%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET018-6 : TPTP v9.0.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n005.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:24:50 AM UTC 2025
% Result : Unsatisfiable 40.14s 40.32s
% Output : Proof 40.17s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.12 % Problem : SET018-6 : TPTP v9.0.0. Bugfixed v2.1.0.
% 0.08/0.12 % Command : run_E %s %d THM
% 0.11/0.33 % Computer : n005.cluster.edu
% 0.11/0.33 % Model : x86_64 x86_64
% 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.33 % Memory : 8042.1875MB
% 0.11/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.33 % CPULimit : 300
% 0.11/0.33 % WCLimit : 300
% 0.11/0.33 % DateTime : Fri Jun 20 04:02:20 EDT 2025
% 0.11/0.33 % CPUTime :
% 40.14/40.32 % SZS status Unsatisfiable
% 40.14/40.32 % SZS output start Proof
% 40.14/40.32 tff(member_type, type, (
% 40.14/40.32 member: ( $i * $i ) > $o)).
% 40.14/40.32 tff(universal_class_type, type, (
% 40.14/40.32 universal_class: $i)).
% 40.14/40.32 tff(w_type, type, (
% 40.14/40.32 w: $i)).
% 40.14/40.32 tff(singleton_type, type, (
% 40.14/40.32 singleton: $i > $i)).
% 40.14/40.32 tff(z_type, type, (
% 40.14/40.32 z: $i)).
% 40.14/40.32 tff(unordered_pair_type, type, (
% 40.14/40.32 unordered_pair: ( $i * $i ) > $i)).
% 40.14/40.32 tff(x_type, type, (
% 40.14/40.32 x: $i)).
% 40.14/40.32 tff(y_type, type, (
% 40.14/40.32 y: $i)).
% 40.14/40.32 tff(ordered_pair_type, type, (
% 40.14/40.32 ordered_pair: ( $i * $i ) > $i)).
% 40.14/40.32 tff(sum_class_type, type, (
% 40.14/40.32 sum_class: $i > $i)).
% 40.14/40.32 tff(image_type, type, (
% 40.14/40.32 image: ( $i * $i ) > $i)).
% 40.14/40.32 tff(choice_type, type, (
% 40.14/40.32 choice: $i)).
% 40.14/40.32 tff(apply_type, type, (
% 40.14/40.32 apply: ( $i * $i ) > $i)).
% 40.14/40.32 tff(null_class_type, type, (
% 40.14/40.32 null_class: $i)).
% 40.14/40.32 tff(cross_product_type, type, (
% 40.14/40.32 cross_product: ( $i * $i ) > $i)).
% 40.14/40.32 tff(function_type, type, (
% 40.14/40.32 function: $i > $o)).
% 40.14/40.32 tff(complement_type, type, (
% 40.14/40.32 complement: $i > $i)).
% 40.14/40.32 tff(1,plain,
% 40.14/40.32 (^[X: $i] : refl((unordered_pair(X, X) = singleton(X)) <=> (unordered_pair(X, X) = singleton(X)))),
% 40.14/40.32 inference(bind,[status(th)],[])).
% 40.14/40.32 tff(2,plain,
% 40.14/40.32 (![X: $i] : (unordered_pair(X, X) = singleton(X)) <=> ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 40.14/40.32 inference(quant_intro,[status(thm)],[1])).
% 40.14/40.32 tff(3,plain,
% 40.14/40.32 (![X: $i] : (unordered_pair(X, X) = singleton(X)) <=> ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 40.14/40.32 inference(rewrite,[status(thm)],[])).
% 40.14/40.32 tff(4,axiom,(![X: $i] : (unordered_pair(X, X) = singleton(X))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','singleton_set')).
% 40.14/40.32 tff(5,plain,
% 40.14/40.32 (![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 40.14/40.32 inference(modus_ponens,[status(thm)],[4, 3])).
% 40.14/40.32 tff(6,plain,(
% 40.14/40.32 ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 40.14/40.32 inference(skolemize,[status(sab)],[5])).
% 40.14/40.32 tff(7,plain,
% 40.14/40.32 (![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 40.14/40.32 inference(modus_ponens,[status(thm)],[6, 2])).
% 40.14/40.32 tff(8,plain,
% 40.14/40.32 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(y, y) = singleton(y))),
% 40.14/40.32 inference(quant_inst,[status(thm)],[])).
% 40.14/40.32 tff(9,plain,
% 40.14/40.32 (unordered_pair(y, y) = singleton(y)),
% 40.14/40.32 inference(unit_resolution,[status(thm)],[8, 7])).
% 40.14/40.32 tff(10,plain,
% 40.14/40.32 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(w, w) = singleton(w))),
% 40.14/40.32 inference(quant_inst,[status(thm)],[])).
% 40.14/40.32 tff(11,plain,
% 40.14/40.32 (unordered_pair(w, w) = singleton(w)),
% 40.14/40.32 inference(unit_resolution,[status(thm)],[10, 7])).
% 40.14/40.32 tff(12,assumption,(~(unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z)))), introduced(assumption)).
% 40.14/40.32 tff(13,plain,
% 40.14/40.32 (^[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)))),
% 40.14/40.32 inference(bind,[status(th)],[])).
% 40.14/40.32 tff(14,plain,
% 40.14/40.32 (![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))),
% 40.14/40.32 inference(quant_intro,[status(thm)],[13])).
% 40.14/40.32 tff(15,plain,
% 40.14/40.32 (![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))),
% 40.14/40.32 inference(rewrite,[status(thm)],[])).
% 40.14/40.32 tff(16,axiom,(![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','ordered_pair')).
% 40.14/40.32 tff(17,plain,
% 40.14/40.32 (![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))),
% 40.14/40.32 inference(modus_ponens,[status(thm)],[16, 15])).
% 40.14/40.32 tff(18,plain,(
% 40.14/40.32 ![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))),
% 40.14/40.32 inference(skolemize,[status(sab)],[17])).
% 40.14/40.32 tff(19,plain,
% 40.14/40.32 (![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))),
% 40.14/40.32 inference(modus_ponens,[status(thm)],[18, 14])).
% 40.14/40.32 tff(20,plain,
% 40.14/40.32 ((~![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))),
% 40.14/40.32 inference(quant_inst,[status(thm)],[])).
% 40.14/40.32 tff(21,plain,
% 40.14/40.32 (unordered_pair(singleton(w), unordered_pair(w, singleton(x))) = ordered_pair(w, x)),
% 40.14/40.32 inference(unit_resolution,[status(thm)],[20, 19])).
% 40.14/40.32 tff(22,plain,
% 40.14/40.32 (ordered_pair(w, x) = unordered_pair(singleton(w), unordered_pair(w, singleton(x)))),
% 40.14/40.32 inference(symmetry,[status(thm)],[21])).
% 40.14/40.32 tff(23,plain,
% 40.14/40.32 ((ordered_pair(w, x) = ordered_pair(y, z)) <=> (ordered_pair(w, x) = ordered_pair(y, z))),
% 40.14/40.32 inference(rewrite,[status(thm)],[])).
% 40.14/40.32 tff(24,axiom,(ordered_pair(w, x) = ordered_pair(y, z)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_ordered_pair_determines_components2_1')).
% 40.14/40.32 tff(25,plain,
% 40.14/40.32 (ordered_pair(w, x) = ordered_pair(y, z)),
% 40.14/40.32 inference(modus_ponens,[status(thm)],[24, 23])).
% 40.14/40.32 tff(26,plain,
% 40.14/40.32 (ordered_pair(y, z) = ordered_pair(w, x)),
% 40.14/40.32 inference(symmetry,[status(thm)],[25])).
% 40.14/40.32 tff(27,plain,
% 40.14/40.32 ((~![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))),
% 40.14/40.32 inference(quant_inst,[status(thm)],[])).
% 40.14/40.32 tff(28,plain,
% 40.14/40.32 (unordered_pair(singleton(y), unordered_pair(y, singleton(z))) = ordered_pair(y, z)),
% 40.14/40.32 inference(unit_resolution,[status(thm)],[27, 19])).
% 40.14/40.32 tff(29,plain,
% 40.14/40.32 (unordered_pair(singleton(y), unordered_pair(y, singleton(z))) = unordered_pair(singleton(w), unordered_pair(w, singleton(x)))),
% 40.14/40.32 inference(transitivity,[status(thm)],[28, 26, 22])).
% 40.14/40.32 tff(30,plain,
% 40.14/40.32 (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))))),
% 40.14/40.32 inference(monotonicity,[status(thm)],[29])).
% 40.14/40.32 tff(31,plain,
% 40.14/40.32 (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))))),
% 40.14/40.32 inference(symmetry,[status(thm)],[30])).
% 40.14/40.32 tff(32,plain,
% 40.14/40.32 (^[Y: $i, X: $i] : refl(member(unordered_pair(X, Y), universal_class) <=> member(unordered_pair(X, Y), universal_class))),
% 40.14/40.32 inference(bind,[status(th)],[])).
% 40.14/40.32 tff(33,plain,
% 40.14/40.32 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class) <=> ![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 40.14/40.32 inference(quant_intro,[status(thm)],[32])).
% 40.14/40.32 tff(34,plain,
% 40.14/40.32 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class) <=> ![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 40.14/40.32 inference(rewrite,[status(thm)],[])).
% 40.14/40.32 tff(35,axiom,(![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','unordered_pairs_in_universal')).
% 40.14/40.32 tff(36,plain,
% 40.14/40.32 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 40.14/40.32 inference(modus_ponens,[status(thm)],[35, 34])).
% 40.14/40.32 tff(37,plain,(
% 40.14/40.32 ![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 40.14/40.32 inference(skolemize,[status(sab)],[36])).
% 40.14/40.32 tff(38,plain,
% 40.14/40.32 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 40.14/40.32 inference(modus_ponens,[status(thm)],[37, 33])).
% 40.14/40.32 tff(39,plain,
% 40.14/40.32 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(w, singleton(x)), universal_class)),
% 40.14/40.32 inference(quant_inst,[status(thm)],[])).
% 40.14/40.32 tff(40,plain,
% 40.14/40.32 (member(unordered_pair(w, singleton(x)), universal_class)),
% 40.14/40.32 inference(unit_resolution,[status(thm)],[39, 38])).
% 40.14/40.32 tff(41,plain,
% 40.14/40.32 (^[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))))),
% 40.14/40.32 inference(bind,[status(th)],[])).
% 40.14/40.32 tff(42,plain,
% 40.14/40.32 (![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)))),
% 40.14/40.33 inference(quant_intro,[status(thm)],[41])).
% 40.14/40.33 tff(43,plain,
% 40.14/40.33 (![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)))),
% 40.14/40.33 inference(rewrite,[status(thm)],[])).
% 40.14/40.33 tff(44,axiom,(![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','unordered_pair3')).
% 40.14/40.33 tff(45,plain,
% 40.14/40.33 (![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[44, 43])).
% 40.14/40.33 tff(46,plain,(
% 40.14/40.33 ![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))),
% 40.14/40.33 inference(skolemize,[status(sab)],[45])).
% 40.14/40.33 tff(47,plain,
% 40.14/40.33 (![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[46, 42])).
% 40.14/40.33 tff(48,plain,
% 40.14/40.33 (((~![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)))))),
% 40.14/40.33 inference(rewrite,[status(thm)],[])).
% 40.14/40.33 tff(49,plain,
% 40.14/40.33 ((~![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)))))),
% 40.14/40.33 inference(quant_inst,[status(thm)],[])).
% 40.14/40.33 tff(50,plain,
% 40.14/40.33 ((~![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))))),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[49, 48])).
% 40.14/40.33 tff(51,plain,
% 40.14/40.33 ((~member(unordered_pair(w, singleton(x)), universal_class)) | member(unordered_pair(w, singleton(x)), unordered_pair(singleton(w), unordered_pair(w, singleton(x))))),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[50, 47])).
% 40.14/40.33 tff(52,plain,
% 40.14/40.33 (member(unordered_pair(w, singleton(x)), unordered_pair(singleton(w), unordered_pair(w, singleton(x))))),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[51, 40])).
% 40.14/40.33 tff(53,plain,
% 40.14/40.33 (member(unordered_pair(w, singleton(x)), unordered_pair(singleton(y), unordered_pair(y, singleton(z))))),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[52, 31])).
% 40.14/40.33 tff(54,plain,
% 40.14/40.33 (^[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)))))),
% 40.14/40.33 inference(bind,[status(th)],[])).
% 40.14/40.33 tff(55,plain,
% 40.14/40.33 (![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))))),
% 40.14/40.33 inference(quant_intro,[status(thm)],[54])).
% 40.14/40.33 tff(56,plain,
% 40.14/40.33 (![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))))),
% 40.14/40.33 inference(rewrite,[status(thm)],[])).
% 40.14/40.33 tff(57,plain,
% 40.14/40.33 (^[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)))))),
% 40.14/40.33 inference(bind,[status(th)],[])).
% 40.14/40.33 tff(58,plain,
% 40.14/40.33 (![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))))),
% 40.14/40.33 inference(quant_intro,[status(thm)],[57])).
% 40.14/40.33 tff(59,axiom,(![Y: $i, U: $i, X: $i] : (((~member(U, unordered_pair(X, Y))) | (U = X)) | (U = Y))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','unordered_pair_member')).
% 40.14/40.33 tff(60,plain,
% 40.14/40.33 (![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[59, 58])).
% 40.14/40.33 tff(61,plain,
% 40.14/40.33 (![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[60, 56])).
% 40.14/40.33 tff(62,plain,(
% 40.14/40.33 ![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 40.14/40.33 inference(skolemize,[status(sab)],[61])).
% 40.14/40.33 tff(63,plain,
% 40.14/40.33 (![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[62, 55])).
% 40.14/40.33 tff(64,plain,
% 40.14/40.33 (((~![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))))))),
% 40.14/40.33 inference(rewrite,[status(thm)],[])).
% 40.14/40.33 tff(65,plain,
% 40.14/40.33 ((~![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))))))),
% 40.14/40.33 inference(quant_inst,[status(thm)],[])).
% 40.14/40.33 tff(66,plain,
% 40.14/40.33 ((~![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)))))),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[65, 64])).
% 40.14/40.33 tff(67,plain,
% 40.14/40.33 ((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)))))),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[66, 63])).
% 40.14/40.33 tff(68,plain,
% 40.14/40.33 ((unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))) | (unordered_pair(w, singleton(x)) = singleton(y))),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[67, 53])).
% 40.14/40.33 tff(69,plain,
% 40.14/40.33 (unordered_pair(w, singleton(x)) = singleton(y)),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[68, 12])).
% 40.14/40.33 tff(70,assumption,(unordered_pair(w, singleton(x)) = singleton(y)), introduced(assumption)).
% 40.14/40.33 tff(71,plain,
% 40.14/40.33 (singleton(y) = unordered_pair(w, singleton(x))),
% 40.14/40.33 inference(symmetry,[status(thm)],[70])).
% 40.14/40.33 tff(72,plain,
% 40.14/40.33 (unordered_pair(y, y) = unordered_pair(w, singleton(x))),
% 40.14/40.33 inference(transitivity,[status(thm)],[9, 71])).
% 40.14/40.33 tff(73,plain,
% 40.14/40.33 (member(singleton(x), unordered_pair(y, y)) <=> member(singleton(x), unordered_pair(w, singleton(x)))),
% 40.14/40.33 inference(monotonicity,[status(thm)],[72])).
% 40.14/40.33 tff(74,plain,
% 40.14/40.33 (member(singleton(x), unordered_pair(w, singleton(x))) <=> member(singleton(x), unordered_pair(y, y))),
% 40.14/40.33 inference(symmetry,[status(thm)],[73])).
% 40.14/40.33 tff(75,plain,
% 40.14/40.33 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(x, x) = singleton(x))),
% 40.14/40.33 inference(quant_inst,[status(thm)],[])).
% 40.14/40.33 tff(76,plain,
% 40.14/40.33 (unordered_pair(x, x) = singleton(x)),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[75, 7])).
% 40.14/40.33 tff(77,plain,
% 40.14/40.33 (member(unordered_pair(x, x), universal_class) <=> member(singleton(x), universal_class)),
% 40.14/40.33 inference(monotonicity,[status(thm)],[76])).
% 40.14/40.33 tff(78,plain,
% 40.14/40.33 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(x, x), universal_class)),
% 40.14/40.33 inference(quant_inst,[status(thm)],[])).
% 40.14/40.33 tff(79,plain,
% 40.14/40.33 (member(unordered_pair(x, x), universal_class)),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[78, 38])).
% 40.14/40.33 tff(80,plain,
% 40.14/40.33 (member(singleton(x), universal_class)),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[79, 77])).
% 40.14/40.33 tff(81,plain,
% 40.14/40.33 (((~![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))))),
% 40.14/40.33 inference(rewrite,[status(thm)],[])).
% 40.14/40.33 tff(82,plain,
% 40.14/40.33 ((~![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))))),
% 40.14/40.33 inference(quant_inst,[status(thm)],[])).
% 40.14/40.33 tff(83,plain,
% 40.14/40.33 ((~![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)))),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[82, 81])).
% 40.14/40.33 tff(84,plain,
% 40.14/40.33 ((~member(singleton(x), universal_class)) | member(singleton(x), unordered_pair(w, singleton(x)))),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[83, 47])).
% 40.14/40.33 tff(85,plain,
% 40.14/40.33 (member(singleton(x), unordered_pair(w, singleton(x)))),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[84, 80])).
% 40.14/40.33 tff(86,plain,
% 40.14/40.33 (member(singleton(x), unordered_pair(y, y))),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[85, 74])).
% 40.14/40.33 tff(87,assumption,(~(singleton(x) = y)), introduced(assumption)).
% 40.14/40.33 tff(88,plain,
% 40.14/40.33 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((~member(singleton(x), unordered_pair(y, y))) | (singleton(x) = y))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(singleton(x), unordered_pair(y, y))) | (singleton(x) = y))),
% 40.14/40.33 inference(rewrite,[status(thm)],[])).
% 40.14/40.33 tff(89,plain,
% 40.14/40.33 (((singleton(x) = y) | (singleton(x) = y) | (~member(singleton(x), unordered_pair(y, y)))) <=> ((~member(singleton(x), unordered_pair(y, y))) | (singleton(x) = y))),
% 40.14/40.33 inference(rewrite,[status(thm)],[])).
% 40.14/40.33 tff(90,plain,
% 40.14/40.33 (((~![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))))) | ((~member(singleton(x), unordered_pair(y, y))) | (singleton(x) = y)))),
% 40.14/40.33 inference(monotonicity,[status(thm)],[89])).
% 40.14/40.33 tff(91,plain,
% 40.14/40.33 (((~![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))))) | (~member(singleton(x), unordered_pair(y, y))) | (singleton(x) = y))),
% 40.14/40.33 inference(transitivity,[status(thm)],[90, 88])).
% 40.14/40.33 tff(92,plain,
% 40.14/40.33 ((~![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))))),
% 40.14/40.33 inference(quant_inst,[status(thm)],[])).
% 40.14/40.33 tff(93,plain,
% 40.14/40.33 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(singleton(x), unordered_pair(y, y))) | (singleton(x) = y)),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[92, 91])).
% 40.14/40.33 tff(94,plain,
% 40.14/40.33 (~member(singleton(x), unordered_pair(y, y))),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[93, 63, 87])).
% 40.14/40.33 tff(95,plain,
% 40.14/40.33 ($false),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[94, 86])).
% 40.14/40.33 tff(96,plain,((~(unordered_pair(w, singleton(x)) = singleton(y))) | (singleton(x) = y)), inference(lemma,lemma(discharge,[]))).
% 40.14/40.33 tff(97,plain,
% 40.14/40.33 (singleton(x) = y),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[96, 69])).
% 40.14/40.33 tff(98,plain,
% 40.14/40.33 (y = singleton(x)),
% 40.14/40.33 inference(symmetry,[status(thm)],[97])).
% 40.14/40.33 tff(99,plain,
% 40.14/40.33 (member(y, unordered_pair(w, w)) <=> member(singleton(x), singleton(w))),
% 40.14/40.33 inference(monotonicity,[status(thm)],[98, 11])).
% 40.14/40.33 tff(100,plain,
% 40.14/40.33 (member(singleton(x), singleton(w)) <=> member(y, unordered_pair(w, w))),
% 40.14/40.33 inference(symmetry,[status(thm)],[99])).
% 40.14/40.33 tff(101,plain,
% 40.14/40.33 ((unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x))) <=> (unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z)))),
% 40.14/40.33 inference(commutativity,[status(thm)],[])).
% 40.14/40.33 tff(102,assumption,(unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x))), introduced(assumption)).
% 40.14/40.33 tff(103,plain,
% 40.14/40.33 (unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[102, 101])).
% 40.14/40.33 tff(104,plain,
% 40.14/40.33 ($false),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[12, 103])).
% 40.14/40.33 tff(105,plain,((~(unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x)))) | (unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z)))), inference(lemma,lemma(discharge,[]))).
% 40.14/40.33 tff(106,plain,
% 40.14/40.33 (~(unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x)))),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[105, 12])).
% 40.14/40.33 tff(107,plain,
% 40.14/40.33 (ordered_pair(y, z) = unordered_pair(singleton(y), unordered_pair(y, singleton(z)))),
% 40.14/40.33 inference(symmetry,[status(thm)],[28])).
% 40.14/40.33 tff(108,plain,
% 40.14/40.33 (unordered_pair(singleton(w), unordered_pair(w, singleton(x))) = unordered_pair(singleton(y), unordered_pair(y, singleton(z)))),
% 40.14/40.33 inference(transitivity,[status(thm)],[21, 25, 107])).
% 40.14/40.33 tff(109,plain,
% 40.14/40.33 (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))))),
% 40.14/40.33 inference(monotonicity,[status(thm)],[108])).
% 40.14/40.33 tff(110,plain,
% 40.14/40.33 (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))))),
% 40.14/40.33 inference(symmetry,[status(thm)],[109])).
% 40.14/40.33 tff(111,plain,
% 40.14/40.33 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(y, singleton(z)), universal_class)),
% 40.14/40.33 inference(quant_inst,[status(thm)],[])).
% 40.14/40.33 tff(112,plain,
% 40.14/40.33 (member(unordered_pair(y, singleton(z)), universal_class)),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[111, 38])).
% 40.14/40.33 tff(113,plain,
% 40.14/40.33 (((~![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)))))),
% 40.14/40.33 inference(rewrite,[status(thm)],[])).
% 40.14/40.33 tff(114,plain,
% 40.14/40.33 ((~![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)))))),
% 40.14/40.33 inference(quant_inst,[status(thm)],[])).
% 40.14/40.33 tff(115,plain,
% 40.14/40.33 ((~![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))))),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[114, 113])).
% 40.14/40.33 tff(116,plain,
% 40.14/40.33 ((~member(unordered_pair(y, singleton(z)), universal_class)) | member(unordered_pair(y, singleton(z)), unordered_pair(singleton(y), unordered_pair(y, singleton(z))))),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[115, 47])).
% 40.14/40.33 tff(117,plain,
% 40.14/40.33 (member(unordered_pair(y, singleton(z)), unordered_pair(singleton(y), unordered_pair(y, singleton(z))))),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[116, 112])).
% 40.14/40.33 tff(118,plain,
% 40.14/40.33 (member(unordered_pair(y, singleton(z)), unordered_pair(singleton(w), unordered_pair(w, singleton(x))))),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[117, 110])).
% 40.14/40.33 tff(119,plain,
% 40.14/40.33 (((~![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))))))),
% 40.14/40.33 inference(rewrite,[status(thm)],[])).
% 40.14/40.33 tff(120,plain,
% 40.14/40.33 ((~![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))))))),
% 40.14/40.33 inference(quant_inst,[status(thm)],[])).
% 40.14/40.33 tff(121,plain,
% 40.14/40.33 ((~![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)))))),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[120, 119])).
% 40.14/40.33 tff(122,plain,
% 40.14/40.33 ((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)))))),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[121, 63])).
% 40.14/40.33 tff(123,plain,
% 40.14/40.33 ((unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x))) | (unordered_pair(y, singleton(z)) = singleton(w))),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[122, 118])).
% 40.14/40.33 tff(124,plain,
% 40.14/40.33 (unordered_pair(y, singleton(z)) = singleton(w)),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[123, 106])).
% 40.14/40.33 tff(125,plain,
% 40.14/40.33 (member(y, unordered_pair(y, singleton(z))) <=> member(singleton(x), singleton(w))),
% 40.14/40.33 inference(monotonicity,[status(thm)],[98, 124])).
% 40.14/40.33 tff(126,plain,
% 40.14/40.33 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(singleton(x), singleton(x)) = singleton(singleton(x)))),
% 40.14/40.33 inference(quant_inst,[status(thm)],[])).
% 40.14/40.33 tff(127,plain,
% 40.14/40.33 (unordered_pair(singleton(x), singleton(x)) = singleton(singleton(x))),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[126, 7])).
% 40.14/40.33 tff(128,plain,
% 40.14/40.33 (singleton(singleton(x)) = unordered_pair(singleton(x), singleton(x))),
% 40.14/40.33 inference(symmetry,[status(thm)],[127])).
% 40.14/40.33 tff(129,plain,
% 40.14/40.33 (member(w, singleton(singleton(x))) <=> member(w, unordered_pair(singleton(x), singleton(x)))),
% 40.14/40.33 inference(monotonicity,[status(thm)],[128])).
% 40.14/40.33 tff(130,plain,
% 40.14/40.33 (member(ordered_pair(y, z), cross_product(singleton(singleton(x)), universal_class)) <=> member(ordered_pair(w, x), cross_product(singleton(singleton(x)), universal_class))),
% 40.14/40.33 inference(monotonicity,[status(thm)],[26])).
% 40.14/40.33 tff(131,plain,
% 40.14/40.33 (singleton(y) = unordered_pair(y, y)),
% 40.14/40.33 inference(symmetry,[status(thm)],[9])).
% 40.14/40.33 tff(132,assumption,(singleton(y) = null_class), introduced(assumption)).
% 40.14/40.33 tff(133,plain,
% 40.14/40.33 (null_class = singleton(y)),
% 40.14/40.33 inference(symmetry,[status(thm)],[132])).
% 40.14/40.33 tff(134,assumption,(member(apply(choice, singleton(z)), singleton(z))), introduced(assumption)).
% 40.14/40.33 tff(135,plain,
% 40.14/40.33 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(singleton(z), singleton(z)) = singleton(singleton(z)))),
% 40.14/40.33 inference(quant_inst,[status(thm)],[])).
% 40.14/40.33 tff(136,plain,
% 40.14/40.33 (unordered_pair(singleton(z), singleton(z)) = singleton(singleton(z))),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[135, 7])).
% 40.14/40.33 tff(137,plain,
% 40.14/40.33 (member(unordered_pair(singleton(z), singleton(z)), universal_class) <=> member(singleton(singleton(z)), universal_class)),
% 40.14/40.33 inference(monotonicity,[status(thm)],[136])).
% 40.14/40.33 tff(138,plain,
% 40.14/40.33 (member(singleton(singleton(z)), universal_class) <=> member(unordered_pair(singleton(z), singleton(z)), universal_class)),
% 40.14/40.33 inference(symmetry,[status(thm)],[137])).
% 40.14/40.33 tff(139,plain,
% 40.14/40.33 ((~member(singleton(singleton(z)), universal_class)) <=> (~member(unordered_pair(singleton(z), singleton(z)), universal_class))),
% 40.14/40.33 inference(monotonicity,[status(thm)],[138])).
% 40.14/40.33 tff(140,plain,
% 40.14/40.33 (^[Y: $i, Xf: $i] : refl((sum_class(image(Xf, singleton(Y))) = apply(Xf, Y)) <=> (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y)))),
% 40.14/40.33 inference(bind,[status(th)],[])).
% 40.14/40.33 tff(141,plain,
% 40.14/40.33 (![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))),
% 40.14/40.33 inference(quant_intro,[status(thm)],[140])).
% 40.14/40.33 tff(142,plain,
% 40.14/40.33 (![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))),
% 40.14/40.33 inference(rewrite,[status(thm)],[])).
% 40.14/40.33 tff(143,axiom,(![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','apply')).
% 40.14/40.33 tff(144,plain,
% 40.14/40.33 (![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[143, 142])).
% 40.14/40.33 tff(145,plain,(
% 40.14/40.33 ![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))),
% 40.14/40.33 inference(skolemize,[status(sab)],[144])).
% 40.14/40.33 tff(146,plain,
% 40.14/40.33 (![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[145, 141])).
% 40.14/40.33 tff(147,plain,
% 40.14/40.33 ((~![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))) | (sum_class(image(choice, singleton(singleton(z)))) = apply(choice, singleton(z)))),
% 40.14/40.33 inference(quant_inst,[status(thm)],[])).
% 40.14/40.33 tff(148,plain,
% 40.14/40.33 (sum_class(image(choice, singleton(singleton(z)))) = apply(choice, singleton(z))),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[147, 146])).
% 40.14/40.33 tff(149,plain,
% 40.14/40.33 (apply(choice, singleton(z)) = sum_class(image(choice, singleton(singleton(z))))),
% 40.14/40.33 inference(symmetry,[status(thm)],[148])).
% 40.14/40.33 tff(150,plain,
% 40.14/40.33 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(z, z) = singleton(z))),
% 40.14/40.33 inference(quant_inst,[status(thm)],[])).
% 40.14/40.33 tff(151,plain,
% 40.14/40.33 (unordered_pair(z, z) = singleton(z)),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[150, 7])).
% 40.14/40.33 tff(152,plain,
% 40.14/40.33 (member(apply(choice, singleton(z)), unordered_pair(z, z)) <=> member(apply(choice, singleton(z)), singleton(z))),
% 40.14/40.33 inference(monotonicity,[status(thm)],[151])).
% 40.14/40.33 tff(153,plain,
% 40.14/40.33 (member(apply(choice, singleton(z)), singleton(z)) <=> member(apply(choice, singleton(z)), unordered_pair(z, z))),
% 40.14/40.33 inference(symmetry,[status(thm)],[152])).
% 40.14/40.33 tff(154,plain,
% 40.14/40.33 (member(apply(choice, singleton(z)), unordered_pair(z, z))),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[134, 153])).
% 40.14/40.33 tff(155,plain,
% 40.14/40.33 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, singleton(z)) = z) | (~member(apply(choice, singleton(z)), unordered_pair(z, z))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, singleton(z)) = z) | (~member(apply(choice, singleton(z)), unordered_pair(z, z))))),
% 40.14/40.33 inference(rewrite,[status(thm)],[])).
% 40.14/40.33 tff(156,plain,
% 40.14/40.33 (((apply(choice, singleton(z)) = z) | (apply(choice, singleton(z)) = z) | (~member(apply(choice, singleton(z)), unordered_pair(z, z)))) <=> ((apply(choice, singleton(z)) = z) | (~member(apply(choice, singleton(z)), unordered_pair(z, z))))),
% 40.14/40.33 inference(rewrite,[status(thm)],[])).
% 40.14/40.33 tff(157,plain,
% 40.14/40.33 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, singleton(z)) = z) | (apply(choice, singleton(z)) = z) | (~member(apply(choice, singleton(z)), unordered_pair(z, z))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, singleton(z)) = z) | (~member(apply(choice, singleton(z)), unordered_pair(z, z)))))),
% 40.14/40.33 inference(monotonicity,[status(thm)],[156])).
% 40.14/40.33 tff(158,plain,
% 40.14/40.33 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, singleton(z)) = z) | (apply(choice, singleton(z)) = z) | (~member(apply(choice, singleton(z)), unordered_pair(z, z))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, singleton(z)) = z) | (~member(apply(choice, singleton(z)), unordered_pair(z, z))))),
% 40.14/40.33 inference(transitivity,[status(thm)],[157, 155])).
% 40.14/40.33 tff(159,plain,
% 40.14/40.33 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, singleton(z)) = z) | (apply(choice, singleton(z)) = z) | (~member(apply(choice, singleton(z)), unordered_pair(z, z))))),
% 40.14/40.33 inference(quant_inst,[status(thm)],[])).
% 40.14/40.33 tff(160,plain,
% 40.14/40.33 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, singleton(z)) = z) | (~member(apply(choice, singleton(z)), unordered_pair(z, z)))),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[159, 158])).
% 40.14/40.33 tff(161,plain,
% 40.14/40.33 ((apply(choice, singleton(z)) = z) | (~member(apply(choice, singleton(z)), unordered_pair(z, z)))),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[160, 63])).
% 40.14/40.33 tff(162,plain,
% 40.14/40.33 (apply(choice, singleton(z)) = z),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[161, 154])).
% 40.14/40.33 tff(163,plain,
% 40.14/40.33 (z = apply(choice, singleton(z))),
% 40.14/40.33 inference(symmetry,[status(thm)],[162])).
% 40.14/40.33 tff(164,plain,
% 40.14/40.33 (z = sum_class(image(choice, singleton(singleton(z))))),
% 40.14/40.33 inference(transitivity,[status(thm)],[163, 149])).
% 40.14/40.33 tff(165,plain,
% 40.14/40.33 (member(z, universal_class) <=> member(sum_class(image(choice, singleton(singleton(z)))), universal_class)),
% 40.14/40.33 inference(monotonicity,[status(thm)],[164])).
% 40.14/40.33 tff(166,plain,
% 40.14/40.33 ((~member(z, universal_class)) <=> (~member(sum_class(image(choice, singleton(singleton(z)))), universal_class))),
% 40.14/40.33 inference(monotonicity,[status(thm)],[165])).
% 40.14/40.33 tff(167,assumption,(~member(z, universal_class)), introduced(assumption)).
% 40.14/40.33 tff(168,plain,
% 40.14/40.33 (~member(sum_class(image(choice, singleton(singleton(z)))), universal_class)),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[167, 166])).
% 40.14/40.33 tff(169,plain,
% 40.14/40.33 (^[X: $i] : refl(((~member(X, universal_class)) | member(sum_class(X), universal_class)) <=> ((~member(X, universal_class)) | member(sum_class(X), universal_class)))),
% 40.14/40.33 inference(bind,[status(th)],[])).
% 40.14/40.33 tff(170,plain,
% 40.14/40.33 (![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class)) <=> ![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))),
% 40.14/40.33 inference(quant_intro,[status(thm)],[169])).
% 40.14/40.33 tff(171,plain,
% 40.14/40.33 (![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class)) <=> ![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))),
% 40.14/40.33 inference(rewrite,[status(thm)],[])).
% 40.14/40.33 tff(172,axiom,(![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','sum_class2')).
% 40.14/40.33 tff(173,plain,
% 40.14/40.33 (![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[172, 171])).
% 40.14/40.33 tff(174,plain,(
% 40.14/40.33 ![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))),
% 40.14/40.33 inference(skolemize,[status(sab)],[173])).
% 40.14/40.33 tff(175,plain,
% 40.14/40.33 (![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[174, 170])).
% 40.14/40.33 tff(176,plain,
% 40.14/40.33 (((~![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))) | ((~member(image(choice, singleton(singleton(z))), universal_class)) | member(sum_class(image(choice, singleton(singleton(z)))), universal_class))) <=> ((~![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))) | (~member(image(choice, singleton(singleton(z))), universal_class)) | member(sum_class(image(choice, singleton(singleton(z)))), universal_class))),
% 40.14/40.33 inference(rewrite,[status(thm)],[])).
% 40.14/40.33 tff(177,plain,
% 40.14/40.33 ((~![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))) | ((~member(image(choice, singleton(singleton(z))), universal_class)) | member(sum_class(image(choice, singleton(singleton(z)))), universal_class))),
% 40.14/40.33 inference(quant_inst,[status(thm)],[])).
% 40.14/40.33 tff(178,plain,
% 40.14/40.33 ((~![X: $i] : ((~member(X, universal_class)) | member(sum_class(X), universal_class))) | (~member(image(choice, singleton(singleton(z))), universal_class)) | member(sum_class(image(choice, singleton(singleton(z)))), universal_class)),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[177, 176])).
% 40.14/40.33 tff(179,plain,
% 40.14/40.33 ((~member(image(choice, singleton(singleton(z))), universal_class)) | member(sum_class(image(choice, singleton(singleton(z)))), universal_class)),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[178, 175])).
% 40.14/40.33 tff(180,plain,
% 40.14/40.33 (~member(image(choice, singleton(singleton(z))), universal_class)),
% 40.14/40.33 inference(unit_resolution,[status(thm)],[179, 168])).
% 40.14/40.33 tff(181,plain,
% 40.14/40.33 (function(choice) <=> function(choice)),
% 40.14/40.33 inference(rewrite,[status(thm)],[])).
% 40.14/40.33 tff(182,axiom,(function(choice)), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','choice1')).
% 40.14/40.33 tff(183,plain,
% 40.14/40.33 (function(choice)),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[182, 181])).
% 40.14/40.33 tff(184,plain,
% 40.14/40.33 (^[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))))),
% 40.14/40.33 inference(bind,[status(th)],[])).
% 40.14/40.33 tff(185,plain,
% 40.14/40.33 (![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)))),
% 40.14/40.33 inference(quant_intro,[status(thm)],[184])).
% 40.14/40.33 tff(186,plain,
% 40.14/40.33 (![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)))),
% 40.14/40.33 inference(rewrite,[status(thm)],[])).
% 40.14/40.33 tff(187,plain,
% 40.14/40.33 (^[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))))),
% 40.14/40.33 inference(bind,[status(th)],[])).
% 40.14/40.33 tff(188,plain,
% 40.14/40.33 (![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)))),
% 40.14/40.33 inference(quant_intro,[status(thm)],[187])).
% 40.14/40.33 tff(189,axiom,(![Xf: $i, X: $i] : (((~function(Xf)) | (~member(X, universal_class))) | member(image(Xf, X), universal_class))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','replacement')).
% 40.14/40.33 tff(190,plain,
% 40.14/40.33 (![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[189, 188])).
% 40.14/40.33 tff(191,plain,
% 40.14/40.33 (![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[190, 186])).
% 40.14/40.33 tff(192,plain,(
% 40.14/40.33 ![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))),
% 40.14/40.33 inference(skolemize,[status(sab)],[191])).
% 40.14/40.33 tff(193,plain,
% 40.14/40.33 (![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))),
% 40.14/40.33 inference(modus_ponens,[status(thm)],[192, 185])).
% 40.14/40.33 tff(194,plain,
% 40.14/40.33 (((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | ((~function(choice)) | member(image(choice, singleton(singleton(z))), universal_class) | (~member(singleton(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(singleton(z))), universal_class) | (~member(singleton(singleton(z)), universal_class)))),
% 40.17/40.34 inference(rewrite,[status(thm)],[])).
% 40.17/40.34 tff(195,plain,
% 40.17/40.34 (((~member(singleton(singleton(z)), universal_class)) | member(image(choice, singleton(singleton(z))), universal_class) | (~function(choice))) <=> ((~function(choice)) | member(image(choice, singleton(singleton(z))), universal_class) | (~member(singleton(singleton(z)), universal_class)))),
% 40.17/40.34 inference(rewrite,[status(thm)],[])).
% 40.17/40.34 tff(196,plain,
% 40.17/40.34 (((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | ((~member(singleton(singleton(z)), universal_class)) | member(image(choice, singleton(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(singleton(z))), universal_class) | (~member(singleton(singleton(z)), universal_class))))),
% 40.17/40.34 inference(monotonicity,[status(thm)],[195])).
% 40.17/40.34 tff(197,plain,
% 40.17/40.34 (((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | ((~member(singleton(singleton(z)), universal_class)) | member(image(choice, singleton(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(singleton(z))), universal_class) | (~member(singleton(singleton(z)), universal_class)))),
% 40.17/40.34 inference(transitivity,[status(thm)],[196, 194])).
% 40.17/40.34 tff(198,plain,
% 40.17/40.34 ((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | ((~member(singleton(singleton(z)), universal_class)) | member(image(choice, singleton(singleton(z))), universal_class) | (~function(choice)))),
% 40.17/40.34 inference(quant_inst,[status(thm)],[])).
% 40.17/40.34 tff(199,plain,
% 40.17/40.34 ((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | (~function(choice)) | member(image(choice, singleton(singleton(z))), universal_class) | (~member(singleton(singleton(z)), universal_class))),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[198, 197])).
% 40.17/40.34 tff(200,plain,
% 40.17/40.34 (member(image(choice, singleton(singleton(z))), universal_class) | (~member(singleton(singleton(z)), universal_class))),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[199, 193, 183])).
% 40.17/40.34 tff(201,plain,
% 40.17/40.34 (~member(singleton(singleton(z)), universal_class)),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[200, 180])).
% 40.17/40.34 tff(202,plain,
% 40.17/40.34 (~member(unordered_pair(singleton(z), singleton(z)), universal_class)),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[201, 139])).
% 40.17/40.34 tff(203,plain,
% 40.17/40.34 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(singleton(z), singleton(z)), universal_class)),
% 40.17/40.34 inference(quant_inst,[status(thm)],[])).
% 40.17/40.34 tff(204,plain,
% 40.17/40.34 (member(unordered_pair(singleton(z), singleton(z)), universal_class)),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[203, 38])).
% 40.17/40.34 tff(205,plain,
% 40.17/40.34 ($false),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[204, 202])).
% 40.17/40.34 tff(206,plain,((~member(apply(choice, singleton(z)), singleton(z))) | member(z, universal_class)), inference(lemma,lemma(discharge,[]))).
% 40.17/40.34 tff(207,plain,
% 40.17/40.34 (member(z, universal_class)),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[206, 134])).
% 40.17/40.34 tff(208,plain,
% 40.17/40.34 (member(ordered_pair(y, z), cross_product(universal_class, universal_class)) <=> member(ordered_pair(w, x), cross_product(universal_class, universal_class))),
% 40.17/40.34 inference(monotonicity,[status(thm)],[26])).
% 40.17/40.34 tff(209,assumption,(~member(y, universal_class)), introduced(assumption)).
% 40.17/40.34 tff(210,assumption,(singleton(x) = y), introduced(assumption)).
% 40.17/40.34 tff(211,plain,
% 40.17/40.34 (y = singleton(x)),
% 40.17/40.34 inference(symmetry,[status(thm)],[210])).
% 40.17/40.34 tff(212,plain,
% 40.17/40.34 (member(y, universal_class) <=> member(singleton(x), universal_class)),
% 40.17/40.34 inference(monotonicity,[status(thm)],[211])).
% 40.17/40.34 tff(213,plain,
% 40.17/40.34 (member(singleton(x), universal_class) <=> member(y, universal_class)),
% 40.17/40.34 inference(symmetry,[status(thm)],[212])).
% 40.17/40.34 tff(214,plain,
% 40.17/40.34 (member(y, universal_class)),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[80, 213])).
% 40.17/40.34 tff(215,plain,
% 40.17/40.34 ($false),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[209, 214])).
% 40.17/40.34 tff(216,plain,((~(singleton(x) = y)) | member(y, universal_class)), inference(lemma,lemma(discharge,[]))).
% 40.17/40.34 tff(217,plain,
% 40.17/40.34 (~(singleton(x) = y)),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[216, 209])).
% 40.17/40.34 tff(218,assumption,(x = y), introduced(assumption)).
% 40.17/40.34 tff(219,plain,
% 40.17/40.34 (member(x, universal_class) <=> member(y, universal_class)),
% 40.17/40.34 inference(monotonicity,[status(thm)],[218])).
% 40.17/40.34 tff(220,plain,
% 40.17/40.34 (member(y, universal_class) <=> member(x, universal_class)),
% 40.17/40.34 inference(symmetry,[status(thm)],[219])).
% 40.17/40.34 tff(221,plain,
% 40.17/40.34 ((~member(y, universal_class)) <=> (~member(x, universal_class))),
% 40.17/40.34 inference(monotonicity,[status(thm)],[220])).
% 40.17/40.34 tff(222,plain,
% 40.17/40.34 (~member(x, universal_class)),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[209, 221])).
% 40.17/40.34 tff(223,plain,
% 40.17/40.34 (member(x, universal_class) <=> member(x, universal_class)),
% 40.17/40.34 inference(rewrite,[status(thm)],[])).
% 40.17/40.34 tff(224,axiom,(member(x, universal_class)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_ordered_pair_determines_components2_2')).
% 40.17/40.34 tff(225,plain,
% 40.17/40.34 (member(x, universal_class)),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[224, 223])).
% 40.17/40.34 tff(226,plain,
% 40.17/40.34 ($false),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[225, 222])).
% 40.17/40.34 tff(227,plain,((~(x = y)) | member(y, universal_class)), inference(lemma,lemma(discharge,[]))).
% 40.17/40.34 tff(228,plain,
% 40.17/40.34 (~(x = y)),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[227, 209])).
% 40.17/40.34 tff(229,plain,
% 40.17/40.34 (singleton(x) = unordered_pair(x, x)),
% 40.17/40.34 inference(symmetry,[status(thm)],[76])).
% 40.17/40.34 tff(230,assumption,(singleton(x) = null_class), introduced(assumption)).
% 40.17/40.34 tff(231,plain,
% 40.17/40.34 (null_class = singleton(x)),
% 40.17/40.34 inference(symmetry,[status(thm)],[230])).
% 40.17/40.34 tff(232,plain,
% 40.17/40.34 (unordered_pair(y, y) = unordered_pair(x, x)),
% 40.17/40.34 inference(transitivity,[status(thm)],[9, 132, 231, 229])).
% 40.17/40.34 tff(233,plain,
% 40.17/40.34 (member(x, unordered_pair(y, y)) <=> member(x, unordered_pair(x, x))),
% 40.17/40.34 inference(monotonicity,[status(thm)],[232])).
% 40.17/40.34 tff(234,plain,
% 40.17/40.34 (member(x, unordered_pair(x, x)) <=> member(x, unordered_pair(y, y))),
% 40.17/40.34 inference(symmetry,[status(thm)],[233])).
% 40.17/40.34 tff(235,plain,
% 40.17/40.34 (^[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))))),
% 40.17/40.34 inference(bind,[status(th)],[])).
% 40.17/40.34 tff(236,plain,
% 40.17/40.34 (![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)))),
% 40.17/40.34 inference(quant_intro,[status(thm)],[235])).
% 40.17/40.34 tff(237,plain,
% 40.17/40.34 (![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)))),
% 40.17/40.34 inference(rewrite,[status(thm)],[])).
% 40.17/40.34 tff(238,axiom,(![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','unordered_pair2')).
% 40.17/40.34 tff(239,plain,
% 40.17/40.34 (![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[238, 237])).
% 40.17/40.34 tff(240,plain,(
% 40.17/40.34 ![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 40.17/40.34 inference(skolemize,[status(sab)],[239])).
% 40.17/40.34 tff(241,plain,
% 40.17/40.34 (![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[240, 236])).
% 40.17/40.34 tff(242,plain,
% 40.17/40.34 (((~![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)))),
% 40.17/40.34 inference(rewrite,[status(thm)],[])).
% 40.17/40.34 tff(243,plain,
% 40.17/40.34 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(x, universal_class)) | member(x, unordered_pair(x, x)))),
% 40.17/40.34 inference(quant_inst,[status(thm)],[])).
% 40.17/40.34 tff(244,plain,
% 40.17/40.34 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(x, universal_class)) | member(x, unordered_pair(x, x))),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[243, 242])).
% 40.17/40.34 tff(245,plain,
% 40.17/40.34 (member(x, unordered_pair(x, x))),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[244, 241, 225])).
% 40.17/40.34 tff(246,plain,
% 40.17/40.34 (member(x, unordered_pair(y, y))),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[245, 234])).
% 40.17/40.34 tff(247,assumption,(~(x = y)), introduced(assumption)).
% 40.17/40.34 tff(248,plain,
% 40.17/40.34 (((~![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))))),
% 40.17/40.34 inference(rewrite,[status(thm)],[])).
% 40.17/40.34 tff(249,plain,
% 40.17/40.34 (((x = y) | (x = y) | (~member(x, unordered_pair(y, y)))) <=> ((x = y) | (~member(x, unordered_pair(y, y))))),
% 40.17/40.34 inference(rewrite,[status(thm)],[])).
% 40.17/40.34 tff(250,plain,
% 40.17/40.34 (((~![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)))))),
% 40.17/40.34 inference(monotonicity,[status(thm)],[249])).
% 40.17/40.34 tff(251,plain,
% 40.17/40.34 (((~![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))))),
% 40.17/40.34 inference(transitivity,[status(thm)],[250, 248])).
% 40.17/40.34 tff(252,plain,
% 40.17/40.34 ((~![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))))),
% 40.17/40.34 inference(quant_inst,[status(thm)],[])).
% 40.17/40.34 tff(253,plain,
% 40.17/40.34 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (x = y) | (~member(x, unordered_pair(y, y)))),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[252, 251])).
% 40.17/40.34 tff(254,plain,
% 40.17/40.34 (~member(x, unordered_pair(y, y))),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[253, 63, 247])).
% 40.17/40.34 tff(255,plain,
% 40.17/40.34 ($false),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[254, 246])).
% 40.17/40.34 tff(256,plain,((~(singleton(x) = null_class)) | (~(singleton(y) = null_class)) | (x = y)), inference(lemma,lemma(discharge,[]))).
% 40.17/40.34 tff(257,plain,
% 40.17/40.34 (~(singleton(x) = null_class)),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[256, 228, 132])).
% 40.17/40.34 tff(258,plain,
% 40.17/40.34 (~(unordered_pair(w, singleton(x)) = singleton(y))),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[96, 217])).
% 40.17/40.34 tff(259,plain,
% 40.17/40.34 (unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[68, 258])).
% 40.17/40.34 tff(260,assumption,(singleton(z) = null_class), introduced(assumption)).
% 40.17/40.34 tff(261,plain,
% 40.17/40.34 ((singleton(x) = singleton(z)) <=> (singleton(x) = null_class)),
% 40.17/40.34 inference(monotonicity,[status(thm)],[260])).
% 40.17/40.34 tff(262,plain,
% 40.17/40.34 ((singleton(x) = null_class) <=> (singleton(x) = singleton(z))),
% 40.17/40.34 inference(symmetry,[status(thm)],[261])).
% 40.17/40.34 tff(263,plain,
% 40.17/40.34 ((~(singleton(x) = null_class)) <=> (~(singleton(x) = singleton(z)))),
% 40.17/40.34 inference(monotonicity,[status(thm)],[262])).
% 40.17/40.34 tff(264,assumption,(~(singleton(x) = null_class)), introduced(assumption)).
% 40.17/40.34 tff(265,plain,
% 40.17/40.34 (~(singleton(x) = singleton(z))),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[264, 263])).
% 40.17/40.34 tff(266,assumption,(unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))), introduced(assumption)).
% 40.17/40.34 tff(267,plain,
% 40.17/40.34 (unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x))),
% 40.17/40.34 inference(symmetry,[status(thm)],[266])).
% 40.17/40.34 tff(268,plain,
% 40.17/40.34 (member(singleton(x), unordered_pair(y, singleton(z))) <=> member(singleton(x), unordered_pair(w, singleton(x)))),
% 40.17/40.34 inference(monotonicity,[status(thm)],[267])).
% 40.17/40.34 tff(269,plain,
% 40.17/40.34 (member(singleton(x), unordered_pair(w, singleton(x))) <=> member(singleton(x), unordered_pair(y, singleton(z)))),
% 40.17/40.34 inference(symmetry,[status(thm)],[268])).
% 40.17/40.34 tff(270,plain,
% 40.17/40.34 (member(singleton(x), unordered_pair(y, singleton(z)))),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[85, 269])).
% 40.17/40.34 tff(271,plain,
% 40.17/40.34 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(x) = y) | (singleton(x) = singleton(z)) | (~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) = y) | (singleton(x) = singleton(z)) | (~member(singleton(x), unordered_pair(y, singleton(z)))))),
% 40.17/40.34 inference(rewrite,[status(thm)],[])).
% 40.17/40.34 tff(272,plain,
% 40.17/40.34 (((singleton(x) = singleton(z)) | (singleton(x) = y) | (~member(singleton(x), unordered_pair(y, singleton(z))))) <=> ((singleton(x) = y) | (singleton(x) = singleton(z)) | (~member(singleton(x), unordered_pair(y, singleton(z)))))),
% 40.17/40.34 inference(rewrite,[status(thm)],[])).
% 40.17/40.34 tff(273,plain,
% 40.17/40.34 (((~![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) = y) | (singleton(x) = singleton(z)) | (~member(singleton(x), unordered_pair(y, singleton(z))))))),
% 40.17/40.34 inference(monotonicity,[status(thm)],[272])).
% 40.17/40.34 tff(274,plain,
% 40.17/40.34 (((~![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) = y) | (singleton(x) = singleton(z)) | (~member(singleton(x), unordered_pair(y, singleton(z)))))),
% 40.17/40.34 inference(transitivity,[status(thm)],[273, 271])).
% 40.17/40.34 tff(275,plain,
% 40.17/40.34 ((~![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)))))),
% 40.17/40.34 inference(quant_inst,[status(thm)],[])).
% 40.17/40.34 tff(276,plain,
% 40.17/40.34 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(x) = y) | (singleton(x) = singleton(z)) | (~member(singleton(x), unordered_pair(y, singleton(z))))),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[275, 274])).
% 40.17/40.34 tff(277,plain,
% 40.17/40.34 ((singleton(x) = singleton(z)) | (~member(singleton(x), unordered_pair(y, singleton(z))))),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[276, 63, 87])).
% 40.17/40.34 tff(278,plain,
% 40.17/40.34 ($false),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[277, 270, 265])).
% 40.17/40.34 tff(279,plain,((~(unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z)))) | (singleton(x) = y) | (singleton(x) = null_class) | (~(singleton(z) = null_class))), inference(lemma,lemma(discharge,[]))).
% 40.17/40.34 tff(280,plain,
% 40.17/40.34 (~(singleton(z) = null_class)),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[279, 259, 257, 217])).
% 40.17/40.34 tff(281,plain,
% 40.17/40.34 (member(unordered_pair(z, z), universal_class) <=> member(singleton(z), universal_class)),
% 40.17/40.34 inference(monotonicity,[status(thm)],[151])).
% 40.17/40.34 tff(282,plain,
% 40.17/40.34 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(z, z), universal_class)),
% 40.17/40.34 inference(quant_inst,[status(thm)],[])).
% 40.17/40.34 tff(283,plain,
% 40.17/40.34 (member(unordered_pair(z, z), universal_class)),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[282, 38])).
% 40.17/40.34 tff(284,plain,
% 40.17/40.34 (member(singleton(z), universal_class)),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[283, 281])).
% 40.17/40.34 tff(285,plain,
% 40.17/40.34 (^[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)))),
% 40.17/40.34 inference(bind,[status(th)],[])).
% 40.17/40.34 tff(286,plain,
% 40.17/40.34 (![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))),
% 40.17/40.34 inference(quant_intro,[status(thm)],[285])).
% 40.17/40.34 tff(287,plain,
% 40.17/40.34 (![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))),
% 40.17/40.34 inference(rewrite,[status(thm)],[])).
% 40.17/40.34 tff(288,plain,
% 40.17/40.34 (^[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)))),
% 40.17/40.34 inference(bind,[status(th)],[])).
% 40.17/40.34 tff(289,plain,
% 40.17/40.34 (![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))),
% 40.17/40.34 inference(quant_intro,[status(thm)],[288])).
% 40.17/40.34 tff(290,axiom,(![Y: $i] : (((~member(Y, universal_class)) | (Y = null_class)) | member(apply(choice, Y), Y))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','choice2')).
% 40.17/40.34 tff(291,plain,
% 40.17/40.34 (![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[290, 289])).
% 40.17/40.34 tff(292,plain,
% 40.17/40.34 (![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[291, 287])).
% 40.17/40.34 tff(293,plain,(
% 40.17/40.34 ![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))),
% 40.17/40.34 inference(skolemize,[status(sab)],[292])).
% 40.17/40.34 tff(294,plain,
% 40.17/40.34 (![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[293, 286])).
% 40.17/40.34 tff(295,plain,
% 40.17/40.34 (((~![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))) | ((~member(singleton(z), universal_class)) | (singleton(z) = null_class) | member(apply(choice, singleton(z)), singleton(z)))) <=> ((~![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))) | (~member(singleton(z), universal_class)) | (singleton(z) = null_class) | member(apply(choice, singleton(z)), singleton(z)))),
% 40.17/40.34 inference(rewrite,[status(thm)],[])).
% 40.17/40.34 tff(296,plain,
% 40.17/40.34 ((~![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))) | ((~member(singleton(z), universal_class)) | (singleton(z) = null_class) | member(apply(choice, singleton(z)), singleton(z)))),
% 40.17/40.34 inference(quant_inst,[status(thm)],[])).
% 40.17/40.34 tff(297,plain,
% 40.17/40.34 ((~![Y: $i] : ((~member(Y, universal_class)) | (Y = null_class) | member(apply(choice, Y), Y))) | (~member(singleton(z), universal_class)) | (singleton(z) = null_class) | member(apply(choice, singleton(z)), singleton(z))),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[296, 295])).
% 40.17/40.34 tff(298,plain,
% 40.17/40.34 ((singleton(z) = null_class) | member(apply(choice, singleton(z)), singleton(z))),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[297, 294, 284])).
% 40.17/40.34 tff(299,plain,
% 40.17/40.34 (member(apply(choice, singleton(z)), singleton(z))),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[298, 280])).
% 40.17/40.34 tff(300,assumption,(~(apply(choice, singleton(z)) = z)), introduced(assumption)).
% 40.17/40.34 tff(301,plain,
% 40.17/40.34 (~member(apply(choice, singleton(z)), unordered_pair(z, z))),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[160, 63, 300])).
% 40.17/40.34 tff(302,plain,
% 40.17/40.34 ($false),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[301, 154])).
% 40.17/40.34 tff(303,plain,((~member(apply(choice, singleton(z)), singleton(z))) | (apply(choice, singleton(z)) = z)), inference(lemma,lemma(discharge,[]))).
% 40.17/40.34 tff(304,plain,
% 40.17/40.34 (apply(choice, singleton(z)) = z),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[303, 299])).
% 40.17/40.34 tff(305,plain,
% 40.17/40.34 (unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x))),
% 40.17/40.34 inference(symmetry,[status(thm)],[259])).
% 40.17/40.34 tff(306,plain,
% 40.17/40.34 (member(singleton(x), unordered_pair(y, singleton(z))) <=> member(singleton(x), unordered_pair(w, singleton(x)))),
% 40.17/40.34 inference(monotonicity,[status(thm)],[305])).
% 40.17/40.34 tff(307,plain,
% 40.17/40.34 (member(singleton(x), unordered_pair(w, singleton(x))) <=> member(singleton(x), unordered_pair(y, singleton(z)))),
% 40.17/40.34 inference(symmetry,[status(thm)],[306])).
% 40.17/40.34 tff(308,plain,
% 40.17/40.34 (member(singleton(x), unordered_pair(y, singleton(z)))),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[85, 307])).
% 40.17/40.34 tff(309,plain,
% 40.17/40.34 ((singleton(x) = singleton(z)) | (~member(singleton(x), unordered_pair(y, singleton(z))))),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[276, 63, 217])).
% 40.17/40.34 tff(310,plain,
% 40.17/40.34 (singleton(x) = singleton(z)),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[309, 308])).
% 40.17/40.34 tff(311,plain,
% 40.17/40.34 (singleton(z) = singleton(x)),
% 40.17/40.34 inference(symmetry,[status(thm)],[310])).
% 40.17/40.34 tff(312,plain,
% 40.17/40.34 (apply(choice, singleton(z)) = apply(choice, singleton(x))),
% 40.17/40.34 inference(monotonicity,[status(thm)],[311])).
% 40.17/40.34 tff(313,plain,
% 40.17/40.34 (apply(choice, singleton(x)) = apply(choice, singleton(z))),
% 40.17/40.34 inference(symmetry,[status(thm)],[312])).
% 40.17/40.34 tff(314,plain,
% 40.17/40.34 (member(apply(choice, singleton(x)), unordered_pair(x, x)) <=> member(apply(choice, singleton(x)), singleton(x))),
% 40.17/40.34 inference(monotonicity,[status(thm)],[76])).
% 40.17/40.34 tff(315,plain,
% 40.17/40.34 (member(apply(choice, singleton(x)), singleton(x)) <=> member(apply(choice, singleton(x)), unordered_pair(x, x))),
% 40.17/40.34 inference(symmetry,[status(thm)],[314])).
% 40.17/40.34 tff(316,plain,
% 40.17/40.34 (((~![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)))),
% 40.17/40.34 inference(rewrite,[status(thm)],[])).
% 40.17/40.34 tff(317,plain,
% 40.17/40.34 ((~![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)))),
% 40.17/40.34 inference(quant_inst,[status(thm)],[])).
% 40.17/40.34 tff(318,plain,
% 40.17/40.34 ((~![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))),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[317, 316])).
% 40.17/40.34 tff(319,plain,
% 40.17/40.34 ((singleton(x) = null_class) | member(apply(choice, singleton(x)), singleton(x))),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[318, 294, 80])).
% 40.17/40.34 tff(320,plain,
% 40.17/40.34 (member(apply(choice, singleton(x)), singleton(x))),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[319, 257])).
% 40.17/40.34 tff(321,plain,
% 40.17/40.34 (member(apply(choice, singleton(x)), unordered_pair(x, x))),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[320, 315])).
% 40.17/40.34 tff(322,plain,
% 40.17/40.34 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((~member(apply(choice, singleton(x)), unordered_pair(x, x))) | (apply(choice, singleton(x)) = x))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(apply(choice, singleton(x)), unordered_pair(x, x))) | (apply(choice, singleton(x)) = x))),
% 40.17/40.34 inference(rewrite,[status(thm)],[])).
% 40.17/40.34 tff(323,plain,
% 40.17/40.34 (((apply(choice, singleton(x)) = x) | (apply(choice, singleton(x)) = x) | (~member(apply(choice, singleton(x)), unordered_pair(x, x)))) <=> ((~member(apply(choice, singleton(x)), unordered_pair(x, x))) | (apply(choice, singleton(x)) = x))),
% 40.17/40.34 inference(rewrite,[status(thm)],[])).
% 40.17/40.34 tff(324,plain,
% 40.17/40.34 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, singleton(x)) = x) | (apply(choice, singleton(x)) = x) | (~member(apply(choice, singleton(x)), unordered_pair(x, x))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((~member(apply(choice, singleton(x)), unordered_pair(x, x))) | (apply(choice, singleton(x)) = x)))),
% 40.17/40.34 inference(monotonicity,[status(thm)],[323])).
% 40.17/40.34 tff(325,plain,
% 40.17/40.34 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, singleton(x)) = x) | (apply(choice, singleton(x)) = x) | (~member(apply(choice, singleton(x)), unordered_pair(x, x))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(apply(choice, singleton(x)), unordered_pair(x, x))) | (apply(choice, singleton(x)) = x))),
% 40.17/40.34 inference(transitivity,[status(thm)],[324, 322])).
% 40.17/40.34 tff(326,plain,
% 40.17/40.34 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((apply(choice, singleton(x)) = x) | (apply(choice, singleton(x)) = x) | (~member(apply(choice, singleton(x)), unordered_pair(x, x))))),
% 40.17/40.34 inference(quant_inst,[status(thm)],[])).
% 40.17/40.34 tff(327,plain,
% 40.17/40.34 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(apply(choice, singleton(x)), unordered_pair(x, x))) | (apply(choice, singleton(x)) = x)),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[326, 325])).
% 40.17/40.34 tff(328,plain,
% 40.17/40.34 ((~member(apply(choice, singleton(x)), unordered_pair(x, x))) | (apply(choice, singleton(x)) = x)),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[327, 63])).
% 40.17/40.34 tff(329,plain,
% 40.17/40.34 (apply(choice, singleton(x)) = x),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[328, 321])).
% 40.17/40.34 tff(330,plain,
% 40.17/40.34 (x = apply(choice, singleton(x))),
% 40.17/40.34 inference(symmetry,[status(thm)],[329])).
% 40.17/40.34 tff(331,plain,
% 40.17/40.34 (x = z),
% 40.17/40.34 inference(transitivity,[status(thm)],[330, 313, 304])).
% 40.17/40.34 tff(332,plain,
% 40.17/40.34 ((~(x = z)) <=> (~(x = z))),
% 40.17/40.34 inference(rewrite,[status(thm)],[])).
% 40.17/40.34 tff(333,axiom,(~(x = z)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_ordered_pair_determines_components2_3')).
% 40.17/40.34 tff(334,plain,
% 40.17/40.34 (~(x = z)),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[333, 332])).
% 40.17/40.34 tff(335,plain,
% 40.17/40.34 ($false),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[334, 331])).
% 40.17/40.34 tff(336,plain,(member(y, universal_class) | (~(singleton(y) = null_class))), inference(lemma,lemma(discharge,[]))).
% 40.17/40.34 tff(337,plain,
% 40.17/40.34 (member(y, universal_class)),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[336, 132])).
% 40.17/40.34 tff(338,plain,
% 40.17/40.34 (^[V: $i, Y: $i, U: $i, X: $i] : refl(((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y))) <=> ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y))))),
% 40.17/40.34 inference(bind,[status(th)],[])).
% 40.17/40.34 tff(339,plain,
% 40.17/40.34 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y))) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))),
% 40.17/40.34 inference(quant_intro,[status(thm)],[338])).
% 40.17/40.34 tff(340,plain,
% 40.17/40.34 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y))) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))),
% 40.17/40.34 inference(rewrite,[status(thm)],[])).
% 40.17/40.34 tff(341,plain,
% 40.17/40.34 (^[V: $i, Y: $i, U: $i, X: $i] : rewrite((((~member(U, X)) | (~member(V, Y))) | member(ordered_pair(U, V), cross_product(X, Y))) <=> ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y))))),
% 40.17/40.34 inference(bind,[status(th)],[])).
% 40.17/40.34 tff(342,plain,
% 40.17/40.34 (![V: $i, Y: $i, U: $i, X: $i] : (((~member(U, X)) | (~member(V, Y))) | member(ordered_pair(U, V), cross_product(X, Y))) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))),
% 40.17/40.34 inference(quant_intro,[status(thm)],[341])).
% 40.17/40.34 tff(343,axiom,(![V: $i, Y: $i, U: $i, X: $i] : (((~member(U, X)) | (~member(V, Y))) | member(ordered_pair(U, V), cross_product(X, Y)))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','cartesian_product3')).
% 40.17/40.34 tff(344,plain,
% 40.17/40.34 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[343, 342])).
% 40.17/40.34 tff(345,plain,
% 40.17/40.34 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[344, 340])).
% 40.17/40.34 tff(346,plain,(
% 40.17/40.34 ![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))),
% 40.17/40.34 inference(skolemize,[status(sab)],[345])).
% 40.17/40.34 tff(347,plain,
% 40.17/40.34 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[346, 339])).
% 40.17/40.34 tff(348,plain,
% 40.17/40.34 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | (member(ordered_pair(y, z), cross_product(universal_class, universal_class)) | (~member(y, universal_class)) | (~member(z, universal_class)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | member(ordered_pair(y, z), cross_product(universal_class, universal_class)) | (~member(y, universal_class)) | (~member(z, universal_class)))),
% 40.17/40.34 inference(rewrite,[status(thm)],[])).
% 40.17/40.34 tff(349,plain,
% 40.17/40.34 (((~member(y, universal_class)) | member(ordered_pair(y, z), cross_product(universal_class, universal_class)) | (~member(z, universal_class))) <=> (member(ordered_pair(y, z), cross_product(universal_class, universal_class)) | (~member(y, universal_class)) | (~member(z, universal_class)))),
% 40.17/40.34 inference(rewrite,[status(thm)],[])).
% 40.17/40.34 tff(350,plain,
% 40.17/40.34 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | ((~member(y, universal_class)) | member(ordered_pair(y, z), cross_product(universal_class, universal_class)) | (~member(z, universal_class)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | (member(ordered_pair(y, z), cross_product(universal_class, universal_class)) | (~member(y, universal_class)) | (~member(z, universal_class))))),
% 40.17/40.34 inference(monotonicity,[status(thm)],[349])).
% 40.17/40.34 tff(351,plain,
% 40.17/40.34 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | ((~member(y, universal_class)) | member(ordered_pair(y, z), cross_product(universal_class, universal_class)) | (~member(z, universal_class)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | member(ordered_pair(y, z), cross_product(universal_class, universal_class)) | (~member(y, universal_class)) | (~member(z, universal_class)))),
% 40.17/40.34 inference(transitivity,[status(thm)],[350, 348])).
% 40.17/40.34 tff(352,plain,
% 40.17/40.34 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | ((~member(y, universal_class)) | member(ordered_pair(y, z), cross_product(universal_class, universal_class)) | (~member(z, universal_class)))),
% 40.17/40.34 inference(quant_inst,[status(thm)],[])).
% 40.17/40.34 tff(353,plain,
% 40.17/40.34 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | member(ordered_pair(y, z), cross_product(universal_class, universal_class)) | (~member(y, universal_class)) | (~member(z, universal_class))),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[352, 351])).
% 40.17/40.34 tff(354,plain,
% 40.17/40.34 (member(ordered_pair(y, z), cross_product(universal_class, universal_class)) | (~member(y, universal_class)) | (~member(z, universal_class))),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[353, 347])).
% 40.17/40.34 tff(355,plain,
% 40.17/40.34 (member(ordered_pair(y, z), cross_product(universal_class, universal_class))),
% 40.17/40.34 inference(unit_resolution,[status(thm)],[354, 207, 337])).
% 40.17/40.34 tff(356,plain,
% 40.17/40.34 (member(ordered_pair(w, x), cross_product(universal_class, universal_class))),
% 40.17/40.34 inference(modus_ponens,[status(thm)],[355, 208])).
% 40.17/40.34 tff(357,plain,
% 40.17/40.34 (^[V: $i, Y: $i, U: $i, X: $i] : refl(((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X)) <=> ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X)))),
% 40.17/40.35 inference(bind,[status(th)],[])).
% 40.17/40.35 tff(358,plain,
% 40.17/40.35 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X)) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))),
% 40.17/40.35 inference(quant_intro,[status(thm)],[357])).
% 40.17/40.35 tff(359,plain,
% 40.17/40.35 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X)) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))),
% 40.17/40.35 inference(rewrite,[status(thm)],[])).
% 40.17/40.35 tff(360,axiom,(![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','cartesian_product1')).
% 40.17/40.35 tff(361,plain,
% 40.17/40.35 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[360, 359])).
% 40.17/40.35 tff(362,plain,(
% 40.17/40.35 ![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))),
% 40.17/40.35 inference(skolemize,[status(sab)],[361])).
% 40.17/40.35 tff(363,plain,
% 40.17/40.35 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[362, 358])).
% 40.17/40.35 tff(364,plain,
% 40.17/40.35 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | ((~member(ordered_pair(w, x), cross_product(universal_class, universal_class))) | member(w, universal_class))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | (~member(ordered_pair(w, x), cross_product(universal_class, universal_class))) | member(w, universal_class))),
% 40.17/40.35 inference(rewrite,[status(thm)],[])).
% 40.17/40.35 tff(365,plain,
% 40.17/40.35 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | ((~member(ordered_pair(w, x), cross_product(universal_class, universal_class))) | member(w, universal_class))),
% 40.17/40.35 inference(quant_inst,[status(thm)],[])).
% 40.17/40.35 tff(366,plain,
% 40.17/40.35 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | (~member(ordered_pair(w, x), cross_product(universal_class, universal_class))) | member(w, universal_class)),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[365, 364])).
% 40.17/40.35 tff(367,plain,
% 40.17/40.35 ((~member(ordered_pair(w, x), cross_product(universal_class, universal_class))) | member(w, universal_class)),
% 40.17/40.35 inference(unit_resolution,[status(thm)],[366, 363])).
% 40.17/40.35 tff(368,plain,
% 40.17/40.35 (member(w, universal_class)),
% 40.17/40.35 inference(unit_resolution,[status(thm)],[367, 356])).
% 40.17/40.35 tff(369,plain,
% 40.17/40.35 (((~![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))))),
% 40.17/40.35 inference(rewrite,[status(thm)],[])).
% 40.17/40.35 tff(370,plain,
% 40.17/40.35 ((~![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))))),
% 40.17/40.35 inference(quant_inst,[status(thm)],[])).
% 40.17/40.35 tff(371,plain,
% 40.17/40.35 ((~![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)))),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[370, 369])).
% 40.17/40.35 tff(372,plain,
% 40.17/40.35 ((~member(w, universal_class)) | member(w, unordered_pair(w, singleton(x)))),
% 40.17/40.35 inference(unit_resolution,[status(thm)],[371, 241])).
% 40.17/40.35 tff(373,plain,
% 40.17/40.35 (member(w, unordered_pair(w, singleton(x)))),
% 40.17/40.35 inference(unit_resolution,[status(thm)],[372, 368])).
% 40.17/40.35 tff(374,plain,
% 40.17/40.35 (singleton(y) = singleton(singleton(x))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[98])).
% 40.17/40.35 tff(375,plain,
% 40.17/40.35 (singleton(singleton(x)) = singleton(y)),
% 40.17/40.35 inference(symmetry,[status(thm)],[374])).
% 40.17/40.35 tff(376,plain,
% 40.17/40.35 (unordered_pair(w, singleton(x)) = unordered_pair(singleton(x), singleton(x))),
% 40.17/40.35 inference(transitivity,[status(thm)],[69, 374, 128])).
% 40.17/40.35 tff(377,plain,
% 40.17/40.35 (member(w, unordered_pair(w, singleton(x))) <=> member(w, unordered_pair(singleton(x), singleton(x)))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[376])).
% 40.17/40.35 tff(378,assumption,(member(w, unordered_pair(w, singleton(x)))), introduced(assumption)).
% 40.17/40.35 tff(379,plain,
% 40.17/40.35 (member(w, unordered_pair(singleton(x), singleton(x)))),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[378, 377])).
% 40.17/40.35 tff(380,plain,
% 40.17/40.35 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((~member(w, unordered_pair(singleton(x), singleton(x)))) | (w = singleton(x)))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(w, unordered_pair(singleton(x), singleton(x)))) | (w = singleton(x)))),
% 40.17/40.35 inference(rewrite,[status(thm)],[])).
% 40.17/40.35 tff(381,plain,
% 40.17/40.35 (((w = singleton(x)) | (w = singleton(x)) | (~member(w, unordered_pair(singleton(x), singleton(x))))) <=> ((~member(w, unordered_pair(singleton(x), singleton(x)))) | (w = singleton(x)))),
% 40.17/40.35 inference(rewrite,[status(thm)],[])).
% 40.17/40.35 tff(382,plain,
% 40.17/40.35 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((w = singleton(x)) | (w = singleton(x)) | (~member(w, unordered_pair(singleton(x), singleton(x)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((~member(w, unordered_pair(singleton(x), singleton(x)))) | (w = singleton(x))))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[381])).
% 40.17/40.35 tff(383,plain,
% 40.17/40.35 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((w = singleton(x)) | (w = singleton(x)) | (~member(w, unordered_pair(singleton(x), singleton(x)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(w, unordered_pair(singleton(x), singleton(x)))) | (w = singleton(x)))),
% 40.17/40.35 inference(transitivity,[status(thm)],[382, 380])).
% 40.17/40.35 tff(384,plain,
% 40.17/40.35 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((w = singleton(x)) | (w = singleton(x)) | (~member(w, unordered_pair(singleton(x), singleton(x)))))),
% 40.17/40.35 inference(quant_inst,[status(thm)],[])).
% 40.17/40.35 tff(385,plain,
% 40.17/40.35 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(w, unordered_pair(singleton(x), singleton(x)))) | (w = singleton(x))),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[384, 383])).
% 40.17/40.35 tff(386,plain,
% 40.17/40.35 (w = singleton(x)),
% 40.17/40.35 inference(unit_resolution,[status(thm)],[385, 63, 379])).
% 40.17/40.35 tff(387,plain,
% 40.17/40.35 (singleton(w) = singleton(singleton(x))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[386])).
% 40.17/40.35 tff(388,plain,
% 40.17/40.35 (singleton(w) = null_class),
% 40.17/40.35 inference(transitivity,[status(thm)],[387, 375, 132])).
% 40.17/40.35 tff(389,plain,
% 40.17/40.35 ((unordered_pair(y, singleton(z)) = singleton(w)) <=> (unordered_pair(y, singleton(z)) = null_class)),
% 40.17/40.35 inference(monotonicity,[status(thm)],[388])).
% 40.17/40.35 tff(390,plain,
% 40.17/40.35 ((unordered_pair(y, singleton(z)) = null_class) <=> (unordered_pair(y, singleton(z)) = singleton(w))),
% 40.17/40.35 inference(symmetry,[status(thm)],[389])).
% 40.17/40.35 tff(391,plain,
% 40.17/40.35 ((~(unordered_pair(y, singleton(z)) = null_class)) <=> (~(unordered_pair(y, singleton(z)) = singleton(w)))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[390])).
% 40.17/40.35 tff(392,plain,
% 40.17/40.35 (singleton(y) = unordered_pair(w, singleton(x))),
% 40.17/40.35 inference(symmetry,[status(thm)],[69])).
% 40.17/40.35 tff(393,plain,
% 40.17/40.35 (null_class = unordered_pair(w, singleton(x))),
% 40.17/40.35 inference(transitivity,[status(thm)],[133, 392])).
% 40.17/40.35 tff(394,plain,
% 40.17/40.35 ((unordered_pair(y, singleton(z)) = null_class) <=> (unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x)))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[393])).
% 40.17/40.35 tff(395,plain,
% 40.17/40.35 ((unordered_pair(y, singleton(z)) = null_class) <=> (unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z)))),
% 40.17/40.35 inference(transitivity,[status(thm)],[394, 101])).
% 40.17/40.35 tff(396,plain,
% 40.17/40.35 ((unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))) <=> (unordered_pair(y, singleton(z)) = null_class)),
% 40.17/40.35 inference(symmetry,[status(thm)],[395])).
% 40.17/40.35 tff(397,plain,
% 40.17/40.35 ((~(unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z)))) <=> (~(unordered_pair(y, singleton(z)) = null_class))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[396])).
% 40.17/40.35 tff(398,plain,
% 40.17/40.35 (~(unordered_pair(y, singleton(z)) = null_class)),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[12, 397])).
% 40.17/40.35 tff(399,plain,
% 40.17/40.35 (~(unordered_pair(y, singleton(z)) = singleton(w))),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[398, 391])).
% 40.17/40.35 tff(400,plain,
% 40.17/40.35 (unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x))),
% 40.17/40.35 inference(unit_resolution,[status(thm)],[123, 399])).
% 40.17/40.35 tff(401,plain,
% 40.17/40.35 (unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[400, 101])).
% 40.17/40.35 tff(402,plain,
% 40.17/40.35 ($false),
% 40.17/40.35 inference(unit_resolution,[status(thm)],[12, 401])).
% 40.17/40.35 tff(403,plain,((unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))) | (~(singleton(y) = null_class)) | (~member(w, unordered_pair(w, singleton(x))))), inference(lemma,lemma(discharge,[]))).
% 40.17/40.35 tff(404,plain,
% 40.17/40.35 (unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))),
% 40.17/40.35 inference(unit_resolution,[status(thm)],[403, 373, 132])).
% 40.17/40.35 tff(405,plain,
% 40.17/40.35 (unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x))),
% 40.17/40.35 inference(symmetry,[status(thm)],[404])).
% 40.17/40.35 tff(406,plain,
% 40.17/40.35 (member(singleton(x), unordered_pair(y, singleton(z))) <=> member(singleton(x), unordered_pair(w, singleton(x)))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[405])).
% 40.17/40.35 tff(407,plain,
% 40.17/40.35 (member(singleton(x), unordered_pair(w, singleton(x))) <=> member(singleton(x), unordered_pair(y, singleton(z)))),
% 40.17/40.35 inference(symmetry,[status(thm)],[406])).
% 40.17/40.35 tff(408,plain,
% 40.17/40.35 (member(singleton(x), unordered_pair(y, singleton(z)))),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[85, 407])).
% 40.17/40.35 tff(409,plain,
% 40.17/40.35 ((singleton(z) = singleton(y)) <=> (singleton(z) = null_class)),
% 40.17/40.35 inference(monotonicity,[status(thm)],[132])).
% 40.17/40.35 tff(410,plain,
% 40.17/40.35 ((singleton(z) = null_class) <=> (singleton(z) = singleton(y))),
% 40.17/40.35 inference(symmetry,[status(thm)],[409])).
% 40.17/40.35 tff(411,plain,
% 40.17/40.35 ((~(singleton(z) = null_class)) <=> (~(singleton(z) = singleton(y)))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[410])).
% 40.17/40.35 tff(412,plain,
% 40.17/40.35 (x = y),
% 40.17/40.35 inference(unit_resolution,[status(thm)],[256, 230, 132])).
% 40.17/40.35 tff(413,plain,
% 40.17/40.35 (y = x),
% 40.17/40.35 inference(symmetry,[status(thm)],[218])).
% 40.17/40.35 tff(414,plain,
% 40.17/40.35 ((y = z) <=> (x = z)),
% 40.17/40.35 inference(monotonicity,[status(thm)],[413])).
% 40.17/40.35 tff(415,plain,
% 40.17/40.35 ((x = z) <=> (y = z)),
% 40.17/40.35 inference(symmetry,[status(thm)],[414])).
% 40.17/40.35 tff(416,plain,
% 40.17/40.35 ((~(x = z)) <=> (~(y = z))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[415])).
% 40.17/40.35 tff(417,plain,
% 40.17/40.35 (~(y = z)),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[334, 416])).
% 40.17/40.35 tff(418,plain,
% 40.17/40.35 (unordered_pair(z, z) = unordered_pair(y, y)),
% 40.17/40.35 inference(transitivity,[status(thm)],[151, 260, 133, 131])).
% 40.17/40.35 tff(419,plain,
% 40.17/40.35 (member(y, unordered_pair(z, z)) <=> member(y, unordered_pair(y, y))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[418])).
% 40.17/40.35 tff(420,plain,
% 40.17/40.35 (member(y, unordered_pair(y, y)) <=> member(y, unordered_pair(z, z))),
% 40.17/40.35 inference(symmetry,[status(thm)],[419])).
% 40.17/40.35 tff(421,plain,
% 40.17/40.35 (((~![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)))),
% 40.17/40.35 inference(rewrite,[status(thm)],[])).
% 40.17/40.35 tff(422,plain,
% 40.17/40.35 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(y, universal_class)) | member(y, unordered_pair(y, y)))),
% 40.17/40.35 inference(quant_inst,[status(thm)],[])).
% 40.17/40.35 tff(423,plain,
% 40.17/40.35 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(y, universal_class)) | member(y, unordered_pair(y, y))),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[422, 421])).
% 40.17/40.35 tff(424,plain,
% 40.17/40.35 ((~member(y, universal_class)) | member(y, unordered_pair(y, y))),
% 40.17/40.35 inference(unit_resolution,[status(thm)],[423, 241])).
% 40.17/40.35 tff(425,plain,
% 40.17/40.35 (member(y, unordered_pair(y, y))),
% 40.17/40.35 inference(unit_resolution,[status(thm)],[424, 337])).
% 40.17/40.35 tff(426,plain,
% 40.17/40.35 (member(y, unordered_pair(z, z))),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[425, 420])).
% 40.17/40.35 tff(427,plain,
% 40.17/40.35 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((y = z) | (~member(y, unordered_pair(z, z))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (y = z) | (~member(y, unordered_pair(z, z))))),
% 40.17/40.35 inference(rewrite,[status(thm)],[])).
% 40.17/40.35 tff(428,plain,
% 40.17/40.35 (((y = z) | (y = z) | (~member(y, unordered_pair(z, z)))) <=> ((y = z) | (~member(y, unordered_pair(z, z))))),
% 40.17/40.35 inference(rewrite,[status(thm)],[])).
% 40.17/40.35 tff(429,plain,
% 40.17/40.35 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((y = z) | (y = z) | (~member(y, unordered_pair(z, z))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((y = z) | (~member(y, unordered_pair(z, z)))))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[428])).
% 40.17/40.35 tff(430,plain,
% 40.17/40.35 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((y = z) | (y = z) | (~member(y, unordered_pair(z, z))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (y = z) | (~member(y, unordered_pair(z, z))))),
% 40.17/40.35 inference(transitivity,[status(thm)],[429, 427])).
% 40.17/40.35 tff(431,plain,
% 40.17/40.35 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((y = z) | (y = z) | (~member(y, unordered_pair(z, z))))),
% 40.17/40.35 inference(quant_inst,[status(thm)],[])).
% 40.17/40.35 tff(432,plain,
% 40.17/40.35 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (y = z) | (~member(y, unordered_pair(z, z)))),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[431, 430])).
% 40.17/40.35 tff(433,plain,
% 40.17/40.35 ((y = z) | (~member(y, unordered_pair(z, z)))),
% 40.17/40.35 inference(unit_resolution,[status(thm)],[432, 63])).
% 40.17/40.35 tff(434,plain,
% 40.17/40.35 ($false),
% 40.17/40.35 inference(unit_resolution,[status(thm)],[433, 426, 417])).
% 40.17/40.35 tff(435,plain,((~(singleton(z) = null_class)) | (~(singleton(y) = null_class)) | (~(x = y))), inference(lemma,lemma(discharge,[]))).
% 40.17/40.35 tff(436,plain,
% 40.17/40.35 (~(singleton(z) = null_class)),
% 40.17/40.35 inference(unit_resolution,[status(thm)],[435, 412, 132])).
% 40.17/40.35 tff(437,plain,
% 40.17/40.35 (~(singleton(z) = singleton(y))),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[436, 411])).
% 40.17/40.35 tff(438,plain,
% 40.17/40.35 (singleton(y) = singleton(x)),
% 40.17/40.35 inference(transitivity,[status(thm)],[132, 231])).
% 40.17/40.35 tff(439,plain,
% 40.17/40.35 (unordered_pair(w, singleton(y)) = unordered_pair(w, singleton(x))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[438])).
% 40.17/40.35 tff(440,plain,
% 40.17/40.35 (unordered_pair(w, singleton(y)) = unordered_pair(y, singleton(z))),
% 40.17/40.35 inference(transitivity,[status(thm)],[439, 266])).
% 40.17/40.35 tff(441,plain,
% 40.17/40.35 (member(singleton(z), unordered_pair(w, singleton(y))) <=> member(singleton(z), unordered_pair(y, singleton(z)))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[440])).
% 40.17/40.35 tff(442,plain,
% 40.17/40.35 (member(singleton(z), unordered_pair(y, singleton(z))) <=> member(singleton(z), unordered_pair(w, singleton(y)))),
% 40.17/40.35 inference(symmetry,[status(thm)],[441])).
% 40.17/40.35 tff(443,plain,
% 40.17/40.35 (((~![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))))),
% 40.17/40.35 inference(rewrite,[status(thm)],[])).
% 40.17/40.35 tff(444,plain,
% 40.17/40.35 ((~![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))))),
% 40.17/40.35 inference(quant_inst,[status(thm)],[])).
% 40.17/40.35 tff(445,plain,
% 40.17/40.35 ((~![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)))),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[444, 443])).
% 40.17/40.35 tff(446,plain,
% 40.17/40.35 ((~member(singleton(z), universal_class)) | member(singleton(z), unordered_pair(y, singleton(z)))),
% 40.17/40.35 inference(unit_resolution,[status(thm)],[445, 47])).
% 40.17/40.35 tff(447,plain,
% 40.17/40.35 (member(singleton(z), unordered_pair(y, singleton(z)))),
% 40.17/40.35 inference(unit_resolution,[status(thm)],[446, 284])).
% 40.17/40.35 tff(448,plain,
% 40.17/40.35 (member(singleton(z), unordered_pair(w, singleton(y)))),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[447, 442])).
% 40.17/40.35 tff(449,plain,
% 40.17/40.35 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = singleton(y)) | (~member(singleton(z), unordered_pair(w, singleton(y)))) | (singleton(z) = w))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(z) = singleton(y)) | (~member(singleton(z), unordered_pair(w, singleton(y)))) | (singleton(z) = w))),
% 40.17/40.35 inference(rewrite,[status(thm)],[])).
% 40.17/40.35 tff(450,plain,
% 40.17/40.35 (((singleton(z) = singleton(y)) | (singleton(z) = w) | (~member(singleton(z), unordered_pair(w, singleton(y))))) <=> ((singleton(z) = singleton(y)) | (~member(singleton(z), unordered_pair(w, singleton(y)))) | (singleton(z) = w))),
% 40.17/40.35 inference(rewrite,[status(thm)],[])).
% 40.17/40.35 tff(451,plain,
% 40.17/40.35 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = singleton(y)) | (singleton(z) = w) | (~member(singleton(z), unordered_pair(w, singleton(y)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = singleton(y)) | (~member(singleton(z), unordered_pair(w, singleton(y)))) | (singleton(z) = w)))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[450])).
% 40.17/40.35 tff(452,plain,
% 40.17/40.35 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = singleton(y)) | (singleton(z) = w) | (~member(singleton(z), unordered_pair(w, singleton(y)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(z) = singleton(y)) | (~member(singleton(z), unordered_pair(w, singleton(y)))) | (singleton(z) = w))),
% 40.17/40.35 inference(transitivity,[status(thm)],[451, 449])).
% 40.17/40.35 tff(453,plain,
% 40.17/40.35 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = singleton(y)) | (singleton(z) = w) | (~member(singleton(z), unordered_pair(w, singleton(y)))))),
% 40.17/40.35 inference(quant_inst,[status(thm)],[])).
% 40.17/40.35 tff(454,plain,
% 40.17/40.35 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(z) = singleton(y)) | (~member(singleton(z), unordered_pair(w, singleton(y)))) | (singleton(z) = w)),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[453, 452])).
% 40.17/40.35 tff(455,plain,
% 40.17/40.35 ((singleton(z) = singleton(y)) | (~member(singleton(z), unordered_pair(w, singleton(y)))) | (singleton(z) = w)),
% 40.17/40.35 inference(unit_resolution,[status(thm)],[454, 63])).
% 40.17/40.35 tff(456,plain,
% 40.17/40.35 (singleton(z) = w),
% 40.17/40.35 inference(unit_resolution,[status(thm)],[455, 448, 437])).
% 40.17/40.35 tff(457,plain,
% 40.17/40.35 (w = singleton(z)),
% 40.17/40.35 inference(symmetry,[status(thm)],[456])).
% 40.17/40.35 tff(458,plain,
% 40.17/40.35 (singleton(w) = singleton(singleton(z))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[457])).
% 40.17/40.35 tff(459,plain,
% 40.17/40.35 (singleton(x) = singleton(y)),
% 40.17/40.35 inference(transitivity,[status(thm)],[230, 133])).
% 40.17/40.35 tff(460,plain,
% 40.17/40.35 (member(singleton(x), singleton(w)) <=> member(singleton(y), singleton(singleton(z)))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[459, 458])).
% 40.17/40.35 tff(461,plain,
% 40.17/40.35 (member(singleton(y), singleton(singleton(z))) <=> member(singleton(x), singleton(w))),
% 40.17/40.35 inference(symmetry,[status(thm)],[460])).
% 40.17/40.35 tff(462,plain,
% 40.17/40.35 (member(singleton(y), unordered_pair(y, singleton(z))) <=> member(singleton(x), unordered_pair(w, singleton(x)))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[438, 267])).
% 40.17/40.35 tff(463,plain,
% 40.17/40.35 (member(singleton(x), unordered_pair(w, singleton(x))) <=> member(singleton(y), unordered_pair(y, singleton(z)))),
% 40.17/40.35 inference(symmetry,[status(thm)],[462])).
% 40.17/40.35 tff(464,plain,
% 40.17/40.35 (member(singleton(y), unordered_pair(y, singleton(z)))),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[85, 463])).
% 40.17/40.35 tff(465,plain,
% 40.17/40.35 ((null_class = singleton(z)) <=> (singleton(z) = null_class)),
% 40.17/40.35 inference(commutativity,[status(thm)],[])).
% 40.17/40.35 tff(466,plain,
% 40.17/40.35 ((singleton(y) = singleton(z)) <=> (null_class = singleton(z))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[132])).
% 40.17/40.35 tff(467,plain,
% 40.17/40.35 ((singleton(y) = singleton(z)) <=> (singleton(z) = null_class)),
% 40.17/40.35 inference(transitivity,[status(thm)],[466, 465])).
% 40.17/40.35 tff(468,plain,
% 40.17/40.35 ((singleton(z) = null_class) <=> (singleton(y) = singleton(z))),
% 40.17/40.35 inference(symmetry,[status(thm)],[467])).
% 40.17/40.35 tff(469,plain,
% 40.17/40.35 ((~(singleton(z) = null_class)) <=> (~(singleton(y) = singleton(z)))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[468])).
% 40.17/40.35 tff(470,plain,
% 40.17/40.35 (~(singleton(y) = singleton(z))),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[436, 469])).
% 40.17/40.35 tff(471,plain,
% 40.17/40.35 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(y) = y) | (singleton(y) = singleton(z)) | (~member(singleton(y), unordered_pair(y, singleton(z)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(y) = y) | (singleton(y) = singleton(z)) | (~member(singleton(y), unordered_pair(y, singleton(z)))))),
% 40.17/40.35 inference(rewrite,[status(thm)],[])).
% 40.17/40.35 tff(472,plain,
% 40.17/40.35 (((singleton(y) = singleton(z)) | (singleton(y) = y) | (~member(singleton(y), unordered_pair(y, singleton(z))))) <=> ((singleton(y) = y) | (singleton(y) = singleton(z)) | (~member(singleton(y), unordered_pair(y, singleton(z)))))),
% 40.17/40.35 inference(rewrite,[status(thm)],[])).
% 40.17/40.35 tff(473,plain,
% 40.17/40.35 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(y) = singleton(z)) | (singleton(y) = y) | (~member(singleton(y), unordered_pair(y, singleton(z)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(y) = y) | (singleton(y) = singleton(z)) | (~member(singleton(y), unordered_pair(y, singleton(z))))))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[472])).
% 40.17/40.35 tff(474,plain,
% 40.17/40.35 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(y) = singleton(z)) | (singleton(y) = y) | (~member(singleton(y), unordered_pair(y, singleton(z)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(y) = y) | (singleton(y) = singleton(z)) | (~member(singleton(y), unordered_pair(y, singleton(z)))))),
% 40.17/40.35 inference(transitivity,[status(thm)],[473, 471])).
% 40.17/40.35 tff(475,plain,
% 40.17/40.35 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(y) = singleton(z)) | (singleton(y) = y) | (~member(singleton(y), unordered_pair(y, singleton(z)))))),
% 40.17/40.35 inference(quant_inst,[status(thm)],[])).
% 40.17/40.35 tff(476,plain,
% 40.17/40.35 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(y) = y) | (singleton(y) = singleton(z)) | (~member(singleton(y), unordered_pair(y, singleton(z))))),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[475, 474])).
% 40.17/40.35 tff(477,plain,
% 40.17/40.35 ((singleton(y) = y) | (singleton(y) = singleton(z)) | (~member(singleton(y), unordered_pair(y, singleton(z))))),
% 40.17/40.35 inference(unit_resolution,[status(thm)],[476, 63])).
% 40.17/40.35 tff(478,plain,
% 40.17/40.35 (singleton(y) = y),
% 40.17/40.35 inference(unit_resolution,[status(thm)],[477, 470, 464])).
% 40.17/40.35 tff(479,plain,
% 40.17/40.35 (y = singleton(y)),
% 40.17/40.35 inference(symmetry,[status(thm)],[478])).
% 40.17/40.35 tff(480,plain,
% 40.17/40.35 (member(y, singleton(singleton(z))) <=> member(singleton(y), singleton(singleton(z)))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[479])).
% 40.17/40.35 tff(481,plain,
% 40.17/40.35 (member(y, singleton(singleton(z))) <=> member(singleton(x), singleton(w))),
% 40.17/40.35 inference(transitivity,[status(thm)],[480, 461])).
% 40.17/40.35 tff(482,plain,
% 40.17/40.35 (y = x),
% 40.17/40.35 inference(symmetry,[status(thm)],[412])).
% 40.17/40.35 tff(483,plain,
% 40.17/40.35 (ordered_pair(w, y) = ordered_pair(w, x)),
% 40.17/40.35 inference(monotonicity,[status(thm)],[482])).
% 40.17/40.35 tff(484,plain,
% 40.17/40.35 (ordered_pair(w, x) = ordered_pair(w, y)),
% 40.17/40.35 inference(symmetry,[status(thm)],[483])).
% 40.17/40.35 tff(485,plain,
% 40.17/40.35 (ordered_pair(y, z) = ordered_pair(w, y)),
% 40.17/40.35 inference(transitivity,[status(thm)],[26, 484])).
% 40.17/40.35 tff(486,plain,
% 40.17/40.35 (member(ordered_pair(y, z), cross_product(singleton(singleton(z)), universal_class)) <=> member(ordered_pair(w, y), cross_product(singleton(singleton(z)), universal_class))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[485])).
% 40.17/40.35 tff(487,plain,
% 40.17/40.35 (member(ordered_pair(w, y), cross_product(singleton(singleton(z)), universal_class)) <=> member(ordered_pair(y, z), cross_product(singleton(singleton(z)), universal_class))),
% 40.17/40.35 inference(symmetry,[status(thm)],[486])).
% 40.17/40.35 tff(488,plain,
% 40.17/40.35 (singleton(singleton(z)) = unordered_pair(singleton(z), singleton(z))),
% 40.17/40.35 inference(symmetry,[status(thm)],[136])).
% 40.17/40.35 tff(489,plain,
% 40.17/40.35 (member(w, singleton(singleton(z))) <=> member(singleton(z), unordered_pair(singleton(z), singleton(z)))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[457, 488])).
% 40.17/40.35 tff(490,plain,
% 40.17/40.35 (member(singleton(z), unordered_pair(singleton(z), singleton(z))) <=> member(w, singleton(singleton(z)))),
% 40.17/40.35 inference(symmetry,[status(thm)],[489])).
% 40.17/40.35 tff(491,plain,
% 40.17/40.35 (((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(singleton(z), universal_class)) | member(singleton(z), unordered_pair(singleton(z), singleton(z))))) <=> ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(singleton(z), universal_class)) | member(singleton(z), unordered_pair(singleton(z), singleton(z))))),
% 40.17/40.35 inference(rewrite,[status(thm)],[])).
% 40.17/40.35 tff(492,plain,
% 40.17/40.35 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(singleton(z), universal_class)) | member(singleton(z), unordered_pair(singleton(z), singleton(z))))),
% 40.17/40.35 inference(quant_inst,[status(thm)],[])).
% 40.17/40.35 tff(493,plain,
% 40.17/40.35 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(singleton(z), universal_class)) | member(singleton(z), unordered_pair(singleton(z), singleton(z)))),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[492, 491])).
% 40.17/40.35 tff(494,plain,
% 40.17/40.35 (member(singleton(z), unordered_pair(singleton(z), singleton(z)))),
% 40.17/40.35 inference(unit_resolution,[status(thm)],[493, 241, 284])).
% 40.17/40.35 tff(495,plain,
% 40.17/40.35 (member(w, singleton(singleton(z)))),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[494, 490])).
% 40.17/40.35 tff(496,plain,
% 40.17/40.35 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | ((~member(y, universal_class)) | (~member(w, singleton(singleton(z)))) | member(ordered_pair(w, y), cross_product(singleton(singleton(z)), universal_class)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | (~member(y, universal_class)) | (~member(w, singleton(singleton(z)))) | member(ordered_pair(w, y), cross_product(singleton(singleton(z)), universal_class)))),
% 40.17/40.35 inference(rewrite,[status(thm)],[])).
% 40.17/40.35 tff(497,plain,
% 40.17/40.35 (((~member(w, singleton(singleton(z)))) | member(ordered_pair(w, y), cross_product(singleton(singleton(z)), universal_class)) | (~member(y, universal_class))) <=> ((~member(y, universal_class)) | (~member(w, singleton(singleton(z)))) | member(ordered_pair(w, y), cross_product(singleton(singleton(z)), universal_class)))),
% 40.17/40.35 inference(rewrite,[status(thm)],[])).
% 40.17/40.35 tff(498,plain,
% 40.17/40.35 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | ((~member(w, singleton(singleton(z)))) | member(ordered_pair(w, y), cross_product(singleton(singleton(z)), universal_class)) | (~member(y, universal_class)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | ((~member(y, universal_class)) | (~member(w, singleton(singleton(z)))) | member(ordered_pair(w, y), cross_product(singleton(singleton(z)), universal_class))))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[497])).
% 40.17/40.35 tff(499,plain,
% 40.17/40.35 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | ((~member(w, singleton(singleton(z)))) | member(ordered_pair(w, y), cross_product(singleton(singleton(z)), universal_class)) | (~member(y, universal_class)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | (~member(y, universal_class)) | (~member(w, singleton(singleton(z)))) | member(ordered_pair(w, y), cross_product(singleton(singleton(z)), universal_class)))),
% 40.17/40.35 inference(transitivity,[status(thm)],[498, 496])).
% 40.17/40.35 tff(500,plain,
% 40.17/40.35 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | ((~member(w, singleton(singleton(z)))) | member(ordered_pair(w, y), cross_product(singleton(singleton(z)), universal_class)) | (~member(y, universal_class)))),
% 40.17/40.35 inference(quant_inst,[status(thm)],[])).
% 40.17/40.35 tff(501,plain,
% 40.17/40.35 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | (~member(y, universal_class)) | (~member(w, singleton(singleton(z)))) | member(ordered_pair(w, y), cross_product(singleton(singleton(z)), universal_class))),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[500, 499])).
% 40.17/40.35 tff(502,plain,
% 40.17/40.35 ((~member(w, singleton(singleton(z)))) | member(ordered_pair(w, y), cross_product(singleton(singleton(z)), universal_class))),
% 40.17/40.35 inference(unit_resolution,[status(thm)],[501, 347, 337])).
% 40.17/40.35 tff(503,plain,
% 40.17/40.35 (member(ordered_pair(w, y), cross_product(singleton(singleton(z)), universal_class))),
% 40.17/40.35 inference(unit_resolution,[status(thm)],[502, 495])).
% 40.17/40.35 tff(504,plain,
% 40.17/40.35 (member(ordered_pair(y, z), cross_product(singleton(singleton(z)), universal_class))),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[503, 487])).
% 40.17/40.35 tff(505,plain,
% 40.17/40.35 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | ((~member(ordered_pair(y, z), cross_product(singleton(singleton(z)), universal_class))) | member(y, singleton(singleton(z))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | (~member(ordered_pair(y, z), cross_product(singleton(singleton(z)), universal_class))) | member(y, singleton(singleton(z))))),
% 40.17/40.35 inference(rewrite,[status(thm)],[])).
% 40.17/40.35 tff(506,plain,
% 40.17/40.35 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | ((~member(ordered_pair(y, z), cross_product(singleton(singleton(z)), universal_class))) | member(y, singleton(singleton(z))))),
% 40.17/40.35 inference(quant_inst,[status(thm)],[])).
% 40.17/40.35 tff(507,plain,
% 40.17/40.35 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | (~member(ordered_pair(y, z), cross_product(singleton(singleton(z)), universal_class))) | member(y, singleton(singleton(z)))),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[506, 505])).
% 40.17/40.35 tff(508,plain,
% 40.17/40.35 ((~member(ordered_pair(y, z), cross_product(singleton(singleton(z)), universal_class))) | member(y, singleton(singleton(z)))),
% 40.17/40.35 inference(unit_resolution,[status(thm)],[507, 363])).
% 40.17/40.35 tff(509,plain,
% 40.17/40.35 (member(y, singleton(singleton(z)))),
% 40.17/40.35 inference(unit_resolution,[status(thm)],[508, 504])).
% 40.17/40.35 tff(510,plain,
% 40.17/40.35 (member(singleton(x), singleton(w))),
% 40.17/40.35 inference(modus_ponens,[status(thm)],[509, 481])).
% 40.17/40.35 tff(511,plain,
% 40.17/40.35 ((singleton(x) = w) <=> (null_class = singleton(z))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[230, 457])).
% 40.17/40.35 tff(512,plain,
% 40.17/40.35 ((singleton(x) = w) <=> (singleton(z) = null_class)),
% 40.17/40.35 inference(transitivity,[status(thm)],[511, 465])).
% 40.17/40.35 tff(513,plain,
% 40.17/40.35 ((singleton(z) = null_class) <=> (singleton(x) = w)),
% 40.17/40.35 inference(symmetry,[status(thm)],[512])).
% 40.17/40.35 tff(514,plain,
% 40.17/40.35 ((~(singleton(z) = null_class)) <=> (~(singleton(x) = w))),
% 40.17/40.35 inference(monotonicity,[status(thm)],[513])).
% 40.17/40.35 tff(515,plain,
% 40.17/40.36 (~(singleton(x) = w)),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[436, 514])).
% 40.17/40.36 tff(516,plain,
% 40.17/40.36 (member(singleton(x), unordered_pair(w, w)) <=> member(singleton(x), singleton(w))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[11])).
% 40.17/40.36 tff(517,plain,
% 40.17/40.36 (member(singleton(x), singleton(w)) <=> member(singleton(x), unordered_pair(w, w))),
% 40.17/40.36 inference(symmetry,[status(thm)],[516])).
% 40.17/40.36 tff(518,assumption,(member(singleton(x), singleton(w))), introduced(assumption)).
% 40.17/40.36 tff(519,plain,
% 40.17/40.36 (member(singleton(x), unordered_pair(w, w))),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[518, 517])).
% 40.17/40.36 tff(520,assumption,(~(singleton(x) = w)), introduced(assumption)).
% 40.17/40.36 tff(521,plain,
% 40.17/40.36 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(x) = w) | (~member(singleton(x), unordered_pair(w, w))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(x) = w) | (~member(singleton(x), unordered_pair(w, w))))),
% 40.17/40.36 inference(rewrite,[status(thm)],[])).
% 40.17/40.36 tff(522,plain,
% 40.17/40.36 (((singleton(x) = w) | (singleton(x) = w) | (~member(singleton(x), unordered_pair(w, w)))) <=> ((singleton(x) = w) | (~member(singleton(x), unordered_pair(w, w))))),
% 40.17/40.36 inference(rewrite,[status(thm)],[])).
% 40.17/40.36 tff(523,plain,
% 40.17/40.36 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(x) = w) | (singleton(x) = w) | (~member(singleton(x), unordered_pair(w, w))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(x) = w) | (~member(singleton(x), unordered_pair(w, w)))))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[522])).
% 40.17/40.36 tff(524,plain,
% 40.17/40.36 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(x) = w) | (singleton(x) = w) | (~member(singleton(x), unordered_pair(w, w))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(x) = w) | (~member(singleton(x), unordered_pair(w, w))))),
% 40.17/40.36 inference(transitivity,[status(thm)],[523, 521])).
% 40.17/40.36 tff(525,plain,
% 40.17/40.36 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(x) = w) | (singleton(x) = w) | (~member(singleton(x), unordered_pair(w, w))))),
% 40.17/40.36 inference(quant_inst,[status(thm)],[])).
% 40.17/40.36 tff(526,plain,
% 40.17/40.36 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(x) = w) | (~member(singleton(x), unordered_pair(w, w)))),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[525, 524])).
% 40.17/40.36 tff(527,plain,
% 40.17/40.36 (~member(singleton(x), unordered_pair(w, w))),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[526, 63, 520])).
% 40.17/40.36 tff(528,plain,
% 40.17/40.36 ($false),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[527, 519])).
% 40.17/40.36 tff(529,plain,((~member(singleton(x), singleton(w))) | (singleton(x) = w)), inference(lemma,lemma(discharge,[]))).
% 40.17/40.36 tff(530,plain,
% 40.17/40.36 (~member(singleton(x), singleton(w))),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[529, 515])).
% 40.17/40.36 tff(531,plain,
% 40.17/40.36 ($false),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[530, 510])).
% 40.17/40.36 tff(532,plain,((~(unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z)))) | (~(singleton(x) = null_class)) | (~(singleton(y) = null_class))), inference(lemma,lemma(discharge,[]))).
% 40.17/40.36 tff(533,plain,
% 40.17/40.36 (~(unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z)))),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[532, 230, 132])).
% 40.17/40.36 tff(534,plain,
% 40.17/40.36 (unordered_pair(w, singleton(x)) = singleton(y)),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[68, 533])).
% 40.17/40.36 tff(535,plain,
% 40.17/40.36 (singleton(y) = unordered_pair(w, singleton(x))),
% 40.17/40.36 inference(symmetry,[status(thm)],[534])).
% 40.17/40.36 tff(536,plain,
% 40.17/40.36 (unordered_pair(y, y) = unordered_pair(w, singleton(x))),
% 40.17/40.36 inference(transitivity,[status(thm)],[9, 535])).
% 40.17/40.36 tff(537,plain,
% 40.17/40.36 (member(w, unordered_pair(y, y)) <=> member(w, unordered_pair(w, singleton(x)))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[536])).
% 40.17/40.36 tff(538,plain,
% 40.17/40.36 (member(w, unordered_pair(w, singleton(x))) <=> member(w, unordered_pair(y, y))),
% 40.17/40.36 inference(symmetry,[status(thm)],[537])).
% 40.17/40.36 tff(539,plain,
% 40.17/40.36 (member(apply(choice, singleton(z)), singleton(z))),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[298, 436])).
% 40.17/40.36 tff(540,plain,
% 40.17/40.36 (member(z, universal_class)),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[206, 539])).
% 40.17/40.36 tff(541,plain,
% 40.17/40.36 (member(ordered_pair(y, z), cross_product(universal_class, universal_class))),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[354, 540, 337])).
% 40.17/40.36 tff(542,plain,
% 40.17/40.36 (member(ordered_pair(w, x), cross_product(universal_class, universal_class))),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[541, 208])).
% 40.17/40.36 tff(543,plain,
% 40.17/40.36 (member(w, universal_class)),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[367, 542])).
% 40.17/40.36 tff(544,plain,
% 40.17/40.36 (member(w, unordered_pair(w, singleton(x)))),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[372, 543])).
% 40.17/40.36 tff(545,plain,
% 40.17/40.36 (member(w, unordered_pair(y, y))),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[544, 538])).
% 40.17/40.36 tff(546,plain,
% 40.17/40.36 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((~member(w, unordered_pair(y, y))) | (w = y))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(w, unordered_pair(y, y))) | (w = y))),
% 40.17/40.36 inference(rewrite,[status(thm)],[])).
% 40.17/40.36 tff(547,plain,
% 40.17/40.36 (((w = y) | (w = y) | (~member(w, unordered_pair(y, y)))) <=> ((~member(w, unordered_pair(y, y))) | (w = y))),
% 40.17/40.36 inference(rewrite,[status(thm)],[])).
% 40.17/40.36 tff(548,plain,
% 40.17/40.36 (((~![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))))) | ((~member(w, unordered_pair(y, y))) | (w = y)))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[547])).
% 40.17/40.36 tff(549,plain,
% 40.17/40.36 (((~![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))))) | (~member(w, unordered_pair(y, y))) | (w = y))),
% 40.17/40.36 inference(transitivity,[status(thm)],[548, 546])).
% 40.17/40.36 tff(550,plain,
% 40.17/40.36 ((~![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))))),
% 40.17/40.36 inference(quant_inst,[status(thm)],[])).
% 40.17/40.36 tff(551,plain,
% 40.17/40.36 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(w, unordered_pair(y, y))) | (w = y)),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[550, 549])).
% 40.17/40.36 tff(552,plain,
% 40.17/40.36 ((~member(w, unordered_pair(y, y))) | (w = y)),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[551, 63])).
% 40.17/40.36 tff(553,plain,
% 40.17/40.36 (w = y),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[552, 545])).
% 40.17/40.36 tff(554,plain,
% 40.17/40.36 (singleton(w) = singleton(y)),
% 40.17/40.36 inference(monotonicity,[status(thm)],[553])).
% 40.17/40.36 tff(555,plain,
% 40.17/40.36 ((unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))) <=> (unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x)))),
% 40.17/40.36 inference(symmetry,[status(thm)],[101])).
% 40.17/40.36 tff(556,plain,
% 40.17/40.36 ((~(unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z)))) <=> (~(unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x))))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[555])).
% 40.17/40.36 tff(557,plain,
% 40.17/40.36 (~(unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x)))),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[533, 556])).
% 40.17/40.36 tff(558,plain,
% 40.17/40.36 (unordered_pair(y, singleton(z)) = singleton(w)),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[123, 557])).
% 40.17/40.36 tff(559,plain,
% 40.17/40.36 ((unordered_pair(y, singleton(z)) = null_class) <=> (singleton(w) = singleton(y))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[558, 133])).
% 40.17/40.36 tff(560,plain,
% 40.17/40.36 ((~(unordered_pair(y, singleton(z)) = null_class)) <=> (~(singleton(w) = singleton(y)))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[559])).
% 40.17/40.36 tff(561,plain,
% 40.17/40.36 ((null_class = unordered_pair(y, singleton(z))) <=> (unordered_pair(y, singleton(z)) = null_class)),
% 40.17/40.36 inference(commutativity,[status(thm)],[])).
% 40.17/40.36 tff(562,plain,
% 40.17/40.36 (unordered_pair(w, singleton(x)) = null_class),
% 40.17/40.36 inference(transitivity,[status(thm)],[534, 132])).
% 40.17/40.36 tff(563,plain,
% 40.17/40.36 ((unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))) <=> (null_class = unordered_pair(y, singleton(z)))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[562])).
% 40.17/40.36 tff(564,plain,
% 40.17/40.36 ((unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))) <=> (unordered_pair(y, singleton(z)) = null_class)),
% 40.17/40.36 inference(transitivity,[status(thm)],[563, 561])).
% 40.17/40.36 tff(565,plain,
% 40.17/40.36 ((~(unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z)))) <=> (~(unordered_pair(y, singleton(z)) = null_class))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[564])).
% 40.17/40.36 tff(566,plain,
% 40.17/40.36 (~(unordered_pair(y, singleton(z)) = null_class)),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[533, 565])).
% 40.17/40.36 tff(567,plain,
% 40.17/40.36 (~(singleton(w) = singleton(y))),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[566, 560])).
% 40.17/40.36 tff(568,plain,
% 40.17/40.36 ($false),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[567, 554])).
% 40.17/40.36 tff(569,plain,((~(singleton(x) = null_class)) | (~(singleton(y) = null_class))), inference(lemma,lemma(discharge,[]))).
% 40.17/40.36 tff(570,plain,
% 40.17/40.36 (~(singleton(x) = null_class)),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[569, 132])).
% 40.17/40.36 tff(571,plain,
% 40.17/40.36 (member(apply(choice, singleton(x)), singleton(x))),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[319, 570])).
% 40.17/40.36 tff(572,assumption,(member(apply(choice, singleton(x)), singleton(x))), introduced(assumption)).
% 40.17/40.36 tff(573,plain,
% 40.17/40.36 (member(apply(choice, singleton(x)), unordered_pair(x, x))),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[572, 315])).
% 40.17/40.36 tff(574,assumption,(~(apply(choice, singleton(x)) = x)), introduced(assumption)).
% 40.17/40.36 tff(575,plain,
% 40.17/40.36 (~member(apply(choice, singleton(x)), unordered_pair(x, x))),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[327, 63, 574])).
% 40.17/40.36 tff(576,plain,
% 40.17/40.36 ($false),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[575, 573])).
% 40.17/40.36 tff(577,plain,((~member(apply(choice, singleton(x)), singleton(x))) | (apply(choice, singleton(x)) = x)), inference(lemma,lemma(discharge,[]))).
% 40.17/40.36 tff(578,plain,
% 40.17/40.36 (apply(choice, singleton(x)) = x),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[577, 571])).
% 40.17/40.36 tff(579,plain,
% 40.17/40.36 (apply(choice, singleton(z)) = z),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[303, 134])).
% 40.17/40.36 tff(580,assumption,(apply(choice, singleton(z)) = z), introduced(assumption)).
% 40.17/40.36 tff(581,assumption,(singleton(x) = singleton(z)), introduced(assumption)).
% 40.17/40.36 tff(582,plain,
% 40.17/40.36 (apply(choice, singleton(x)) = apply(choice, singleton(z))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[581])).
% 40.17/40.36 tff(583,assumption,(apply(choice, singleton(x)) = x), introduced(assumption)).
% 40.17/40.36 tff(584,plain,
% 40.17/40.36 (x = apply(choice, singleton(x))),
% 40.17/40.36 inference(symmetry,[status(thm)],[583])).
% 40.17/40.36 tff(585,plain,
% 40.17/40.36 (x = z),
% 40.17/40.36 inference(transitivity,[status(thm)],[584, 582, 580])).
% 40.17/40.36 tff(586,plain,
% 40.17/40.36 ($false),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[334, 585])).
% 40.17/40.36 tff(587,plain,((~(singleton(x) = singleton(z))) | (~(apply(choice, singleton(x)) = x)) | (~(apply(choice, singleton(z)) = z))), inference(lemma,lemma(discharge,[]))).
% 40.17/40.36 tff(588,plain,
% 40.17/40.36 (~(singleton(x) = singleton(z))),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[587, 579, 578])).
% 40.17/40.36 tff(589,plain,
% 40.17/40.36 ((singleton(x) = y) | (singleton(x) = singleton(z)) | (~member(singleton(x), unordered_pair(y, singleton(z))))),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[276, 63])).
% 40.17/40.36 tff(590,plain,
% 40.17/40.36 (singleton(x) = y),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[589, 588, 408])).
% 40.17/40.36 tff(591,plain,
% 40.17/40.36 (y = singleton(x)),
% 40.17/40.36 inference(symmetry,[status(thm)],[590])).
% 40.17/40.36 tff(592,plain,
% 40.17/40.36 (member(y, singleton(singleton(x))) <=> member(singleton(x), unordered_pair(singleton(x), singleton(x)))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[591, 128])).
% 40.17/40.36 tff(593,plain,
% 40.17/40.36 (member(singleton(x), unordered_pair(singleton(x), singleton(x))) <=> member(y, singleton(singleton(x)))),
% 40.17/40.36 inference(symmetry,[status(thm)],[592])).
% 40.17/40.36 tff(594,plain,
% 40.17/40.36 (((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(singleton(x), universal_class)) | member(singleton(x), unordered_pair(singleton(x), singleton(x))))) <=> ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(singleton(x), universal_class)) | member(singleton(x), unordered_pair(singleton(x), singleton(x))))),
% 40.17/40.36 inference(rewrite,[status(thm)],[])).
% 40.17/40.36 tff(595,plain,
% 40.17/40.36 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(singleton(x), universal_class)) | member(singleton(x), unordered_pair(singleton(x), singleton(x))))),
% 40.17/40.36 inference(quant_inst,[status(thm)],[])).
% 40.17/40.36 tff(596,plain,
% 40.17/40.36 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(singleton(x), universal_class)) | member(singleton(x), unordered_pair(singleton(x), singleton(x)))),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[595, 594])).
% 40.17/40.36 tff(597,plain,
% 40.17/40.36 (member(singleton(x), unordered_pair(singleton(x), singleton(x)))),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[596, 241, 80])).
% 40.17/40.36 tff(598,plain,
% 40.17/40.36 (member(y, singleton(singleton(x)))),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[597, 593])).
% 40.17/40.36 tff(599,plain,
% 40.17/40.36 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | ((~member(z, universal_class)) | (~member(y, singleton(singleton(x)))) | member(ordered_pair(y, z), cross_product(singleton(singleton(x)), universal_class)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | (~member(z, universal_class)) | (~member(y, singleton(singleton(x)))) | member(ordered_pair(y, z), cross_product(singleton(singleton(x)), universal_class)))),
% 40.17/40.36 inference(rewrite,[status(thm)],[])).
% 40.17/40.36 tff(600,plain,
% 40.17/40.36 (((~member(y, singleton(singleton(x)))) | member(ordered_pair(y, z), cross_product(singleton(singleton(x)), universal_class)) | (~member(z, universal_class))) <=> ((~member(z, universal_class)) | (~member(y, singleton(singleton(x)))) | member(ordered_pair(y, z), cross_product(singleton(singleton(x)), universal_class)))),
% 40.17/40.36 inference(rewrite,[status(thm)],[])).
% 40.17/40.36 tff(601,plain,
% 40.17/40.36 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | ((~member(y, singleton(singleton(x)))) | member(ordered_pair(y, z), cross_product(singleton(singleton(x)), universal_class)) | (~member(z, universal_class)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | ((~member(z, universal_class)) | (~member(y, singleton(singleton(x)))) | member(ordered_pair(y, z), cross_product(singleton(singleton(x)), universal_class))))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[600])).
% 40.17/40.36 tff(602,plain,
% 40.17/40.36 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | ((~member(y, singleton(singleton(x)))) | member(ordered_pair(y, z), cross_product(singleton(singleton(x)), universal_class)) | (~member(z, universal_class)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | (~member(z, universal_class)) | (~member(y, singleton(singleton(x)))) | member(ordered_pair(y, z), cross_product(singleton(singleton(x)), universal_class)))),
% 40.17/40.36 inference(transitivity,[status(thm)],[601, 599])).
% 40.17/40.36 tff(603,plain,
% 40.17/40.36 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | ((~member(y, singleton(singleton(x)))) | member(ordered_pair(y, z), cross_product(singleton(singleton(x)), universal_class)) | (~member(z, universal_class)))),
% 40.17/40.36 inference(quant_inst,[status(thm)],[])).
% 40.17/40.36 tff(604,plain,
% 40.17/40.36 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | (~member(z, universal_class)) | (~member(y, singleton(singleton(x)))) | member(ordered_pair(y, z), cross_product(singleton(singleton(x)), universal_class))),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[603, 602])).
% 40.17/40.36 tff(605,plain,
% 40.17/40.36 ((~member(z, universal_class)) | (~member(y, singleton(singleton(x)))) | member(ordered_pair(y, z), cross_product(singleton(singleton(x)), universal_class))),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[604, 347])).
% 40.17/40.36 tff(606,plain,
% 40.17/40.36 (member(ordered_pair(y, z), cross_product(singleton(singleton(x)), universal_class))),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[605, 598, 207])).
% 40.17/40.36 tff(607,plain,
% 40.17/40.36 (member(ordered_pair(w, x), cross_product(singleton(singleton(x)), universal_class))),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[606, 130])).
% 40.17/40.36 tff(608,plain,
% 40.17/40.36 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | ((~member(ordered_pair(w, x), cross_product(singleton(singleton(x)), universal_class))) | member(w, singleton(singleton(x))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | (~member(ordered_pair(w, x), cross_product(singleton(singleton(x)), universal_class))) | member(w, singleton(singleton(x))))),
% 40.17/40.36 inference(rewrite,[status(thm)],[])).
% 40.17/40.36 tff(609,plain,
% 40.17/40.36 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | ((~member(ordered_pair(w, x), cross_product(singleton(singleton(x)), universal_class))) | member(w, singleton(singleton(x))))),
% 40.17/40.36 inference(quant_inst,[status(thm)],[])).
% 40.17/40.36 tff(610,plain,
% 40.17/40.36 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | (~member(ordered_pair(w, x), cross_product(singleton(singleton(x)), universal_class))) | member(w, singleton(singleton(x)))),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[609, 608])).
% 40.17/40.36 tff(611,plain,
% 40.17/40.36 ((~member(ordered_pair(w, x), cross_product(singleton(singleton(x)), universal_class))) | member(w, singleton(singleton(x)))),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[610, 363])).
% 40.17/40.36 tff(612,plain,
% 40.17/40.36 (member(w, singleton(singleton(x)))),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[611, 607])).
% 40.17/40.36 tff(613,plain,
% 40.17/40.36 (member(w, unordered_pair(singleton(x), singleton(x)))),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[612, 129])).
% 40.17/40.36 tff(614,plain,
% 40.17/40.36 (member(singleton(z), unordered_pair(w, singleton(x))) <=> member(singleton(z), unordered_pair(y, singleton(z)))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[404])).
% 40.17/40.36 tff(615,plain,
% 40.17/40.36 (member(singleton(z), unordered_pair(y, singleton(z))) <=> member(singleton(z), unordered_pair(w, singleton(x)))),
% 40.17/40.36 inference(symmetry,[status(thm)],[614])).
% 40.17/40.36 tff(616,plain,
% 40.17/40.36 (member(singleton(z), unordered_pair(w, singleton(x)))),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[447, 615])).
% 40.17/40.36 tff(617,plain,
% 40.17/40.36 ((singleton(z) = singleton(x)) <=> (singleton(x) = singleton(z))),
% 40.17/40.36 inference(commutativity,[status(thm)],[])).
% 40.17/40.36 tff(618,plain,
% 40.17/40.36 ((singleton(x) = singleton(z)) <=> (singleton(z) = singleton(x))),
% 40.17/40.36 inference(symmetry,[status(thm)],[617])).
% 40.17/40.36 tff(619,plain,
% 40.17/40.36 ((~(singleton(x) = singleton(z))) <=> (~(singleton(z) = singleton(x)))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[618])).
% 40.17/40.36 tff(620,plain,
% 40.17/40.36 (~(singleton(z) = singleton(x))),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[588, 619])).
% 40.17/40.36 tff(621,plain,
% 40.17/40.36 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = w) | (singleton(z) = singleton(x)) | (~member(singleton(z), unordered_pair(w, singleton(x)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(z) = w) | (singleton(z) = singleton(x)) | (~member(singleton(z), unordered_pair(w, singleton(x)))))),
% 40.17/40.36 inference(rewrite,[status(thm)],[])).
% 40.17/40.36 tff(622,plain,
% 40.17/40.36 (((singleton(z) = singleton(x)) | (singleton(z) = w) | (~member(singleton(z), unordered_pair(w, singleton(x))))) <=> ((singleton(z) = w) | (singleton(z) = singleton(x)) | (~member(singleton(z), unordered_pair(w, singleton(x)))))),
% 40.17/40.36 inference(rewrite,[status(thm)],[])).
% 40.17/40.36 tff(623,plain,
% 40.17/40.36 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = singleton(x)) | (singleton(z) = w) | (~member(singleton(z), unordered_pair(w, singleton(x)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = w) | (singleton(z) = singleton(x)) | (~member(singleton(z), unordered_pair(w, singleton(x))))))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[622])).
% 40.17/40.36 tff(624,plain,
% 40.17/40.36 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = singleton(x)) | (singleton(z) = w) | (~member(singleton(z), unordered_pair(w, singleton(x)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(z) = w) | (singleton(z) = singleton(x)) | (~member(singleton(z), unordered_pair(w, singleton(x)))))),
% 40.17/40.36 inference(transitivity,[status(thm)],[623, 621])).
% 40.17/40.36 tff(625,plain,
% 40.17/40.36 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(z) = singleton(x)) | (singleton(z) = w) | (~member(singleton(z), unordered_pair(w, singleton(x)))))),
% 40.17/40.36 inference(quant_inst,[status(thm)],[])).
% 40.17/40.36 tff(626,plain,
% 40.17/40.36 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(z) = w) | (singleton(z) = singleton(x)) | (~member(singleton(z), unordered_pair(w, singleton(x))))),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[625, 624])).
% 40.17/40.36 tff(627,plain,
% 40.17/40.36 ((singleton(z) = w) | (singleton(z) = singleton(x)) | (~member(singleton(z), unordered_pair(w, singleton(x))))),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[626, 63])).
% 40.17/40.36 tff(628,plain,
% 40.17/40.36 (singleton(z) = w),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[627, 620, 616])).
% 40.17/40.36 tff(629,plain,
% 40.17/40.36 (w = singleton(z)),
% 40.17/40.36 inference(symmetry,[status(thm)],[628])).
% 40.17/40.36 tff(630,plain,
% 40.17/40.36 ((singleton(x) = w) <=> (singleton(x) = singleton(z))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[629])).
% 40.17/40.36 tff(631,plain,
% 40.17/40.36 ((singleton(x) = singleton(z)) <=> (singleton(x) = w)),
% 40.17/40.36 inference(symmetry,[status(thm)],[630])).
% 40.17/40.36 tff(632,plain,
% 40.17/40.36 ((~(singleton(x) = singleton(z))) <=> (~(singleton(x) = w))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[631])).
% 40.17/40.36 tff(633,plain,
% 40.17/40.36 (~(singleton(x) = w)),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[588, 632])).
% 40.17/40.36 tff(634,assumption,(member(w, unordered_pair(singleton(x), singleton(x)))), introduced(assumption)).
% 40.17/40.36 tff(635,plain,
% 40.17/40.36 ((w = singleton(x)) <=> (singleton(x) = w)),
% 40.17/40.36 inference(commutativity,[status(thm)],[])).
% 40.17/40.36 tff(636,plain,
% 40.17/40.36 ((singleton(x) = w) <=> (w = singleton(x))),
% 40.17/40.36 inference(symmetry,[status(thm)],[635])).
% 40.17/40.36 tff(637,plain,
% 40.17/40.36 ((~(singleton(x) = w)) <=> (~(w = singleton(x)))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[636])).
% 40.17/40.36 tff(638,plain,
% 40.17/40.36 (~(w = singleton(x))),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[520, 637])).
% 40.17/40.36 tff(639,plain,
% 40.17/40.36 ((~member(w, unordered_pair(singleton(x), singleton(x)))) | (w = singleton(x))),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[385, 63])).
% 40.17/40.36 tff(640,plain,
% 40.17/40.36 ($false),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[639, 638, 634])).
% 40.17/40.36 tff(641,plain,((~member(w, unordered_pair(singleton(x), singleton(x)))) | (singleton(x) = w)), inference(lemma,lemma(discharge,[]))).
% 40.17/40.36 tff(642,plain,
% 40.17/40.36 (~member(w, unordered_pair(singleton(x), singleton(x)))),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[641, 633])).
% 40.17/40.36 tff(643,plain,
% 40.17/40.36 ($false),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[642, 613])).
% 40.17/40.36 tff(644,plain,((~member(apply(choice, singleton(z)), singleton(z))) | (~(singleton(y) = null_class))), inference(lemma,lemma(discharge,[]))).
% 40.17/40.36 tff(645,plain,
% 40.17/40.36 (~member(apply(choice, singleton(z)), singleton(z))),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[644, 132])).
% 40.17/40.36 tff(646,plain,
% 40.17/40.36 (singleton(z) = null_class),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[298, 645])).
% 40.17/40.36 tff(647,plain,
% 40.17/40.36 (unordered_pair(z, z) = unordered_pair(y, y)),
% 40.17/40.36 inference(transitivity,[status(thm)],[151, 646, 133, 131])).
% 40.17/40.36 tff(648,plain,
% 40.17/40.36 (member(y, unordered_pair(z, z)) <=> member(y, unordered_pair(y, y))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[647])).
% 40.17/40.36 tff(649,plain,
% 40.17/40.36 (member(y, unordered_pair(y, y)) <=> member(y, unordered_pair(z, z))),
% 40.17/40.36 inference(symmetry,[status(thm)],[648])).
% 40.17/40.36 tff(650,plain,
% 40.17/40.36 (member(y, unordered_pair(z, z))),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[425, 649])).
% 40.17/40.36 tff(651,plain,
% 40.17/40.36 (y = z),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[433, 650])).
% 40.17/40.36 tff(652,plain,
% 40.17/40.36 (member(y, universal_class) <=> member(z, universal_class)),
% 40.17/40.36 inference(monotonicity,[status(thm)],[651])).
% 40.17/40.36 tff(653,plain,
% 40.17/40.36 (member(z, universal_class)),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[337, 652])).
% 40.17/40.36 tff(654,plain,
% 40.17/40.36 (z = y),
% 40.17/40.36 inference(symmetry,[status(thm)],[651])).
% 40.17/40.36 tff(655,plain,
% 40.17/40.36 ((singleton(x) = z) <=> (singleton(x) = y)),
% 40.17/40.36 inference(monotonicity,[status(thm)],[654])).
% 40.17/40.36 tff(656,plain,
% 40.17/40.36 ((singleton(x) = y) <=> (singleton(x) = z)),
% 40.17/40.36 inference(symmetry,[status(thm)],[655])).
% 40.17/40.36 tff(657,plain,
% 40.17/40.36 (null_class = singleton(z)),
% 40.17/40.36 inference(symmetry,[status(thm)],[646])).
% 40.17/40.36 tff(658,plain,
% 40.17/40.36 (singleton(y) = singleton(z)),
% 40.17/40.36 inference(transitivity,[status(thm)],[132, 657])).
% 40.17/40.36 tff(659,plain,
% 40.17/40.36 (unordered_pair(y, singleton(y)) = unordered_pair(y, singleton(z))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[658])).
% 40.17/40.36 tff(660,plain,
% 40.17/40.36 (unordered_pair(singleton(y), unordered_pair(y, singleton(y))) = unordered_pair(singleton(y), unordered_pair(y, singleton(z)))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[659])).
% 40.17/40.36 tff(661,plain,
% 40.17/40.36 ((~![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(y))) = ordered_pair(y, y))),
% 40.17/40.36 inference(quant_inst,[status(thm)],[])).
% 40.17/40.36 tff(662,plain,
% 40.17/40.36 (unordered_pair(singleton(y), unordered_pair(y, singleton(y))) = ordered_pair(y, y)),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[661, 19])).
% 40.17/40.36 tff(663,plain,
% 40.17/40.36 (ordered_pair(y, y) = unordered_pair(singleton(y), unordered_pair(y, singleton(y)))),
% 40.17/40.36 inference(symmetry,[status(thm)],[662])).
% 40.17/40.36 tff(664,plain,
% 40.17/40.36 (ordered_pair(y, y) = ordered_pair(w, x)),
% 40.17/40.36 inference(transitivity,[status(thm)],[663, 660, 28, 26])).
% 40.17/40.36 tff(665,plain,
% 40.17/40.36 (member(ordered_pair(y, y), cross_product(universal_class, universal_class)) <=> member(ordered_pair(w, x), cross_product(universal_class, universal_class))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[664])).
% 40.17/40.36 tff(666,plain,
% 40.17/40.36 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | ((~member(y, universal_class)) | member(ordered_pair(y, y), cross_product(universal_class, universal_class)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | (~member(y, universal_class)) | member(ordered_pair(y, y), cross_product(universal_class, universal_class)))),
% 40.17/40.36 inference(rewrite,[status(thm)],[])).
% 40.17/40.36 tff(667,plain,
% 40.17/40.36 (((~member(y, universal_class)) | member(ordered_pair(y, y), cross_product(universal_class, universal_class)) | (~member(y, universal_class))) <=> ((~member(y, universal_class)) | member(ordered_pair(y, y), cross_product(universal_class, universal_class)))),
% 40.17/40.36 inference(rewrite,[status(thm)],[])).
% 40.17/40.36 tff(668,plain,
% 40.17/40.36 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | ((~member(y, universal_class)) | member(ordered_pair(y, y), cross_product(universal_class, universal_class)) | (~member(y, universal_class)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | ((~member(y, universal_class)) | member(ordered_pair(y, y), cross_product(universal_class, universal_class))))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[667])).
% 40.17/40.36 tff(669,plain,
% 40.17/40.36 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | ((~member(y, universal_class)) | member(ordered_pair(y, y), cross_product(universal_class, universal_class)) | (~member(y, universal_class)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | (~member(y, universal_class)) | member(ordered_pair(y, y), cross_product(universal_class, universal_class)))),
% 40.17/40.36 inference(transitivity,[status(thm)],[668, 666])).
% 40.17/40.36 tff(670,plain,
% 40.17/40.36 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | ((~member(y, universal_class)) | member(ordered_pair(y, y), cross_product(universal_class, universal_class)) | (~member(y, universal_class)))),
% 40.17/40.36 inference(quant_inst,[status(thm)],[])).
% 40.17/40.36 tff(671,plain,
% 40.17/40.36 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | (~member(y, universal_class)) | member(ordered_pair(y, y), cross_product(universal_class, universal_class))),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[670, 669])).
% 40.17/40.36 tff(672,plain,
% 40.17/40.36 (member(ordered_pair(y, y), cross_product(universal_class, universal_class))),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[671, 347, 337])).
% 40.17/40.36 tff(673,plain,
% 40.17/40.36 (member(ordered_pair(w, x), cross_product(universal_class, universal_class))),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[672, 665])).
% 40.17/40.36 tff(674,plain,
% 40.17/40.36 (member(w, universal_class)),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[367, 673])).
% 40.17/40.36 tff(675,plain,
% 40.17/40.36 (member(w, unordered_pair(w, singleton(x)))),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[372, 674])).
% 40.17/40.36 tff(676,plain,
% 40.17/40.36 (unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[403, 675, 132])).
% 40.17/40.36 tff(677,plain,
% 40.17/40.36 (singleton(x) = y),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[279, 676, 646, 570])).
% 40.17/40.36 tff(678,plain,
% 40.17/40.36 (singleton(x) = z),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[677, 656])).
% 40.17/40.36 tff(679,plain,
% 40.17/40.36 (z = singleton(x)),
% 40.17/40.36 inference(symmetry,[status(thm)],[678])).
% 40.17/40.36 tff(680,plain,
% 40.17/40.36 (y = singleton(x)),
% 40.17/40.36 inference(transitivity,[status(thm)],[651, 679])).
% 40.17/40.36 tff(681,plain,
% 40.17/40.36 (member(y, singleton(singleton(x))) <=> member(singleton(x), unordered_pair(singleton(x), singleton(x)))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[680, 128])).
% 40.17/40.36 tff(682,plain,
% 40.17/40.36 (member(singleton(x), unordered_pair(singleton(x), singleton(x))) <=> member(y, singleton(singleton(x)))),
% 40.17/40.36 inference(symmetry,[status(thm)],[681])).
% 40.17/40.36 tff(683,plain,
% 40.17/40.36 (member(y, singleton(singleton(x)))),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[597, 682])).
% 40.17/40.36 tff(684,plain,
% 40.17/40.36 (member(ordered_pair(y, z), cross_product(singleton(singleton(x)), universal_class))),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[605, 683, 653])).
% 40.17/40.36 tff(685,plain,
% 40.17/40.36 (member(ordered_pair(w, x), cross_product(singleton(singleton(x)), universal_class))),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[684, 130])).
% 40.17/40.36 tff(686,plain,
% 40.17/40.36 (member(w, singleton(singleton(x)))),
% 40.17/40.36 inference(unit_resolution,[status(thm)],[611, 685])).
% 40.17/40.36 tff(687,plain,
% 40.17/40.36 (member(w, unordered_pair(singleton(x), singleton(x)))),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[686, 129])).
% 40.17/40.36 tff(688,plain,
% 40.17/40.36 ((null_class = singleton(x)) <=> (singleton(x) = null_class)),
% 40.17/40.36 inference(commutativity,[status(thm)],[])).
% 40.17/40.36 tff(689,plain,
% 40.17/40.36 ((singleton(y) = singleton(x)) <=> (null_class = singleton(x))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[132])).
% 40.17/40.36 tff(690,plain,
% 40.17/40.36 ((singleton(y) = singleton(x)) <=> (singleton(x) = null_class)),
% 40.17/40.36 inference(transitivity,[status(thm)],[689, 688])).
% 40.17/40.36 tff(691,plain,
% 40.17/40.36 ((singleton(x) = null_class) <=> (singleton(y) = singleton(x))),
% 40.17/40.36 inference(symmetry,[status(thm)],[690])).
% 40.17/40.36 tff(692,plain,
% 40.17/40.36 ((~(singleton(x) = null_class)) <=> (~(singleton(y) = singleton(x)))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[691])).
% 40.17/40.36 tff(693,plain,
% 40.17/40.36 (~(singleton(y) = singleton(x))),
% 40.17/40.36 inference(modus_ponens,[status(thm)],[570, 692])).
% 40.17/40.36 tff(694,plain,
% 40.17/40.36 (member(singleton(y), unordered_pair(w, singleton(x))) <=> member(singleton(z), unordered_pair(y, singleton(z)))),
% 40.17/40.36 inference(monotonicity,[status(thm)],[658, 676])).
% 40.17/40.36 tff(695,plain,
% 40.17/40.36 (member(singleton(z), unordered_pair(y, singleton(z))) <=> member(singleton(y), unordered_pair(w, singleton(x)))),
% 40.17/40.36 inference(symmetry,[status(thm)],[694])).
% 40.17/40.36 tff(696,plain,
% 40.17/40.36 (member(singleton(y), unordered_pair(w, singleton(x)))),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[447, 695])).
% 40.17/40.37 tff(697,plain,
% 40.17/40.37 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(y) = w) | (~member(singleton(y), unordered_pair(w, singleton(x)))) | (singleton(y) = singleton(x)))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(y) = w) | (~member(singleton(y), unordered_pair(w, singleton(x)))) | (singleton(y) = singleton(x)))),
% 40.17/40.37 inference(rewrite,[status(thm)],[])).
% 40.17/40.37 tff(698,plain,
% 40.17/40.37 (((singleton(y) = singleton(x)) | (singleton(y) = w) | (~member(singleton(y), unordered_pair(w, singleton(x))))) <=> ((singleton(y) = w) | (~member(singleton(y), unordered_pair(w, singleton(x)))) | (singleton(y) = singleton(x)))),
% 40.17/40.37 inference(rewrite,[status(thm)],[])).
% 40.17/40.37 tff(699,plain,
% 40.17/40.37 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(y) = singleton(x)) | (singleton(y) = w) | (~member(singleton(y), unordered_pair(w, singleton(x)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(y) = w) | (~member(singleton(y), unordered_pair(w, singleton(x)))) | (singleton(y) = singleton(x))))),
% 40.17/40.37 inference(monotonicity,[status(thm)],[698])).
% 40.17/40.37 tff(700,plain,
% 40.17/40.37 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(y) = singleton(x)) | (singleton(y) = w) | (~member(singleton(y), unordered_pair(w, singleton(x)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(y) = w) | (~member(singleton(y), unordered_pair(w, singleton(x)))) | (singleton(y) = singleton(x)))),
% 40.17/40.37 inference(transitivity,[status(thm)],[699, 697])).
% 40.17/40.37 tff(701,plain,
% 40.17/40.37 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(y) = singleton(x)) | (singleton(y) = w) | (~member(singleton(y), unordered_pair(w, singleton(x)))))),
% 40.17/40.37 inference(quant_inst,[status(thm)],[])).
% 40.17/40.37 tff(702,plain,
% 40.17/40.37 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(y) = w) | (~member(singleton(y), unordered_pair(w, singleton(x)))) | (singleton(y) = singleton(x))),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[701, 700])).
% 40.17/40.37 tff(703,plain,
% 40.17/40.37 ((singleton(y) = w) | (~member(singleton(y), unordered_pair(w, singleton(x)))) | (singleton(y) = singleton(x))),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[702, 63])).
% 40.17/40.37 tff(704,plain,
% 40.17/40.37 (singleton(y) = w),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[703, 696, 693])).
% 40.17/40.37 tff(705,plain,
% 40.17/40.37 (w = singleton(y)),
% 40.17/40.37 inference(symmetry,[status(thm)],[704])).
% 40.17/40.37 tff(706,plain,
% 40.17/40.37 (w = null_class),
% 40.17/40.37 inference(transitivity,[status(thm)],[705, 132])).
% 40.17/40.37 tff(707,plain,
% 40.17/40.37 ((singleton(x) = w) <=> (singleton(x) = null_class)),
% 40.17/40.37 inference(monotonicity,[status(thm)],[706])).
% 40.17/40.37 tff(708,plain,
% 40.17/40.37 ((singleton(x) = null_class) <=> (singleton(x) = w)),
% 40.17/40.37 inference(symmetry,[status(thm)],[707])).
% 40.17/40.37 tff(709,plain,
% 40.17/40.37 ((~(singleton(x) = null_class)) <=> (~(singleton(x) = w))),
% 40.17/40.37 inference(monotonicity,[status(thm)],[708])).
% 40.17/40.37 tff(710,plain,
% 40.17/40.37 (~(singleton(x) = w)),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[570, 709])).
% 40.17/40.37 tff(711,plain,
% 40.17/40.37 (~member(w, unordered_pair(singleton(x), singleton(x)))),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[641, 710])).
% 40.17/40.37 tff(712,plain,
% 40.17/40.37 ($false),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[711, 687])).
% 40.17/40.37 tff(713,plain,(~(singleton(y) = null_class)), inference(lemma,lemma(discharge,[]))).
% 40.17/40.37 tff(714,plain,
% 40.17/40.37 (member(unordered_pair(y, y), universal_class) <=> member(singleton(y), universal_class)),
% 40.17/40.37 inference(monotonicity,[status(thm)],[9])).
% 40.17/40.37 tff(715,plain,
% 40.17/40.37 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(y, y), universal_class)),
% 40.17/40.37 inference(quant_inst,[status(thm)],[])).
% 40.17/40.37 tff(716,plain,
% 40.17/40.37 (member(unordered_pair(y, y), universal_class)),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[715, 38])).
% 40.17/40.37 tff(717,plain,
% 40.17/40.37 (member(singleton(y), universal_class)),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[716, 714])).
% 40.17/40.37 tff(718,plain,
% 40.17/40.37 (((~![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)))),
% 40.17/40.37 inference(rewrite,[status(thm)],[])).
% 40.17/40.37 tff(719,plain,
% 40.17/40.37 ((~![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)))),
% 40.17/40.37 inference(quant_inst,[status(thm)],[])).
% 40.17/40.37 tff(720,plain,
% 40.17/40.37 ((~![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))),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[719, 718])).
% 40.17/40.37 tff(721,plain,
% 40.17/40.37 ((singleton(y) = null_class) | member(apply(choice, singleton(y)), singleton(y))),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[720, 294, 717])).
% 40.17/40.37 tff(722,plain,
% 40.17/40.37 (member(apply(choice, singleton(y)), singleton(y))),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[721, 713])).
% 40.17/40.37 tff(723,plain,
% 40.17/40.37 (member(apply(choice, singleton(y)), unordered_pair(y, y)) <=> member(apply(choice, singleton(y)), singleton(y))),
% 40.17/40.37 inference(monotonicity,[status(thm)],[9])).
% 40.17/40.37 tff(724,plain,
% 40.17/40.37 (member(apply(choice, singleton(y)), singleton(y)) <=> member(apply(choice, singleton(y)), unordered_pair(y, y))),
% 40.17/40.37 inference(symmetry,[status(thm)],[723])).
% 40.17/40.37 tff(725,assumption,(member(apply(choice, singleton(y)), singleton(y))), introduced(assumption)).
% 40.17/40.37 tff(726,plain,
% 40.17/40.37 (member(apply(choice, singleton(y)), unordered_pair(y, y))),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[725, 724])).
% 40.17/40.37 tff(727,assumption,(~(apply(choice, singleton(y)) = y)), introduced(assumption)).
% 40.17/40.37 tff(728,plain,
% 40.17/40.37 (((~![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))))),
% 40.17/40.37 inference(rewrite,[status(thm)],[])).
% 40.17/40.37 tff(729,plain,
% 40.17/40.37 (((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))))),
% 40.17/40.37 inference(rewrite,[status(thm)],[])).
% 40.17/40.37 tff(730,plain,
% 40.17/40.37 (((~![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)))))),
% 40.17/40.37 inference(monotonicity,[status(thm)],[729])).
% 40.17/40.37 tff(731,plain,
% 40.17/40.37 (((~![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))))),
% 40.17/40.37 inference(transitivity,[status(thm)],[730, 728])).
% 40.17/40.37 tff(732,plain,
% 40.17/40.37 ((~![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))))),
% 40.17/40.37 inference(quant_inst,[status(thm)],[])).
% 40.17/40.37 tff(733,plain,
% 40.17/40.37 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (apply(choice, singleton(y)) = y) | (~member(apply(choice, singleton(y)), unordered_pair(y, y)))),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[732, 731])).
% 40.17/40.37 tff(734,plain,
% 40.17/40.37 (~member(apply(choice, singleton(y)), unordered_pair(y, y))),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[733, 63, 727])).
% 40.17/40.37 tff(735,plain,
% 40.17/40.37 ($false),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[734, 726])).
% 40.17/40.37 tff(736,plain,((~member(apply(choice, singleton(y)), singleton(y))) | (apply(choice, singleton(y)) = y)), inference(lemma,lemma(discharge,[]))).
% 40.17/40.37 tff(737,plain,
% 40.17/40.37 (apply(choice, singleton(y)) = y),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[736, 722])).
% 40.17/40.37 tff(738,plain,
% 40.17/40.37 ((~![Y: $i, Xf: $i] : (sum_class(image(Xf, singleton(Y))) = apply(Xf, Y))) | (sum_class(image(choice, singleton(singleton(y)))) = apply(choice, singleton(y)))),
% 40.17/40.37 inference(quant_inst,[status(thm)],[])).
% 40.17/40.37 tff(739,plain,
% 40.17/40.37 (sum_class(image(choice, singleton(singleton(y)))) = apply(choice, singleton(y))),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[738, 146])).
% 40.17/40.37 tff(740,plain,
% 40.17/40.37 (sum_class(image(choice, singleton(singleton(y)))) = y),
% 40.17/40.37 inference(transitivity,[status(thm)],[739, 737])).
% 40.17/40.37 tff(741,plain,
% 40.17/40.37 (member(sum_class(image(choice, singleton(singleton(y)))), universal_class) <=> member(y, universal_class)),
% 40.17/40.37 inference(monotonicity,[status(thm)],[740])).
% 40.17/40.37 tff(742,plain,
% 40.17/40.37 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(singleton(y), singleton(y)) = singleton(singleton(y)))),
% 40.17/40.37 inference(quant_inst,[status(thm)],[])).
% 40.17/40.37 tff(743,plain,
% 40.17/40.37 (unordered_pair(singleton(y), singleton(y)) = singleton(singleton(y))),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[742, 7])).
% 40.17/40.37 tff(744,plain,
% 40.17/40.37 (member(unordered_pair(singleton(y), singleton(y)), universal_class) <=> member(singleton(singleton(y)), universal_class)),
% 40.17/40.37 inference(monotonicity,[status(thm)],[743])).
% 40.17/40.37 tff(745,plain,
% 40.17/40.37 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(singleton(y), singleton(y)), universal_class)),
% 40.17/40.37 inference(quant_inst,[status(thm)],[])).
% 40.17/40.37 tff(746,plain,
% 40.17/40.37 (member(unordered_pair(singleton(y), singleton(y)), universal_class)),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[745, 38])).
% 40.17/40.37 tff(747,plain,
% 40.17/40.37 (member(singleton(singleton(y)), universal_class)),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[746, 744])).
% 40.17/40.37 tff(748,plain,
% 40.17/40.37 (((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | ((~function(choice)) | member(image(choice, singleton(singleton(y))), universal_class) | (~member(singleton(singleton(y)), universal_class)))) <=> ((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | (~function(choice)) | member(image(choice, singleton(singleton(y))), universal_class) | (~member(singleton(singleton(y)), universal_class)))),
% 40.17/40.37 inference(rewrite,[status(thm)],[])).
% 40.17/40.37 tff(749,plain,
% 40.17/40.37 (((~member(singleton(singleton(y)), universal_class)) | member(image(choice, singleton(singleton(y))), universal_class) | (~function(choice))) <=> ((~function(choice)) | member(image(choice, singleton(singleton(y))), universal_class) | (~member(singleton(singleton(y)), universal_class)))),
% 40.17/40.37 inference(rewrite,[status(thm)],[])).
% 40.17/40.37 tff(750,plain,
% 40.17/40.37 (((~![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(image(choice, singleton(singleton(y))), universal_class) | (~member(singleton(singleton(y)), universal_class))))),
% 40.17/40.37 inference(monotonicity,[status(thm)],[749])).
% 40.17/40.37 tff(751,plain,
% 40.17/40.37 (((~![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(image(choice, singleton(singleton(y))), universal_class) | (~member(singleton(singleton(y)), universal_class)))),
% 40.17/40.37 inference(transitivity,[status(thm)],[750, 748])).
% 40.17/40.37 tff(752,plain,
% 40.17/40.37 ((~![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)))),
% 40.17/40.37 inference(quant_inst,[status(thm)],[])).
% 40.17/40.37 tff(753,plain,
% 40.17/40.37 ((~![Xf: $i, X: $i] : ((~member(X, universal_class)) | member(image(Xf, X), universal_class) | (~function(Xf)))) | (~function(choice)) | member(image(choice, singleton(singleton(y))), universal_class) | (~member(singleton(singleton(y)), universal_class))),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[752, 751])).
% 40.17/40.37 tff(754,plain,
% 40.17/40.37 (member(image(choice, singleton(singleton(y))), universal_class) | (~member(singleton(singleton(y)), universal_class))),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[753, 193, 183])).
% 40.17/40.37 tff(755,plain,
% 40.17/40.37 (member(image(choice, singleton(singleton(y))), universal_class)),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[754, 747])).
% 40.17/40.37 tff(756,plain,
% 40.17/40.37 (((~![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))),
% 40.17/40.37 inference(rewrite,[status(thm)],[])).
% 40.17/40.37 tff(757,plain,
% 40.17/40.37 ((~![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))),
% 40.17/40.37 inference(quant_inst,[status(thm)],[])).
% 40.17/40.37 tff(758,plain,
% 40.17/40.37 ((~![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)),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[757, 756])).
% 40.17/40.37 tff(759,plain,
% 40.17/40.37 ((~member(image(choice, singleton(singleton(y))), universal_class)) | member(sum_class(image(choice, singleton(singleton(y)))), universal_class)),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[758, 175])).
% 40.17/40.37 tff(760,plain,
% 40.17/40.37 (member(sum_class(image(choice, singleton(singleton(y)))), universal_class)),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[759, 755])).
% 40.17/40.37 tff(761,plain,
% 40.17/40.37 (member(y, universal_class)),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[760, 741])).
% 40.17/40.37 tff(762,plain,
% 40.17/40.37 (((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(y, universal_class)) | member(y, unordered_pair(y, singleton(z))))) <=> ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(y, universal_class)) | member(y, unordered_pair(y, singleton(z))))),
% 40.17/40.37 inference(rewrite,[status(thm)],[])).
% 40.17/40.37 tff(763,plain,
% 40.17/40.37 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(y, universal_class)) | member(y, unordered_pair(y, singleton(z))))),
% 40.17/40.37 inference(quant_inst,[status(thm)],[])).
% 40.17/40.37 tff(764,plain,
% 40.17/40.37 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(y, universal_class)) | member(y, unordered_pair(y, singleton(z)))),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[763, 762])).
% 40.17/40.37 tff(765,plain,
% 40.17/40.37 ((~member(y, universal_class)) | member(y, unordered_pair(y, singleton(z)))),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[764, 241])).
% 40.17/40.37 tff(766,plain,
% 40.17/40.37 (member(y, unordered_pair(y, singleton(z)))),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[765, 761])).
% 40.17/40.37 tff(767,plain,
% 40.17/40.37 (member(singleton(x), singleton(w))),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[766, 125])).
% 40.17/40.37 tff(768,plain,
% 40.17/40.37 (member(y, unordered_pair(w, w))),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[767, 100])).
% 40.17/40.37 tff(769,plain,
% 40.17/40.37 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((y = w) | (~member(y, unordered_pair(w, w))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (y = w) | (~member(y, unordered_pair(w, w))))),
% 40.17/40.37 inference(rewrite,[status(thm)],[])).
% 40.17/40.37 tff(770,plain,
% 40.17/40.37 (((y = w) | (y = w) | (~member(y, unordered_pair(w, w)))) <=> ((y = w) | (~member(y, unordered_pair(w, w))))),
% 40.17/40.37 inference(rewrite,[status(thm)],[])).
% 40.17/40.37 tff(771,plain,
% 40.17/40.37 (((~![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))))) | ((y = w) | (~member(y, unordered_pair(w, w)))))),
% 40.17/40.37 inference(monotonicity,[status(thm)],[770])).
% 40.17/40.37 tff(772,plain,
% 40.17/40.37 (((~![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))))) | (y = w) | (~member(y, unordered_pair(w, w))))),
% 40.17/40.37 inference(transitivity,[status(thm)],[771, 769])).
% 40.17/40.37 tff(773,plain,
% 40.17/40.37 ((~![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))))),
% 40.17/40.37 inference(quant_inst,[status(thm)],[])).
% 40.17/40.37 tff(774,plain,
% 40.17/40.37 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (y = w) | (~member(y, unordered_pair(w, w)))),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[773, 772])).
% 40.17/40.37 tff(775,plain,
% 40.17/40.37 ((y = w) | (~member(y, unordered_pair(w, w)))),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[774, 63])).
% 40.17/40.37 tff(776,plain,
% 40.17/40.37 (y = w),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[775, 768])).
% 40.17/40.37 tff(777,plain,
% 40.17/40.37 (w = y),
% 40.17/40.37 inference(symmetry,[status(thm)],[776])).
% 40.17/40.37 tff(778,plain,
% 40.17/40.37 (singleton(w) = unordered_pair(y, singleton(z))),
% 40.17/40.37 inference(symmetry,[status(thm)],[124])).
% 40.17/40.37 tff(779,plain,
% 40.17/40.37 (unordered_pair(w, w) = unordered_pair(y, singleton(z))),
% 40.17/40.37 inference(transitivity,[status(thm)],[11, 778])).
% 40.17/40.37 tff(780,plain,
% 40.17/40.37 (member(singleton(z), unordered_pair(w, w)) <=> member(singleton(z), unordered_pair(y, singleton(z)))),
% 40.17/40.37 inference(monotonicity,[status(thm)],[779])).
% 40.17/40.37 tff(781,plain,
% 40.17/40.37 (member(singleton(z), unordered_pair(y, singleton(z))) <=> member(singleton(z), unordered_pair(w, w))),
% 40.17/40.37 inference(symmetry,[status(thm)],[780])).
% 40.17/40.37 tff(782,plain,
% 40.17/40.37 (member(singleton(z), unordered_pair(w, w))),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[447, 781])).
% 40.17/40.37 tff(783,plain,
% 40.17/40.37 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((~member(singleton(z), unordered_pair(w, w))) | (singleton(z) = w))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(singleton(z), unordered_pair(w, w))) | (singleton(z) = w))),
% 40.17/40.37 inference(rewrite,[status(thm)],[])).
% 40.17/40.37 tff(784,plain,
% 40.17/40.37 (((singleton(z) = w) | (singleton(z) = w) | (~member(singleton(z), unordered_pair(w, w)))) <=> ((~member(singleton(z), unordered_pair(w, w))) | (singleton(z) = w))),
% 40.17/40.37 inference(rewrite,[status(thm)],[])).
% 40.17/40.37 tff(785,plain,
% 40.17/40.37 (((~![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))))) | ((~member(singleton(z), unordered_pair(w, w))) | (singleton(z) = w)))),
% 40.17/40.37 inference(monotonicity,[status(thm)],[784])).
% 40.17/40.37 tff(786,plain,
% 40.17/40.37 (((~![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))))) | (~member(singleton(z), unordered_pair(w, w))) | (singleton(z) = w))),
% 40.17/40.37 inference(transitivity,[status(thm)],[785, 783])).
% 40.17/40.37 tff(787,plain,
% 40.17/40.37 ((~![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))))),
% 40.17/40.37 inference(quant_inst,[status(thm)],[])).
% 40.17/40.37 tff(788,plain,
% 40.17/40.37 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(singleton(z), unordered_pair(w, w))) | (singleton(z) = w)),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[787, 786])).
% 40.17/40.37 tff(789,plain,
% 40.17/40.37 ((~member(singleton(z), unordered_pair(w, w))) | (singleton(z) = w)),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[788, 63])).
% 40.17/40.37 tff(790,plain,
% 40.17/40.37 (singleton(z) = w),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[789, 782])).
% 40.17/40.37 tff(791,plain,
% 40.17/40.37 (singleton(z) = y),
% 40.17/40.37 inference(transitivity,[status(thm)],[790, 777])).
% 40.17/40.37 tff(792,plain,
% 40.17/40.37 (unordered_pair(y, singleton(z)) = unordered_pair(y, y)),
% 40.17/40.37 inference(monotonicity,[status(thm)],[791])).
% 40.17/40.37 tff(793,plain,
% 40.17/40.37 (singleton(w) = singleton(y)),
% 40.17/40.37 inference(transitivity,[status(thm)],[778, 792, 9])).
% 40.17/40.37 tff(794,plain,
% 40.17/40.37 ((singleton(y) = singleton(w)) <=> (singleton(w) = singleton(y))),
% 40.17/40.37 inference(commutativity,[status(thm)],[])).
% 40.17/40.37 tff(795,plain,
% 40.17/40.37 ((unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))) <=> (singleton(y) = singleton(w))),
% 40.17/40.37 inference(monotonicity,[status(thm)],[69, 124])).
% 40.17/40.37 tff(796,plain,
% 40.17/40.37 ((unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))) <=> (singleton(w) = singleton(y))),
% 40.17/40.37 inference(transitivity,[status(thm)],[795, 794])).
% 40.17/40.37 tff(797,plain,
% 40.17/40.37 ((~(unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z)))) <=> (~(singleton(w) = singleton(y)))),
% 40.17/40.37 inference(monotonicity,[status(thm)],[796])).
% 40.17/40.37 tff(798,plain,
% 40.17/40.37 (~(singleton(w) = singleton(y))),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[12, 797])).
% 40.17/40.37 tff(799,plain,
% 40.17/40.37 ($false),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[798, 793])).
% 40.17/40.37 tff(800,plain,(unordered_pair(w, singleton(x)) = unordered_pair(y, singleton(z))), inference(lemma,lemma(discharge,[]))).
% 40.17/40.37 tff(801,plain,
% 40.17/40.37 (member(singleton(z), unordered_pair(w, singleton(x))) <=> member(singleton(z), unordered_pair(y, singleton(z)))),
% 40.17/40.37 inference(monotonicity,[status(thm)],[800])).
% 40.17/40.37 tff(802,plain,
% 40.17/40.37 (member(singleton(z), unordered_pair(y, singleton(z))) <=> member(singleton(z), unordered_pair(w, singleton(x)))),
% 40.17/40.37 inference(symmetry,[status(thm)],[801])).
% 40.17/40.37 tff(803,plain,
% 40.17/40.37 (member(singleton(z), unordered_pair(w, singleton(x)))),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[447, 802])).
% 40.17/40.37 tff(804,plain,
% 40.17/40.37 (y = apply(choice, singleton(y))),
% 40.17/40.37 inference(symmetry,[status(thm)],[737])).
% 40.17/40.37 tff(805,plain,
% 40.17/40.37 ((w = y) <=> (w = apply(choice, singleton(y)))),
% 40.17/40.37 inference(monotonicity,[status(thm)],[804])).
% 40.17/40.37 tff(806,plain,
% 40.17/40.37 ((w = apply(choice, singleton(y))) <=> (w = y)),
% 40.17/40.37 inference(symmetry,[status(thm)],[805])).
% 40.17/40.37 tff(807,plain,
% 40.17/40.37 ((~(w = apply(choice, singleton(y)))) <=> (~(w = y))),
% 40.17/40.37 inference(monotonicity,[status(thm)],[806])).
% 40.17/40.37 tff(808,assumption,(~(singleton(z) = null_class)), introduced(assumption)).
% 40.17/40.37 tff(809,plain,
% 40.17/40.37 ((singleton(z) = null_class) <=> (singleton(z) = singleton(x))),
% 40.17/40.37 inference(monotonicity,[status(thm)],[231])).
% 40.17/40.37 tff(810,plain,
% 40.17/40.37 ((singleton(z) = null_class) <=> (singleton(x) = singleton(z))),
% 40.17/40.37 inference(transitivity,[status(thm)],[809, 617])).
% 40.17/40.37 tff(811,plain,
% 40.17/40.37 ((singleton(x) = singleton(z)) <=> (singleton(z) = null_class)),
% 40.17/40.37 inference(symmetry,[status(thm)],[810])).
% 40.17/40.37 tff(812,plain,
% 40.17/40.37 (singleton(z) = null_class),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[581, 811])).
% 40.17/40.37 tff(813,plain,
% 40.17/40.37 ($false),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[808, 812])).
% 40.17/40.37 tff(814,plain,((~(singleton(x) = singleton(z))) | (singleton(z) = null_class) | (~(singleton(x) = null_class))), inference(lemma,lemma(discharge,[]))).
% 40.17/40.37 tff(815,plain,
% 40.17/40.37 (~(singleton(x) = singleton(z))),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[814, 808, 230])).
% 40.17/40.37 tff(816,plain,
% 40.17/40.37 (~(singleton(z) = singleton(x))),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[815, 619])).
% 40.17/40.37 tff(817,plain,
% 40.17/40.37 (member(w, unordered_pair(y, y)) <=> member(w, singleton(y))),
% 40.17/40.37 inference(monotonicity,[status(thm)],[9])).
% 40.17/40.37 tff(818,plain,
% 40.17/40.37 (member(w, singleton(y)) <=> member(w, unordered_pair(y, y))),
% 40.17/40.37 inference(symmetry,[status(thm)],[817])).
% 40.17/40.37 tff(819,plain,
% 40.17/40.37 (member(apply(choice, singleton(z)), singleton(z))),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[298, 808])).
% 40.17/40.37 tff(820,plain,
% 40.17/40.37 (member(z, universal_class)),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[206, 819])).
% 40.17/40.37 tff(821,plain,
% 40.17/40.37 (member(ordered_pair(y, z), cross_product(universal_class, universal_class)) | (~member(z, universal_class))),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[354, 761])).
% 40.17/40.37 tff(822,plain,
% 40.17/40.37 (member(ordered_pair(y, z), cross_product(universal_class, universal_class))),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[821, 820])).
% 40.17/40.37 tff(823,plain,
% 40.17/40.37 (member(ordered_pair(w, x), cross_product(universal_class, universal_class))),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[822, 208])).
% 40.17/40.37 tff(824,plain,
% 40.17/40.37 (member(w, universal_class)),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[367, 823])).
% 40.17/40.37 tff(825,plain,
% 40.17/40.37 (member(ordered_pair(y, z), cross_product(complement(singleton(y)), universal_class)) <=> member(ordered_pair(w, x), cross_product(complement(singleton(y)), universal_class))),
% 40.17/40.37 inference(monotonicity,[status(thm)],[26])).
% 40.17/40.37 tff(826,plain,
% 40.17/40.37 ((~member(ordered_pair(y, z), cross_product(complement(singleton(y)), universal_class))) <=> (~member(ordered_pair(w, x), cross_product(complement(singleton(y)), universal_class)))),
% 40.17/40.37 inference(monotonicity,[status(thm)],[825])).
% 40.17/40.37 tff(827,plain,
% 40.17/40.37 (member(apply(choice, singleton(y)), complement(singleton(y))) <=> member(y, complement(singleton(y)))),
% 40.17/40.37 inference(monotonicity,[status(thm)],[737])).
% 40.17/40.37 tff(828,plain,
% 40.17/40.37 ((~member(apply(choice, singleton(y)), complement(singleton(y)))) <=> (~member(y, complement(singleton(y))))),
% 40.17/40.37 inference(monotonicity,[status(thm)],[827])).
% 40.17/40.37 tff(829,assumption,(member(apply(choice, singleton(y)), complement(singleton(y)))), introduced(assumption)).
% 40.17/40.37 tff(830,plain,
% 40.17/40.37 (^[Z: $i, X: $i] : refl(((~member(Z, X)) | (~member(Z, complement(X)))) <=> ((~member(Z, X)) | (~member(Z, complement(X)))))),
% 40.17/40.37 inference(bind,[status(th)],[])).
% 40.17/40.37 tff(831,plain,
% 40.17/40.37 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X)))) <=> ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 40.17/40.37 inference(quant_intro,[status(thm)],[830])).
% 40.17/40.37 tff(832,plain,
% 40.17/40.37 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X)))) <=> ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 40.17/40.37 inference(rewrite,[status(thm)],[])).
% 40.17/40.37 tff(833,plain,
% 40.17/40.37 (^[Z: $i, X: $i] : rewrite(((~member(Z, complement(X))) | (~member(Z, X))) <=> ((~member(Z, X)) | (~member(Z, complement(X)))))),
% 40.17/40.37 inference(bind,[status(th)],[])).
% 40.17/40.37 tff(834,plain,
% 40.17/40.37 (![Z: $i, X: $i] : ((~member(Z, complement(X))) | (~member(Z, X))) <=> ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 40.17/40.37 inference(quant_intro,[status(thm)],[833])).
% 40.17/40.37 tff(835,axiom,(![Z: $i, X: $i] : ((~member(Z, complement(X))) | (~member(Z, X)))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','complement1')).
% 40.17/40.37 tff(836,plain,
% 40.17/40.37 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[835, 834])).
% 40.17/40.37 tff(837,plain,
% 40.17/40.37 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[836, 832])).
% 40.17/40.37 tff(838,plain,(
% 40.17/40.37 ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 40.17/40.37 inference(skolemize,[status(sab)],[837])).
% 40.17/40.37 tff(839,plain,
% 40.17/40.37 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[838, 831])).
% 40.17/40.37 tff(840,plain,
% 40.17/40.37 (((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | ((~member(apply(choice, singleton(y)), singleton(y))) | (~member(apply(choice, singleton(y)), complement(singleton(y)))))) <=> ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | (~member(apply(choice, singleton(y)), singleton(y))) | (~member(apply(choice, singleton(y)), complement(singleton(y)))))),
% 40.17/40.37 inference(rewrite,[status(thm)],[])).
% 40.17/40.37 tff(841,plain,
% 40.17/40.37 ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | ((~member(apply(choice, singleton(y)), singleton(y))) | (~member(apply(choice, singleton(y)), complement(singleton(y)))))),
% 40.17/40.37 inference(quant_inst,[status(thm)],[])).
% 40.17/40.37 tff(842,plain,
% 40.17/40.37 ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | (~member(apply(choice, singleton(y)), singleton(y))) | (~member(apply(choice, singleton(y)), complement(singleton(y))))),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[841, 840])).
% 40.17/40.37 tff(843,plain,
% 40.17/40.37 ($false),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[842, 839, 725, 829])).
% 40.17/40.37 tff(844,plain,((~member(apply(choice, singleton(y)), complement(singleton(y)))) | (~member(apply(choice, singleton(y)), singleton(y)))), inference(lemma,lemma(discharge,[]))).
% 40.17/40.37 tff(845,plain,
% 40.17/40.37 (~member(apply(choice, singleton(y)), complement(singleton(y)))),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[844, 722])).
% 40.17/40.37 tff(846,plain,
% 40.17/40.37 (~member(y, complement(singleton(y)))),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[845, 828])).
% 40.17/40.37 tff(847,plain,
% 40.17/40.37 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | ((~member(ordered_pair(y, z), cross_product(complement(singleton(y)), universal_class))) | member(y, complement(singleton(y))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | (~member(ordered_pair(y, z), cross_product(complement(singleton(y)), universal_class))) | member(y, complement(singleton(y))))),
% 40.17/40.37 inference(rewrite,[status(thm)],[])).
% 40.17/40.37 tff(848,plain,
% 40.17/40.37 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | ((~member(ordered_pair(y, z), cross_product(complement(singleton(y)), universal_class))) | member(y, complement(singleton(y))))),
% 40.17/40.37 inference(quant_inst,[status(thm)],[])).
% 40.17/40.37 tff(849,plain,
% 40.17/40.37 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | (~member(ordered_pair(y, z), cross_product(complement(singleton(y)), universal_class))) | member(y, complement(singleton(y)))),
% 40.17/40.37 inference(modus_ponens,[status(thm)],[848, 847])).
% 40.17/40.37 tff(850,plain,
% 40.17/40.37 ((~member(ordered_pair(y, z), cross_product(complement(singleton(y)), universal_class))) | member(y, complement(singleton(y)))),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[849, 363])).
% 40.17/40.37 tff(851,plain,
% 40.17/40.37 (~member(ordered_pair(y, z), cross_product(complement(singleton(y)), universal_class))),
% 40.17/40.37 inference(unit_resolution,[status(thm)],[850, 846])).
% 40.17/40.37 tff(852,plain,
% 40.17/40.37 (~member(ordered_pair(w, x), cross_product(complement(singleton(y)), universal_class))),
% 40.17/40.38 inference(modus_ponens,[status(thm)],[851, 826])).
% 40.17/40.38 tff(853,plain,
% 40.17/40.38 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | ((~member(x, universal_class)) | (~member(w, complement(singleton(y)))) | member(ordered_pair(w, x), cross_product(complement(singleton(y)), universal_class)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | (~member(x, universal_class)) | (~member(w, complement(singleton(y)))) | member(ordered_pair(w, x), cross_product(complement(singleton(y)), universal_class)))),
% 40.17/40.38 inference(rewrite,[status(thm)],[])).
% 40.17/40.38 tff(854,plain,
% 40.17/40.38 (((~member(w, complement(singleton(y)))) | member(ordered_pair(w, x), cross_product(complement(singleton(y)), universal_class)) | (~member(x, universal_class))) <=> ((~member(x, universal_class)) | (~member(w, complement(singleton(y)))) | member(ordered_pair(w, x), cross_product(complement(singleton(y)), universal_class)))),
% 40.17/40.38 inference(rewrite,[status(thm)],[])).
% 40.17/40.38 tff(855,plain,
% 40.17/40.38 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | ((~member(w, complement(singleton(y)))) | member(ordered_pair(w, x), cross_product(complement(singleton(y)), universal_class)) | (~member(x, universal_class)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | ((~member(x, universal_class)) | (~member(w, complement(singleton(y)))) | member(ordered_pair(w, x), cross_product(complement(singleton(y)), universal_class))))),
% 40.17/40.38 inference(monotonicity,[status(thm)],[854])).
% 40.17/40.38 tff(856,plain,
% 40.17/40.38 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | ((~member(w, complement(singleton(y)))) | member(ordered_pair(w, x), cross_product(complement(singleton(y)), universal_class)) | (~member(x, universal_class)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | (~member(x, universal_class)) | (~member(w, complement(singleton(y)))) | member(ordered_pair(w, x), cross_product(complement(singleton(y)), universal_class)))),
% 40.17/40.38 inference(transitivity,[status(thm)],[855, 853])).
% 40.17/40.38 tff(857,plain,
% 40.17/40.38 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | ((~member(w, complement(singleton(y)))) | member(ordered_pair(w, x), cross_product(complement(singleton(y)), universal_class)) | (~member(x, universal_class)))),
% 40.17/40.38 inference(quant_inst,[status(thm)],[])).
% 40.17/40.38 tff(858,plain,
% 40.17/40.38 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(U, X)) | member(ordered_pair(U, V), cross_product(X, Y)) | (~member(V, Y)))) | (~member(x, universal_class)) | (~member(w, complement(singleton(y)))) | member(ordered_pair(w, x), cross_product(complement(singleton(y)), universal_class))),
% 40.17/40.38 inference(modus_ponens,[status(thm)],[857, 856])).
% 40.17/40.38 tff(859,plain,
% 40.17/40.38 ((~member(w, complement(singleton(y)))) | member(ordered_pair(w, x), cross_product(complement(singleton(y)), universal_class))),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[858, 347, 225])).
% 40.17/40.38 tff(860,plain,
% 40.17/40.38 (~member(w, complement(singleton(y)))),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[859, 852])).
% 40.17/40.38 tff(861,plain,
% 40.17/40.38 (^[Z: $i, X: $i] : refl((member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X))) <=> (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X))))),
% 40.17/40.38 inference(bind,[status(th)],[])).
% 40.17/40.38 tff(862,plain,
% 40.17/40.38 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X))) <=> ![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 40.17/40.38 inference(quant_intro,[status(thm)],[861])).
% 40.17/40.38 tff(863,plain,
% 40.17/40.38 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X))) <=> ![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 40.17/40.38 inference(rewrite,[status(thm)],[])).
% 40.17/40.38 tff(864,plain,
% 40.17/40.38 (^[Z: $i, X: $i] : rewrite((((~member(Z, universal_class)) | member(Z, complement(X))) | member(Z, X)) <=> (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X))))),
% 40.17/40.38 inference(bind,[status(th)],[])).
% 40.17/40.38 tff(865,plain,
% 40.17/40.38 (![Z: $i, X: $i] : (((~member(Z, universal_class)) | member(Z, complement(X))) | member(Z, X)) <=> ![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 40.17/40.38 inference(quant_intro,[status(thm)],[864])).
% 40.17/40.38 tff(866,axiom,(![Z: $i, X: $i] : (((~member(Z, universal_class)) | member(Z, complement(X))) | member(Z, X))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','complement2')).
% 40.17/40.38 tff(867,plain,
% 40.17/40.38 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 40.17/40.38 inference(modus_ponens,[status(thm)],[866, 865])).
% 40.17/40.38 tff(868,plain,
% 40.17/40.38 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 40.17/40.38 inference(modus_ponens,[status(thm)],[867, 863])).
% 40.17/40.38 tff(869,plain,(
% 40.17/40.38 ![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 40.17/40.38 inference(skolemize,[status(sab)],[868])).
% 40.17/40.38 tff(870,plain,
% 40.17/40.38 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 40.17/40.38 inference(modus_ponens,[status(thm)],[869, 862])).
% 40.17/40.38 tff(871,plain,
% 40.17/40.38 (((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | ((~member(w, universal_class)) | member(w, complement(singleton(y))) | member(w, singleton(y)))) <=> ((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | (~member(w, universal_class)) | member(w, complement(singleton(y))) | member(w, singleton(y)))),
% 40.17/40.38 inference(rewrite,[status(thm)],[])).
% 40.17/40.38 tff(872,plain,
% 40.17/40.38 ((member(w, singleton(y)) | (~member(w, universal_class)) | member(w, complement(singleton(y)))) <=> ((~member(w, universal_class)) | member(w, complement(singleton(y))) | member(w, singleton(y)))),
% 40.17/40.38 inference(rewrite,[status(thm)],[])).
% 40.17/40.38 tff(873,plain,
% 40.17/40.38 (((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | (member(w, singleton(y)) | (~member(w, universal_class)) | member(w, complement(singleton(y))))) <=> ((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | ((~member(w, universal_class)) | member(w, complement(singleton(y))) | member(w, singleton(y))))),
% 40.17/40.38 inference(monotonicity,[status(thm)],[872])).
% 40.17/40.38 tff(874,plain,
% 40.17/40.38 (((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | (member(w, singleton(y)) | (~member(w, universal_class)) | member(w, complement(singleton(y))))) <=> ((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | (~member(w, universal_class)) | member(w, complement(singleton(y))) | member(w, singleton(y)))),
% 40.17/40.38 inference(transitivity,[status(thm)],[873, 871])).
% 40.17/40.38 tff(875,plain,
% 40.17/40.38 ((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | (member(w, singleton(y)) | (~member(w, universal_class)) | member(w, complement(singleton(y))))),
% 40.17/40.38 inference(quant_inst,[status(thm)],[])).
% 40.17/40.38 tff(876,plain,
% 40.17/40.38 ((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | (~member(w, universal_class)) | member(w, complement(singleton(y))) | member(w, singleton(y))),
% 40.17/40.38 inference(modus_ponens,[status(thm)],[875, 874])).
% 40.17/40.38 tff(877,plain,
% 40.17/40.38 ((~member(w, universal_class)) | member(w, singleton(y))),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[876, 870, 860])).
% 40.17/40.38 tff(878,plain,
% 40.17/40.38 (member(w, singleton(y))),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[877, 824])).
% 40.17/40.38 tff(879,plain,
% 40.17/40.38 (member(w, unordered_pair(y, y))),
% 40.17/40.38 inference(modus_ponens,[status(thm)],[878, 818])).
% 40.17/40.38 tff(880,plain,
% 40.17/40.38 (w = y),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[552, 879])).
% 40.17/40.38 tff(881,plain,
% 40.17/40.38 ((singleton(z) = w) <=> (singleton(z) = y)),
% 40.17/40.38 inference(monotonicity,[status(thm)],[880])).
% 40.17/40.38 tff(882,plain,
% 40.17/40.38 ((singleton(z) = y) <=> (singleton(z) = w)),
% 40.17/40.38 inference(symmetry,[status(thm)],[881])).
% 40.17/40.38 tff(883,plain,
% 40.17/40.38 ((y = singleton(z)) <=> (singleton(z) = y)),
% 40.17/40.38 inference(commutativity,[status(thm)],[])).
% 40.17/40.38 tff(884,plain,
% 40.17/40.38 (unordered_pair(y, singleton(z)) = unordered_pair(w, singleton(x))),
% 40.17/40.38 inference(symmetry,[status(thm)],[800])).
% 40.17/40.38 tff(885,plain,
% 40.17/40.38 (member(singleton(x), unordered_pair(y, singleton(z))) <=> member(singleton(x), unordered_pair(w, singleton(x)))),
% 40.17/40.38 inference(monotonicity,[status(thm)],[884])).
% 40.17/40.38 tff(886,plain,
% 40.17/40.38 (member(singleton(x), unordered_pair(w, singleton(x))) <=> member(singleton(x), unordered_pair(y, singleton(z)))),
% 40.17/40.38 inference(symmetry,[status(thm)],[885])).
% 40.17/40.38 tff(887,plain,
% 40.17/40.38 (member(singleton(x), unordered_pair(y, singleton(z)))),
% 40.17/40.38 inference(modus_ponens,[status(thm)],[85, 886])).
% 40.17/40.38 tff(888,plain,
% 40.17/40.38 (singleton(x) = y),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[589, 815, 887])).
% 40.17/40.38 tff(889,plain,
% 40.17/40.38 ((singleton(x) = singleton(z)) <=> (y = singleton(z))),
% 40.17/40.38 inference(monotonicity,[status(thm)],[888])).
% 40.17/40.38 tff(890,plain,
% 40.17/40.38 ((singleton(x) = singleton(z)) <=> (singleton(z) = y)),
% 40.17/40.38 inference(transitivity,[status(thm)],[889, 883])).
% 40.17/40.38 tff(891,plain,
% 40.17/40.38 ((singleton(x) = singleton(z)) <=> (singleton(z) = w)),
% 40.17/40.38 inference(transitivity,[status(thm)],[890, 882])).
% 40.17/40.38 tff(892,plain,
% 40.17/40.38 ((~(singleton(x) = singleton(z))) <=> (~(singleton(z) = w))),
% 40.17/40.38 inference(monotonicity,[status(thm)],[891])).
% 40.17/40.38 tff(893,plain,
% 40.17/40.38 (~(singleton(z) = w)),
% 40.17/40.38 inference(modus_ponens,[status(thm)],[815, 892])).
% 40.17/40.38 tff(894,plain,
% 40.17/40.38 ($false),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[627, 893, 816, 803])).
% 40.17/40.38 tff(895,plain,((singleton(z) = null_class) | (~(singleton(x) = null_class))), inference(lemma,lemma(discharge,[]))).
% 40.17/40.38 tff(896,plain,
% 40.17/40.38 (singleton(z) = null_class),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[895, 230])).
% 40.17/40.38 tff(897,plain,
% 40.17/40.38 (unordered_pair(z, z) = unordered_pair(x, x)),
% 40.17/40.38 inference(transitivity,[status(thm)],[151, 896, 231, 229])).
% 40.17/40.38 tff(898,plain,
% 40.17/40.38 (member(x, unordered_pair(z, z)) <=> member(x, unordered_pair(x, x))),
% 40.17/40.38 inference(monotonicity,[status(thm)],[897])).
% 40.17/40.38 tff(899,plain,
% 40.17/40.38 (member(x, unordered_pair(x, x)) <=> member(x, unordered_pair(z, z))),
% 40.17/40.38 inference(symmetry,[status(thm)],[898])).
% 40.17/40.38 tff(900,plain,
% 40.17/40.38 (member(x, unordered_pair(z, z))),
% 40.17/40.38 inference(modus_ponens,[status(thm)],[245, 899])).
% 40.17/40.38 tff(901,plain,
% 40.17/40.38 (((~![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))))),
% 40.17/40.38 inference(rewrite,[status(thm)],[])).
% 40.17/40.38 tff(902,plain,
% 40.17/40.38 (((x = z) | (x = z) | (~member(x, unordered_pair(z, z)))) <=> ((x = z) | (~member(x, unordered_pair(z, z))))),
% 40.17/40.38 inference(rewrite,[status(thm)],[])).
% 40.17/40.38 tff(903,plain,
% 40.17/40.38 (((~![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)))))),
% 40.17/40.38 inference(monotonicity,[status(thm)],[902])).
% 40.17/40.38 tff(904,plain,
% 40.17/40.38 (((~![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))))),
% 40.17/40.38 inference(transitivity,[status(thm)],[903, 901])).
% 40.17/40.38 tff(905,plain,
% 40.17/40.38 ((~![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))))),
% 40.17/40.38 inference(quant_inst,[status(thm)],[])).
% 40.17/40.38 tff(906,plain,
% 40.17/40.38 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (x = z) | (~member(x, unordered_pair(z, z)))),
% 40.17/40.38 inference(modus_ponens,[status(thm)],[905, 904])).
% 40.17/40.38 tff(907,plain,
% 40.17/40.38 (~member(x, unordered_pair(z, z))),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[906, 63, 334])).
% 40.17/40.38 tff(908,plain,
% 40.17/40.38 ($false),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[907, 900])).
% 40.17/40.38 tff(909,plain,(~(singleton(x) = null_class)), inference(lemma,lemma(discharge,[]))).
% 40.17/40.38 tff(910,plain,
% 40.17/40.38 (member(apply(choice, singleton(x)), singleton(x))),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[319, 909])).
% 40.17/40.38 tff(911,plain,
% 40.17/40.38 (apply(choice, singleton(x)) = x),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[577, 910])).
% 40.17/40.38 tff(912,plain,
% 40.17/40.38 ((singleton(x) = apply(choice, singleton(y))) <=> (singleton(x) = y)),
% 40.17/40.38 inference(monotonicity,[status(thm)],[737])).
% 40.17/40.38 tff(913,plain,
% 40.17/40.38 ((singleton(x) = y) <=> (singleton(x) = apply(choice, singleton(y)))),
% 40.17/40.38 inference(symmetry,[status(thm)],[912])).
% 40.17/40.38 tff(914,plain,
% 40.17/40.38 (~(singleton(x) = singleton(z))),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[587, 579, 911])).
% 40.17/40.38 tff(915,plain,
% 40.17/40.38 (singleton(x) = y),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[589, 914, 887])).
% 40.17/40.38 tff(916,plain,
% 40.17/40.38 (singleton(x) = apply(choice, singleton(y))),
% 40.17/40.38 inference(modus_ponens,[status(thm)],[915, 913])).
% 40.17/40.38 tff(917,assumption,(apply(choice, singleton(y)) = y), introduced(assumption)).
% 40.17/40.38 tff(918,assumption,(singleton(x) = apply(choice, singleton(y))), introduced(assumption)).
% 40.17/40.38 tff(919,plain,
% 40.17/40.38 (singleton(x) = y),
% 40.17/40.38 inference(transitivity,[status(thm)],[918, 917])).
% 40.17/40.38 tff(920,plain,
% 40.17/40.38 (singleton(singleton(x)) = singleton(y)),
% 40.17/40.38 inference(monotonicity,[status(thm)],[919])).
% 40.17/40.38 tff(921,plain,
% 40.17/40.38 (singleton(y) = singleton(singleton(x))),
% 40.17/40.38 inference(symmetry,[status(thm)],[920])).
% 40.17/40.38 tff(922,plain,
% 40.17/40.38 (unordered_pair(apply(choice, singleton(y)), apply(choice, singleton(y))) = unordered_pair(y, y)),
% 40.17/40.38 inference(monotonicity,[status(thm)],[917, 917])).
% 40.17/40.38 tff(923,plain,
% 40.17/40.38 (unordered_pair(apply(choice, singleton(y)), apply(choice, singleton(y))) = singleton(singleton(x))),
% 40.17/40.38 inference(transitivity,[status(thm)],[922, 9, 921])).
% 40.17/40.38 tff(924,plain,
% 40.17/40.38 (member(singleton(z), unordered_pair(apply(choice, singleton(y)), apply(choice, singleton(y)))) <=> member(singleton(z), singleton(singleton(x)))),
% 40.17/40.38 inference(monotonicity,[status(thm)],[923])).
% 40.17/40.38 tff(925,plain,
% 40.17/40.38 (member(singleton(z), singleton(singleton(x))) <=> member(singleton(z), unordered_pair(apply(choice, singleton(y)), apply(choice, singleton(y))))),
% 40.17/40.38 inference(symmetry,[status(thm)],[924])).
% 40.17/40.38 tff(926,assumption,(w = apply(choice, singleton(y))), introduced(assumption)).
% 40.17/40.38 tff(927,plain,
% 40.17/40.38 (w = y),
% 40.17/40.38 inference(transitivity,[status(thm)],[926, 917])).
% 40.17/40.38 tff(928,plain,
% 40.17/40.38 (unordered_pair(w, singleton(x)) = unordered_pair(y, y)),
% 40.17/40.38 inference(monotonicity,[status(thm)],[927, 919])).
% 40.17/40.38 tff(929,plain,
% 40.17/40.38 (unordered_pair(y, y) = unordered_pair(w, singleton(x))),
% 40.17/40.38 inference(symmetry,[status(thm)],[928])).
% 40.17/40.38 tff(930,plain,
% 40.17/40.38 (singleton(y) = unordered_pair(w, singleton(x))),
% 40.17/40.38 inference(transitivity,[status(thm)],[131, 929])).
% 40.17/40.38 tff(931,plain,
% 40.17/40.38 (singleton(w) = singleton(y)),
% 40.17/40.38 inference(monotonicity,[status(thm)],[927])).
% 40.17/40.38 tff(932,plain,
% 40.17/40.38 (singleton(y) = singleton(w)),
% 40.17/40.38 inference(symmetry,[status(thm)],[931])).
% 40.17/40.38 tff(933,plain,
% 40.17/40.38 (unordered_pair(singleton(y), singleton(y)) = unordered_pair(singleton(w), unordered_pair(w, singleton(x)))),
% 40.17/40.38 inference(monotonicity,[status(thm)],[932, 930])).
% 40.17/40.38 tff(934,plain,
% 40.17/40.38 (unordered_pair(singleton(y), singleton(y)) = unordered_pair(singleton(y), unordered_pair(y, singleton(z)))),
% 40.17/40.38 inference(transitivity,[status(thm)],[933, 21, 25, 107])).
% 40.17/40.38 tff(935,plain,
% 40.17/40.38 (member(unordered_pair(y, singleton(z)), unordered_pair(singleton(y), singleton(y))) <=> member(unordered_pair(y, singleton(z)), unordered_pair(singleton(y), unordered_pair(y, singleton(z))))),
% 40.17/40.38 inference(monotonicity,[status(thm)],[934])).
% 40.17/40.38 tff(936,plain,
% 40.17/40.38 (member(unordered_pair(y, singleton(z)), unordered_pair(singleton(y), unordered_pair(y, singleton(z)))) <=> member(unordered_pair(y, singleton(z)), unordered_pair(singleton(y), singleton(y)))),
% 40.17/40.38 inference(symmetry,[status(thm)],[935])).
% 40.17/40.38 tff(937,plain,
% 40.17/40.38 (member(unordered_pair(y, singleton(z)), unordered_pair(singleton(y), singleton(y)))),
% 40.17/40.38 inference(modus_ponens,[status(thm)],[117, 936])).
% 40.17/40.38 tff(938,plain,
% 40.17/40.38 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((~member(unordered_pair(y, singleton(z)), unordered_pair(singleton(y), singleton(y)))) | (unordered_pair(y, singleton(z)) = singleton(y)))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(unordered_pair(y, singleton(z)), unordered_pair(singleton(y), singleton(y)))) | (unordered_pair(y, singleton(z)) = singleton(y)))),
% 40.17/40.38 inference(rewrite,[status(thm)],[])).
% 40.17/40.38 tff(939,plain,
% 40.17/40.38 (((unordered_pair(y, singleton(z)) = singleton(y)) | (unordered_pair(y, singleton(z)) = singleton(y)) | (~member(unordered_pair(y, singleton(z)), unordered_pair(singleton(y), singleton(y))))) <=> ((~member(unordered_pair(y, singleton(z)), unordered_pair(singleton(y), singleton(y)))) | (unordered_pair(y, singleton(z)) = singleton(y)))),
% 40.17/40.38 inference(rewrite,[status(thm)],[])).
% 40.17/40.38 tff(940,plain,
% 40.17/40.38 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((unordered_pair(y, singleton(z)) = singleton(y)) | (unordered_pair(y, singleton(z)) = singleton(y)) | (~member(unordered_pair(y, singleton(z)), unordered_pair(singleton(y), singleton(y)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((~member(unordered_pair(y, singleton(z)), unordered_pair(singleton(y), singleton(y)))) | (unordered_pair(y, singleton(z)) = singleton(y))))),
% 40.17/40.38 inference(monotonicity,[status(thm)],[939])).
% 40.17/40.38 tff(941,plain,
% 40.17/40.38 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((unordered_pair(y, singleton(z)) = singleton(y)) | (unordered_pair(y, singleton(z)) = singleton(y)) | (~member(unordered_pair(y, singleton(z)), unordered_pair(singleton(y), singleton(y)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(unordered_pair(y, singleton(z)), unordered_pair(singleton(y), singleton(y)))) | (unordered_pair(y, singleton(z)) = singleton(y)))),
% 40.17/40.38 inference(transitivity,[status(thm)],[940, 938])).
% 40.17/40.38 tff(942,plain,
% 40.17/40.38 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((unordered_pair(y, singleton(z)) = singleton(y)) | (unordered_pair(y, singleton(z)) = singleton(y)) | (~member(unordered_pair(y, singleton(z)), unordered_pair(singleton(y), singleton(y)))))),
% 40.17/40.38 inference(quant_inst,[status(thm)],[])).
% 40.17/40.38 tff(943,plain,
% 40.17/40.38 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(unordered_pair(y, singleton(z)), unordered_pair(singleton(y), singleton(y)))) | (unordered_pair(y, singleton(z)) = singleton(y))),
% 40.17/40.38 inference(modus_ponens,[status(thm)],[942, 941])).
% 40.17/40.38 tff(944,plain,
% 40.17/40.38 ((~member(unordered_pair(y, singleton(z)), unordered_pair(singleton(y), singleton(y)))) | (unordered_pair(y, singleton(z)) = singleton(y))),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[943, 63])).
% 40.17/40.38 tff(945,plain,
% 40.17/40.38 (unordered_pair(y, singleton(z)) = singleton(y)),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[944, 937])).
% 40.17/40.38 tff(946,plain,
% 40.17/40.38 (unordered_pair(y, singleton(z)) = singleton(singleton(x))),
% 40.17/40.38 inference(transitivity,[status(thm)],[945, 921])).
% 40.17/40.38 tff(947,plain,
% 40.17/40.38 (member(singleton(z), unordered_pair(y, singleton(z))) <=> member(singleton(z), singleton(singleton(x)))),
% 40.17/40.38 inference(monotonicity,[status(thm)],[946])).
% 40.17/40.38 tff(948,plain,
% 40.17/40.38 (member(singleton(z), singleton(singleton(x)))),
% 40.17/40.38 inference(modus_ponens,[status(thm)],[447, 947])).
% 40.17/40.38 tff(949,plain,
% 40.17/40.38 (member(singleton(z), unordered_pair(apply(choice, singleton(y)), apply(choice, singleton(y))))),
% 40.17/40.38 inference(modus_ponens,[status(thm)],[948, 925])).
% 40.17/40.38 tff(950,plain,
% 40.17/40.38 (((~![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))))))),
% 40.17/40.38 inference(rewrite,[status(thm)],[])).
% 40.17/40.38 tff(951,plain,
% 40.17/40.38 (((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))))))),
% 40.17/40.38 inference(rewrite,[status(thm)],[])).
% 40.17/40.38 tff(952,plain,
% 40.17/40.38 (((~![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)))))))),
% 40.17/40.38 inference(monotonicity,[status(thm)],[951])).
% 40.17/40.38 tff(953,plain,
% 40.17/40.38 (((~![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))))))),
% 40.17/40.38 inference(transitivity,[status(thm)],[952, 950])).
% 40.17/40.38 tff(954,plain,
% 40.17/40.38 ((~![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))))))),
% 40.17/40.38 inference(quant_inst,[status(thm)],[])).
% 40.17/40.38 tff(955,plain,
% 40.17/40.38 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(z) = apply(choice, singleton(y))) | (~member(singleton(z), unordered_pair(apply(choice, singleton(y)), apply(choice, singleton(y)))))),
% 40.17/40.38 inference(modus_ponens,[status(thm)],[954, 953])).
% 40.17/40.38 tff(956,plain,
% 40.17/40.38 ((singleton(z) = apply(choice, singleton(y))) | (~member(singleton(z), unordered_pair(apply(choice, singleton(y)), apply(choice, singleton(y)))))),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[955, 63])).
% 40.17/40.38 tff(957,plain,
% 40.17/40.38 (singleton(z) = apply(choice, singleton(y))),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[956, 949])).
% 40.17/40.38 tff(958,plain,
% 40.17/40.38 (apply(choice, singleton(z)) = apply(choice, apply(choice, singleton(y)))),
% 40.17/40.38 inference(monotonicity,[status(thm)],[957])).
% 40.17/40.38 tff(959,plain,
% 40.17/40.38 (apply(choice, apply(choice, singleton(y))) = apply(choice, singleton(z))),
% 40.17/40.38 inference(symmetry,[status(thm)],[958])).
% 40.17/40.38 tff(960,plain,
% 40.17/40.38 (apply(choice, singleton(y)) = singleton(x)),
% 40.17/40.38 inference(symmetry,[status(thm)],[918])).
% 40.17/40.38 tff(961,plain,
% 40.17/40.38 (apply(choice, apply(choice, singleton(y))) = apply(choice, singleton(x))),
% 40.17/40.38 inference(monotonicity,[status(thm)],[960])).
% 40.17/40.38 tff(962,plain,
% 40.17/40.38 (apply(choice, singleton(x)) = apply(choice, apply(choice, singleton(y)))),
% 40.17/40.38 inference(symmetry,[status(thm)],[961])).
% 40.17/40.38 tff(963,plain,
% 40.17/40.38 (x = z),
% 40.17/40.38 inference(transitivity,[status(thm)],[584, 962, 959, 580])).
% 40.17/40.38 tff(964,plain,
% 40.17/40.38 ($false),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[334, 963])).
% 40.17/40.38 tff(965,plain,((~(singleton(x) = apply(choice, singleton(y)))) | (~(apply(choice, singleton(z)) = z)) | (~(apply(choice, singleton(x)) = x)) | (~(apply(choice, singleton(y)) = y)) | (~(w = apply(choice, singleton(y))))), inference(lemma,lemma(discharge,[]))).
% 40.17/40.38 tff(966,plain,
% 40.17/40.38 ((~(singleton(x) = apply(choice, singleton(y)))) | (~(apply(choice, singleton(z)) = z)) | (~(apply(choice, singleton(x)) = x)) | (~(w = apply(choice, singleton(y))))),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[965, 737])).
% 40.17/40.38 tff(967,plain,
% 40.17/40.38 (~(w = apply(choice, singleton(y)))),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[966, 916, 911, 579])).
% 40.17/40.38 tff(968,plain,
% 40.17/40.38 (~(w = y)),
% 40.17/40.38 inference(modus_ponens,[status(thm)],[967, 807])).
% 40.17/40.38 tff(969,plain,
% 40.17/40.38 (member(ordered_pair(y, z), cross_product(universal_class, universal_class))),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[821, 207])).
% 40.17/40.38 tff(970,plain,
% 40.17/40.38 (member(ordered_pair(w, x), cross_product(universal_class, universal_class))),
% 40.17/40.38 inference(modus_ponens,[status(thm)],[969, 208])).
% 40.17/40.38 tff(971,plain,
% 40.17/40.38 (member(w, universal_class)),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[367, 970])).
% 40.17/40.38 tff(972,plain,
% 40.17/40.38 (member(w, singleton(y))),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[877, 971])).
% 40.17/40.38 tff(973,plain,
% 40.17/40.38 (member(w, unordered_pair(y, y))),
% 40.17/40.38 inference(modus_ponens,[status(thm)],[972, 818])).
% 40.17/40.38 tff(974,plain,
% 40.17/40.38 ($false),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[552, 973, 968])).
% 40.17/40.38 tff(975,plain,(~member(apply(choice, singleton(z)), singleton(z))), inference(lemma,lemma(discharge,[]))).
% 40.17/40.38 tff(976,plain,
% 40.17/40.38 (singleton(z) = null_class),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[298, 975])).
% 40.17/40.38 tff(977,plain,
% 40.17/40.38 (null_class = singleton(z)),
% 40.17/40.38 inference(symmetry,[status(thm)],[976])).
% 40.17/40.38 tff(978,plain,
% 40.17/40.38 ((singleton(x) = null_class) <=> (singleton(x) = singleton(z))),
% 40.17/40.38 inference(monotonicity,[status(thm)],[977])).
% 40.17/40.38 tff(979,plain,
% 40.17/40.38 ((singleton(x) = null_class) <=> (singleton(z) = singleton(x))),
% 40.17/40.38 inference(transitivity,[status(thm)],[978, 618])).
% 40.17/40.38 tff(980,plain,
% 40.17/40.38 ((~(singleton(x) = null_class)) <=> (~(singleton(z) = singleton(x)))),
% 40.17/40.38 inference(monotonicity,[status(thm)],[979])).
% 40.17/40.38 tff(981,plain,
% 40.17/40.38 (~(singleton(z) = singleton(x))),
% 40.17/40.38 inference(modus_ponens,[status(thm)],[909, 980])).
% 40.17/40.38 tff(982,plain,
% 40.17/40.38 (singleton(z) = w),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[627, 981, 803])).
% 40.17/40.38 tff(983,plain,
% 40.17/40.38 (w = singleton(z)),
% 40.17/40.38 inference(symmetry,[status(thm)],[982])).
% 40.17/40.38 tff(984,plain,
% 40.17/40.38 (member(w, universal_class) <=> member(singleton(z), universal_class)),
% 40.17/40.38 inference(monotonicity,[status(thm)],[983])).
% 40.17/40.38 tff(985,plain,
% 40.17/40.38 (member(singleton(z), universal_class) <=> member(w, universal_class)),
% 40.17/40.38 inference(symmetry,[status(thm)],[984])).
% 40.17/40.38 tff(986,plain,
% 40.17/40.38 (member(w, universal_class)),
% 40.17/40.38 inference(modus_ponens,[status(thm)],[284, 985])).
% 40.17/40.38 tff(987,plain,
% 40.17/40.38 (member(singleton(z), singleton(y)) <=> member(w, singleton(y))),
% 40.17/40.38 inference(monotonicity,[status(thm)],[982])).
% 40.17/40.38 tff(988,plain,
% 40.17/40.38 ((~member(singleton(z), singleton(y))) <=> (~member(w, singleton(y)))),
% 40.17/40.38 inference(monotonicity,[status(thm)],[987])).
% 40.17/40.38 tff(989,plain,
% 40.17/40.38 (singleton(x) = y),
% 40.17/40.38 inference(unit_resolution,[status(thm)],[279, 976, 800, 909])).
% 40.17/40.38 tff(990,plain,
% 40.17/40.38 (y = singleton(x)),
% 40.17/40.38 inference(symmetry,[status(thm)],[989])).
% 40.17/40.38 tff(991,plain,
% 40.17/40.38 ((singleton(z) = y) <=> (singleton(z) = singleton(x))),
% 40.17/40.38 inference(monotonicity,[status(thm)],[990])).
% 40.17/40.38 tff(992,plain,
% 40.17/40.38 ((singleton(z) = y) <=> (singleton(x) = singleton(z))),
% 40.17/40.38 inference(transitivity,[status(thm)],[991, 617])).
% 40.17/40.38 tff(993,plain,
% 40.17/40.38 ((singleton(x) = singleton(z)) <=> (singleton(z) = y)),
% 40.17/40.38 inference(symmetry,[status(thm)],[992])).
% 40.17/40.38 tff(994,plain,
% 40.17/40.38 ((~(singleton(x) = singleton(z))) <=> (~(singleton(z) = y))),
% 40.17/40.38 inference(monotonicity,[status(thm)],[993])).
% 40.17/40.38 tff(995,plain,
% 40.17/40.38 ((~(singleton(x) = null_class)) <=> (~(singleton(x) = singleton(z)))),
% 40.17/40.38 inference(monotonicity,[status(thm)],[978])).
% 40.17/40.38 tff(996,plain,
% 40.17/40.38 (~(singleton(x) = singleton(z))),
% 40.17/40.38 inference(modus_ponens,[status(thm)],[909, 995])).
% 40.17/40.38 tff(997,plain,
% 40.17/40.38 (~(singleton(z) = y)),
% 40.17/40.38 inference(modus_ponens,[status(thm)],[996, 994])).
% 40.17/40.38 tff(998,plain,
% 40.17/40.38 (member(singleton(z), unordered_pair(y, y)) <=> member(singleton(z), singleton(y))),
% 40.17/40.38 inference(monotonicity,[status(thm)],[9])).
% 40.17/40.38 tff(999,plain,
% 40.17/40.38 (member(singleton(z), singleton(y)) <=> member(singleton(z), unordered_pair(y, y))),
% 40.17/40.38 inference(symmetry,[status(thm)],[998])).
% 40.17/40.38 tff(1000,assumption,(member(singleton(z), singleton(y))), introduced(assumption)).
% 40.17/40.38 tff(1001,plain,
% 40.17/40.38 (member(singleton(z), unordered_pair(y, y))),
% 40.17/40.38 inference(modus_ponens,[status(thm)],[1000, 999])).
% 40.17/40.38 tff(1002,assumption,(~(singleton(z) = y)), introduced(assumption)).
% 40.17/40.38 tff(1003,plain,
% 40.17/40.38 (((~![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))))),
% 40.17/40.38 inference(rewrite,[status(thm)],[])).
% 40.17/40.38 tff(1004,plain,
% 40.17/40.38 (((singleton(z) = y) | (singleton(z) = y) | (~member(singleton(z), unordered_pair(y, y)))) <=> ((singleton(z) = y) | (~member(singleton(z), unordered_pair(y, y))))),
% 40.17/40.38 inference(rewrite,[status(thm)],[])).
% 40.17/40.38 tff(1005,plain,
% 40.17/40.38 (((~![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)))))),
% 40.17/40.40 inference(monotonicity,[status(thm)],[1004])).
% 40.17/40.40 tff(1006,plain,
% 40.17/40.40 (((~![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))))),
% 40.17/40.40 inference(transitivity,[status(thm)],[1005, 1003])).
% 40.17/40.40 tff(1007,plain,
% 40.17/40.40 ((~![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))))),
% 40.17/40.40 inference(quant_inst,[status(thm)],[])).
% 40.17/40.40 tff(1008,plain,
% 40.17/40.40 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(z) = y) | (~member(singleton(z), unordered_pair(y, y)))),
% 40.17/40.40 inference(modus_ponens,[status(thm)],[1007, 1006])).
% 40.17/40.40 tff(1009,plain,
% 40.17/40.40 (~member(singleton(z), unordered_pair(y, y))),
% 40.17/40.40 inference(unit_resolution,[status(thm)],[1008, 63, 1002])).
% 40.17/40.40 tff(1010,plain,
% 40.17/40.40 ($false),
% 40.17/40.40 inference(unit_resolution,[status(thm)],[1009, 1001])).
% 40.17/40.40 tff(1011,plain,((~member(singleton(z), singleton(y))) | (singleton(z) = y)), inference(lemma,lemma(discharge,[]))).
% 40.17/40.40 tff(1012,plain,
% 40.17/40.40 (~member(singleton(z), singleton(y))),
% 40.17/40.40 inference(unit_resolution,[status(thm)],[1011, 997])).
% 40.17/40.40 tff(1013,plain,
% 40.17/40.40 (~member(w, singleton(y))),
% 40.17/40.40 inference(modus_ponens,[status(thm)],[1012, 988])).
% 40.17/40.40 tff(1014,plain,
% 40.17/40.40 ($false),
% 40.17/40.40 inference(unit_resolution,[status(thm)],[877, 1013, 986])).
% 40.17/40.40 % SZS output end Proof
% 40.17/40.40 % E exiting
%------------------------------------------------------------------------------