),
% 0.55/0.60 inference(symmetry,[status(thm)],[421])).
% 0.55/0.60 tff(423,plain,
% 0.55/0.60 (w = sum_class(image(choice, singleton(singleton(y))))),
% 0.55/0.60 inference(transitivity,[status(thm)],[422, 407])).
% 0.55/0.60 tff(424,plain,
% 0.55/0.60 (member(w, universal_class) <=> member(sum_class(image(choice, singleton(singleton(y)))), universal_class)),
% 0.55/0.60 inference(monotonicity,[status(thm)],[423])).
% 0.55/0.60 tff(425,plain,
% 0.55/0.60 ((~member(w, universal_class)) <=> (~member(sum_class(image(choice, singleton(singleton(y)))), universal_class))),
% 0.55/0.60 inference(monotonicity,[status(thm)],[424])).
% 0.55/0.60 tff(426,plain,
% 0.55/0.60 (~member(sum_class(image(choice, singleton(singleton(y)))), universal_class)),
% 0.55/0.60 inference(modus_ponens,[status(thm)],[386, 425])).
% 0.55/0.60 tff(427,plain,
% 0.55/0.60 (^[X: $i] : refl(((~member(X, universal_class)) | member(sum_class(X), universal_class)) <=> ((~member(X, universal_class)) | member(sum_class(X), universal_class)))),
% 0.55/0.60 inference(bind,[status(th)],[])).
% 0.55/0.60 tff(428,plain,
% 0.55/0.60 (![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class)) <=> ![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))),
% 0.55/0.60 inference(quant_intro,[status(thm)],[427])).
% 0.55/0.60 tff(429,plain,
% 0.55/0.60 (![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class)) <=> ![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))),
% 0.55/0.60 inference(rewrite,[status(thm)],[])).
% 0.55/0.60 tff(430,axiom,(![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','sum_class2')).
% 0.55/0.60 tff(431,plain,
% 0.55/0.60 (![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))),
% 0.55/0.60 inference(modus_ponens,[status(thm)],[430, 429])).
% 0.55/0.60 tff(432,plain,(
% 0.55/0.60 ![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))),
% 0.55/0.60 inference(skolemize,[status(sab)],[431])).
% 0.55/0.60 tff(433,plain,
% 0.55/0.60 (![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))),
% 0.55/0.60 inference(modus_ponens,[status(thm)],[432, 428])).
% 0.55/0.60 tff(434,plain,
% 0.55/0.60 (((~![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))) | ((~member(image(choice, singleton(singleton(y))), universal_class)) | member(sum_class(image(choice, singleton(singleton(y)))), universal_class))) <=> ((~![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))) | (~member(image(choice, singleton(singleton(y))), universal_class)) | member(sum_class(image(choice, singleton(singleton(y)))), universal_class))),
% 0.55/0.60 inference(rewrite,[status(thm)],[])).
% 0.55/0.60 tff(435,plain,
% 0.55/0.60 ((~![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))) | ((~member(image(choice, singleton(singleton(y))), universal_class)) | member(sum_class(image(choice, singleton(singleton(y)))), universal_class))),
% 0.55/0.60 inference(quant_inst,[status(thm)],[])).
% 0.55/0.60 tff(436,plain,
% 0.55/0.60 ((~![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))) | (~member(image(choice, singleton(singleton(y))), universal_class)) | member(sum_class(image(choice, singleton(singleton(y)))), universal_class)),
% 0.55/0.60 inference(modus_ponens,[status(thm)],[435, 434])).
% 0.55/0.60 tff(437,plain,
% 0.55/0.60 ((~member(image(choice, singleton(singleton(y))), universal_class)) | member(sum_class(image(choice, singleton(singleton(y)))), universal_class)),
% 0.55/0.60 inference(unit_resolution,[status(thm)],[436, 433])).
% 0.55/0.60 tff(438,plain,
% 0.55/0.60 (~member(image(choice, singleton(singleton(y))), universal_class)),
% 0.55/0.60 inference(unit_resolution,[status(thm)],[437, 426])).
% 0.55/0.60 tff(439,plain,
% 0.55/0.60 (function(choice) <=> function(choice)),
% 0.55/0.60 inference(rewrite,[status(thm)],[])).
% 0.55/0.60 tff(440,axiom,(function(choice)), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','choice1')).
% 0.55/0.60 tff(441,plain,
% 0.55/0.60 (function(choice)),
% 0.55/0.60 inference(modus_ponens,[status(thm)],[440, 439])).
% 0.55/0.60 tff(442,plain,
% 0.55/0.60 (^[Xf: $i, X: $i] : refl(((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf))) <=> ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf))))),
% 0.55/0.60 inference(bind,[status(th)],[])).
% 0.55/0.60 tff(443,plain,
% 0.55/0.60 (![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf))) <=> ![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))),
% 0.55/0.60 inference(quant_intro,[status(thm)],[442])).
% 0.55/0.60 tff(444,plain,
% 0.55/0.60 (![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf))) <=> ![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))),
% 0.55/0.60 inference(rewrite,[status(thm)],[])).
% 0.55/0.60 tff(445,plain,
% 0.55/0.60 (^[Xf: $i, X: $i] : rewrite((((~function(Xf)) | (~member(X, universal_class))) | member(image(Xf, X), universal_class)) <=> ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf))))),
% 0.55/0.60 inference(bind,[status(th)],[])).
% 0.55/0.60 tff(446,plain,
% 0.55/0.60 (![Xf: $i, X: $i] : (((~function(Xf)) | (~member(X, universal_class))) | member(image(Xf, X), universal_class)) <=> ![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))),
% 0.55/0.60 inference(quant_intro,[status(thm)],[445])).
% 0.55/0.60 tff(447,axiom,(![Xf: $i, X: $i] : (((~function(Xf)) | (~member(X, universal_class))) | member(image(Xf, X), universal_class))), file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax','replacement')).
% 0.55/0.61 tff(448,plain,
% 0.55/0.61 (![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))),
% 0.55/0.61 inference(modus_ponens,[status(thm)],[447, 446])).
% 0.55/0.61 tff(449,plain,
% 0.55/0.61 (![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))),
% 0.55/0.61 inference(modus_ponens,[status(thm)],[448, 444])).
% 0.55/0.61 tff(450,plain,(
% 0.55/0.61 ![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))),
% 0.55/0.61 inference(skolemize,[status(sab)],[449])).
% 0.55/0.61 tff(451,plain,
% 0.55/0.61 (![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))),
% 0.55/0.61 inference(modus_ponens,[status(thm)],[450, 443])).
% 0.55/0.61 tff(452,plain,
% 0.55/0.61 (((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | ((~function(choice)) | (~member(singleton(singleton(y)), universal_class)) | member(image(choice, singleton(singleton(y))), universal_class))) <=> ((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | (~function(choice)) | (~member(singleton(singleton(y)), universal_class)) | member(image(choice, singleton(singleton(y))), universal_class))),
% 0.55/0.61 inference(rewrite,[status(thm)],[])).
% 0.55/0.61 tff(453,plain,
% 0.55/0.61 (((~member(singleton(singleton(y)), universal_class)) | member(image(choice, singleton(singleton(y))), universal_class) | (~function(choice))) <=> ((~function(choice)) | (~member(singleton(singleton(y)), universal_class)) | member(image(choice, singleton(singleton(y))), universal_class))),
% 0.55/0.61 inference(rewrite,[status(thm)],[])).
% 0.55/0.61 tff(454,plain,
% 0.55/0.61 (((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | ((~member(singleton(singleton(y)), universal_class)) | member(image(choice, singleton(singleton(y))), universal_class) | (~function(choice)))) <=> ((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | ((~function(choice)) | (~member(singleton(singleton(y)), universal_class)) | member(image(choice, singleton(singleton(y))), universal_class)))),
% 0.55/0.61 inference(monotonicity,[status(thm)],[453])).
% 0.55/0.61 tff(455,plain,
% 0.55/0.61 (((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | ((~member(singleton(singleton(y)), universal_class)) | member(image(choice, singleton(singleton(y))), universal_class) | (~function(choice)))) <=> ((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | (~function(choice)) | (~member(singleton(singleton(y)), universal_class)) | member(image(choice, singleton(singleton(y))), universal_class))),
% 0.55/0.61 inference(transitivity,[status(thm)],[454, 452])).
% 0.55/0.61 tff(456,plain,
% 0.55/0.61 ((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | ((~member(singleton(singleton(y)), universal_class)) | member(image(choice, singleton(singleton(y))), universal_class) | (~function(choice)))),
% 0.55/0.61 inference(quant_inst,[status(thm)],[])).
% 0.55/0.61 tff(457,plain,
% 0.55/0.61 ((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | (~function(choice)) | (~member(singleton(singleton(y)), universal_class)) | member(image(choice, singleton(singleton(y))), universal_class)),
% 0.55/0.61 inference(modus_ponens,[status(thm)],[456, 455])).
% 0.55/0.61 tff(458,plain,
% 0.55/0.61 ((~member(singleton(singleton(y)), universal_class)) | member(image(choice, singleton(singleton(y))), universal_class)),
% 0.55/0.61 inference(unit_resolution,[status(thm)],[457, 451, 441])).
% 0.55/0.61 tff(459,plain,
% 0.55/0.61 (~member(singleton(singleton(y)), universal_class)),
% 0.55/0.61 inference(unit_resolution,[status(thm)],[458, 438])).
% 0.55/0.61 tff(460,plain,
% 0.55/0.61 (~member(unordered_pair(singleton(y), singleton(y)), universal_class)),
% 0.55/0.61 inference(modus_ponens,[status(thm)],[459, 397])).
% 0.55/0.61 tff(461,plain,
% 0.55/0.61 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(singleton(y), singleton(y)), universal_class)),
% 0.55/0.61 inference(quant_inst,[status(thm)],[])).
% 0.55/0.61 tff(462,plain,
% 0.55/0.61 (member(unordered_pair(singleton(y), singleton(y)), universal_class)),
% 0.55/0.61 inference(unit_resolution,[status(thm)],[461, 94])).
% 0.55/0.61 tff(463,plain,
% 0.55/0.61 ($false),
% 0.55/0.61 inference(unit_resolution,[status(thm)],[462, 460])).
% 0.55/0.61 tff(464,plain,((~member(apply(choice, singleton(y)), singleton(y))) | member(w, universal_class) | (~(singleton(w) = singleton(y)))), inference(lemma,lemma(discharge,[]))).
% 0.55/0.61 tff(465,plain,
% 0.55/0.61 (~member(apply(choice, singleton(y)), singleton(y))),
% 0.55/0.61 inference(unit_resolution,[status(thm)],[464, 386, 389])).
% 0.55/0.61 tff(466,plain,
% 0.55/0.61 (singleton(y) = null_class),
% 0.55/0.61 inference(unit_resolution,[status(thm)],[303, 465])).
% 0.55/0.61 tff(467,plain,
% 0.55/0.61 (unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))),
% 0.55/0.61 inference(symmetry,[status(thm)],[388])).
% 0.55/0.61 tff(468,plain,
% 0.55/0.61 (member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(w, singleton(x))) <=> member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, singleton(z)))),
% 0.55/0.61 inference(monotonicity,[status(thm)],[467])).
% 0.55/0.61 tff(469,plain,
% 0.55/0.61 (member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, singleton(z))) <=> member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(w, singleton(x)))),
% 0.55/0.61 inference(symmetry,[status(thm)],[468])).
% 0.55/0.61 tff(470,plain,
% 0.55/0.61 (member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(w, singleton(x)))),
% 0.55/0.61 inference(modus_ponens,[status(thm)],[219, 469])).
% 0.55/0.61 tff(471,plain,
% 0.55/0.61 ((~![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))) | (sum_class(image(choice, singleton(unordered_pair(y, singleton(z))))) = apply(choice, unordered_pair(y, singleton(z))))),
% 0.55/0.61 inference(quant_inst,[status(thm)],[])).
% 0.55/0.61 tff(472,plain,
% 0.55/0.61 (sum_class(image(choice, singleton(unordered_pair(y, singleton(z))))) = apply(choice, unordered_pair(y, singleton(z)))),
% 0.55/0.61 inference(unit_resolution,[status(thm)],[471, 404])).
% 0.55/0.61 tff(473,plain,
% 0.55/0.61 (apply(choice, unordered_pair(y, singleton(z))) = sum_class(image(choice, singleton(unordered_pair(y, singleton(z)))))),
% 0.55/0.61 inference(symmetry,[status(thm)],[472])).
% 0.55/0.61 tff(474,plain,
% 0.55/0.61 (w = apply(choice, unordered_pair(y, singleton(z)))),
% 0.55/0.61 inference(symmetry,[status(thm)],[229])).
% 0.55/0.61 tff(475,plain,
% 0.55/0.61 (w = sum_class(image(choice, singleton(unordered_pair(y, singleton(z)))))),
% 0.55/0.61 inference(transitivity,[status(thm)],[474, 473])).
% 0.55/0.61 tff(476,plain,
% 0.55/0.61 (member(w, universal_class) <=> member(sum_class(image(choice, singleton(unordered_pair(y, singleton(z))))), universal_class)),
% 0.55/0.61 inference(monotonicity,[status(thm)],[475])).
% 0.55/0.61 tff(477,plain,
% 0.55/0.61 (member(sum_class(image(choice, singleton(unordered_pair(y, singleton(z))))), universal_class) <=> member(w, universal_class)),
% 0.55/0.61 inference(symmetry,[status(thm)],[476])).
% 0.55/0.61 tff(478,plain,
% 0.55/0.61 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(unordered_pair(y, singleton(z)), unordered_pair(y, singleton(z))) = singleton(unordered_pair(y, singleton(z))))),
% 0.55/0.61 inference(quant_inst,[status(thm)],[])).
% 0.55/0.61 tff(479,plain,
% 0.55/0.61 (unordered_pair(unordered_pair(y, singleton(z)), unordered_pair(y, singleton(z))) = singleton(unordered_pair(y, singleton(z)))),
% 0.55/0.61 inference(unit_resolution,[status(thm)],[478, 7])).
% 0.55/0.61 tff(480,plain,
% 0.55/0.61 (member(unordered_pair(unordered_pair(y, singleton(z)), unordered_pair(y, singleton(z))), universal_class) <=> member(singleton(unordered_pair(y, singleton(z))), universal_class)),
% 0.55/0.61 inference(monotonicity,[status(thm)],[479])).
% 0.55/0.61 tff(481,plain,
% 0.55/0.61 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(unordered_pair(y, singleton(z)), unordered_pair(y, singleton(z))), universal_class)),
% 0.55/0.61 inference(quant_inst,[status(thm)],[])).
% 0.55/0.61 tff(482,plain,
% 0.55/0.61 (member(unordered_pair(unordered_pair(y, singleton(z)), unordered_pair(y, singleton(z))), universal_class)),
% 0.55/0.61 inference(unit_resolution,[status(thm)],[481, 94])).
% 0.55/0.61 tff(483,plain,
% 0.55/0.61 (member(singleton(unordered_pair(y, singleton(z))), universal_class)),
% 0.55/0.61 inference(modus_ponens,[status(thm)],[482, 480])).
% 0.55/0.61 tff(484,plain,
% 0.55/0.61 (((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | ((~function(choice)) | member(image(choice, singleton(unordered_pair(y, singleton(z)))), universal_class) | (~member(singleton(unordered_pair(y, singleton(z))), universal_class)))) <=> ((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | (~function(choice)) | member(image(choice, singleton(unordered_pair(y, singleton(z)))), universal_class) | (~member(singleton(unordered_pair(y, singleton(z))), universal_class)))),
% 0.55/0.61 inference(rewrite,[status(thm)],[])).
% 0.55/0.61 tff(485,plain,
% 0.55/0.61 (((~member(singleton(unordered_pair(y, singleton(z))), universal_class)) | member(image(choice, singleton(unordered_pair(y, singleton(z)))), universal_class) | (~function(choice))) <=> ((~function(choice)) | member(image(choice, singleton(unordered_pair(y, singleton(z)))), universal_class) | (~member(singleton(unordered_pair(y, singleton(z))), universal_class)))),
% 0.55/0.61 inference(rewrite,[status(thm)],[])).
% 0.55/0.61 tff(486,plain,
% 0.55/0.61 (((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | ((~member(singleton(unordered_pair(y, singleton(z))), universal_class)) | member(image(choice, singleton(unordered_pair(y, singleton(z)))), universal_class) | (~function(choice)))) <=> ((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | ((~function(choice)) | member(image(choice, singleton(unordered_pair(y, singleton(z)))), universal_class) | (~member(singleton(unordered_pair(y, singleton(z))), universal_class))))),
% 0.55/0.61 inference(monotonicity,[status(thm)],[485])).
% 0.55/0.61 tff(487,plain,
% 0.55/0.61 (((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | ((~member(singleton(unordered_pair(y, singleton(z))), universal_class)) | member(image(choice, singleton(unordered_pair(y, singleton(z)))), universal_class) | (~function(choice)))) <=> ((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | (~function(choice)) | member(image(choice, singleton(unordered_pair(y, singleton(z)))), universal_class) | (~member(singleton(unordered_pair(y, singleton(z))), universal_class)))),
% 0.55/0.61 inference(transitivity,[status(thm)],[486, 484])).
% 0.55/0.61 tff(488,plain,
% 0.55/0.61 ((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | ((~member(singleton(unordered_pair(y, singleton(z))), universal_class)) | member(image(choice, singleton(unordered_pair(y, singleton(z)))), universal_class) | (~function(choice)))),
% 0.55/0.61 inference(quant_inst,[status(thm)],[])).
% 0.55/0.61 tff(489,plain,
% 0.55/0.61 ((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | (~function(choice)) | member(image(choice, singleton(unordered_pair(y, singleton(z)))), universal_class) | (~member(singleton(unordered_pair(y, singleton(z))), universal_class))),
% 0.55/0.61 inference(modus_ponens,[status(thm)],[488, 487])).
% 0.55/0.61 tff(490,plain,
% 0.55/0.61 (member(image(choice, singleton(unordered_pair(y, singleton(z)))), universal_class) | (~member(singleton(unordered_pair(y, singleton(z))), universal_class))),
% 0.55/0.61 inference(unit_resolution,[status(thm)],[489, 451, 441])).
% 0.55/0.61 tff(491,plain,
% 0.55/0.61 (member(image(choice, singleton(unordered_pair(y, singleton(z)))), universal_class)),
% 0.55/0.61 inference(unit_resolution,[status(thm)],[490, 483])).
% 0.55/0.61 tff(492,plain,
% 0.55/0.61 (((~![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))) | ((~member(image(choice, singleton(unordered_pair(y, singleton(z)))), universal_class)) | member(sum_class(image(choice, singleton(unordered_pair(y, singleton(z))))), universal_class))) <=> ((~![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))) | (~member(image(choice, singleton(unordered_pair(y, singleton(z)))), universal_class)) | member(sum_class(image(choice, singleton(unordered_pair(y, singleton(z))))), universal_class))),
% 0.55/0.61 inference(rewrite,[status(thm)],[])).
% 0.55/0.61 tff(493,plain,
% 0.55/0.61 ((~![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))) | ((~member(image(choice, singleton(unordered_pair(y, singleton(z)))), universal_class)) | member(sum_class(image(choice, singleton(unordered_pair(y, singleton(z))))), universal_class))),
% 0.55/0.61 inference(quant_inst,[status(thm)],[])).
% 0.55/0.61 tff(494,plain,
% 0.55/0.61 ((~![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))) | (~member(image(choice, singleton(unordered_pair(y, singleton(z)))), universal_class)) | member(sum_class(image(choice, singleton(unordered_pair(y, singleton(z))))), universal_class)),
% 0.55/0.61 inference(modus_ponens,[status(thm)],[493, 492])).
% 0.55/0.61 tff(495,plain,
% 0.55/0.61 ((~member(image(choice, singleton(unordered_pair(y, singleton(z)))), universal_class)) | member(sum_class(image(choice, singleton(unordered_pair(y, singleton(z))))), universal_class)),
% 0.55/0.61 inference(unit_resolution,[status(thm)],[494, 433])).
% 0.55/0.61 tff(496,plain,
% 0.55/0.61 (member(sum_class(image(choice, singleton(unordered_pair(y, singleton(z))))), universal_class)),
% 0.55/0.61 inference(unit_resolution,[status(thm)],[495, 491])).
% 0.55/0.61 tff(497,plain,
% 0.55/0.61 (member(w, universal_class)),
% 0.55/0.61 inference(modus_ponens,[status(thm)],[496, 477])).
% 0.55/0.61 tff(498,plain,
% 0.55/0.61 ($false),
% 0.55/0.61 inference(unit_resolution,[status(thm)],[386, 497])).
% 0.55/0.61 tff(499,plain,((~(apply(choice, unordered_pair(y, singleton(z))) = w)) | member(w, universal_class)), inference(lemma,lemma(discharge,[]))).
% 0.55/0.61 tff(500,plain,
% 0.55/0.61 (~(apply(choice, unordered_pair(y, singleton(z))) = w)),
% 0.55/0.61 inference(unit_resolution,[status(thm)],[499, 386])).
% 0.55/0.61 tff(501,plain,
% 0.55/0.61 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, unordered_pair(y, singleton(z))) = singleton(x)) | (apply(choice, unordered_pair(y, singleton(z))) = w) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(w, singleton(x)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, unordered_pair(y, singleton(z))) = singleton(x)) | (apply(choice, unordered_pair(y, singleton(z))) = w) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(w, singleton(x)))))),
% 0.55/0.61 inference(rewrite,[status(thm)],[])).
% 0.55/0.61 tff(502,plain,
% 0.55/0.61 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, unordered_pair(y, singleton(z))) = singleton(x)) | (apply(choice, unordered_pair(y, singleton(z))) = w) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(w, singleton(x)))))),
% 0.55/0.61 inference(quant_inst,[status(thm)],[])).
% 0.55/0.61 tff(503,plain,
% 0.55/0.61 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, unordered_pair(y, singleton(z))) = singleton(x)) | (apply(choice, unordered_pair(y, singleton(z))) = w) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(w, singleton(x))))),
% 0.55/0.61 inference(modus_ponens,[status(thm)],[502, 501])).
% 0.55/0.61 tff(504,plain,
% 0.55/0.61 ((apply(choice, unordered_pair(y, singleton(z))) = singleton(x)) | (apply(choice, unordered_pair(y, singleton(z))) = w) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(w, singleton(x))))),
% 0.55/0.61 inference(unit_resolution,[status(thm)],[503, 49])).
% 0.55/0.61 tff(505,plain,
% 0.55/0.61 (apply(choice, unordered_pair(y, singleton(z))) = singleton(x)),
% 0.55/0.61 inference(unit_resolution,[status(thm)],[504, 500, 470])).
% 0.55/0.61 tff(506,assumption,(apply(choice, unordered_pair(y, singleton(z))) = singleton(x)), introduced(assumption)).
% 0.55/0.61 tff(507,plain,
% 0.55/0.61 (singleton(x) = apply(choice, unordered_pair(y, singleton(z)))),
% 0.55/0.61 inference(symmetry,[status(thm)],[506])).
% 0.55/0.61 tff(508,plain,
% 0.55/0.61 (unordered_pair(x, x) = apply(choice, unordered_pair(y, singleton(z)))),
% 0.55/0.61 inference(transitivity,[status(thm)],[14, 507])).
% 0.55/0.61 tff(509,plain,
% 0.55/0.61 (member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(x, x)) <=> member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), apply(choice, unordered_pair(y, singleton(z))))),
% 0.55/0.61 inference(monotonicity,[status(thm)],[508])).
% 0.55/0.61 tff(510,plain,
% 0.55/0.61 (member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), apply(choice, unordered_pair(y, singleton(z)))) <=> member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(x, x))),
% 0.55/0.61 inference(symmetry,[status(thm)],[509])).
% 0.55/0.61 tff(511,plain,
% 0.55/0.61 (apply(choice, singleton(x)) = apply(choice, apply(choice, unordered_pair(y, singleton(z))))),
% 0.55/0.61 inference(monotonicity,[status(thm)],[507])).
% 0.55/0.61 tff(512,plain,
% 0.55/0.61 (member(apply(choice, singleton(x)), singleton(x)) <=> member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), apply(choice, unordered_pair(y, singleton(z))))),
% 0.55/0.61 inference(monotonicity,[status(thm)],[511, 507])).
% 0.55/0.61 tff(513,assumption,(singleton(y) = null_class), introduced(assumption)).
% 0.55/0.61 tff(514,assumption,(~member(y, universal_class)), introduced(assumption)).
% 0.55/0.61 tff(515,assumption,(x = y), introduced(assumption)).
% 0.55/0.61 tff(516,plain,
% 0.55/0.61 (member(x, universal_class) <=> member(y, universal_class)),
% 0.55/0.61 inference(monotonicity,[status(thm)],[515])).
% 0.55/0.61 tff(517,plain,
% 0.55/0.61 (member(y, universal_class) <=> member(x, universal_class)),
% 0.55/0.61 inference(symmetry,[status(thm)],[516])).
% 0.55/0.61 tff(518,plain,
% 0.55/0.61 ((~member(y, universal_class)) <=> (~member(x, universal_class))),
% 0.55/0.61 inference(monotonicity,[status(thm)],[517])).
% 0.55/0.61 tff(519,plain,
% 0.55/0.61 (~member(x, universal_class)),
% 0.55/0.61 inference(modus_ponens,[status(thm)],[514, 518])).
% 0.55/0.61 tff(520,plain,
% 0.55/0.61 ($false),
% 0.55/0.61 inference(unit_resolution,[status(thm)],[24, 519])).
% 0.55/0.61 tff(521,plain,((~(x = y)) | member(y, universal_class)), inference(lemma,lemma(discharge,[]))).
% 0.55/0.61 tff(522,plain,
% 0.55/0.61 (~(x = y)),
% 0.55/0.61 inference(unit_resolution,[status(thm)],[521, 514])).
% 0.55/0.61 tff(523,assumption,(singleton(x) = null_class), introduced(assumption)).
% 0.55/0.61 tff(524,plain,
% 0.55/0.61 (null_class = singleton(x)),
% 0.55/0.61 inference(symmetry,[status(thm)],[523])).
% 0.55/0.61 tff(525,plain,
% 0.55/0.61 (unordered_pair(y, y) = unordered_pair(x, x)),
% 0.55/0.61 inference(transitivity,[status(thm)],[9, 513, 524, 15])).
% 0.55/0.61 tff(526,plain,
% 0.55/0.61 (member(x, unordered_pair(y, y)) <=> member(x, unordered_pair(x, x))),
% 0.55/0.61 inference(monotonicity,[status(thm)],[525])).
% 0.55/0.61 tff(527,plain,
% 0.55/0.61 (member(x, unordered_pair(x, x)) <=> member(x, unordered_pair(y, y))),
% 0.55/0.61 inference(symmetry,[status(thm)],[526])).
% 0.55/0.61 tff(528,plain,
% 0.55/0.61 (member(x, unordered_pair(y, y))),
% 0.55/0.61 inference(modus_ponens,[status(thm)],[35, 527])).
% 0.55/0.61 tff(529,assumption,(~(x = y)), introduced(assumption)).
% 0.55/0.61 tff(530,plain,
% 0.55/0.61 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((x = y) | (~member(x, unordered_pair(y, y))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (x = y) | (~member(x, unordered_pair(y, y))))),
% 0.55/0.61 inference(rewrite,[status(thm)],[])).
% 0.55/0.61 tff(531,plain,
% 0.55/0.61 (((x = y) | (x = y) | (~member(x, unordered_pair(y, y)))) <=> ((x = y) | (~member(x, unordered_pair(y, y))))),
% 0.55/0.61 inference(rewrite,[status(thm)],[])).
% 0.55/0.61 tff(532,plain,
% 0.55/0.61 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((x = y) | (x = y) | (~member(x, unordered_pair(y, y))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((x = y) | (~member(x, unordered_pair(y, y)))))),
% 0.55/0.61 inference(monotonicity,[status(thm)],[531])).
% 0.55/0.61 tff(533,plain,
% 0.55/0.61 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((x = y) | (x = y) | (~member(x, unordered_pair(y, y))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (x = y) | (~member(x, unordered_pair(y, y))))),
% 0.55/0.61 inference(transitivity,[status(thm)],[532, 530])).
% 0.55/0.61 tff(534,plain,
% 0.55/0.61 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((x = y) | (x = y) | (~member(x, unordered_pair(y, y))))),
% 0.55/0.61 inference(quant_inst,[status(thm)],[])).
% 0.55/0.61 tff(535,plain,
% 0.55/0.61 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (x = y) | (~member(x, unordered_pair(y, y)))),
% 0.55/0.61 inference(modus_ponens,[status(thm)],[534, 533])).
% 0.55/0.61 tff(536,plain,
% 0.55/0.61 (~member(x, unordered_pair(y, y))),
% 0.55/0.61 inference(unit_resolution,[status(thm)],[535, 49, 529])).
% 0.55/0.61 tff(537,plain,
% 0.55/0.61 ($false),
% 0.55/0.61 inference(unit_resolution,[status(thm)],[536, 528])).
% 0.55/0.61 tff(538,plain,((~(singleton(x) = null_class)) | (~(singleton(y) = null_class)) | (x = y)), inference(lemma,lemma(discharge,[]))).
% 0.55/0.61 tff(539,plain,
% 0.55/0.61 (~(singleton(x) = null_class)),
% 0.55/0.61 inference(unit_resolution,[status(thm)],[538, 522, 513])).
% 0.55/0.61 tff(540,plain,
% 0.55/0.61 (((~![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))) | ((~member(singleton(x), universal_class)) | (singleton(x) = null_class) | member(apply(choice, singleton(x)), singleton(x)))) <=> ((~![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))) | (~member(singleton(x), universal_class)) | (singleton(x) = null_class) | member(apply(choice, singleton(x)), singleton(x)))),
% 0.55/0.61 inference(rewrite,[status(thm)],[])).
% 0.55/0.61 tff(541,plain,
% 0.55/0.61 ((~![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))) | ((~member(singleton(x), universal_class)) | (singleton(x) = null_class) | member(apply(choice, singleton(x)), singleton(x)))),
% 0.55/0.61 inference(quant_inst,[status(thm)],[])).
% 0.55/0.61 tff(542,plain,
% 0.55/0.61 ((~![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))) | (~member(singleton(x), universal_class)) | (singleton(x) = null_class) | member(apply(choice, singleton(x)), singleton(x))),
% 0.55/0.61 inference(modus_ponens,[status(thm)],[541, 540])).
% 0.55/0.61 tff(543,plain,
% 0.55/0.61 ((singleton(x) = null_class) | member(apply(choice, singleton(x)), singleton(x))),
% 0.55/0.61 inference(unit_resolution,[status(thm)],[542, 287, 320])).
% 0.55/0.61 tff(544,plain,
% 0.55/0.61 (member(apply(choice, singleton(x)), singleton(x))),
% 0.55/0.61 inference(unit_resolution,[status(thm)],[543, 539])).
% 0.55/0.61 tff(545,plain,
% 0.55/0.61 (member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), apply(choice, unordered_pair(y, singleton(z))))),
% 0.55/0.61 inference(modus_ponens,[status(thm)],[544, 512])).
% 0.55/0.61 tff(546,plain,
% 0.55/0.61 (member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(x, x))),
% 0.55/0.61 inference(modus_ponens,[status(thm)],[545, 510])).
% 0.55/0.61 tff(547,plain,
% 0.55/0.61 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = x) | (~member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(x, x))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = x) | (~member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(x, x))))),
% 0.55/0.61 inference(rewrite,[status(thm)],[])).
% 0.55/0.61 tff(548,plain,
% 0.55/0.61 (((apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = x) | (apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = x) | (~member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(x, x)))) <=> ((apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = x) | (~member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(x, x))))),
% 0.55/0.61 inference(rewrite,[status(thm)],[])).
% 0.55/0.61 tff(549,plain,
% 0.55/0.61 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = x) | (apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = x) | (~member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(x, x))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = x) | (~member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(x, x)))))),
% 0.55/0.61 inference(monotonicity,[status(thm)],[548])).
% 0.55/0.61 tff(550,plain,
% 0.55/0.61 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = x) | (apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = x) | (~member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(x, x))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = x) | (~member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(x, x))))),
% 0.58/0.62 inference(transitivity,[status(thm)],[549, 547])).
% 0.58/0.62 tff(551,plain,
% 0.58/0.62 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = x) | (apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = x) | (~member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(x, x))))),
% 0.58/0.62 inference(quant_inst,[status(thm)],[])).
% 0.58/0.62 tff(552,plain,
% 0.58/0.62 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = x) | (~member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(x, x)))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[551, 550])).
% 0.58/0.62 tff(553,plain,
% 0.58/0.62 ((apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = x) | (~member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(x, x)))),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[552, 49])).
% 0.58/0.62 tff(554,plain,
% 0.58/0.62 (apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = x),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[553, 546])).
% 0.58/0.62 tff(555,assumption,(apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = x), introduced(assumption)).
% 0.58/0.62 tff(556,plain,
% 0.58/0.62 ((apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = z) <=> (x = z)),
% 0.58/0.62 inference(monotonicity,[status(thm)],[555])).
% 0.58/0.62 tff(557,plain,
% 0.58/0.62 ((x = z) <=> (apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = z)),
% 0.58/0.62 inference(symmetry,[status(thm)],[556])).
% 0.58/0.62 tff(558,plain,
% 0.58/0.62 ((~(x = z)) <=> (~(apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = z))),
% 0.58/0.62 inference(monotonicity,[status(thm)],[557])).
% 0.58/0.62 tff(559,plain,
% 0.58/0.62 (~(apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = z)),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[39, 558])).
% 0.58/0.62 tff(560,assumption,(apply(choice, unordered_pair(y, singleton(z))) = singleton(z)), introduced(assumption)).
% 0.58/0.62 tff(561,plain,
% 0.58/0.62 (singleton(z) = apply(choice, unordered_pair(y, singleton(z)))),
% 0.58/0.62 inference(symmetry,[status(thm)],[560])).
% 0.58/0.62 tff(562,plain,
% 0.58/0.62 (unordered_pair(z, z) = apply(choice, unordered_pair(y, singleton(z)))),
% 0.58/0.62 inference(transitivity,[status(thm)],[18, 561])).
% 0.58/0.62 tff(563,plain,
% 0.58/0.62 (member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(z, z)) <=> member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), apply(choice, unordered_pair(y, singleton(z))))),
% 0.58/0.62 inference(monotonicity,[status(thm)],[562])).
% 0.58/0.62 tff(564,plain,
% 0.58/0.62 (member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), apply(choice, unordered_pair(y, singleton(z)))) <=> member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(z, z))),
% 0.58/0.62 inference(symmetry,[status(thm)],[563])).
% 0.58/0.62 tff(565,assumption,(member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), apply(choice, unordered_pair(y, singleton(z))))), introduced(assumption)).
% 0.58/0.62 tff(566,plain,
% 0.58/0.62 (member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(z, z))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[565, 564])).
% 0.58/0.62 tff(567,plain,
% 0.58/0.62 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = z) | (~member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(z, z))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = z) | (~member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(z, z))))),
% 0.58/0.62 inference(rewrite,[status(thm)],[])).
% 0.58/0.62 tff(568,plain,
% 0.58/0.62 (((apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = z) | (apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = z) | (~member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(z, z)))) <=> ((apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = z) | (~member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(z, z))))),
% 0.58/0.62 inference(rewrite,[status(thm)],[])).
% 0.58/0.62 tff(569,plain,
% 0.58/0.62 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = z) | (apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = z) | (~member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(z, z))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = z) | (~member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(z, z)))))),
% 0.58/0.62 inference(monotonicity,[status(thm)],[568])).
% 0.58/0.62 tff(570,plain,
% 0.58/0.62 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = z) | (apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = z) | (~member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(z, z))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = z) | (~member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(z, z))))),
% 0.58/0.62 inference(transitivity,[status(thm)],[569, 567])).
% 0.58/0.62 tff(571,plain,
% 0.58/0.62 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = z) | (apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = z) | (~member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(z, z))))),
% 0.58/0.62 inference(quant_inst,[status(thm)],[])).
% 0.58/0.62 tff(572,plain,
% 0.58/0.62 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = z) | (~member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(z, z)))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[571, 570])).
% 0.58/0.62 tff(573,plain,
% 0.58/0.62 ((apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = z) | (~member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), unordered_pair(z, z)))),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[572, 49])).
% 0.58/0.62 tff(574,plain,
% 0.58/0.62 ($false),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[573, 566, 559])).
% 0.58/0.62 tff(575,plain,((~(apply(choice, unordered_pair(y, singleton(z))) = singleton(z))) | (~member(apply(choice, apply(choice, unordered_pair(y, singleton(z)))), apply(choice, unordered_pair(y, singleton(z))))) | (~(apply(choice, apply(choice, unordered_pair(y, singleton(z)))) = x))), inference(lemma,lemma(discharge,[]))).
% 0.58/0.62 tff(576,plain,
% 0.58/0.62 (~(apply(choice, unordered_pair(y, singleton(z))) = singleton(z))),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[575, 545, 554])).
% 0.58/0.62 tff(577,plain,
% 0.58/0.62 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, unordered_pair(y, singleton(z))) = singleton(z)) | (apply(choice, unordered_pair(y, singleton(z))) = y) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, singleton(z)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, unordered_pair(y, singleton(z))) = singleton(z)) | (apply(choice, unordered_pair(y, singleton(z))) = y) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, singleton(z)))))),
% 0.58/0.62 inference(rewrite,[status(thm)],[])).
% 0.58/0.62 tff(578,plain,
% 0.58/0.62 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, unordered_pair(y, singleton(z))) = singleton(z)) | (apply(choice, unordered_pair(y, singleton(z))) = y) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, singleton(z)))))),
% 0.58/0.62 inference(quant_inst,[status(thm)],[])).
% 0.58/0.62 tff(579,plain,
% 0.58/0.62 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, unordered_pair(y, singleton(z))) = singleton(z)) | (apply(choice, unordered_pair(y, singleton(z))) = y) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, singleton(z))))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[578, 577])).
% 0.58/0.62 tff(580,plain,
% 0.58/0.62 ((apply(choice, unordered_pair(y, singleton(z))) = singleton(z)) | (apply(choice, unordered_pair(y, singleton(z))) = y)),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[579, 49, 219])).
% 0.58/0.62 tff(581,plain,
% 0.58/0.62 (apply(choice, unordered_pair(y, singleton(z))) = y),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[580, 576])).
% 0.58/0.62 tff(582,plain,
% 0.58/0.62 (y = apply(choice, unordered_pair(y, singleton(z)))),
% 0.58/0.62 inference(symmetry,[status(thm)],[581])).
% 0.58/0.62 tff(583,plain,
% 0.58/0.62 (y = sum_class(image(choice, singleton(unordered_pair(y, singleton(z)))))),
% 0.58/0.62 inference(transitivity,[status(thm)],[582, 473])).
% 0.58/0.62 tff(584,plain,
% 0.58/0.62 (member(y, universal_class) <=> member(sum_class(image(choice, singleton(unordered_pair(y, singleton(z))))), universal_class)),
% 0.58/0.62 inference(monotonicity,[status(thm)],[583])).
% 0.58/0.62 tff(585,plain,
% 0.58/0.62 (member(sum_class(image(choice, singleton(unordered_pair(y, singleton(z))))), universal_class) <=> member(y, universal_class)),
% 0.58/0.62 inference(symmetry,[status(thm)],[584])).
% 0.58/0.62 tff(586,plain,
% 0.58/0.62 (member(y, universal_class)),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[496, 585])).
% 0.58/0.62 tff(587,plain,
% 0.58/0.62 ($false),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[514, 586])).
% 0.58/0.62 tff(588,plain,(member(y, universal_class) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, singleton(z)))) | (~(apply(choice, unordered_pair(y, singleton(z))) = singleton(x))) | (~(singleton(y) = null_class))), inference(lemma,lemma(discharge,[]))).
% 0.58/0.62 tff(589,plain,
% 0.58/0.62 (member(y, universal_class)),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[588, 505, 219, 466])).
% 0.58/0.62 tff(590,plain,
% 0.58/0.62 (((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(y, universal_class)) | member(y, unordered_pair(y, y)))) <=> ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(y, universal_class)) | member(y, unordered_pair(y, y)))),
% 0.58/0.62 inference(rewrite,[status(thm)],[])).
% 0.58/0.62 tff(591,plain,
% 0.58/0.62 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(y, universal_class)) | member(y, unordered_pair(y, y)))),
% 0.58/0.62 inference(quant_inst,[status(thm)],[])).
% 0.58/0.62 tff(592,plain,
% 0.58/0.62 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(y, universal_class)) | member(y, unordered_pair(y, y))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[591, 590])).
% 0.58/0.62 tff(593,plain,
% 0.58/0.62 ((~member(y, universal_class)) | member(y, unordered_pair(y, y))),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[592, 31])).
% 0.58/0.62 tff(594,plain,
% 0.58/0.62 (member(y, unordered_pair(y, y))),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[593, 589])).
% 0.58/0.62 tff(595,plain,
% 0.58/0.62 (member(y, unordered_pair(w, w))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[594, 392])).
% 0.58/0.62 tff(596,plain,
% 0.58/0.62 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((~member(y, unordered_pair(w, w))) | (y = w))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(y, unordered_pair(w, w))) | (y = w))),
% 0.58/0.62 inference(rewrite,[status(thm)],[])).
% 0.58/0.62 tff(597,plain,
% 0.58/0.62 (((y = w) | (y = w) | (~member(y, unordered_pair(w, w)))) <=> ((~member(y, unordered_pair(w, w))) | (y = w))),
% 0.58/0.62 inference(rewrite,[status(thm)],[])).
% 0.58/0.62 tff(598,plain,
% 0.58/0.62 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((y = w) | (y = w) | (~member(y, unordered_pair(w, w))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((~member(y, unordered_pair(w, w))) | (y = w)))),
% 0.58/0.62 inference(monotonicity,[status(thm)],[597])).
% 0.58/0.62 tff(599,plain,
% 0.58/0.62 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((y = w) | (y = w) | (~member(y, unordered_pair(w, w))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(y, unordered_pair(w, w))) | (y = w))),
% 0.58/0.62 inference(transitivity,[status(thm)],[598, 596])).
% 0.58/0.62 tff(600,plain,
% 0.58/0.62 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((y = w) | (y = w) | (~member(y, unordered_pair(w, w))))),
% 0.58/0.62 inference(quant_inst,[status(thm)],[])).
% 0.58/0.62 tff(601,plain,
% 0.58/0.62 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(y, unordered_pair(w, w))) | (y = w)),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[600, 599])).
% 0.58/0.62 tff(602,plain,
% 0.58/0.62 ((~member(y, unordered_pair(w, w))) | (y = w)),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[601, 49])).
% 0.58/0.62 tff(603,plain,
% 0.58/0.62 (y = w),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[602, 595])).
% 0.58/0.62 tff(604,plain,
% 0.58/0.62 (member(y, universal_class) <=> member(w, universal_class)),
% 0.58/0.62 inference(monotonicity,[status(thm)],[603])).
% 0.58/0.62 tff(605,plain,
% 0.58/0.62 (member(w, universal_class) <=> member(y, universal_class)),
% 0.58/0.62 inference(symmetry,[status(thm)],[604])).
% 0.58/0.62 tff(606,plain,
% 0.58/0.62 ((~member(w, universal_class)) <=> (~member(y, universal_class))),
% 0.58/0.62 inference(monotonicity,[status(thm)],[605])).
% 0.58/0.62 tff(607,plain,
% 0.58/0.62 (~member(y, universal_class)),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[386, 606])).
% 0.58/0.62 tff(608,plain,
% 0.58/0.62 ($false),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[589, 607])).
% 0.58/0.62 tff(609,plain,(member(w, universal_class) | (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, singleton(z))))), inference(lemma,lemma(discharge,[]))).
% 0.58/0.62 tff(610,plain,
% 0.58/0.62 (~member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, singleton(z)))),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[609, 386])).
% 0.58/0.62 tff(611,plain,
% 0.58/0.62 (unordered_pair(y, singleton(z)) = null_class),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[291, 610])).
% 0.58/0.62 tff(612,plain,
% 0.58/0.62 (null_class = unordered_pair(y, singleton(z))),
% 0.58/0.62 inference(symmetry,[status(thm)],[611])).
% 0.58/0.62 tff(613,plain,
% 0.58/0.62 (~member(apply(choice, singleton(y)), singleton(y))),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[464, 386, 359])).
% 0.58/0.62 tff(614,plain,
% 0.58/0.62 (singleton(y) = null_class),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[303, 613])).
% 0.58/0.62 tff(615,plain,
% 0.58/0.62 (unordered_pair(w, w) = unordered_pair(y, singleton(z))),
% 0.58/0.62 inference(transitivity,[status(thm)],[86, 359, 614, 612])).
% 0.58/0.62 tff(616,plain,
% 0.58/0.62 (member(singleton(z), unordered_pair(w, w)) <=> member(singleton(z), unordered_pair(y, singleton(z)))),
% 0.58/0.62 inference(monotonicity,[status(thm)],[615])).
% 0.58/0.62 tff(617,plain,
% 0.58/0.62 (member(singleton(z), unordered_pair(y, singleton(z))) <=> member(singleton(z), unordered_pair(w, w))),
% 0.58/0.62 inference(symmetry,[status(thm)],[616])).
% 0.58/0.62 tff(618,plain,
% 0.58/0.62 (member(singleton(z), unordered_pair(w, w))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[128, 617])).
% 0.58/0.62 tff(619,plain,
% 0.58/0.62 (singleton(z) = w),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[136, 618])).
% 0.58/0.62 tff(620,plain,
% 0.58/0.62 (w = singleton(z)),
% 0.58/0.62 inference(symmetry,[status(thm)],[619])).
% 0.58/0.62 tff(621,plain,
% 0.58/0.62 (member(w, universal_class) <=> member(singleton(z), universal_class)),
% 0.58/0.62 inference(monotonicity,[status(thm)],[620])).
% 0.58/0.62 tff(622,plain,
% 0.58/0.62 (member(singleton(z), universal_class) <=> member(w, universal_class)),
% 0.58/0.62 inference(symmetry,[status(thm)],[621])).
% 0.58/0.62 tff(623,plain,
% 0.58/0.62 (member(w, universal_class)),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[116, 622])).
% 0.58/0.62 tff(624,plain,
% 0.58/0.62 ($false),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[386, 623])).
% 0.58/0.62 tff(625,plain,(member(w, universal_class)), inference(lemma,lemma(discharge,[]))).
% 0.58/0.62 tff(626,plain,
% 0.58/0.62 (((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(w, universal_class)) | member(w, unordered_pair(w, w)))) <=> ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(w, universal_class)) | member(w, unordered_pair(w, w)))),
% 0.58/0.62 inference(rewrite,[status(thm)],[])).
% 0.58/0.62 tff(627,plain,
% 0.58/0.62 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(w, universal_class)) | member(w, unordered_pair(w, w)))),
% 0.58/0.62 inference(quant_inst,[status(thm)],[])).
% 0.58/0.62 tff(628,plain,
% 0.58/0.62 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(w, universal_class)) | member(w, unordered_pair(w, w))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[627, 626])).
% 0.58/0.62 tff(629,plain,
% 0.58/0.62 ((~member(w, universal_class)) | member(w, unordered_pair(w, w))),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[628, 31])).
% 0.58/0.62 tff(630,plain,
% 0.58/0.62 (member(w, unordered_pair(w, w))),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[629, 625])).
% 0.58/0.62 tff(631,plain,
% 0.58/0.62 (member(w, unordered_pair(y, y))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[630, 385])).
% 0.58/0.62 tff(632,plain,
% 0.58/0.62 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((w = y) | (~member(w, unordered_pair(y, y))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (w = y) | (~member(w, unordered_pair(y, y))))),
% 0.58/0.62 inference(rewrite,[status(thm)],[])).
% 0.58/0.62 tff(633,plain,
% 0.58/0.62 (((w = y) | (w = y) | (~member(w, unordered_pair(y, y)))) <=> ((w = y) | (~member(w, unordered_pair(y, y))))),
% 0.58/0.62 inference(rewrite,[status(thm)],[])).
% 0.58/0.62 tff(634,plain,
% 0.58/0.62 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((w = y) | (w = y) | (~member(w, unordered_pair(y, y))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((w = y) | (~member(w, unordered_pair(y, y)))))),
% 0.58/0.62 inference(monotonicity,[status(thm)],[633])).
% 0.58/0.62 tff(635,plain,
% 0.58/0.62 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((w = y) | (w = y) | (~member(w, unordered_pair(y, y))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (w = y) | (~member(w, unordered_pair(y, y))))),
% 0.58/0.62 inference(transitivity,[status(thm)],[634, 632])).
% 0.58/0.62 tff(636,plain,
% 0.58/0.62 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((w = y) | (w = y) | (~member(w, unordered_pair(y, y))))),
% 0.58/0.62 inference(quant_inst,[status(thm)],[])).
% 0.58/0.62 tff(637,plain,
% 0.58/0.62 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (w = y) | (~member(w, unordered_pair(y, y)))),
% 0.58/0.62 inference(modus_ponens,[status(thm)],[636, 635])).
% 0.58/0.62 tff(638,plain,
% 0.58/0.62 ((w = y) | (~member(w, unordered_pair(y, y)))),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[637, 49])).
% 0.58/0.62 tff(639,plain,
% 0.58/0.62 (w = y),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[638, 631])).
% 0.58/0.62 tff(640,plain,
% 0.58/0.62 (unordered_pair(w, singleton(x)) = unordered_pair(y, y)),
% 0.58/0.62 inference(monotonicity,[status(thm)],[639, 380])).
% 0.58/0.62 tff(641,plain,
% 0.58/0.62 (unordered_pair(y, y) = unordered_pair(w, singleton(x))),
% 0.58/0.62 inference(symmetry,[status(thm)],[640])).
% 0.58/0.62 tff(642,plain,
% 0.58/0.62 (singleton(y) = null_class),
% 0.58/0.62 inference(transitivity,[status(thm)],[387, 641, 366, 60])).
% 0.58/0.62 tff(643,plain,
% 0.58/0.62 ($false),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[62, 642])).
% 0.58/0.62 tff(644,plain,((~(unordered_pair(y, singleton(z)) = null_class)) | (singleton(y) = null_class)), inference(lemma,lemma(discharge,[]))).
% 0.58/0.62 tff(645,plain,
% 0.58/0.62 (~(unordered_pair(y, singleton(z)) = null_class)),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[644, 62])).
% 0.58/0.62 tff(646,plain,
% 0.58/0.62 (member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, singleton(z)))),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[291, 645])).
% 0.58/0.62 tff(647,plain,
% 0.58/0.62 (unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x))),
% 0.58/0.62 inference(unit_resolution,[status(thm)],[276, 646])).
% 0.58/0.63 tff(648,plain,
% 0.58/0.63 (unordered_pair(apply(choice, singleton(y)), singleton(z)) = unordered_pair(w, singleton(x))),
% 0.58/0.63 inference(transitivity,[status(thm)],[370, 647])).
% 0.58/0.63 tff(649,plain,
% 0.58/0.63 (member(singleton(x), unordered_pair(apply(choice, singleton(y)), singleton(z))) <=> member(singleton(x), unordered_pair(w, singleton(x)))),
% 0.58/0.63 inference(monotonicity,[status(thm)],[648])).
% 0.58/0.63 tff(650,plain,
% 0.58/0.63 (member(singleton(x), unordered_pair(w, singleton(x))) <=> member(singleton(x), unordered_pair(apply(choice, singleton(y)), singleton(z)))),
% 0.58/0.63 inference(symmetry,[status(thm)],[649])).
% 0.58/0.63 tff(651,plain,
% 0.58/0.63 (member(singleton(x), unordered_pair(apply(choice, singleton(y)), singleton(z)))),
% 0.58/0.63 inference(modus_ponens,[status(thm)],[325, 650])).
% 0.58/0.63 tff(652,plain,
% 0.58/0.63 (singleton(x) = apply(choice, singleton(y))),
% 0.58/0.63 inference(unit_resolution,[status(thm)],[378, 59, 651])).
% 0.58/0.63 tff(653,plain,
% 0.58/0.63 (apply(choice, singleton(y)) = singleton(x)),
% 0.58/0.63 inference(symmetry,[status(thm)],[652])).
% 0.58/0.63 tff(654,plain,
% 0.58/0.63 ((singleton(z) = apply(choice, singleton(y))) <=> (singleton(z) = singleton(x))),
% 0.58/0.63 inference(monotonicity,[status(thm)],[653])).
% 0.58/0.63 tff(655,plain,
% 0.58/0.63 ((singleton(z) = singleton(x)) <=> (singleton(z) = apply(choice, singleton(y)))),
% 0.58/0.63 inference(symmetry,[status(thm)],[654])).
% 0.58/0.63 tff(656,plain,
% 0.58/0.63 ((~(singleton(z) = singleton(x))) <=> (~(singleton(z) = apply(choice, singleton(y))))),
% 0.58/0.63 inference(monotonicity,[status(thm)],[655])).
% 0.58/0.63 tff(657,plain,
% 0.58/0.63 (~(singleton(z) = apply(choice, singleton(y)))),
% 0.58/0.63 inference(modus_ponens,[status(thm)],[58, 656])).
% 0.58/0.63 tff(658,plain,
% 0.58/0.63 (unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))),
% 0.58/0.63 inference(symmetry,[status(thm)],[647])).
% 0.58/0.63 tff(659,plain,
% 0.58/0.63 (y = w),
% 0.58/0.63 inference(symmetry,[status(thm)],[639])).
% 0.58/0.63 tff(660,plain,
% 0.58/0.63 (apply(choice, singleton(y)) = w),
% 0.58/0.63 inference(transitivity,[status(thm)],[369, 659])).
% 0.58/0.63 tff(661,plain,
% 0.58/0.63 (unordered_pair(apply(choice, singleton(y)), singleton(x)) = unordered_pair(w, singleton(x))),
% 0.58/0.63 inference(monotonicity,[status(thm)],[660])).
% 0.58/0.63 tff(662,plain,
% 0.58/0.63 (unordered_pair(apply(choice, singleton(y)), singleton(x)) = unordered_pair(y, singleton(z))),
% 0.58/0.63 inference(transitivity,[status(thm)],[661, 658])).
% 0.58/0.63 tff(663,plain,
% 0.58/0.63 (member(singleton(z), unordered_pair(apply(choice, singleton(y)), singleton(x))) <=> member(singleton(z), unordered_pair(y, singleton(z)))),
% 0.58/0.63 inference(monotonicity,[status(thm)],[662])).
% 0.58/0.63 tff(664,plain,
% 0.58/0.63 (member(singleton(z), unordered_pair(y, singleton(z))) <=> member(singleton(z), unordered_pair(apply(choice, singleton(y)), singleton(x)))),
% 0.58/0.63 inference(symmetry,[status(thm)],[663])).
% 0.58/0.63 tff(665,plain,
% 0.58/0.63 (member(singleton(z), unordered_pair(apply(choice, singleton(y)), singleton(x)))),
% 0.58/0.63 inference(modus_ponens,[status(thm)],[128, 664])).
% 0.58/0.63 tff(666,plain,
% 0.58/0.63 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = apply(choice, singleton(y))) | (singleton(z) = singleton(x)) | (~member(singleton(z), unordered_pair(apply(choice, singleton(y)), singleton(x)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(z) = apply(choice, singleton(y))) | (singleton(z) = singleton(x)) | (~member(singleton(z), unordered_pair(apply(choice, singleton(y)), singleton(x)))))),
% 0.58/0.63 inference(rewrite,[status(thm)],[])).
% 0.58/0.63 tff(667,plain,
% 0.58/0.63 (((singleton(z) = singleton(x)) | (singleton(z) = apply(choice, singleton(y))) | (~member(singleton(z), unordered_pair(apply(choice, singleton(y)), singleton(x))))) <=> ((singleton(z) = apply(choice, singleton(y))) | (singleton(z) = singleton(x)) | (~member(singleton(z), unordered_pair(apply(choice, singleton(y)), singleton(x)))))),
% 0.58/0.63 inference(rewrite,[status(thm)],[])).
% 0.58/0.63 tff(668,plain,
% 0.58/0.63 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = singleton(x)) | (singleton(z) = apply(choice, singleton(y))) | (~member(singleton(z), unordered_pair(apply(choice, singleton(y)), singleton(x)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = apply(choice, singleton(y))) | (singleton(z) = singleton(x)) | (~member(singleton(z), unordered_pair(apply(choice, singleton(y)), singleton(x))))))),
% 0.58/0.63 inference(monotonicity,[status(thm)],[667])).
% 0.58/0.63 tff(669,plain,
% 0.58/0.63 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = singleton(x)) | (singleton(z) = apply(choice, singleton(y))) | (~member(singleton(z), unordered_pair(apply(choice, singleton(y)), singleton(x)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(z) = apply(choice, singleton(y))) | (singleton(z) = singleton(x)) | (~member(singleton(z), unordered_pair(apply(choice, singleton(y)), singleton(x)))))),
% 0.58/0.63 inference(transitivity,[status(thm)],[668, 666])).
% 0.58/0.63 tff(670,plain,
% 0.58/0.63 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = singleton(x)) | (singleton(z) = apply(choice, singleton(y))) | (~member(singleton(z), unordered_pair(apply(choice, singleton(y)), singleton(x)))))),
% 0.58/0.63 inference(quant_inst,[status(thm)],[])).
% 0.58/0.63 tff(671,plain,
% 0.58/0.63 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(z) = apply(choice, singleton(y))) | (singleton(z) = singleton(x)) | (~member(singleton(z), unordered_pair(apply(choice, singleton(y)), singleton(x))))),
% 0.58/0.63 inference(modus_ponens,[status(thm)],[670, 669])).
% 0.58/0.63 tff(672,plain,
% 0.58/0.63 ((singleton(z) = apply(choice, singleton(y))) | (~member(singleton(z), unordered_pair(apply(choice, singleton(y)), singleton(x))))),
% 0.58/0.63 inference(unit_resolution,[status(thm)],[671, 49, 58])).
% 0.58/0.63 tff(673,plain,
% 0.58/0.63 (singleton(z) = apply(choice, singleton(y))),
% 0.58/0.63 inference(unit_resolution,[status(thm)],[672, 665])).
% 0.58/0.63 tff(674,plain,
% 0.58/0.63 ($false),
% 0.58/0.63 inference(unit_resolution,[status(thm)],[673, 657])).
% 0.58/0.63 tff(675,plain,(singleton(y) = null_class), inference(lemma,lemma(discharge,[]))).
% 0.58/0.63 tff(676,plain,
% 0.58/0.63 (unordered_pair(y, y) = unordered_pair(y, singleton(z))),
% 0.58/0.63 inference(transitivity,[status(thm)],[9, 675, 61])).
% 0.58/0.63 tff(677,plain,
% 0.58/0.63 (member(singleton(z), unordered_pair(y, y)) <=> member(singleton(z), unordered_pair(y, singleton(z)))),
% 0.58/0.63 inference(monotonicity,[status(thm)],[676])).
% 0.58/0.63 tff(678,plain,
% 0.58/0.63 (member(singleton(z), unordered_pair(y, singleton(z))) <=> member(singleton(z), unordered_pair(y, y))),
% 0.58/0.63 inference(symmetry,[status(thm)],[677])).
% 0.58/0.63 tff(679,plain,
% 0.58/0.63 (member(singleton(z), unordered_pair(y, y))),
% 0.58/0.63 inference(modus_ponens,[status(thm)],[128, 678])).
% 0.58/0.63 tff(680,plain,
% 0.58/0.63 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = y) | (~member(singleton(z), unordered_pair(y, y))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(z) = y) | (~member(singleton(z), unordered_pair(y, y))))),
% 0.58/0.63 inference(rewrite,[status(thm)],[])).
% 0.58/0.63 tff(681,plain,
% 0.58/0.63 (((singleton(z) = y) | (singleton(z) = y) | (~member(singleton(z), unordered_pair(y, y)))) <=> ((singleton(z) = y) | (~member(singleton(z), unordered_pair(y, y))))),
% 0.58/0.63 inference(rewrite,[status(thm)],[])).
% 0.58/0.63 tff(682,plain,
% 0.58/0.63 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = y) | (singleton(z) = y) | (~member(singleton(z), unordered_pair(y, y))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = y) | (~member(singleton(z), unordered_pair(y, y)))))),
% 0.58/0.63 inference(monotonicity,[status(thm)],[681])).
% 0.58/0.63 tff(683,plain,
% 0.58/0.63 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = y) | (singleton(z) = y) | (~member(singleton(z), unordered_pair(y, y))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(z) = y) | (~member(singleton(z), unordered_pair(y, y))))),
% 0.58/0.63 inference(transitivity,[status(thm)],[682, 680])).
% 0.58/0.63 tff(684,plain,
% 0.58/0.63 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = y) | (singleton(z) = y) | (~member(singleton(z), unordered_pair(y, y))))),
% 0.58/0.63 inference(quant_inst,[status(thm)],[])).
% 0.58/0.63 tff(685,plain,
% 0.58/0.63 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(z) = y) | (~member(singleton(z), unordered_pair(y, y)))),
% 0.58/0.63 inference(modus_ponens,[status(thm)],[684, 683])).
% 0.58/0.63 tff(686,plain,
% 0.58/0.63 ((singleton(z) = y) | (~member(singleton(z), unordered_pair(y, y)))),
% 0.58/0.63 inference(unit_resolution,[status(thm)],[685, 49])).
% 0.58/0.63 tff(687,plain,
% 0.58/0.63 (singleton(z) = y),
% 0.58/0.63 inference(unit_resolution,[status(thm)],[686, 679])).
% 0.58/0.63 tff(688,plain,
% 0.58/0.63 (y = singleton(z)),
% 0.58/0.63 inference(symmetry,[status(thm)],[687])).
% 0.58/0.63 tff(689,plain,
% 0.58/0.63 ((singleton(x) = y) <=> (singleton(x) = singleton(z))),
% 0.58/0.63 inference(monotonicity,[status(thm)],[688])).
% 0.58/0.63 tff(690,plain,
% 0.58/0.63 ((singleton(x) = y) <=> (singleton(z) = singleton(x))),
% 0.58/0.63 inference(transitivity,[status(thm)],[689, 10])).
% 0.58/0.63 tff(691,plain,
% 0.58/0.63 ((singleton(z) = singleton(x)) <=> (singleton(x) = y)),
% 0.58/0.63 inference(symmetry,[status(thm)],[690])).
% 0.58/0.63 tff(692,plain,
% 0.58/0.63 ((~(singleton(z) = singleton(x))) <=> (~(singleton(x) = y))),
% 0.58/0.63 inference(monotonicity,[status(thm)],[691])).
% 0.58/0.63 tff(693,plain,
% 0.58/0.63 (~(singleton(x) = y)),
% 0.58/0.63 inference(modus_ponens,[status(thm)],[58, 692])).
% 0.58/0.63 tff(694,plain,
% 0.58/0.63 ((~(singleton(y) = unordered_pair(w, singleton(x)))) <=> (~(unordered_pair(w, singleton(x)) = singleton(y)))),
% 0.58/0.63 inference(monotonicity,[status(thm)],[150])).
% 0.58/0.63 tff(695,assumption,(~(singleton(y) = unordered_pair(w, singleton(x)))), introduced(assumption)).
% 0.58/0.63 tff(696,plain,
% 0.58/0.63 (~(unordered_pair(w, singleton(x)) = singleton(y))),
% 0.58/0.63 inference(modus_ponens,[status(thm)],[695, 694])).
% 0.58/0.63 tff(697,plain,
% 0.58/0.63 ((unordered_pair(w, singleton(x)) = singleton(y)) <=> (singleton(y) = unordered_pair(w, singleton(x)))),
% 0.58/0.63 inference(symmetry,[status(thm)],[150])).
% 0.58/0.63 tff(698,plain,
% 0.58/0.63 (null_class = singleton(y)),
% 0.58/0.63 inference(symmetry,[status(thm)],[675])).
% 0.58/0.63 tff(699,plain,
% 0.58/0.63 (unordered_pair(y, singleton(z)) = singleton(y)),
% 0.58/0.63 inference(transitivity,[status(thm)],[60, 698])).
% 0.58/0.63 tff(700,plain,
% 0.58/0.63 ((unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))) <=> (unordered_pair(w, singleton(x)) = singleton(y))),
% 0.58/0.63 inference(monotonicity,[status(thm)],[699])).
% 0.58/0.63 tff(701,plain,
% 0.58/0.63 ((unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))) <=> (singleton(y) = unordered_pair(w, singleton(x)))),
% 0.58/0.63 inference(transitivity,[status(thm)],[700, 697])).
% 0.58/0.63 tff(702,assumption,(unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))), introduced(assumption)).
% 0.58/0.63 tff(703,plain,
% 0.58/0.63 (singleton(y) = unordered_pair(w, singleton(x))),
% 0.58/0.63 inference(modus_ponens,[status(thm)],[702, 701])).
% 0.58/0.63 tff(704,plain,
% 0.58/0.63 ($false),
% 0.58/0.63 inference(unit_resolution,[status(thm)],[695, 703])).
% 0.58/0.63 tff(705,plain,((~(unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z)))) | (singleton(y) = unordered_pair(w, singleton(x))) | (~(unordered_pair(y, singleton(z)) = null_class))), inference(lemma,lemma(discharge,[]))).
% 0.58/0.63 tff(706,plain,
% 0.58/0.63 (~(unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z)))),
% 0.58/0.63 inference(unit_resolution,[status(thm)],[705, 695, 60])).
% 0.58/0.63 tff(707,plain,
% 0.58/0.63 (member(unordered_pair(w, singleton(x)), unordered_pair(singleton(y), unordered_pair(y, singleton(z)))) <=> member(unordered_pair(w, singleton(x)), unordered_pair(singleton(w), unordered_pair(w, singleton(x))))),
% 0.58/0.63 inference(monotonicity,[status(thm)],[82])).
% 0.58/0.63 tff(708,plain,
% 0.58/0.63 (member(unordered_pair(w, singleton(x)), unordered_pair(singleton(w), unordered_pair(w, singleton(x)))) <=> member(unordered_pair(w, singleton(x)), unordered_pair(singleton(y), unordered_pair(y, singleton(z))))),
% 0.58/0.63 inference(symmetry,[status(thm)],[707])).
% 0.58/0.63 tff(709,plain,
% 0.58/0.63 (member(unordered_pair(w, singleton(x)), unordered_pair(singleton(y), unordered_pair(y, singleton(z))))),
% 0.58/0.63 inference(modus_ponens,[status(thm)],[264, 708])).
% 0.58/0.63 tff(710,plain,
% 0.58/0.63 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))) | (unordered_pair(w, singleton(x)) = singleton(y)) | (~member(unordered_pair(w, singleton(x)), unordered_pair(singleton(y), unordered_pair(y, singleton(z))))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))) | (unordered_pair(w, singleton(x)) = singleton(y)) | (~member(unordered_pair(w, singleton(x)), unordered_pair(singleton(y), unordered_pair(y, singleton(z))))))),
% 0.58/0.63 inference(rewrite,[status(thm)],[])).
% 0.58/0.63 tff(711,plain,
% 0.58/0.63 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))) | (unordered_pair(w, singleton(x)) = singleton(y)) | (~member(unordered_pair(w, singleton(x)), unordered_pair(singleton(y), unordered_pair(y, singleton(z))))))),
% 0.58/0.63 inference(quant_inst,[status(thm)],[])).
% 0.58/0.63 tff(712,plain,
% 0.58/0.63 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))) | (unordered_pair(w, singleton(x)) = singleton(y)) | (~member(unordered_pair(w, singleton(x)), unordered_pair(singleton(y), unordered_pair(y, singleton(z)))))),
% 0.58/0.63 inference(modus_ponens,[status(thm)],[711, 710])).
% 0.58/0.63 tff(713,plain,
% 0.58/0.63 ((unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))) | (unordered_pair(w, singleton(x)) = singleton(y)) | (~member(unordered_pair(w, singleton(x)), unordered_pair(singleton(y), unordered_pair(y, singleton(z)))))),
% 0.58/0.63 inference(unit_resolution,[status(thm)],[712, 49])).
% 0.58/0.63 tff(714,plain,
% 0.58/0.63 ((unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))) | (unordered_pair(w, singleton(x)) = singleton(y))),
% 0.58/0.63 inference(unit_resolution,[status(thm)],[713, 709])).
% 0.58/0.63 tff(715,plain,
% 0.58/0.63 (unordered_pair(w, singleton(x)) = singleton(y)),
% 0.58/0.63 inference(unit_resolution,[status(thm)],[714, 706])).
% 0.58/0.63 tff(716,plain,
% 0.58/0.63 ($false),
% 0.58/0.63 inference(unit_resolution,[status(thm)],[715, 696])).
% 0.58/0.63 tff(717,plain,((singleton(y) = unordered_pair(w, singleton(x))) | (~(unordered_pair(y, singleton(z)) = null_class))), inference(lemma,lemma(discharge,[]))).
% 0.58/0.63 tff(718,plain,
% 0.58/0.63 (singleton(y) = unordered_pair(w, singleton(x))),
% 0.58/0.63 inference(unit_resolution,[status(thm)],[717, 60])).
% 0.58/0.63 tff(719,plain,
% 0.58/0.63 (unordered_pair(y, y) = unordered_pair(w, singleton(x))),
% 0.58/0.63 inference(transitivity,[status(thm)],[9, 718])).
% 0.58/0.63 tff(720,plain,
% 0.58/0.63 (member(singleton(x), unordered_pair(y, y)) <=> member(singleton(x), unordered_pair(w, singleton(x)))),
% 0.58/0.63 inference(monotonicity,[status(thm)],[719])).
% 0.58/0.63 tff(721,plain,
% 0.58/0.63 (member(singleton(x), unordered_pair(w, singleton(x))) <=> member(singleton(x), unordered_pair(y, y))),
% 0.58/0.63 inference(symmetry,[status(thm)],[720])).
% 0.58/0.63 tff(722,plain,
% 0.58/0.63 (member(singleton(x), unordered_pair(y, y))),
% 0.58/0.63 inference(modus_ponens,[status(thm)],[325, 721])).
% 0.58/0.63 tff(723,plain,
% 0.58/0.63 ($false),
% 0.58/0.63 inference(unit_resolution,[status(thm)],[333, 722, 693])).
% 0.58/0.63 tff(724,plain,(~(unordered_pair(y, singleton(z)) = null_class)), inference(lemma,lemma(discharge,[]))).
% 0.58/0.63 tff(725,plain,
% 0.58/0.63 (member(apply(choice, unordered_pair(y, singleton(z))), unordered_pair(y, singleton(z)))),
% 0.58/0.63 inference(unit_resolution,[status(thm)],[291, 724])).
% 0.58/0.63 tff(726,plain,
% 0.58/0.63 (unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x))),
% 0.58/0.63 inference(unit_resolution,[status(thm)],[276, 725])).
% 0.58/0.63 tff(727,plain,
% 0.58/0.63 (member(singleton(x), unordered_pair(y, singleton(z))) <=> member(singleton(x), unordered_pair(w, singleton(x)))),
% 0.58/0.63 inference(monotonicity,[status(thm)],[726])).
% 0.58/0.63 tff(728,plain,
% 0.58/0.63 (member(singleton(x), unordered_pair(w, singleton(x))) <=> member(singleton(x), unordered_pair(y, singleton(z)))),
% 0.58/0.63 inference(symmetry,[status(thm)],[727])).
% 0.58/0.63 tff(729,plain,
% 0.58/0.63 (member(singleton(x), unordered_pair(y, singleton(z)))),
% 0.58/0.63 inference(modus_ponens,[status(thm)],[325, 728])).
% 0.58/0.63 tff(730,plain,
% 0.58/0.63 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(x) = singleton(z)) | (~member(singleton(x), unordered_pair(y, singleton(z)))) | (singleton(x) = y))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(x) = singleton(z)) | (~member(singleton(x), unordered_pair(y, singleton(z)))) | (singleton(x) = y))),
% 0.59/0.64 inference(rewrite,[status(thm)],[])).
% 0.59/0.64 tff(731,plain,
% 0.59/0.64 (((singleton(x) = singleton(z)) | (singleton(x) = y) | (~member(singleton(x), unordered_pair(y, singleton(z))))) <=> ((singleton(x) = singleton(z)) | (~member(singleton(x), unordered_pair(y, singleton(z)))) | (singleton(x) = y))),
% 0.59/0.64 inference(rewrite,[status(thm)],[])).
% 0.59/0.64 tff(732,plain,
% 0.59/0.64 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(x) = singleton(z)) | (singleton(x) = y) | (~member(singleton(x), unordered_pair(y, singleton(z)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(x) = singleton(z)) | (~member(singleton(x), unordered_pair(y, singleton(z)))) | (singleton(x) = y)))),
% 0.59/0.64 inference(monotonicity,[status(thm)],[731])).
% 0.59/0.64 tff(733,plain,
% 0.59/0.64 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(x) = singleton(z)) | (singleton(x) = y) | (~member(singleton(x), unordered_pair(y, singleton(z)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(x) = singleton(z)) | (~member(singleton(x), unordered_pair(y, singleton(z)))) | (singleton(x) = y))),
% 0.59/0.64 inference(transitivity,[status(thm)],[732, 730])).
% 0.59/0.64 tff(734,plain,
% 0.59/0.64 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(x) = singleton(z)) | (singleton(x) = y) | (~member(singleton(x), unordered_pair(y, singleton(z)))))),
% 0.59/0.64 inference(quant_inst,[status(thm)],[])).
% 0.59/0.64 tff(735,plain,
% 0.59/0.64 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(x) = singleton(z)) | (~member(singleton(x), unordered_pair(y, singleton(z)))) | (singleton(x) = y)),
% 0.59/0.64 inference(modus_ponens,[status(thm)],[734, 733])).
% 0.59/0.64 tff(736,plain,
% 0.59/0.64 ((singleton(x) = singleton(z)) | (~member(singleton(x), unordered_pair(y, singleton(z)))) | (singleton(x) = y)),
% 0.59/0.64 inference(unit_resolution,[status(thm)],[735, 49])).
% 0.59/0.64 tff(737,plain,
% 0.59/0.64 (singleton(x) = y),
% 0.59/0.64 inference(unit_resolution,[status(thm)],[736, 729, 59])).
% 0.59/0.64 tff(738,plain,
% 0.59/0.64 (unordered_pair(w, singleton(x)) = unordered_pair(y, y)),
% 0.59/0.64 inference(monotonicity,[status(thm)],[639, 737])).
% 0.59/0.64 tff(739,plain,
% 0.59/0.64 (unordered_pair(w, singleton(x)) = singleton(y)),
% 0.59/0.64 inference(transitivity,[status(thm)],[738, 9])).
% 0.59/0.64 tff(740,plain,
% 0.59/0.64 ((unordered_pair(y, singleton(z)) = singleton(w)) <=> (unordered_pair(w, singleton(x)) = singleton(y))),
% 0.59/0.64 inference(monotonicity,[status(thm)],[726, 359])).
% 0.59/0.64 tff(741,plain,
% 0.59/0.64 ((~(unordered_pair(y, singleton(z)) = singleton(w))) <=> (~(unordered_pair(w, singleton(x)) = singleton(y)))),
% 0.59/0.64 inference(monotonicity,[status(thm)],[740])).
% 0.59/0.64 tff(742,plain,
% 0.59/0.64 (null_class = singleton(w)),
% 0.59/0.64 inference(transitivity,[status(thm)],[698, 382])).
% 0.59/0.64 tff(743,plain,
% 0.59/0.64 ((unordered_pair(y, singleton(z)) = null_class) <=> (unordered_pair(y, singleton(z)) = singleton(w))),
% 0.59/0.64 inference(monotonicity,[status(thm)],[742])).
% 0.59/0.64 tff(744,plain,
% 0.59/0.64 ((~(unordered_pair(y, singleton(z)) = null_class)) <=> (~(unordered_pair(y, singleton(z)) = singleton(w)))),
% 0.59/0.64 inference(monotonicity,[status(thm)],[743])).
% 0.59/0.64 tff(745,plain,
% 0.59/0.64 (~(unordered_pair(y, singleton(z)) = singleton(w))),
% 0.59/0.64 inference(modus_ponens,[status(thm)],[724, 744])).
% 0.59/0.64 tff(746,plain,
% 0.59/0.64 (~(unordered_pair(w, singleton(x)) = singleton(y))),
% 0.59/0.64 inference(modus_ponens,[status(thm)],[745, 741])).
% 0.59/0.64 tff(747,plain,
% 0.59/0.64 ($false),
% 0.59/0.64 inference(unit_resolution,[status(thm)],[746, 739])).
% 0.59/0.64 % SZS output end Proof
%------------------------------------------------------------------------------