%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET021-6 : TPTP v9.0.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n007.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 05:24:51 AM UTC 2025
% Result : Unsatisfiable 16.21s 16.44s
% Output : Proof 16.31s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.11 % Problem : SET021-6 : TPTP v9.0.0. Bugfixed v2.1.0.
% 0.10/0.11 % Command : run_E %s %d THM
% 0.10/0.32 % Computer : n007.cluster.edu
% 0.10/0.32 % Model : x86_64 x86_64
% 0.10/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.32 % Memory : 8042.1875MB
% 0.10/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.10/0.32 % CPULimit : 300
% 0.10/0.32 % WCLimit : 300
% 0.10/0.32 % DateTime : Fri Jun 20 04:03:20 EDT 2025
% 0.10/0.32 % CPUTime :
% 16.21/16.44 % SZS status Unsatisfiable
% 16.21/16.44 % SZS output start Proof
% 16.21/16.44 tff(singleton_type, type, (
% 16.21/16.44 singleton: $i > $i)).
% 16.21/16.44 tff(second_type, type, (
% 16.21/16.44 second: $i > $i)).
% 16.21/16.44 tff(ordered_pair_type, type, (
% 16.21/16.44 ordered_pair: ( $i * $i ) > $i)).
% 16.21/16.44 tff(v_type, type, (
% 16.21/16.44 v: $i)).
% 16.21/16.44 tff(u_type, type, (
% 16.21/16.44 u: $i)).
% 16.21/16.44 tff(member_type, type, (
% 16.21/16.44 member: ( $i * $i ) > $o)).
% 16.21/16.44 tff(unordered_pair_type, type, (
% 16.21/16.44 unordered_pair: ( $i * $i ) > $i)).
% 16.21/16.44 tff(universal_class_type, type, (
% 16.21/16.44 universal_class: $i)).
% 16.21/16.44 tff(cross_product_type, type, (
% 16.21/16.44 cross_product: ( $i * $i ) > $i)).
% 16.21/16.44 tff(first_type, type, (
% 16.21/16.44 first: $i > $i)).
% 16.21/16.44 tff(1,plain,
% 16.21/16.44 ((singleton(v) = singleton(second(ordered_pair(u, v)))) <=> (singleton(second(ordered_pair(u, v))) = singleton(v))),
% 16.21/16.44 inference(commutativity,[status(thm)],[])).
% 16.21/16.44 tff(2,plain,
% 16.21/16.44 ((singleton(second(ordered_pair(u, v))) = singleton(v)) <=> (singleton(v) = singleton(second(ordered_pair(u, v))))),
% 16.21/16.44 inference(symmetry,[status(thm)],[1])).
% 16.21/16.44 tff(3,plain,
% 16.21/16.44 ((~(singleton(second(ordered_pair(u, v))) = singleton(v))) <=> (~(singleton(v) = singleton(second(ordered_pair(u, v)))))),
% 16.21/16.44 inference(monotonicity,[status(thm)],[2])).
% 16.21/16.44 tff(4,plain,
% 16.21/16.44 (^[X: $i] : refl((unordered_pair(X, X) = singleton(X)) <=> (unordered_pair(X, X) = singleton(X)))),
% 16.21/16.44 inference(bind,[status(th)],[])).
% 16.21/16.44 tff(5,plain,
% 16.21/16.44 (![X: $i] : (unordered_pair(X, X) = singleton(X)) <=> ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 16.21/16.44 inference(quant_intro,[status(thm)],[4])).
% 16.21/16.44 tff(6,plain,
% 16.21/16.44 (![X: $i] : (unordered_pair(X, X) = singleton(X)) <=> ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 16.21/16.44 inference(rewrite,[status(thm)],[])).
% 16.21/16.44 tff(7,axiom,(![X: $i] : (unordered_pair(X, X) = singleton(X))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','singleton_set')).
% 16.21/16.44 tff(8,plain,
% 16.21/16.44 (![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 16.21/16.44 inference(modus_ponens,[status(thm)],[7, 6])).
% 16.21/16.44 tff(9,plain,(
% 16.21/16.44 ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 16.21/16.44 inference(skolemize,[status(sab)],[8])).
% 16.21/16.44 tff(10,plain,
% 16.21/16.44 (![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 16.21/16.44 inference(modus_ponens,[status(thm)],[9, 5])).
% 16.21/16.44 tff(11,plain,
% 16.21/16.44 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v))) = singleton(second(ordered_pair(u, v))))),
% 16.21/16.44 inference(quant_inst,[status(thm)],[])).
% 16.21/16.44 tff(12,plain,
% 16.21/16.44 (unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v))) = singleton(second(ordered_pair(u, v)))),
% 16.21/16.44 inference(unit_resolution,[status(thm)],[11, 10])).
% 16.21/16.44 tff(13,plain,
% 16.21/16.44 (singleton(second(ordered_pair(u, v))) = unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v)))),
% 16.21/16.44 inference(symmetry,[status(thm)],[12])).
% 16.21/16.44 tff(14,assumption,(singleton(second(ordered_pair(u, v))) = singleton(v)), introduced(assumption)).
% 16.21/16.44 tff(15,plain,
% 16.21/16.44 (singleton(v) = singleton(second(ordered_pair(u, v)))),
% 16.21/16.44 inference(symmetry,[status(thm)],[14])).
% 16.21/16.44 tff(16,plain,
% 16.21/16.44 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(v, v) = singleton(v))),
% 16.21/16.44 inference(quant_inst,[status(thm)],[])).
% 16.21/16.44 tff(17,plain,
% 16.21/16.44 (unordered_pair(v, v) = singleton(v)),
% 16.21/16.44 inference(unit_resolution,[status(thm)],[16, 10])).
% 16.21/16.44 tff(18,plain,
% 16.21/16.44 (unordered_pair(v, v) = unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v)))),
% 16.21/16.44 inference(transitivity,[status(thm)],[17, 15, 13])).
% 16.21/16.44 tff(19,plain,
% 16.21/16.44 (member(second(ordered_pair(u, v)), unordered_pair(v, v)) <=> member(second(ordered_pair(u, v)), unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v))))),
% 16.21/16.44 inference(monotonicity,[status(thm)],[18])).
% 16.21/16.44 tff(20,plain,
% 16.21/16.44 (member(second(ordered_pair(u, v)), unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v)))) <=> member(second(ordered_pair(u, v)), unordered_pair(v, v))),
% 16.21/16.44 inference(symmetry,[status(thm)],[19])).
% 16.21/16.44 tff(21,plain,
% 16.21/16.44 (member(ordered_pair(u, v), cross_product(universal_class, universal_class)) <=> member(ordered_pair(u, v), cross_product(universal_class, universal_class))),
% 16.21/16.44 inference(rewrite,[status(thm)],[])).
% 16.21/16.44 tff(22,axiom,(member(ordered_pair(u, v), cross_product(universal_class, universal_class))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_unique_1st_and_2nd_in_pair_of_sets2_1')).
% 16.21/16.44 tff(23,plain,
% 16.21/16.44 (member(ordered_pair(u, v), cross_product(universal_class, universal_class))),
% 16.21/16.44 inference(modus_ponens,[status(thm)],[22, 21])).
% 16.21/16.44 tff(24,plain,
% 16.21/16.44 (^[Z: $i, Y: $i, X: $i] : refl(((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z)) <=> ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z)))),
% 16.21/16.44 inference(bind,[status(th)],[])).
% 16.21/16.44 tff(25,plain,
% 16.21/16.44 (![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z)) <=> ![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))),
% 16.21/16.44 inference(quant_intro,[status(thm)],[24])).
% 16.21/16.44 tff(26,plain,
% 16.21/16.44 (![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z)) <=> ![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))),
% 16.21/16.44 inference(rewrite,[status(thm)],[])).
% 16.21/16.44 tff(27,axiom,(![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','cartesian_product4')).
% 16.21/16.44 tff(28,plain,
% 16.21/16.44 (![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))),
% 16.21/16.44 inference(modus_ponens,[status(thm)],[27, 26])).
% 16.21/16.44 tff(29,plain,(
% 16.21/16.44 ![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))),
% 16.21/16.44 inference(skolemize,[status(sab)],[28])).
% 16.21/16.44 tff(30,plain,
% 16.21/16.44 (![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))),
% 16.21/16.44 inference(modus_ponens,[status(thm)],[29, 25])).
% 16.21/16.44 tff(31,plain,
% 16.21/16.44 (((~![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))) | ((~member(ordered_pair(u, v), cross_product(universal_class, universal_class))) | (ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))) = ordered_pair(u, v)))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))) | (~member(ordered_pair(u, v), cross_product(universal_class, universal_class))) | (ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))) = ordered_pair(u, v)))),
% 16.21/16.44 inference(rewrite,[status(thm)],[])).
% 16.21/16.44 tff(32,plain,
% 16.21/16.44 ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))) | ((~member(ordered_pair(u, v), cross_product(universal_class, universal_class))) | (ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))) = ordered_pair(u, v)))),
% 16.21/16.44 inference(quant_inst,[status(thm)],[])).
% 16.21/16.44 tff(33,plain,
% 16.21/16.44 ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, cross_product(X, Y))) | (ordered_pair(first(Z), second(Z)) = Z))) | (~member(ordered_pair(u, v), cross_product(universal_class, universal_class))) | (ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))) = ordered_pair(u, v))),
% 16.21/16.44 inference(modus_ponens,[status(thm)],[32, 31])).
% 16.21/16.44 tff(34,plain,
% 16.21/16.44 (ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))) = ordered_pair(u, v)),
% 16.21/16.44 inference(unit_resolution,[status(thm)],[33, 30, 23])).
% 16.21/16.44 tff(35,plain,
% 16.21/16.44 (member(ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), cross_product(universal_class, universal_class)) <=> member(ordered_pair(u, v), cross_product(universal_class, universal_class))),
% 16.21/16.44 inference(monotonicity,[status(thm)],[34])).
% 16.21/16.44 tff(36,plain,
% 16.21/16.44 (member(ordered_pair(u, v), cross_product(universal_class, universal_class)) <=> member(ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), cross_product(universal_class, universal_class))),
% 16.21/16.44 inference(symmetry,[status(thm)],[35])).
% 16.21/16.44 tff(37,plain,
% 16.21/16.44 (member(ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), cross_product(universal_class, universal_class))),
% 16.21/16.44 inference(modus_ponens,[status(thm)],[23, 36])).
% 16.21/16.44 tff(38,plain,
% 16.21/16.44 (^[V: $i, Y: $i, U: $i, X: $i] : refl(((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y)) <=> ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y)))),
% 16.21/16.44 inference(bind,[status(th)],[])).
% 16.21/16.44 tff(39,plain,
% 16.21/16.44 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y)) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))),
% 16.21/16.44 inference(quant_intro,[status(thm)],[38])).
% 16.21/16.44 tff(40,plain,
% 16.21/16.44 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y)) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))),
% 16.21/16.44 inference(rewrite,[status(thm)],[])).
% 16.21/16.44 tff(41,axiom,(![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','cartesian_product2')).
% 16.21/16.44 tff(42,plain,
% 16.21/16.44 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))),
% 16.21/16.44 inference(modus_ponens,[status(thm)],[41, 40])).
% 16.21/16.44 tff(43,plain,(
% 16.21/16.44 ![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))),
% 16.21/16.44 inference(skolemize,[status(sab)],[42])).
% 16.21/16.44 tff(44,plain,
% 16.21/16.44 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))),
% 16.21/16.44 inference(modus_ponens,[status(thm)],[43, 39])).
% 16.21/16.44 tff(45,plain,
% 16.21/16.44 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))) | ((~member(ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), cross_product(universal_class, universal_class))) | member(second(ordered_pair(u, v)), universal_class))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))) | (~member(ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), cross_product(universal_class, universal_class))) | member(second(ordered_pair(u, v)), universal_class))),
% 16.21/16.44 inference(rewrite,[status(thm)],[])).
% 16.21/16.44 tff(46,plain,
% 16.21/16.44 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))) | ((~member(ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), cross_product(universal_class, universal_class))) | member(second(ordered_pair(u, v)), universal_class))),
% 16.21/16.44 inference(quant_inst,[status(thm)],[])).
% 16.21/16.44 tff(47,plain,
% 16.21/16.44 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(V, Y))) | (~member(ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), cross_product(universal_class, universal_class))) | member(second(ordered_pair(u, v)), universal_class)),
% 16.21/16.44 inference(modus_ponens,[status(thm)],[46, 45])).
% 16.21/16.44 tff(48,plain,
% 16.21/16.44 ((~member(ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), cross_product(universal_class, universal_class))) | member(second(ordered_pair(u, v)), universal_class)),
% 16.21/16.44 inference(unit_resolution,[status(thm)],[47, 44])).
% 16.21/16.44 tff(49,plain,
% 16.21/16.44 (member(second(ordered_pair(u, v)), universal_class)),
% 16.21/16.44 inference(unit_resolution,[status(thm)],[48, 37])).
% 16.21/16.44 tff(50,plain,
% 16.21/16.44 (^[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))))),
% 16.21/16.44 inference(bind,[status(th)],[])).
% 16.21/16.44 tff(51,plain,
% 16.21/16.44 (![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)))),
% 16.21/16.44 inference(quant_intro,[status(thm)],[50])).
% 16.21/16.44 tff(52,plain,
% 16.21/16.44 (![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)))),
% 16.21/16.44 inference(rewrite,[status(thm)],[])).
% 16.21/16.44 tff(53,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')).
% 16.21/16.44 tff(54,plain,
% 16.21/16.44 (![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 16.21/16.44 inference(modus_ponens,[status(thm)],[53, 52])).
% 16.21/16.44 tff(55,plain,(
% 16.21/16.44 ![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 16.21/16.44 inference(skolemize,[status(sab)],[54])).
% 16.21/16.44 tff(56,plain,
% 16.21/16.44 (![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 16.21/16.44 inference(modus_ponens,[status(thm)],[55, 51])).
% 16.21/16.44 tff(57,plain,
% 16.21/16.44 (((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(second(ordered_pair(u, v)), universal_class)) | member(second(ordered_pair(u, v)), unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v)))))) <=> ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(second(ordered_pair(u, v)), universal_class)) | member(second(ordered_pair(u, v)), unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v)))))),
% 16.21/16.44 inference(rewrite,[status(thm)],[])).
% 16.21/16.44 tff(58,plain,
% 16.21/16.44 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(second(ordered_pair(u, v)), universal_class)) | member(second(ordered_pair(u, v)), unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v)))))),
% 16.21/16.44 inference(quant_inst,[status(thm)],[])).
% 16.21/16.44 tff(59,plain,
% 16.21/16.44 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(second(ordered_pair(u, v)), universal_class)) | member(second(ordered_pair(u, v)), unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v))))),
% 16.21/16.44 inference(modus_ponens,[status(thm)],[58, 57])).
% 16.21/16.44 tff(60,plain,
% 16.21/16.44 (member(second(ordered_pair(u, v)), unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v))))),
% 16.21/16.44 inference(unit_resolution,[status(thm)],[59, 56, 49])).
% 16.21/16.44 tff(61,plain,
% 16.21/16.44 (member(second(ordered_pair(u, v)), unordered_pair(v, v))),
% 16.21/16.44 inference(modus_ponens,[status(thm)],[60, 20])).
% 16.21/16.44 tff(62,plain,
% 16.21/16.44 ((~(second(ordered_pair(u, v)) = v)) <=> (~(second(ordered_pair(u, v)) = v))),
% 16.21/16.44 inference(rewrite,[status(thm)],[])).
% 16.21/16.44 tff(63,axiom,(~(second(ordered_pair(u, v)) = v)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_unique_1st_and_2nd_in_pair_of_sets2_2')).
% 16.21/16.44 tff(64,plain,
% 16.21/16.44 (~(second(ordered_pair(u, v)) = v)),
% 16.21/16.44 inference(modus_ponens,[status(thm)],[63, 62])).
% 16.21/16.44 tff(65,plain,
% 16.21/16.44 (^[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)))))),
% 16.21/16.44 inference(bind,[status(th)],[])).
% 16.21/16.44 tff(66,plain,
% 16.21/16.44 (![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))))),
% 16.21/16.44 inference(quant_intro,[status(thm)],[65])).
% 16.21/16.44 tff(67,plain,
% 16.21/16.44 (![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))))),
% 16.21/16.44 inference(rewrite,[status(thm)],[])).
% 16.21/16.44 tff(68,plain,
% 16.21/16.44 (^[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)))))),
% 16.21/16.44 inference(bind,[status(th)],[])).
% 16.21/16.44 tff(69,plain,
% 16.21/16.44 (![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))))),
% 16.21/16.44 inference(quant_intro,[status(thm)],[68])).
% 16.21/16.44 tff(70,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')).
% 16.21/16.44 tff(71,plain,
% 16.21/16.44 (![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 16.21/16.44 inference(modus_ponens,[status(thm)],[70, 69])).
% 16.21/16.44 tff(72,plain,
% 16.21/16.44 (![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 16.21/16.44 inference(modus_ponens,[status(thm)],[71, 67])).
% 16.21/16.44 tff(73,plain,(
% 16.21/16.45 ![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 16.21/16.45 inference(skolemize,[status(sab)],[72])).
% 16.21/16.45 tff(74,plain,
% 16.21/16.45 (![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))),
% 16.21/16.45 inference(modus_ponens,[status(thm)],[73, 66])).
% 16.21/16.45 tff(75,plain,
% 16.21/16.45 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((second(ordered_pair(u, v)) = v) | (~member(second(ordered_pair(u, v)), unordered_pair(v, v))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (second(ordered_pair(u, v)) = v) | (~member(second(ordered_pair(u, v)), unordered_pair(v, v))))),
% 16.21/16.45 inference(rewrite,[status(thm)],[])).
% 16.21/16.45 tff(76,plain,
% 16.21/16.45 (((second(ordered_pair(u, v)) = v) | (second(ordered_pair(u, v)) = v) | (~member(second(ordered_pair(u, v)), unordered_pair(v, v)))) <=> ((second(ordered_pair(u, v)) = v) | (~member(second(ordered_pair(u, v)), unordered_pair(v, v))))),
% 16.21/16.45 inference(rewrite,[status(thm)],[])).
% 16.21/16.45 tff(77,plain,
% 16.21/16.45 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((second(ordered_pair(u, v)) = v) | (second(ordered_pair(u, v)) = v) | (~member(second(ordered_pair(u, v)), unordered_pair(v, v))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((second(ordered_pair(u, v)) = v) | (~member(second(ordered_pair(u, v)), unordered_pair(v, v)))))),
% 16.21/16.45 inference(monotonicity,[status(thm)],[76])).
% 16.21/16.45 tff(78,plain,
% 16.21/16.45 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((second(ordered_pair(u, v)) = v) | (second(ordered_pair(u, v)) = v) | (~member(second(ordered_pair(u, v)), unordered_pair(v, v))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (second(ordered_pair(u, v)) = v) | (~member(second(ordered_pair(u, v)), unordered_pair(v, v))))),
% 16.21/16.45 inference(transitivity,[status(thm)],[77, 75])).
% 16.21/16.45 tff(79,plain,
% 16.21/16.45 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((second(ordered_pair(u, v)) = v) | (second(ordered_pair(u, v)) = v) | (~member(second(ordered_pair(u, v)), unordered_pair(v, v))))),
% 16.21/16.45 inference(quant_inst,[status(thm)],[])).
% 16.21/16.45 tff(80,plain,
% 16.21/16.45 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (second(ordered_pair(u, v)) = v) | (~member(second(ordered_pair(u, v)), unordered_pair(v, v)))),
% 16.21/16.45 inference(modus_ponens,[status(thm)],[79, 78])).
% 16.21/16.45 tff(81,plain,
% 16.21/16.45 (~member(second(ordered_pair(u, v)), unordered_pair(v, v))),
% 16.21/16.45 inference(unit_resolution,[status(thm)],[80, 74, 64])).
% 16.21/16.45 tff(82,plain,
% 16.21/16.45 ($false),
% 16.21/16.45 inference(unit_resolution,[status(thm)],[81, 61])).
% 16.21/16.45 tff(83,plain,(~(singleton(second(ordered_pair(u, v))) = singleton(v))), inference(lemma,lemma(discharge,[]))).
% 16.21/16.45 tff(84,plain,
% 16.21/16.45 (~(singleton(v) = singleton(second(ordered_pair(u, v))))),
% 16.21/16.45 inference(modus_ponens,[status(thm)],[83, 3])).
% 16.21/16.45 tff(85,plain,
% 16.21/16.45 ((unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = unordered_pair(u, singleton(v))) <=> (unordered_pair(u, singleton(v)) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))),
% 16.21/16.45 inference(commutativity,[status(thm)],[])).
% 16.21/16.45 tff(86,plain,
% 16.21/16.45 ((~(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = unordered_pair(u, singleton(v)))) <=> (~(unordered_pair(u, singleton(v)) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))),
% 16.21/16.45 inference(monotonicity,[status(thm)],[85])).
% 16.21/16.45 tff(87,assumption,(~(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = unordered_pair(u, singleton(v)))), introduced(assumption)).
% 16.21/16.45 tff(88,plain,
% 16.21/16.45 (~(unordered_pair(u, singleton(v)) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))),
% 16.21/16.45 inference(modus_ponens,[status(thm)],[87, 86])).
% 16.21/16.45 tff(89,plain,
% 16.21/16.45 ((singleton(first(ordered_pair(u, v))) = singleton(u)) <=> (singleton(u) = singleton(first(ordered_pair(u, v))))),
% 16.21/16.45 inference(commutativity,[status(thm)],[])).
% 16.21/16.45 tff(90,plain,
% 16.21/16.45 ((singleton(u) = singleton(first(ordered_pair(u, v)))) <=> (singleton(first(ordered_pair(u, v))) = singleton(u))),
% 16.21/16.45 inference(symmetry,[status(thm)],[89])).
% 16.21/16.45 tff(91,assumption,(~(singleton(first(ordered_pair(u, v))) = singleton(u))), introduced(assumption)).
% 16.21/16.45 tff(92,plain,
% 16.21/16.45 (^[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)))),
% 16.21/16.45 inference(bind,[status(th)],[])).
% 16.21/16.45 tff(93,plain,
% 16.21/16.45 (![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))),
% 16.21/16.45 inference(quant_intro,[status(thm)],[92])).
% 16.21/16.45 tff(94,plain,
% 16.21/16.45 (![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))),
% 16.21/16.45 inference(rewrite,[status(thm)],[])).
% 16.21/16.45 tff(95,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')).
% 16.21/16.45 tff(96,plain,
% 16.21/16.45 (![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))),
% 16.21/16.45 inference(modus_ponens,[status(thm)],[95, 94])).
% 16.21/16.45 tff(97,plain,(
% 16.21/16.45 ![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))),
% 16.21/16.45 inference(skolemize,[status(sab)],[96])).
% 16.21/16.45 tff(98,plain,
% 16.21/16.45 (![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))),
% 16.21/16.45 inference(modus_ponens,[status(thm)],[97, 93])).
% 16.21/16.45 tff(99,plain,
% 16.21/16.45 ((~![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))) | (unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) = ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))))),
% 16.21/16.45 inference(quant_inst,[status(thm)],[])).
% 16.21/16.45 tff(100,plain,
% 16.21/16.45 (unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) = ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v)))),
% 16.21/16.45 inference(unit_resolution,[status(thm)],[99, 98])).
% 16.21/16.45 tff(101,plain,
% 16.21/16.45 (ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))) = unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))),
% 16.21/16.45 inference(symmetry,[status(thm)],[100])).
% 16.21/16.45 tff(102,plain,
% 16.21/16.45 (ordered_pair(u, v) = ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v)))),
% 16.21/16.45 inference(symmetry,[status(thm)],[34])).
% 16.21/16.45 tff(103,plain,
% 16.21/16.45 ((~![Y: $i, X: $i] : (unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y))) | (unordered_pair(singleton(u), unordered_pair(u, singleton(v))) = ordered_pair(u, v))),
% 16.21/16.45 inference(quant_inst,[status(thm)],[])).
% 16.21/16.45 tff(104,plain,
% 16.21/16.45 (unordered_pair(singleton(u), unordered_pair(u, singleton(v))) = ordered_pair(u, v)),
% 16.21/16.45 inference(unit_resolution,[status(thm)],[103, 98])).
% 16.21/16.45 tff(105,plain,
% 16.21/16.45 (unordered_pair(singleton(u), unordered_pair(u, singleton(v))) = unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))),
% 16.21/16.45 inference(transitivity,[status(thm)],[104, 102, 101])).
% 16.21/16.45 tff(106,plain,
% 16.21/16.45 (member(singleton(first(ordered_pair(u, v))), unordered_pair(singleton(u), unordered_pair(u, singleton(v)))) <=> member(singleton(first(ordered_pair(u, v))), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))),
% 16.21/16.45 inference(monotonicity,[status(thm)],[105])).
% 16.21/16.45 tff(107,plain,
% 16.21/16.45 (member(singleton(first(ordered_pair(u, v))), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))) <=> member(singleton(first(ordered_pair(u, v))), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))),
% 16.21/16.45 inference(symmetry,[status(thm)],[106])).
% 16.21/16.45 tff(108,plain,
% 16.21/16.45 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))) = singleton(first(ordered_pair(u, v))))),
% 16.21/16.45 inference(quant_inst,[status(thm)],[])).
% 16.21/16.45 tff(109,plain,
% 16.21/16.45 (unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))) = singleton(first(ordered_pair(u, v)))),
% 16.21/16.45 inference(unit_resolution,[status(thm)],[108, 10])).
% 16.21/16.45 tff(110,plain,
% 16.21/16.45 (member(unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))), universal_class) <=> member(singleton(first(ordered_pair(u, v))), universal_class)),
% 16.21/16.45 inference(monotonicity,[status(thm)],[109])).
% 16.21/16.45 tff(111,plain,
% 16.21/16.45 (^[Y: $i, X: $i] : refl(member(unordered_pair(X, Y), universal_class) <=> member(unordered_pair(X, Y), universal_class))),
% 16.21/16.45 inference(bind,[status(th)],[])).
% 16.21/16.45 tff(112,plain,
% 16.21/16.45 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class) <=> ![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 16.21/16.45 inference(quant_intro,[status(thm)],[111])).
% 16.21/16.45 tff(113,plain,
% 16.21/16.45 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class) <=> ![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 16.21/16.45 inference(rewrite,[status(thm)],[])).
% 16.21/16.45 tff(114,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')).
% 16.21/16.45 tff(115,plain,
% 16.21/16.45 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 16.21/16.45 inference(modus_ponens,[status(thm)],[114, 113])).
% 16.21/16.45 tff(116,plain,(
% 16.21/16.45 ![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 16.21/16.45 inference(skolemize,[status(sab)],[115])).
% 16.21/16.45 tff(117,plain,
% 16.21/16.45 (![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)),
% 16.21/16.45 inference(modus_ponens,[status(thm)],[116, 112])).
% 16.21/16.45 tff(118,plain,
% 16.21/16.45 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))), universal_class)),
% 16.21/16.45 inference(quant_inst,[status(thm)],[])).
% 16.21/16.45 tff(119,plain,
% 16.21/16.45 (member(unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))), universal_class)),
% 16.21/16.45 inference(unit_resolution,[status(thm)],[118, 117])).
% 16.21/16.45 tff(120,plain,
% 16.21/16.45 (member(singleton(first(ordered_pair(u, v))), universal_class)),
% 16.21/16.45 inference(modus_ponens,[status(thm)],[119, 110])).
% 16.21/16.45 tff(121,plain,
% 16.21/16.45 (((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(singleton(first(ordered_pair(u, v))), universal_class)) | member(singleton(first(ordered_pair(u, v))), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))))) <=> ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(singleton(first(ordered_pair(u, v))), universal_class)) | member(singleton(first(ordered_pair(u, v))), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))))),
% 16.21/16.45 inference(rewrite,[status(thm)],[])).
% 16.21/16.45 tff(122,plain,
% 16.21/16.45 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(singleton(first(ordered_pair(u, v))), universal_class)) | member(singleton(first(ordered_pair(u, v))), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))))),
% 16.21/16.45 inference(quant_inst,[status(thm)],[])).
% 16.21/16.45 tff(123,plain,
% 16.21/16.45 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(singleton(first(ordered_pair(u, v))), universal_class)) | member(singleton(first(ordered_pair(u, v))), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))),
% 16.21/16.45 inference(modus_ponens,[status(thm)],[122, 121])).
% 16.21/16.45 tff(124,plain,
% 16.21/16.45 ((~member(singleton(first(ordered_pair(u, v))), universal_class)) | member(singleton(first(ordered_pair(u, v))), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))),
% 16.21/16.45 inference(unit_resolution,[status(thm)],[123, 56])).
% 16.21/16.45 tff(125,plain,
% 16.21/16.45 (member(singleton(first(ordered_pair(u, v))), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))),
% 16.21/16.45 inference(unit_resolution,[status(thm)],[124, 120])).
% 16.21/16.45 tff(126,plain,
% 16.21/16.45 (member(singleton(first(ordered_pair(u, v))), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))),
% 16.21/16.45 inference(modus_ponens,[status(thm)],[125, 107])).
% 16.21/16.45 tff(127,plain,
% 16.21/16.45 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(first(ordered_pair(u, v))) = unordered_pair(u, singleton(v))) | (singleton(first(ordered_pair(u, v))) = singleton(u)) | (~member(singleton(first(ordered_pair(u, v))), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(first(ordered_pair(u, v))) = unordered_pair(u, singleton(v))) | (singleton(first(ordered_pair(u, v))) = singleton(u)) | (~member(singleton(first(ordered_pair(u, v))), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))))),
% 16.21/16.45 inference(rewrite,[status(thm)],[])).
% 16.21/16.45 tff(128,plain,
% 16.21/16.45 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(first(ordered_pair(u, v))) = unordered_pair(u, singleton(v))) | (singleton(first(ordered_pair(u, v))) = singleton(u)) | (~member(singleton(first(ordered_pair(u, v))), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))))),
% 16.21/16.45 inference(quant_inst,[status(thm)],[])).
% 16.21/16.45 tff(129,plain,
% 16.21/16.45 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(first(ordered_pair(u, v))) = unordered_pair(u, singleton(v))) | (singleton(first(ordered_pair(u, v))) = singleton(u)) | (~member(singleton(first(ordered_pair(u, v))), unordered_pair(singleton(u), unordered_pair(u, singleton(v)))))),
% 16.21/16.45 inference(modus_ponens,[status(thm)],[128, 127])).
% 16.21/16.45 tff(130,plain,
% 16.21/16.45 ((singleton(first(ordered_pair(u, v))) = unordered_pair(u, singleton(v))) | (singleton(first(ordered_pair(u, v))) = singleton(u)) | (~member(singleton(first(ordered_pair(u, v))), unordered_pair(singleton(u), unordered_pair(u, singleton(v)))))),
% 16.21/16.45 inference(unit_resolution,[status(thm)],[129, 74])).
% 16.21/16.45 tff(131,plain,
% 16.21/16.45 ((singleton(first(ordered_pair(u, v))) = unordered_pair(u, singleton(v))) | (singleton(first(ordered_pair(u, v))) = singleton(u))),
% 16.21/16.45 inference(unit_resolution,[status(thm)],[130, 126])).
% 16.21/16.45 tff(132,plain,
% 16.21/16.45 (singleton(first(ordered_pair(u, v))) = unordered_pair(u, singleton(v))),
% 16.21/16.45 inference(unit_resolution,[status(thm)],[131, 91])).
% 16.21/16.45 tff(133,plain,
% 16.21/16.45 (unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))) = unordered_pair(u, singleton(v))),
% 16.21/16.45 inference(transitivity,[status(thm)],[109, 132])).
% 16.21/16.45 tff(134,plain,
% 16.21/16.45 (member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v)))) <=> member(u, unordered_pair(u, singleton(v)))),
% 16.21/16.45 inference(monotonicity,[status(thm)],[133])).
% 16.21/16.45 tff(135,plain,
% 16.21/16.45 (member(u, unordered_pair(u, singleton(v))) <=> member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))))),
% 16.21/16.45 inference(symmetry,[status(thm)],[134])).
% 16.21/16.45 tff(136,plain,
% 16.21/16.45 (^[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)))),
% 16.21/16.45 inference(bind,[status(th)],[])).
% 16.21/16.45 tff(137,plain,
% 16.21/16.45 (![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))),
% 16.21/16.45 inference(quant_intro,[status(thm)],[136])).
% 16.21/16.45 tff(138,plain,
% 16.21/16.45 (![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))),
% 16.21/16.45 inference(rewrite,[status(thm)],[])).
% 16.21/16.45 tff(139,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')).
% 16.21/16.45 tff(140,plain,
% 16.21/16.45 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))),
% 16.21/16.45 inference(modus_ponens,[status(thm)],[139, 138])).
% 16.21/16.45 tff(141,plain,(
% 16.21/16.45 ![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))),
% 16.21/16.45 inference(skolemize,[status(sab)],[140])).
% 16.21/16.45 tff(142,plain,
% 16.21/16.45 (![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))),
% 16.21/16.45 inference(modus_ponens,[status(thm)],[141, 137])).
% 16.21/16.45 tff(143,plain,
% 16.21/16.45 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | ((~member(ordered_pair(u, v), cross_product(universal_class, universal_class))) | member(u, universal_class))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | (~member(ordered_pair(u, v), cross_product(universal_class, universal_class))) | member(u, universal_class))),
% 16.21/16.45 inference(rewrite,[status(thm)],[])).
% 16.21/16.45 tff(144,plain,
% 16.21/16.45 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | ((~member(ordered_pair(u, v), cross_product(universal_class, universal_class))) | member(u, universal_class))),
% 16.21/16.45 inference(quant_inst,[status(thm)],[])).
% 16.21/16.45 tff(145,plain,
% 16.21/16.45 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | (~member(ordered_pair(u, v), cross_product(universal_class, universal_class))) | member(u, universal_class)),
% 16.21/16.45 inference(modus_ponens,[status(thm)],[144, 143])).
% 16.21/16.45 tff(146,plain,
% 16.21/16.45 (member(u, universal_class)),
% 16.21/16.45 inference(unit_resolution,[status(thm)],[145, 142, 23])).
% 16.21/16.45 tff(147,plain,
% 16.21/16.45 (((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(u, universal_class)) | member(u, unordered_pair(u, singleton(v))))) <=> ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(u, universal_class)) | member(u, unordered_pair(u, singleton(v))))),
% 16.21/16.45 inference(rewrite,[status(thm)],[])).
% 16.21/16.45 tff(148,plain,
% 16.21/16.45 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(u, universal_class)) | member(u, unordered_pair(u, singleton(v))))),
% 16.21/16.45 inference(quant_inst,[status(thm)],[])).
% 16.21/16.45 tff(149,plain,
% 16.21/16.45 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(u, universal_class)) | member(u, unordered_pair(u, singleton(v)))),
% 16.21/16.45 inference(modus_ponens,[status(thm)],[148, 147])).
% 16.21/16.45 tff(150,plain,
% 16.21/16.45 (member(u, unordered_pair(u, singleton(v)))),
% 16.21/16.45 inference(unit_resolution,[status(thm)],[149, 56, 146])).
% 16.21/16.45 tff(151,plain,
% 16.21/16.45 (member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))))),
% 16.21/16.45 inference(modus_ponens,[status(thm)],[150, 135])).
% 16.21/16.45 tff(152,plain,
% 16.21/16.45 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((u = first(ordered_pair(u, v))) | (~member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (u = first(ordered_pair(u, v))) | (~member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))))))),
% 16.21/16.45 inference(rewrite,[status(thm)],[])).
% 16.21/16.45 tff(153,plain,
% 16.21/16.45 (((u = first(ordered_pair(u, v))) | (u = first(ordered_pair(u, v))) | (~member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v)))))) <=> ((u = first(ordered_pair(u, v))) | (~member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))))))),
% 16.21/16.45 inference(rewrite,[status(thm)],[])).
% 16.21/16.45 tff(154,plain,
% 16.21/16.45 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((u = first(ordered_pair(u, v))) | (u = first(ordered_pair(u, v))) | (~member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((u = first(ordered_pair(u, v))) | (~member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v)))))))),
% 16.21/16.45 inference(monotonicity,[status(thm)],[153])).
% 16.21/16.45 tff(155,plain,
% 16.21/16.45 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((u = first(ordered_pair(u, v))) | (u = first(ordered_pair(u, v))) | (~member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (u = first(ordered_pair(u, v))) | (~member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))))))),
% 16.21/16.45 inference(transitivity,[status(thm)],[154, 152])).
% 16.21/16.45 tff(156,plain,
% 16.21/16.45 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((u = first(ordered_pair(u, v))) | (u = first(ordered_pair(u, v))) | (~member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))))))),
% 16.21/16.45 inference(quant_inst,[status(thm)],[])).
% 16.21/16.45 tff(157,plain,
% 16.21/16.45 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (u = first(ordered_pair(u, v))) | (~member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v)))))),
% 16.21/16.45 inference(modus_ponens,[status(thm)],[156, 155])).
% 16.21/16.45 tff(158,plain,
% 16.21/16.45 ((u = first(ordered_pair(u, v))) | (~member(u, unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v)))))),
% 16.21/16.45 inference(unit_resolution,[status(thm)],[157, 74])).
% 16.21/16.45 tff(159,plain,
% 16.21/16.45 (u = first(ordered_pair(u, v))),
% 16.21/16.45 inference(unit_resolution,[status(thm)],[158, 151])).
% 16.21/16.45 tff(160,plain,
% 16.21/16.45 (singleton(u) = singleton(first(ordered_pair(u, v)))),
% 16.21/16.45 inference(monotonicity,[status(thm)],[159])).
% 16.21/16.45 tff(161,plain,
% 16.21/16.45 (singleton(first(ordered_pair(u, v))) = singleton(u)),
% 16.21/16.45 inference(modus_ponens,[status(thm)],[160, 90])).
% 16.21/16.45 tff(162,plain,
% 16.21/16.45 ($false),
% 16.21/16.45 inference(unit_resolution,[status(thm)],[91, 161])).
% 16.21/16.45 tff(163,plain,(singleton(first(ordered_pair(u, v))) = singleton(u)), inference(lemma,lemma(discharge,[]))).
% 16.21/16.45 tff(164,plain,
% 16.21/16.45 (singleton(u) = singleton(first(ordered_pair(u, v)))),
% 16.21/16.45 inference(symmetry,[status(thm)],[163])).
% 16.21/16.45 tff(165,plain,
% 16.21/16.45 (member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(u), unordered_pair(u, singleton(v)))) <=> member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))),
% 16.21/16.45 inference(monotonicity,[status(thm)],[105])).
% 16.21/16.45 tff(166,plain,
% 16.21/16.45 (member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))) <=> member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))),
% 16.21/16.45 inference(symmetry,[status(thm)],[165])).
% 16.21/16.45 tff(167,plain,
% 16.21/16.45 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), universal_class)),
% 16.21/16.45 inference(quant_inst,[status(thm)],[])).
% 16.21/16.45 tff(168,plain,
% 16.21/16.45 (member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), universal_class)),
% 16.21/16.45 inference(unit_resolution,[status(thm)],[167, 117])).
% 16.21/16.45 tff(169,plain,
% 16.21/16.45 (^[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))))),
% 16.21/16.45 inference(bind,[status(th)],[])).
% 16.21/16.45 tff(170,plain,
% 16.21/16.45 (![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)))),
% 16.21/16.45 inference(quant_intro,[status(thm)],[169])).
% 16.21/16.45 tff(171,plain,
% 16.21/16.45 (![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)))),
% 16.21/16.45 inference(rewrite,[status(thm)],[])).
% 16.21/16.45 tff(172,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')).
% 16.21/16.45 tff(173,plain,
% 16.21/16.45 (![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))),
% 16.21/16.45 inference(modus_ponens,[status(thm)],[172, 171])).
% 16.21/16.45 tff(174,plain,(
% 16.21/16.45 ![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))),
% 16.21/16.45 inference(skolemize,[status(sab)],[173])).
% 16.21/16.45 tff(175,plain,
% 16.21/16.45 (![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))),
% 16.21/16.45 inference(modus_ponens,[status(thm)],[174, 170])).
% 16.21/16.45 tff(176,plain,
% 16.21/16.45 (((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | ((~member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), universal_class)) | member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))))) <=> ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | (~member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), universal_class)) | member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))))),
% 16.21/16.45 inference(rewrite,[status(thm)],[])).
% 16.21/16.45 tff(177,plain,
% 16.21/16.45 ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | ((~member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), universal_class)) | member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))))),
% 16.21/16.45 inference(quant_inst,[status(thm)],[])).
% 16.21/16.45 tff(178,plain,
% 16.21/16.45 ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | (~member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), universal_class)) | member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))),
% 16.21/16.45 inference(modus_ponens,[status(thm)],[177, 176])).
% 16.21/16.45 tff(179,plain,
% 16.21/16.45 ((~member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), universal_class)) | member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))),
% 16.21/16.45 inference(unit_resolution,[status(thm)],[178, 175])).
% 16.21/16.45 tff(180,plain,
% 16.21/16.45 (member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))),
% 16.21/16.45 inference(unit_resolution,[status(thm)],[179, 168])).
% 16.21/16.45 tff(181,plain,
% 16.21/16.45 (member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))),
% 16.21/16.45 inference(modus_ponens,[status(thm)],[180, 166])).
% 16.21/16.45 tff(182,plain,
% 16.21/16.45 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = unordered_pair(u, singleton(v))) | (unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = singleton(u)) | (~member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = unordered_pair(u, singleton(v))) | (unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = singleton(u)) | (~member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))))),
% 16.21/16.45 inference(rewrite,[status(thm)],[])).
% 16.21/16.45 tff(183,plain,
% 16.21/16.45 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = unordered_pair(u, singleton(v))) | (unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = singleton(u)) | (~member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))))),
% 16.21/16.45 inference(quant_inst,[status(thm)],[])).
% 16.21/16.45 tff(184,plain,
% 16.21/16.45 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = unordered_pair(u, singleton(v))) | (unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = singleton(u)) | (~member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(u), unordered_pair(u, singleton(v)))))),
% 16.21/16.45 inference(modus_ponens,[status(thm)],[183, 182])).
% 16.21/16.45 tff(185,plain,
% 16.21/16.45 ((unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = unordered_pair(u, singleton(v))) | (unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = singleton(u)) | (~member(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))), unordered_pair(singleton(u), unordered_pair(u, singleton(v)))))),
% 16.21/16.45 inference(unit_resolution,[status(thm)],[184, 74])).
% 16.21/16.45 tff(186,plain,
% 16.21/16.45 ((unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = unordered_pair(u, singleton(v))) | (unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = singleton(u))),
% 16.21/16.45 inference(unit_resolution,[status(thm)],[185, 181])).
% 16.21/16.45 tff(187,plain,
% 16.21/16.45 (unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = singleton(u)),
% 16.21/16.45 inference(unit_resolution,[status(thm)],[186, 87])).
% 16.21/16.45 tff(188,plain,
% 16.21/16.45 (unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = singleton(first(ordered_pair(u, v)))),
% 16.21/16.45 inference(transitivity,[status(thm)],[187, 164])).
% 16.21/16.45 tff(189,plain,
% 16.21/16.45 ((unordered_pair(u, singleton(v)) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) <=> (unordered_pair(u, singleton(v)) = singleton(first(ordered_pair(u, v))))),
% 16.21/16.45 inference(monotonicity,[status(thm)],[188])).
% 16.21/16.45 tff(190,plain,
% 16.21/16.45 ((~(unordered_pair(u, singleton(v)) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))) <=> (~(unordered_pair(u, singleton(v)) = singleton(first(ordered_pair(u, v)))))),
% 16.21/16.45 inference(monotonicity,[status(thm)],[189])).
% 16.21/16.45 tff(191,plain,
% 16.21/16.45 (~(unordered_pair(u, singleton(v)) = singleton(first(ordered_pair(u, v))))),
% 16.21/16.45 inference(modus_ponens,[status(thm)],[88, 190])).
% 16.21/16.45 tff(192,plain,
% 16.21/16.45 (ordered_pair(u, v) = unordered_pair(singleton(u), unordered_pair(u, singleton(v)))),
% 16.21/16.45 inference(symmetry,[status(thm)],[104])).
% 16.21/16.45 tff(193,plain,
% 16.21/16.45 (unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) = unordered_pair(singleton(u), unordered_pair(u, singleton(v)))),
% 16.21/16.46 inference(transitivity,[status(thm)],[100, 34, 192])).
% 16.21/16.46 tff(194,plain,
% 16.21/16.46 (member(unordered_pair(u, singleton(v)), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))) <=> member(unordered_pair(u, singleton(v)), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))),
% 16.21/16.46 inference(monotonicity,[status(thm)],[193])).
% 16.21/16.46 tff(195,plain,
% 16.21/16.46 (member(unordered_pair(u, singleton(v)), unordered_pair(singleton(u), unordered_pair(u, singleton(v)))) <=> member(unordered_pair(u, singleton(v)), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))),
% 16.21/16.46 inference(symmetry,[status(thm)],[194])).
% 16.21/16.46 tff(196,plain,
% 16.21/16.46 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(u, singleton(v)), universal_class)),
% 16.21/16.46 inference(quant_inst,[status(thm)],[])).
% 16.21/16.46 tff(197,plain,
% 16.21/16.46 (member(unordered_pair(u, singleton(v)), universal_class)),
% 16.21/16.46 inference(unit_resolution,[status(thm)],[196, 117])).
% 16.21/16.46 tff(198,plain,
% 16.21/16.46 (((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | ((~member(unordered_pair(u, singleton(v)), universal_class)) | member(unordered_pair(u, singleton(v)), unordered_pair(singleton(u), unordered_pair(u, singleton(v)))))) <=> ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | (~member(unordered_pair(u, singleton(v)), universal_class)) | member(unordered_pair(u, singleton(v)), unordered_pair(singleton(u), unordered_pair(u, singleton(v)))))),
% 16.21/16.46 inference(rewrite,[status(thm)],[])).
% 16.21/16.46 tff(199,plain,
% 16.21/16.46 ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | ((~member(unordered_pair(u, singleton(v)), universal_class)) | member(unordered_pair(u, singleton(v)), unordered_pair(singleton(u), unordered_pair(u, singleton(v)))))),
% 16.21/16.46 inference(quant_inst,[status(thm)],[])).
% 16.21/16.46 tff(200,plain,
% 16.21/16.46 ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | (~member(unordered_pair(u, singleton(v)), universal_class)) | member(unordered_pair(u, singleton(v)), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))),
% 16.21/16.46 inference(modus_ponens,[status(thm)],[199, 198])).
% 16.21/16.46 tff(201,plain,
% 16.21/16.46 ((~member(unordered_pair(u, singleton(v)), universal_class)) | member(unordered_pair(u, singleton(v)), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))),
% 16.21/16.46 inference(unit_resolution,[status(thm)],[200, 175])).
% 16.21/16.46 tff(202,plain,
% 16.21/16.46 (member(unordered_pair(u, singleton(v)), unordered_pair(singleton(u), unordered_pair(u, singleton(v))))),
% 16.21/16.46 inference(unit_resolution,[status(thm)],[201, 197])).
% 16.21/16.46 tff(203,plain,
% 16.21/16.46 (member(unordered_pair(u, singleton(v)), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))),
% 16.21/16.46 inference(modus_ponens,[status(thm)],[202, 195])).
% 16.21/16.46 tff(204,plain,
% 16.21/16.46 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((unordered_pair(u, singleton(v)) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) | (unordered_pair(u, singleton(v)) = singleton(first(ordered_pair(u, v)))) | (~member(unordered_pair(u, singleton(v)), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (unordered_pair(u, singleton(v)) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) | (unordered_pair(u, singleton(v)) = singleton(first(ordered_pair(u, v)))) | (~member(unordered_pair(u, singleton(v)), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))))),
% 16.21/16.46 inference(rewrite,[status(thm)],[])).
% 16.21/16.46 tff(205,plain,
% 16.21/16.46 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((unordered_pair(u, singleton(v)) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) | (unordered_pair(u, singleton(v)) = singleton(first(ordered_pair(u, v)))) | (~member(unordered_pair(u, singleton(v)), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))))),
% 16.21/16.46 inference(quant_inst,[status(thm)],[])).
% 16.21/16.46 tff(206,plain,
% 16.21/16.46 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (unordered_pair(u, singleton(v)) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) | (unordered_pair(u, singleton(v)) = singleton(first(ordered_pair(u, v)))) | (~member(unordered_pair(u, singleton(v)), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))))),
% 16.21/16.46 inference(modus_ponens,[status(thm)],[205, 204])).
% 16.21/16.46 tff(207,plain,
% 16.21/16.46 ((unordered_pair(u, singleton(v)) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) | (unordered_pair(u, singleton(v)) = singleton(first(ordered_pair(u, v)))) | (~member(unordered_pair(u, singleton(v)), unordered_pair(singleton(first(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))))),
% 16.21/16.46 inference(unit_resolution,[status(thm)],[206, 74])).
% 16.21/16.46 tff(208,plain,
% 16.21/16.46 ((unordered_pair(u, singleton(v)) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) | (unordered_pair(u, singleton(v)) = singleton(first(ordered_pair(u, v))))),
% 16.21/16.46 inference(unit_resolution,[status(thm)],[207, 203])).
% 16.21/16.46 tff(209,plain,
% 16.21/16.46 ($false),
% 16.21/16.46 inference(unit_resolution,[status(thm)],[208, 191, 88])).
% 16.21/16.46 tff(210,plain,(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = unordered_pair(u, singleton(v))), inference(lemma,lemma(discharge,[]))).
% 16.21/16.46 tff(211,assumption,(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = unordered_pair(u, singleton(v))), introduced(assumption)).
% 16.21/16.46 tff(212,plain,
% 16.21/16.46 (unordered_pair(u, singleton(v)) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))),
% 16.21/16.46 inference(symmetry,[status(thm)],[211])).
% 16.21/16.46 tff(213,plain,
% 16.21/16.46 (member(singleton(second(ordered_pair(u, v))), unordered_pair(u, singleton(v))) <=> member(singleton(second(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))),
% 16.21/16.46 inference(monotonicity,[status(thm)],[212])).
% 16.21/16.46 tff(214,plain,
% 16.21/16.46 (member(singleton(second(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))) <=> member(singleton(second(ordered_pair(u, v))), unordered_pair(u, singleton(v)))),
% 16.21/16.46 inference(symmetry,[status(thm)],[213])).
% 16.21/16.46 tff(215,plain,
% 16.21/16.46 (member(unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v))), universal_class) <=> member(singleton(second(ordered_pair(u, v))), universal_class)),
% 16.21/16.46 inference(monotonicity,[status(thm)],[12])).
% 16.21/16.46 tff(216,plain,
% 16.21/16.46 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v))), universal_class)),
% 16.21/16.46 inference(quant_inst,[status(thm)],[])).
% 16.21/16.46 tff(217,plain,
% 16.21/16.46 (member(unordered_pair(second(ordered_pair(u, v)), second(ordered_pair(u, v))), universal_class)),
% 16.21/16.46 inference(unit_resolution,[status(thm)],[216, 117])).
% 16.21/16.46 tff(218,plain,
% 16.21/16.46 (member(singleton(second(ordered_pair(u, v))), universal_class)),
% 16.21/16.46 inference(modus_ponens,[status(thm)],[217, 215])).
% 16.21/16.46 tff(219,plain,
% 16.21/16.46 (((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | ((~member(singleton(second(ordered_pair(u, v))), universal_class)) | member(singleton(second(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))) <=> ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | (~member(singleton(second(ordered_pair(u, v))), universal_class)) | member(singleton(second(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))),
% 16.21/16.46 inference(rewrite,[status(thm)],[])).
% 16.21/16.46 tff(220,plain,
% 16.21/16.46 ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | ((~member(singleton(second(ordered_pair(u, v))), universal_class)) | member(singleton(second(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))))),
% 16.21/16.46 inference(quant_inst,[status(thm)],[])).
% 16.21/16.46 tff(221,plain,
% 16.21/16.46 ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | (~member(singleton(second(ordered_pair(u, v))), universal_class)) | member(singleton(second(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))),
% 16.21/16.46 inference(modus_ponens,[status(thm)],[220, 219])).
% 16.21/16.46 tff(222,plain,
% 16.21/16.46 ((~member(singleton(second(ordered_pair(u, v))), universal_class)) | member(singleton(second(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))),
% 16.21/16.46 inference(unit_resolution,[status(thm)],[221, 175])).
% 16.21/16.46 tff(223,plain,
% 16.21/16.46 (member(singleton(second(ordered_pair(u, v))), unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))))),
% 16.21/16.46 inference(unit_resolution,[status(thm)],[222, 218])).
% 16.21/16.46 tff(224,plain,
% 16.21/16.46 (member(singleton(second(ordered_pair(u, v))), unordered_pair(u, singleton(v)))),
% 16.21/16.46 inference(modus_ponens,[status(thm)],[223, 214])).
% 16.21/16.46 tff(225,assumption,(~member(singleton(second(ordered_pair(u, v))), unordered_pair(u, singleton(v)))), introduced(assumption)).
% 16.21/16.46 tff(226,plain,
% 16.21/16.46 ($false),
% 16.21/16.46 inference(unit_resolution,[status(thm)],[225, 224])).
% 16.21/16.46 tff(227,plain,((~(unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v)))) = unordered_pair(u, singleton(v)))) | member(singleton(second(ordered_pair(u, v))), unordered_pair(u, singleton(v)))), inference(lemma,lemma(discharge,[]))).
% 16.21/16.46 tff(228,plain,
% 16.21/16.46 (member(singleton(second(ordered_pair(u, v))), unordered_pair(u, singleton(v)))),
% 16.21/16.46 inference(unit_resolution,[status(thm)],[227, 210])).
% 16.21/16.46 tff(229,assumption,(member(singleton(second(ordered_pair(u, v))), unordered_pair(u, singleton(v)))), introduced(assumption)).
% 16.21/16.46 tff(230,assumption,(~(singleton(second(ordered_pair(u, v))) = u)), introduced(assumption)).
% 16.21/16.46 tff(231,plain,
% 16.21/16.46 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(second(ordered_pair(u, v))) = singleton(v)) | (singleton(second(ordered_pair(u, v))) = u) | (~member(singleton(second(ordered_pair(u, v))), unordered_pair(u, singleton(v)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(second(ordered_pair(u, v))) = singleton(v)) | (singleton(second(ordered_pair(u, v))) = u) | (~member(singleton(second(ordered_pair(u, v))), unordered_pair(u, singleton(v)))))),
% 16.21/16.46 inference(rewrite,[status(thm)],[])).
% 16.21/16.46 tff(232,plain,
% 16.21/16.46 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(second(ordered_pair(u, v))) = singleton(v)) | (singleton(second(ordered_pair(u, v))) = u) | (~member(singleton(second(ordered_pair(u, v))), unordered_pair(u, singleton(v)))))),
% 16.21/16.46 inference(quant_inst,[status(thm)],[])).
% 16.21/16.46 tff(233,plain,
% 16.21/16.46 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (singleton(second(ordered_pair(u, v))) = singleton(v)) | (singleton(second(ordered_pair(u, v))) = u) | (~member(singleton(second(ordered_pair(u, v))), unordered_pair(u, singleton(v))))),
% 16.21/16.46 inference(modus_ponens,[status(thm)],[232, 231])).
% 16.21/16.46 tff(234,plain,
% 16.21/16.46 ($false),
% 16.21/16.46 inference(unit_resolution,[status(thm)],[233, 74, 230, 229, 83])).
% 16.21/16.46 tff(235,plain,((~member(singleton(second(ordered_pair(u, v))), unordered_pair(u, singleton(v)))) | (singleton(second(ordered_pair(u, v))) = u)), inference(lemma,lemma(discharge,[]))).
% 16.29/16.46 tff(236,plain,
% 16.29/16.46 (singleton(second(ordered_pair(u, v))) = u),
% 16.29/16.46 inference(unit_resolution,[status(thm)],[235, 228])).
% 16.29/16.46 tff(237,plain,
% 16.29/16.46 (u = singleton(second(ordered_pair(u, v)))),
% 16.29/16.46 inference(symmetry,[status(thm)],[236])).
% 16.29/16.46 tff(238,plain,
% 16.29/16.46 ((singleton(v) = u) <=> (singleton(v) = singleton(second(ordered_pair(u, v))))),
% 16.29/16.46 inference(monotonicity,[status(thm)],[237])).
% 16.29/16.46 tff(239,plain,
% 16.29/16.46 ((singleton(v) = u) <=> (singleton(second(ordered_pair(u, v))) = singleton(v))),
% 16.29/16.46 inference(transitivity,[status(thm)],[238, 1])).
% 16.29/16.46 tff(240,plain,
% 16.29/16.46 ((singleton(second(ordered_pair(u, v))) = singleton(v)) <=> (singleton(v) = u)),
% 16.29/16.46 inference(symmetry,[status(thm)],[239])).
% 16.29/16.46 tff(241,plain,
% 16.29/16.46 ((~(singleton(second(ordered_pair(u, v))) = singleton(v))) <=> (~(singleton(v) = u))),
% 16.29/16.46 inference(monotonicity,[status(thm)],[240])).
% 16.29/16.46 tff(242,plain,
% 16.29/16.46 (~(singleton(v) = u)),
% 16.29/16.46 inference(modus_ponens,[status(thm)],[83, 241])).
% 16.29/16.46 tff(243,plain,
% 16.29/16.46 (singleton(first(ordered_pair(u, v))) = unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v)))),
% 16.29/16.46 inference(symmetry,[status(thm)],[109])).
% 16.29/16.46 tff(244,plain,
% 16.29/16.46 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(u, u) = singleton(u))),
% 16.29/16.46 inference(quant_inst,[status(thm)],[])).
% 16.29/16.46 tff(245,plain,
% 16.29/16.46 (unordered_pair(u, u) = singleton(u)),
% 16.29/16.46 inference(unit_resolution,[status(thm)],[244, 10])).
% 16.29/16.46 tff(246,plain,
% 16.29/16.46 (unordered_pair(u, u) = unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v)))),
% 16.29/16.46 inference(transitivity,[status(thm)],[245, 164, 243])).
% 16.29/16.46 tff(247,plain,
% 16.29/16.46 (member(first(ordered_pair(u, v)), unordered_pair(u, u)) <=> member(first(ordered_pair(u, v)), unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))))),
% 16.29/16.46 inference(monotonicity,[status(thm)],[246])).
% 16.29/16.46 tff(248,plain,
% 16.29/16.46 (member(first(ordered_pair(u, v)), unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v)))) <=> member(first(ordered_pair(u, v)), unordered_pair(u, u))),
% 16.29/16.46 inference(symmetry,[status(thm)],[247])).
% 16.29/16.46 tff(249,plain,
% 16.29/16.46 (((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | ((~member(ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), cross_product(universal_class, universal_class))) | member(first(ordered_pair(u, v)), universal_class))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | (~member(ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), cross_product(universal_class, universal_class))) | member(first(ordered_pair(u, v)), universal_class))),
% 16.29/16.46 inference(rewrite,[status(thm)],[])).
% 16.29/16.46 tff(250,plain,
% 16.29/16.46 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | ((~member(ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), cross_product(universal_class, universal_class))) | member(first(ordered_pair(u, v)), universal_class))),
% 16.29/16.46 inference(quant_inst,[status(thm)],[])).
% 16.29/16.46 tff(251,plain,
% 16.29/16.46 ((~![V: $i, Y: $i, U: $i, X: $i] : ((~member(ordered_pair(U, V), cross_product(X, Y))) | member(U, X))) | (~member(ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), cross_product(universal_class, universal_class))) | member(first(ordered_pair(u, v)), universal_class)),
% 16.29/16.46 inference(modus_ponens,[status(thm)],[250, 249])).
% 16.29/16.46 tff(252,plain,
% 16.29/16.46 ((~member(ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), cross_product(universal_class, universal_class))) | member(first(ordered_pair(u, v)), universal_class)),
% 16.29/16.46 inference(unit_resolution,[status(thm)],[251, 142])).
% 16.29/16.46 tff(253,plain,
% 16.29/16.46 (member(first(ordered_pair(u, v)), universal_class)),
% 16.29/16.46 inference(unit_resolution,[status(thm)],[252, 37])).
% 16.29/16.46 tff(254,plain,
% 16.29/16.46 (((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(first(ordered_pair(u, v)), universal_class)) | member(first(ordered_pair(u, v)), unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v)))))) <=> ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(first(ordered_pair(u, v)), universal_class)) | member(first(ordered_pair(u, v)), unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v)))))),
% 16.29/16.46 inference(rewrite,[status(thm)],[])).
% 16.29/16.46 tff(255,plain,
% 16.29/16.46 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(first(ordered_pair(u, v)), universal_class)) | member(first(ordered_pair(u, v)), unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v)))))),
% 16.29/16.46 inference(quant_inst,[status(thm)],[])).
% 16.29/16.46 tff(256,plain,
% 16.29/16.46 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(first(ordered_pair(u, v)), universal_class)) | member(first(ordered_pair(u, v)), unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))))),
% 16.29/16.46 inference(modus_ponens,[status(thm)],[255, 254])).
% 16.29/16.46 tff(257,plain,
% 16.29/16.46 (member(first(ordered_pair(u, v)), unordered_pair(first(ordered_pair(u, v)), first(ordered_pair(u, v))))),
% 16.29/16.46 inference(unit_resolution,[status(thm)],[256, 56, 253])).
% 16.29/16.46 tff(258,plain,
% 16.29/16.46 (member(first(ordered_pair(u, v)), unordered_pair(u, u))),
% 16.29/16.46 inference(modus_ponens,[status(thm)],[257, 248])).
% 16.29/16.46 tff(259,plain,
% 16.29/16.46 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((first(ordered_pair(u, v)) = u) | (~member(first(ordered_pair(u, v)), unordered_pair(u, u))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (first(ordered_pair(u, v)) = u) | (~member(first(ordered_pair(u, v)), unordered_pair(u, u))))),
% 16.29/16.46 inference(rewrite,[status(thm)],[])).
% 16.29/16.46 tff(260,plain,
% 16.29/16.46 (((first(ordered_pair(u, v)) = u) | (first(ordered_pair(u, v)) = u) | (~member(first(ordered_pair(u, v)), unordered_pair(u, u)))) <=> ((first(ordered_pair(u, v)) = u) | (~member(first(ordered_pair(u, v)), unordered_pair(u, u))))),
% 16.29/16.46 inference(rewrite,[status(thm)],[])).
% 16.29/16.46 tff(261,plain,
% 16.29/16.46 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((first(ordered_pair(u, v)) = u) | (first(ordered_pair(u, v)) = u) | (~member(first(ordered_pair(u, v)), unordered_pair(u, u))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((first(ordered_pair(u, v)) = u) | (~member(first(ordered_pair(u, v)), unordered_pair(u, u)))))),
% 16.29/16.46 inference(monotonicity,[status(thm)],[260])).
% 16.29/16.46 tff(262,plain,
% 16.29/16.46 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((first(ordered_pair(u, v)) = u) | (first(ordered_pair(u, v)) = u) | (~member(first(ordered_pair(u, v)), unordered_pair(u, u))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (first(ordered_pair(u, v)) = u) | (~member(first(ordered_pair(u, v)), unordered_pair(u, u))))),
% 16.29/16.46 inference(transitivity,[status(thm)],[261, 259])).
% 16.29/16.46 tff(263,plain,
% 16.29/16.46 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((first(ordered_pair(u, v)) = u) | (first(ordered_pair(u, v)) = u) | (~member(first(ordered_pair(u, v)), unordered_pair(u, u))))),
% 16.29/16.46 inference(quant_inst,[status(thm)],[])).
% 16.29/16.46 tff(264,plain,
% 16.29/16.46 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (first(ordered_pair(u, v)) = u) | (~member(first(ordered_pair(u, v)), unordered_pair(u, u)))),
% 16.29/16.46 inference(modus_ponens,[status(thm)],[263, 262])).
% 16.29/16.46 tff(265,plain,
% 16.29/16.46 ((first(ordered_pair(u, v)) = u) | (~member(first(ordered_pair(u, v)), unordered_pair(u, u)))),
% 16.29/16.46 inference(unit_resolution,[status(thm)],[264, 74])).
% 16.29/16.46 tff(266,plain,
% 16.29/16.46 (first(ordered_pair(u, v)) = u),
% 16.29/16.46 inference(unit_resolution,[status(thm)],[265, 258])).
% 16.29/16.46 tff(267,plain,
% 16.29/16.46 (u = first(ordered_pair(u, v))),
% 16.29/16.46 inference(symmetry,[status(thm)],[266])).
% 16.29/16.46 tff(268,plain,
% 16.29/16.46 (unordered_pair(u, singleton(second(ordered_pair(u, v)))) = unordered_pair(first(ordered_pair(u, v)), singleton(second(ordered_pair(u, v))))),
% 16.29/16.46 inference(monotonicity,[status(thm)],[267])).
% 16.29/16.46 tff(269,plain,
% 16.29/16.46 (unordered_pair(u, singleton(second(ordered_pair(u, v)))) = unordered_pair(u, singleton(v))),
% 16.29/16.46 inference(transitivity,[status(thm)],[268, 210])).
% 16.29/16.46 tff(270,plain,
% 16.29/16.46 (member(singleton(v), unordered_pair(u, singleton(second(ordered_pair(u, v))))) <=> member(singleton(v), unordered_pair(u, singleton(v)))),
% 16.29/16.46 inference(monotonicity,[status(thm)],[269])).
% 16.29/16.46 tff(271,plain,
% 16.29/16.46 (member(singleton(v), unordered_pair(u, singleton(v))) <=> member(singleton(v), unordered_pair(u, singleton(second(ordered_pair(u, v)))))),
% 16.29/16.46 inference(symmetry,[status(thm)],[270])).
% 16.29/16.46 tff(272,plain,
% 16.29/16.46 (member(unordered_pair(v, v), universal_class) <=> member(singleton(v), universal_class)),
% 16.29/16.46 inference(monotonicity,[status(thm)],[17])).
% 16.29/16.46 tff(273,plain,
% 16.29/16.46 ((~![Y: $i, X: $i] : member(unordered_pair(X, Y), universal_class)) | member(unordered_pair(v, v), universal_class)),
% 16.29/16.46 inference(quant_inst,[status(thm)],[])).
% 16.29/16.46 tff(274,plain,
% 16.29/16.46 (member(unordered_pair(v, v), universal_class)),
% 16.29/16.46 inference(unit_resolution,[status(thm)],[273, 117])).
% 16.29/16.46 tff(275,plain,
% 16.29/16.46 (member(singleton(v), universal_class)),
% 16.29/16.46 inference(modus_ponens,[status(thm)],[274, 272])).
% 16.29/16.46 tff(276,plain,
% 16.29/16.46 (((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | ((~member(singleton(v), universal_class)) | member(singleton(v), unordered_pair(u, singleton(v))))) <=> ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | (~member(singleton(v), universal_class)) | member(singleton(v), unordered_pair(u, singleton(v))))),
% 16.29/16.46 inference(rewrite,[status(thm)],[])).
% 16.29/16.46 tff(277,plain,
% 16.29/16.46 ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | ((~member(singleton(v), universal_class)) | member(singleton(v), unordered_pair(u, singleton(v))))),
% 16.29/16.46 inference(quant_inst,[status(thm)],[])).
% 16.29/16.46 tff(278,plain,
% 16.29/16.46 ((~![Y: $i, X: $i] : ((~member(Y, universal_class)) | member(Y, unordered_pair(X, Y)))) | (~member(singleton(v), universal_class)) | member(singleton(v), unordered_pair(u, singleton(v)))),
% 16.29/16.46 inference(modus_ponens,[status(thm)],[277, 276])).
% 16.29/16.46 tff(279,plain,
% 16.29/16.46 ((~member(singleton(v), universal_class)) | member(singleton(v), unordered_pair(u, singleton(v)))),
% 16.29/16.46 inference(unit_resolution,[status(thm)],[278, 175])).
% 16.29/16.46 tff(280,plain,
% 16.29/16.46 (member(singleton(v), unordered_pair(u, singleton(v)))),
% 16.29/16.46 inference(unit_resolution,[status(thm)],[279, 275])).
% 16.29/16.46 tff(281,plain,
% 16.29/16.46 (member(singleton(v), unordered_pair(u, singleton(second(ordered_pair(u, v)))))),
% 16.29/16.46 inference(modus_ponens,[status(thm)],[280, 271])).
% 16.29/16.46 tff(282,plain,
% 16.29/16.46 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((~member(singleton(v), unordered_pair(u, singleton(second(ordered_pair(u, v)))))) | (singleton(v) = u) | (singleton(v) = singleton(second(ordered_pair(u, v)))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(singleton(v), unordered_pair(u, singleton(second(ordered_pair(u, v)))))) | (singleton(v) = u) | (singleton(v) = singleton(second(ordered_pair(u, v)))))),
% 16.29/16.46 inference(rewrite,[status(thm)],[])).
% 16.29/16.46 tff(283,plain,
% 16.29/16.46 (((singleton(v) = singleton(second(ordered_pair(u, v)))) | (singleton(v) = u) | (~member(singleton(v), unordered_pair(u, singleton(second(ordered_pair(u, v))))))) <=> ((~member(singleton(v), unordered_pair(u, singleton(second(ordered_pair(u, v)))))) | (singleton(v) = u) | (singleton(v) = singleton(second(ordered_pair(u, v)))))),
% 16.29/16.46 inference(rewrite,[status(thm)],[])).
% 16.29/16.46 tff(284,plain,
% 16.29/16.46 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(v) = singleton(second(ordered_pair(u, v)))) | (singleton(v) = u) | (~member(singleton(v), unordered_pair(u, singleton(second(ordered_pair(u, v)))))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((~member(singleton(v), unordered_pair(u, singleton(second(ordered_pair(u, v)))))) | (singleton(v) = u) | (singleton(v) = singleton(second(ordered_pair(u, v))))))),
% 16.31/16.48 inference(monotonicity,[status(thm)],[283])).
% 16.31/16.48 tff(285,plain,
% 16.31/16.48 (((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(v) = singleton(second(ordered_pair(u, v)))) | (singleton(v) = u) | (~member(singleton(v), unordered_pair(u, singleton(second(ordered_pair(u, v)))))))) <=> ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(singleton(v), unordered_pair(u, singleton(second(ordered_pair(u, v)))))) | (singleton(v) = u) | (singleton(v) = singleton(second(ordered_pair(u, v)))))),
% 16.31/16.48 inference(transitivity,[status(thm)],[284, 282])).
% 16.31/16.48 tff(286,plain,
% 16.31/16.48 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | ((singleton(v) = singleton(second(ordered_pair(u, v)))) | (singleton(v) = u) | (~member(singleton(v), unordered_pair(u, singleton(second(ordered_pair(u, v)))))))),
% 16.31/16.48 inference(quant_inst,[status(thm)],[])).
% 16.31/16.48 tff(287,plain,
% 16.31/16.48 ((~![Y: $i, U: $i, X: $i] : ((U = Y) | (U = X) | (~member(U, unordered_pair(X, Y))))) | (~member(singleton(v), unordered_pair(u, singleton(second(ordered_pair(u, v)))))) | (singleton(v) = u) | (singleton(v) = singleton(second(ordered_pair(u, v))))),
% 16.31/16.48 inference(modus_ponens,[status(thm)],[286, 285])).
% 16.31/16.48 tff(288,plain,
% 16.31/16.48 ((~member(singleton(v), unordered_pair(u, singleton(second(ordered_pair(u, v)))))) | (singleton(v) = u) | (singleton(v) = singleton(second(ordered_pair(u, v))))),
% 16.31/16.48 inference(unit_resolution,[status(thm)],[287, 74])).
% 16.31/16.48 tff(289,plain,
% 16.31/16.48 ($false),
% 16.31/16.48 inference(unit_resolution,[status(thm)],[288, 281, 242, 84])).
% 16.31/16.48 % SZS output end Proof
% 16.31/16.49 % E exiting
%------------------------------------------------------------------------------