TPTP Problem File: SLH0243^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Undirected_Graph_Theory/0015_Graph_Triangles/prob_00086_003122__13142174_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1449 ( 655 unt; 174 typ; 0 def)
% Number of atoms : 3550 (1367 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9996 ( 366 ~; 55 |; 312 &;7797 @)
% ( 0 <=>;1466 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 22 ( 21 usr)
% Number of type conns : 404 ( 404 >; 0 *; 0 +; 0 <<)
% Number of symbols : 154 ( 153 usr; 21 con; 0-4 aty)
% Number of variables : 3352 ( 196 ^;3022 !; 134 ?;3352 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:33:03.613
%------------------------------------------------------------------------------
% Could-be-implicit typings (21)
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thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
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thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_Itf__a_J,type,
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (153)
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thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
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thf(sy_c_Graph__Triangles_Osgraph_Otriangle__triples_001tf__a,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Real__Oreal,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Real__Oreal,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
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undire8732585234338801206od_a_a: produc4044097585999906000od_a_a > set_Pr5530083903271594800od_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire3369688177417741453od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident_001t__Set__Oset_Itf__a_J,type,
undire2320338297334612420_set_a: set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident_001tf__a,type,
undire1521409233611534436dent_a: a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident__edges_001tf__a,type,
undire3231912044278729248dges_a: set_set_a > a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oinduced__edges_001tf__a,type,
undire7777452895879145676dges_a: set_set_a > set_a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Omk__triangle__set_001t__Nat__Onat,type,
undire4970100481470743719et_nat: produc7248412053542808358at_nat > set_nat ).
thf(sy_c_Undirected__Graph__Basics_Omk__triangle__set_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire2459242765783757584od_a_a: produc8857593507947890343od_a_a > set_Product_prod_a_a ).
thf(sy_c_Undirected__Graph__Basics_Omk__triangle__set_001t__Set__Oset_Itf__a_J,type,
undire4638465864238448455_set_a: produc3364680560414100336_set_a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Omk__triangle__set_001tf__a,type,
undire8536760333753235943_set_a: produc4044097585999906000od_a_a > set_a ).
thf(sy_c_Undirected__Graph__Basics_Osgraph_001t__Nat__Onat,type,
undire7290660292559394354ph_nat: set_nat > set_set_nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Osgraph_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire8797707285729112389od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Osgraph_001t__Set__Oset_Itf__a_J,type,
undire6035205377725458044_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Osgraph_001tf__a,type,
undire3507641187627840796raph_a: set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Osgraph_Ocomplement__edges_001tf__a,type,
undire4625228487420481630dges_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Osgraph_Ois__complement_001tf__a,type,
undire8013100667316154652ment_a: set_a > set_set_a > produc7943277765024757383_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Osgraph_Ois__complete__n__graph_001tf__a,type,
undire6087271738840788937raph_a: set_a > set_set_a > nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Osubgraph_001tf__a,type,
undire7103218114511261257raph_a: set_a > set_set_a > set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_001tf__a,type,
undire7251896706689453996raph_a: set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oall__edges__between_001tf__a,type,
undire8383842906760478443ween_a: set_set_a > set_a > set_a > set_Product_prod_a_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Odegree_001tf__a,type,
undire8867928226783802224gree_a: set_set_a > a > nat ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oedge__density_001tf__a,type,
undire297304480579013331sity_a: set_set_a > set_a > set_a > real ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ohas__loop_001t__Nat__Onat,type,
undire5005864372999571214op_nat: set_set_nat > nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ohas__loop_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire7777398424729533289od_a_a: set_se5735800977113168103od_a_a > product_prod_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ohas__loop_001t__Set__Oset_Itf__a_J,type,
undire5774735625301615776_set_a: set_set_set_a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ohas__loop_001tf__a,type,
undire3617971648856834880loop_a: set_set_a > a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oincident__loops_001tf__a,type,
undire4753905205749729249oops_a: set_set_a > a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oincident__sedges_001tf__a,type,
undire1270416042309875431dges_a: set_set_a > a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__edge__between_001t__Nat__Onat,type,
undire6814325412647357297en_nat: set_nat > set_nat > set_nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__edge__between_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire7011261089604658374od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__edge__between_001t__Set__Oset_Itf__a_J,type,
undire2578756059399487229_set_a: set_set_a > set_set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__edge__between_001tf__a,type,
undire8544646567961481629ween_a: set_a > set_a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__isolated__vertex_001tf__a,type,
undire8931668460104145173rtex_a: set_a > set_set_a > a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__loop_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
undire2959850631191844736od_a_a: set_Pr5530083903271594800od_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__loop_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire3428022325429088215od_a_a: set_Product_prod_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__loop_001t__Set__Oset_Itf__a_J,type,
undire3618949687197220622_set_a: set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__loop_001tf__a,type,
undire2905028936066782638loop_a: set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__sedge_001tf__a,type,
undire4917966558017083288edge_a: set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oneighborhood_001tf__a,type,
undire8504279938402040014hood_a: set_a > set_set_a > a > set_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Overt__adj_001t__Nat__Onat,type,
undire1083030068171319366dj_nat: set_set_nat > nat > nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Overt__adj_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire6135774327024169009od_a_a: set_se5735800977113168103od_a_a > product_prod_a_a > product_prod_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Overt__adj_001t__Set__Oset_Itf__a_J,type,
undire3510646817838285160_set_a: set_set_set_a > set_a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Overt__adj_001tf__a,type,
undire397441198561214472_adj_a: set_set_a > a > a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member1816616512716248880od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_X,type,
x: set_a ).
thf(sy_v_Y,type,
y: set_a ).
thf(sy_v_Z,type,
z: set_a ).
thf(sy_v_edges,type,
edges: set_set_a ).
thf(sy_v_vertices,type,
vertices: set_a ).
thf(sy_v_x,type,
x2: a ).
thf(sy_v_y,type,
y2: a ).
thf(sy_v_z,type,
z2: a ).
% Relevant facts (1274)
thf(fact_0_triangle__commu1,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z )
=> ( graph_4582152751571636272raph_a @ edges @ Y @ X @ Z ) ) ).
% triangle_commu1
thf(fact_1_triangle__vertices__distinct1,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z )
=> ( X != Y ) ) ).
% triangle_vertices_distinct1
thf(fact_2_triangle__vertices__distinct2,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z )
=> ( Y != Z ) ) ).
% triangle_vertices_distinct2
thf(fact_3_triangle__vertices__distinct3,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z )
=> ( Z != X ) ) ).
% triangle_vertices_distinct3
thf(fact_4_edge__adj__inE,axiom,
! [E1: set_a,E2: set_a] :
( ( undire4022703626023482010_adj_a @ edges @ E1 @ E2 )
=> ( ( member_set_a @ E1 @ edges )
& ( member_set_a @ E2 @ edges ) ) ) ).
% edge_adj_inE
thf(fact_5_sgraph_Otriangle__in__graph_Ocong,axiom,
graph_4582152751571636272raph_a = graph_4582152751571636272raph_a ).
% sgraph.triangle_in_graph.cong
thf(fact_6_triangle__in__graph__edge__empty,axiom,
! [X: a,Y: a,Z: a] :
( ( edges = bot_bot_set_set_a )
=> ~ ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z ) ) ).
% triangle_in_graph_edge_empty
thf(fact_7_edge__density__commute,axiom,
! [X2: set_a,Y2: set_a] :
( ( undire297304480579013331sity_a @ edges @ X2 @ Y2 )
= ( undire297304480579013331sity_a @ edges @ Y2 @ X2 ) ) ).
% edge_density_commute
thf(fact_8_vert__adj__sym,axiom,
! [V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
= ( undire397441198561214472_adj_a @ edges @ V2 @ V1 ) ) ).
% vert_adj_sym
thf(fact_9_empty__not__edge,axiom,
~ ( member_set_a @ bot_bot_set_a @ edges ) ).
% empty_not_edge
thf(fact_10_assms_I3_J,axiom,
member1426531477525435216od_a_a @ ( product_Pair_a_a @ y2 @ z2 ) @ ( undire8383842906760478443ween_a @ edges @ y @ z ) ).
% assms(3)
thf(fact_11_assms_I2_J,axiom,
member1426531477525435216od_a_a @ ( product_Pair_a_a @ x2 @ z2 ) @ ( undire8383842906760478443ween_a @ edges @ x @ z ) ).
% assms(2)
thf(fact_12_assms_I1_J,axiom,
member1426531477525435216od_a_a @ ( product_Pair_a_a @ x2 @ y2 ) @ ( undire8383842906760478443ween_a @ edges @ x @ y ) ).
% assms(1)
thf(fact_13_triangle__in__graph__def,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z )
= ( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges )
& ( member_set_a @ ( insert_a @ Y @ ( insert_a @ Z @ bot_bot_set_a ) ) @ edges )
& ( member_set_a @ ( insert_a @ X @ ( insert_a @ Z @ bot_bot_set_a ) ) @ edges ) ) ) ).
% triangle_in_graph_def
thf(fact_14_is__edge__or__loop,axiom,
! [E: set_a] :
( ( member_set_a @ E @ edges )
=> ( ( undire2905028936066782638loop_a @ E )
| ( undire4917966558017083288edge_a @ E ) ) ) ).
% is_edge_or_loop
thf(fact_15_card__triangle__triples__rotate,axiom,
! [X2: set_a,Y2: set_a,Z2: set_a] :
( ( finite6893194910719049976od_a_a @ ( graph_4774508486909600516ples_a @ edges @ X2 @ Y2 @ Z2 ) )
= ( finite6893194910719049976od_a_a @ ( graph_4774508486909600516ples_a @ edges @ Y2 @ Z2 @ X2 ) ) ) ).
% card_triangle_triples_rotate
thf(fact_16_sgraph_Otriangle__commu1,axiom,
! [Vertices: set_a,Edges: set_set_a,X: a,Y: a,Z: a] :
( ( undire3507641187627840796raph_a @ Vertices @ Edges )
=> ( ( graph_4582152751571636272raph_a @ Edges @ X @ Y @ Z )
=> ( graph_4582152751571636272raph_a @ Edges @ Y @ X @ Z ) ) ) ).
% sgraph.triangle_commu1
thf(fact_17_sgraph_Otriangle__vertices__distinct1,axiom,
! [Vertices: set_a,Edges: set_set_a,X: a,Y: a,Z: a] :
( ( undire3507641187627840796raph_a @ Vertices @ Edges )
=> ( ( graph_4582152751571636272raph_a @ Edges @ X @ Y @ Z )
=> ( X != Y ) ) ) ).
% sgraph.triangle_vertices_distinct1
thf(fact_18_singleton__not__edge,axiom,
! [X: a] :
~ ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ edges ) ).
% singleton_not_edge
thf(fact_19_edge__vertices__not__equal,axiom,
! [X: a,Y: a] :
( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges )
=> ( X != Y ) ) ).
% edge_vertices_not_equal
thf(fact_20_edge__btw__vertices__not__equal,axiom,
! [X: a,Y: a,X2: set_a,Y2: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y2 ) )
=> ( X != Y ) ) ).
% edge_btw_vertices_not_equal
thf(fact_21_vert__adj__def,axiom,
! [V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
= ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) @ edges ) ) ).
% vert_adj_def
thf(fact_22_not__vert__adj,axiom,
! [V: a,U: a] :
( ~ ( undire397441198561214472_adj_a @ edges @ V @ U )
=> ~ ( member_set_a @ ( insert_a @ V @ ( insert_a @ U @ bot_bot_set_a ) ) @ edges ) ) ).
% not_vert_adj
thf(fact_23_is__edge__between__def,axiom,
( undire8544646567961481629ween_a
= ( ^ [X3: set_a,Y3: set_a,E3: set_a] :
? [X4: a,Y4: a] :
( ( E3
= ( insert_a @ X4 @ ( insert_a @ Y4 @ bot_bot_set_a ) ) )
& ( member_a @ X4 @ X3 )
& ( member_a @ Y4 @ Y3 ) ) ) ) ).
% is_edge_between_def
thf(fact_24_all__edges__betw__I,axiom,
! [X: a,X2: set_a,Y: a,Y2: set_a] :
( ( member_a @ X @ X2 )
=> ( ( member_a @ Y @ Y2 )
=> ( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y2 ) ) ) ) ) ).
% all_edges_betw_I
thf(fact_25_all__edges__betw__D3,axiom,
! [X: a,Y: a,X2: set_a,Y2: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y2 ) )
=> ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges ) ) ).
% all_edges_betw_D3
thf(fact_26_triangle__in__graph__edge__point,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z )
= ( ( member_set_a @ ( insert_a @ Y @ ( insert_a @ Z @ bot_bot_set_a ) ) @ edges )
& ( undire397441198561214472_adj_a @ edges @ X @ Y )
& ( undire397441198561214472_adj_a @ edges @ X @ Z ) ) ) ).
% triangle_in_graph_edge_point
thf(fact_27_sgraph_Otriangle__in__graph__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X: set_a,Y: set_a,Z: set_a] :
( ( undire6035205377725458044_set_a @ Vertices @ Edges )
=> ( ( graph_3840782946058334608_set_a @ Edges @ X @ Y @ Z )
= ( ( member_set_set_a @ ( insert_set_a @ X @ ( insert_set_a @ Y @ bot_bot_set_set_a ) ) @ Edges )
& ( member_set_set_a @ ( insert_set_a @ Y @ ( insert_set_a @ Z @ bot_bot_set_set_a ) ) @ Edges )
& ( member_set_set_a @ ( insert_set_a @ X @ ( insert_set_a @ Z @ bot_bot_set_set_a ) ) @ Edges ) ) ) ) ).
% sgraph.triangle_in_graph_def
thf(fact_28_sgraph_Otriangle__in__graph__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
( ( undire8797707285729112389od_a_a @ Vertices @ Edges )
=> ( ( graph_4803287029668059225od_a_a @ Edges @ X @ Y @ Z )
= ( ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ X @ ( insert4534936382041156343od_a_a @ Y @ bot_bo3357376287454694259od_a_a ) ) @ Edges )
& ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ Y @ ( insert4534936382041156343od_a_a @ Z @ bot_bo3357376287454694259od_a_a ) ) @ Edges )
& ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ X @ ( insert4534936382041156343od_a_a @ Z @ bot_bo3357376287454694259od_a_a ) ) @ Edges ) ) ) ) ).
% sgraph.triangle_in_graph_def
thf(fact_29_sgraph_Otriangle__in__graph__def,axiom,
! [Vertices: set_nat,Edges: set_set_nat,X: nat,Y: nat,Z: nat] :
( ( undire7290660292559394354ph_nat @ Vertices @ Edges )
=> ( ( graph_2911189250448956958ph_nat @ Edges @ X @ Y @ Z )
= ( ( member_set_nat @ ( insert_nat @ X @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ Edges )
& ( member_set_nat @ ( insert_nat @ Y @ ( insert_nat @ Z @ bot_bot_set_nat ) ) @ Edges )
& ( member_set_nat @ ( insert_nat @ X @ ( insert_nat @ Z @ bot_bot_set_nat ) ) @ Edges ) ) ) ) ).
% sgraph.triangle_in_graph_def
thf(fact_30_sgraph_Otriangle__in__graph__def,axiom,
! [Vertices: set_a,Edges: set_set_a,X: a,Y: a,Z: a] :
( ( undire3507641187627840796raph_a @ Vertices @ Edges )
=> ( ( graph_4582152751571636272raph_a @ Edges @ X @ Y @ Z )
= ( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ Edges )
& ( member_set_a @ ( insert_a @ Y @ ( insert_a @ Z @ bot_bot_set_a ) ) @ Edges )
& ( member_set_a @ ( insert_a @ X @ ( insert_a @ Z @ bot_bot_set_a ) ) @ Edges ) ) ) ) ).
% sgraph.triangle_in_graph_def
thf(fact_31_sgraph_Oedge__vertices__not__equal,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X: set_a,Y: set_a] :
( ( undire6035205377725458044_set_a @ Vertices @ Edges )
=> ( ( member_set_set_a @ ( insert_set_a @ X @ ( insert_set_a @ Y @ bot_bot_set_set_a ) ) @ Edges )
=> ( X != Y ) ) ) ).
% sgraph.edge_vertices_not_equal
thf(fact_32_sgraph_Oedge__vertices__not__equal,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
( ( undire8797707285729112389od_a_a @ Vertices @ Edges )
=> ( ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ X @ ( insert4534936382041156343od_a_a @ Y @ bot_bo3357376287454694259od_a_a ) ) @ Edges )
=> ( X != Y ) ) ) ).
% sgraph.edge_vertices_not_equal
thf(fact_33_sgraph_Oedge__vertices__not__equal,axiom,
! [Vertices: set_nat,Edges: set_set_nat,X: nat,Y: nat] :
( ( undire7290660292559394354ph_nat @ Vertices @ Edges )
=> ( ( member_set_nat @ ( insert_nat @ X @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ Edges )
=> ( X != Y ) ) ) ).
% sgraph.edge_vertices_not_equal
thf(fact_34_sgraph_Oedge__vertices__not__equal,axiom,
! [Vertices: set_a,Edges: set_set_a,X: a,Y: a] :
( ( undire3507641187627840796raph_a @ Vertices @ Edges )
=> ( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ Edges )
=> ( X != Y ) ) ) ).
% sgraph.edge_vertices_not_equal
thf(fact_35_sgraph_Oedge__btw__vertices__not__equal,axiom,
! [Vertices: set_a,Edges: set_set_a,X: a,Y: a,X2: set_a,Y2: set_a] :
( ( undire3507641187627840796raph_a @ Vertices @ Edges )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( undire8383842906760478443ween_a @ Edges @ X2 @ Y2 ) )
=> ( X != Y ) ) ) ).
% sgraph.edge_btw_vertices_not_equal
thf(fact_36_sgraph_Ocard__triangle__triples__rotate,axiom,
! [Vertices: set_a,Edges: set_set_a,X2: set_a,Y2: set_a,Z2: set_a] :
( ( undire3507641187627840796raph_a @ Vertices @ Edges )
=> ( ( finite6893194910719049976od_a_a @ ( graph_4774508486909600516ples_a @ Edges @ X2 @ Y2 @ Z2 ) )
= ( finite6893194910719049976od_a_a @ ( graph_4774508486909600516ples_a @ Edges @ Y2 @ Z2 @ X2 ) ) ) ) ).
% sgraph.card_triangle_triples_rotate
thf(fact_37_sgraph_Otriangle__in__graph__edge__point,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X: set_a,Y: set_a,Z: set_a] :
( ( undire6035205377725458044_set_a @ Vertices @ Edges )
=> ( ( graph_3840782946058334608_set_a @ Edges @ X @ Y @ Z )
= ( ( member_set_set_a @ ( insert_set_a @ Y @ ( insert_set_a @ Z @ bot_bot_set_set_a ) ) @ Edges )
& ( undire3510646817838285160_set_a @ Edges @ X @ Y )
& ( undire3510646817838285160_set_a @ Edges @ X @ Z ) ) ) ) ).
% sgraph.triangle_in_graph_edge_point
thf(fact_38_sgraph_Otriangle__in__graph__edge__point,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
( ( undire8797707285729112389od_a_a @ Vertices @ Edges )
=> ( ( graph_4803287029668059225od_a_a @ Edges @ X @ Y @ Z )
= ( ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ Y @ ( insert4534936382041156343od_a_a @ Z @ bot_bo3357376287454694259od_a_a ) ) @ Edges )
& ( undire6135774327024169009od_a_a @ Edges @ X @ Y )
& ( undire6135774327024169009od_a_a @ Edges @ X @ Z ) ) ) ) ).
% sgraph.triangle_in_graph_edge_point
thf(fact_39_sgraph_Otriangle__in__graph__edge__point,axiom,
! [Vertices: set_nat,Edges: set_set_nat,X: nat,Y: nat,Z: nat] :
( ( undire7290660292559394354ph_nat @ Vertices @ Edges )
=> ( ( graph_2911189250448956958ph_nat @ Edges @ X @ Y @ Z )
= ( ( member_set_nat @ ( insert_nat @ Y @ ( insert_nat @ Z @ bot_bot_set_nat ) ) @ Edges )
& ( undire1083030068171319366dj_nat @ Edges @ X @ Y )
& ( undire1083030068171319366dj_nat @ Edges @ X @ Z ) ) ) ) ).
% sgraph.triangle_in_graph_edge_point
thf(fact_40_sgraph_Otriangle__in__graph__edge__point,axiom,
! [Vertices: set_a,Edges: set_set_a,X: a,Y: a,Z: a] :
( ( undire3507641187627840796raph_a @ Vertices @ Edges )
=> ( ( graph_4582152751571636272raph_a @ Edges @ X @ Y @ Z )
= ( ( member_set_a @ ( insert_a @ Y @ ( insert_a @ Z @ bot_bot_set_a ) ) @ Edges )
& ( undire397441198561214472_adj_a @ Edges @ X @ Y )
& ( undire397441198561214472_adj_a @ Edges @ X @ Z ) ) ) ) ).
% sgraph.triangle_in_graph_edge_point
thf(fact_41_sgraph_Otriangle__triples_Ocong,axiom,
graph_4774508486909600516ples_a = graph_4774508486909600516ples_a ).
% sgraph.triangle_triples.cong
thf(fact_42_sgraph_Otriangle__in__graph__edge__empty,axiom,
! [Vertices: set_a,Edges: set_set_a,X: a,Y: a,Z: a] :
( ( undire3507641187627840796raph_a @ Vertices @ Edges )
=> ( ( Edges = bot_bot_set_set_a )
=> ~ ( graph_4582152751571636272raph_a @ Edges @ X @ Y @ Z ) ) ) ).
% sgraph.triangle_in_graph_edge_empty
thf(fact_43_sgraph_Otriangle__vertices__distinct3,axiom,
! [Vertices: set_a,Edges: set_set_a,X: a,Y: a,Z: a] :
( ( undire3507641187627840796raph_a @ Vertices @ Edges )
=> ( ( graph_4582152751571636272raph_a @ Edges @ X @ Y @ Z )
=> ( Z != X ) ) ) ).
% sgraph.triangle_vertices_distinct3
thf(fact_44_sgraph_Otriangle__vertices__distinct2,axiom,
! [Vertices: set_a,Edges: set_set_a,X: a,Y: a,Z: a] :
( ( undire3507641187627840796raph_a @ Vertices @ Edges )
=> ( ( graph_4582152751571636272raph_a @ Edges @ X @ Y @ Z )
=> ( Y != Z ) ) ) ).
% sgraph.triangle_vertices_distinct2
thf(fact_45_vert__adj__inc__edge__iff,axiom,
! [V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
= ( ( undire1521409233611534436dent_a @ V1 @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) )
& ( undire1521409233611534436dent_a @ V2 @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) )
& ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) @ edges ) ) ) ).
% vert_adj_inc_edge_iff
thf(fact_46_has__loop__def,axiom,
! [V: a] :
( ( undire3617971648856834880loop_a @ edges @ V )
= ( member_set_a @ ( insert_a @ V @ bot_bot_set_a ) @ edges ) ) ).
% has_loop_def
thf(fact_47_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_48_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_49_singletonI,axiom,
! [A: product_prod_a_a] : ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ) ).
% singletonI
thf(fact_50_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_51_edge__density__zero,axiom,
! [Y2: set_a,X2: set_a] :
( ( Y2 = bot_bot_set_a )
=> ( ( undire297304480579013331sity_a @ edges @ X2 @ Y2 )
= zero_zero_real ) ) ).
% edge_density_zero
thf(fact_52_vert__adj__edge__iff2,axiom,
! [V1: a,V2: a] :
( ( V1 != V2 )
=> ( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
= ( ? [X4: set_a] :
( ( member_set_a @ X4 @ edges )
& ( undire1521409233611534436dent_a @ V1 @ X4 )
& ( undire1521409233611534436dent_a @ V2 @ X4 ) ) ) ) ) ).
% vert_adj_edge_iff2
thf(fact_53_edge__adj__def,axiom,
! [E1: set_a,E2: set_a] :
( ( undire4022703626023482010_adj_a @ edges @ E1 @ E2 )
= ( ( ( inf_inf_set_a @ E1 @ E2 )
!= bot_bot_set_a )
& ( member_set_a @ E1 @ edges )
& ( member_set_a @ E2 @ edges ) ) ) ).
% edge_adj_def
thf(fact_54_sgraph_Osingleton__not__edge,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X: set_a] :
( ( undire6035205377725458044_set_a @ Vertices @ Edges )
=> ~ ( member_set_set_a @ ( insert_set_a @ X @ bot_bot_set_set_a ) @ Edges ) ) ).
% sgraph.singleton_not_edge
thf(fact_55_sgraph_Osingleton__not__edge,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,X: product_prod_a_a] :
( ( undire8797707285729112389od_a_a @ Vertices @ Edges )
=> ~ ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) @ Edges ) ) ).
% sgraph.singleton_not_edge
thf(fact_56_sgraph_Osingleton__not__edge,axiom,
! [Vertices: set_nat,Edges: set_set_nat,X: nat] :
( ( undire7290660292559394354ph_nat @ Vertices @ Edges )
=> ~ ( member_set_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ Edges ) ) ).
% sgraph.singleton_not_edge
thf(fact_57_sgraph_Osingleton__not__edge,axiom,
! [Vertices: set_a,Edges: set_set_a,X: a] :
( ( undire3507641187627840796raph_a @ Vertices @ Edges )
=> ~ ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ Edges ) ) ).
% sgraph.singleton_not_edge
thf(fact_58_insertCI,axiom,
! [A: set_a,B: set_set_a,B2: set_a] :
( ( ~ ( member_set_a @ A @ B )
=> ( A = B2 ) )
=> ( member_set_a @ A @ ( insert_set_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_59_insertCI,axiom,
! [A: product_prod_a_a,B: set_Product_prod_a_a,B2: product_prod_a_a] :
( ( ~ ( member1426531477525435216od_a_a @ A @ B )
=> ( A = B2 ) )
=> ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_60_insertCI,axiom,
! [A: a,B: set_a,B2: a] :
( ( ~ ( member_a @ A @ B )
=> ( A = B2 ) )
=> ( member_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_61_insertCI,axiom,
! [A: nat,B: set_nat,B2: nat] :
( ( ~ ( member_nat @ A @ B )
=> ( A = B2 ) )
=> ( member_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_62_insert__iff,axiom,
! [A: set_a,B2: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_set_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_63_insert__iff,axiom,
! [A: product_prod_a_a,B2: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member1426531477525435216od_a_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_64_insert__iff,axiom,
! [A: a,B2: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_65_insert__iff,axiom,
! [A: nat,B2: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_66_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_67_mem__Collect__eq,axiom,
! [A: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A @ ( collec3336397797384452498od_a_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_68_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_69_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_70_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X4: set_a] : ( member_set_a @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_71_Collect__mem__eq,axiom,
! [A2: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_72_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_73_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_74_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_75_insert__absorb2,axiom,
! [X: set_a,A2: set_set_a] :
( ( insert_set_a @ X @ ( insert_set_a @ X @ A2 ) )
= ( insert_set_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_76_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_77_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_78_empty__iff,axiom,
! [C: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ C @ bot_bo3357376287454694259od_a_a ) ).
% empty_iff
thf(fact_79_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_80_all__not__in__conv,axiom,
! [A2: set_set_a] :
( ( ! [X4: set_a] :
~ ( member_set_a @ X4 @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_81_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X4: a] :
~ ( member_a @ X4 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_82_all__not__in__conv,axiom,
! [A2: set_Product_prod_a_a] :
( ( ! [X4: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X4 @ A2 ) )
= ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% all_not_in_conv
thf(fact_83_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X4: nat] :
~ ( member_nat @ X4 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_84_incident__def,axiom,
undire1521409233611534436dent_a = member_a ).
% incident_def
thf(fact_85_empty__Collect__eq,axiom,
! [P: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P ) )
= ( ! [X4: set_a] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_86_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X4: a] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_87_empty__Collect__eq,axiom,
! [P: product_prod_a_a > $o] :
( ( bot_bo3357376287454694259od_a_a
= ( collec3336397797384452498od_a_a @ P ) )
= ( ! [X4: product_prod_a_a] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_88_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X4: nat] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_89_Collect__empty__eq,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( ! [X4: set_a] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_90_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X4: a] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_91_Collect__empty__eq,axiom,
! [P: product_prod_a_a > $o] :
( ( ( collec3336397797384452498od_a_a @ P )
= bot_bo3357376287454694259od_a_a )
= ( ! [X4: product_prod_a_a] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_92_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X4: nat] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_93_Int__iff,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) )
= ( ( member1426531477525435216od_a_a @ C @ A2 )
& ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_94_Int__iff,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B ) )
= ( ( member_nat @ C @ A2 )
& ( member_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_95_Int__iff,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( member_a @ C @ A2 )
& ( member_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_96_Int__iff,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B ) )
= ( ( member_set_a @ C @ A2 )
& ( member_set_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_97_IntI,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A2 )
=> ( ( member1426531477525435216od_a_a @ C @ B )
=> ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_98_IntI,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_99_IntI,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ A2 )
=> ( ( member_a @ C @ B )
=> ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_100_IntI,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A2 )
=> ( ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_101_Int__insert__right__if1,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ A2 )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ A @ B ) )
= ( insert4534936382041156343od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_102_Int__insert__right__if1,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_103_Int__insert__right__if1,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_104_Int__insert__right__if1,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_105_Int__insert__right__if0,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A @ A2 )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ A @ B ) )
= ( inf_in8905007599844390133od_a_a @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_106_Int__insert__right__if0,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( inf_inf_set_nat @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_107_Int__insert__right__if0,axiom,
! [A: a,A2: set_a,B: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_108_Int__insert__right__if0,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_109_insert__inter__insert,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_110_insert__inter__insert,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ ( insert_set_a @ A @ B ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_111_Int__insert__left__if1,axiom,
! [A: product_prod_a_a,C2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ B ) @ C2 )
= ( insert4534936382041156343od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_112_Int__insert__left__if1,axiom,
! [A: nat,C2: set_nat,B: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B ) @ C2 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_113_Int__insert__left__if1,axiom,
! [A: a,C2: set_a,B: set_a] :
( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_114_Int__insert__left__if1,axiom,
! [A: set_a,C2: set_set_a,B: set_set_a] :
( ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C2 )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_115_Int__insert__left__if0,axiom,
! [A: product_prod_a_a,C2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ B ) @ C2 )
= ( inf_in8905007599844390133od_a_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_116_Int__insert__left__if0,axiom,
! [A: nat,C2: set_nat,B: set_nat] :
( ~ ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_117_Int__insert__left__if0,axiom,
! [A: a,C2: set_a,B: set_a] :
( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_118_Int__insert__left__if0,axiom,
! [A: set_a,C2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_119_insert__disjoint_I1_J,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ B )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B )
& ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_120_insert__disjoint_I1_J,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B )
& ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_121_insert__disjoint_I1_J,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ A2 ) @ B )
= bot_bo3357376287454694259od_a_a )
= ( ~ ( member1426531477525435216od_a_a @ A @ B )
& ( ( inf_in8905007599844390133od_a_a @ A2 @ B )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% insert_disjoint(1)
thf(fact_122_insert__disjoint_I1_J,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B )
& ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_123_insert__disjoint_I2_J,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ B ) )
= ( ~ ( member_set_a @ A @ B )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_124_insert__disjoint_I2_J,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B ) )
= ( ~ ( member_a @ A @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_125_insert__disjoint_I2_J,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ A2 ) @ B ) )
= ( ~ ( member1426531477525435216od_a_a @ A @ B )
& ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_126_insert__disjoint_I2_J,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B ) )
= ( ~ ( member_nat @ A @ B )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_127_disjoint__insert_I1_J,axiom,
! [B: set_set_a,A: set_a,A2: set_set_a] :
( ( ( inf_inf_set_set_a @ B @ ( insert_set_a @ A @ A2 ) )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B )
& ( ( inf_inf_set_set_a @ B @ A2 )
= bot_bot_set_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_128_disjoint__insert_I1_J,axiom,
! [B: set_a,A: a,A2: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A @ A2 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B )
& ( ( inf_inf_set_a @ B @ A2 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_129_disjoint__insert_I1_J,axiom,
! [B: set_Product_prod_a_a,A: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ B @ ( insert4534936382041156343od_a_a @ A @ A2 ) )
= bot_bo3357376287454694259od_a_a )
= ( ~ ( member1426531477525435216od_a_a @ A @ B )
& ( ( inf_in8905007599844390133od_a_a @ B @ A2 )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% disjoint_insert(1)
thf(fact_130_disjoint__insert_I1_J,axiom,
! [B: set_nat,A: nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ B @ ( insert_nat @ A @ A2 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B )
& ( ( inf_inf_set_nat @ B @ A2 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_131_disjoint__insert_I2_J,axiom,
! [A2: set_set_a,B2: set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ B2 @ B ) ) )
= ( ~ ( member_set_a @ B2 @ A2 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_132_disjoint__insert_I2_J,axiom,
! [A2: set_a,B2: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ ( insert_a @ B2 @ B ) ) )
= ( ~ ( member_a @ B2 @ A2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_133_disjoint__insert_I2_J,axiom,
! [A2: set_Product_prod_a_a,B2: product_prod_a_a,B: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B2 @ B ) ) )
= ( ~ ( member1426531477525435216od_a_a @ B2 @ A2 )
& ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_134_disjoint__insert_I2_J,axiom,
! [A2: set_nat,B2: nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ ( insert_nat @ B2 @ B ) ) )
= ( ~ ( member_nat @ B2 @ A2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_135_all__edges__between__empty_I2_J,axiom,
! [Z2: set_a] :
( ( undire8383842906760478443ween_a @ edges @ Z2 @ bot_bot_set_a )
= bot_bo3357376287454694259od_a_a ) ).
% all_edges_between_empty(2)
thf(fact_136_all__edges__between__empty_I1_J,axiom,
! [Z2: set_a] :
( ( undire8383842906760478443ween_a @ edges @ bot_bot_set_a @ Z2 )
= bot_bo3357376287454694259od_a_a ) ).
% all_edges_between_empty(1)
thf(fact_137_Int__left__commute,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_138_Int__left__commute,axiom,
! [A2: set_set_a,B: set_set_a,C2: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C2 ) )
= ( inf_inf_set_set_a @ B @ ( inf_inf_set_set_a @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_139_Int__left__absorb,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ).
% Int_left_absorb
thf(fact_140_Int__left__absorb,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ ( inf_inf_set_set_a @ A2 @ B ) )
= ( inf_inf_set_set_a @ A2 @ B ) ) ).
% Int_left_absorb
thf(fact_141_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A3: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_142_Int__commute,axiom,
( inf_inf_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] : ( inf_inf_set_set_a @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_143_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_144_Int__absorb,axiom,
! [A2: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_145_Int__assoc,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_146_Int__assoc,axiom,
! [A2: set_set_a,B: set_set_a,C2: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_147_IntD2,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) )
=> ( member1426531477525435216od_a_a @ C @ B ) ) ).
% IntD2
thf(fact_148_IntD2,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B ) )
=> ( member_nat @ C @ B ) ) ).
% IntD2
thf(fact_149_IntD2,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C @ B ) ) ).
% IntD2
thf(fact_150_IntD2,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ( member_set_a @ C @ B ) ) ).
% IntD2
thf(fact_151_IntD1,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) )
=> ( member1426531477525435216od_a_a @ C @ A2 ) ) ).
% IntD1
thf(fact_152_IntD1,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B ) )
=> ( member_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_153_IntD1,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C @ A2 ) ) ).
% IntD1
thf(fact_154_IntD1,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ( member_set_a @ C @ A2 ) ) ).
% IntD1
thf(fact_155_IntE,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) )
=> ~ ( ( member1426531477525435216od_a_a @ C @ A2 )
=> ~ ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% IntE
thf(fact_156_IntE,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ~ ( member_nat @ C @ B ) ) ) ).
% IntE
thf(fact_157_IntE,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
=> ~ ( ( member_a @ C @ A2 )
=> ~ ( member_a @ C @ B ) ) ) ).
% IntE
thf(fact_158_IntE,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ~ ( ( member_set_a @ C @ A2 )
=> ~ ( member_set_a @ C @ B ) ) ) ).
% IntE
thf(fact_159_comp__sgraph_Oincident__def,axiom,
undire2320338297334612420_set_a = member_set_a ).
% comp_sgraph.incident_def
thf(fact_160_comp__sgraph_Oincident__def,axiom,
undire3369688177417741453od_a_a = member1426531477525435216od_a_a ).
% comp_sgraph.incident_def
thf(fact_161_comp__sgraph_Oincident__def,axiom,
undire7858122600432113898nt_nat = member_nat ).
% comp_sgraph.incident_def
thf(fact_162_comp__sgraph_Oincident__def,axiom,
undire1521409233611534436dent_a = member_a ).
% comp_sgraph.incident_def
thf(fact_163_ulgraph_Ohas__loop_Ocong,axiom,
undire3617971648856834880loop_a = undire3617971648856834880loop_a ).
% ulgraph.has_loop.cong
thf(fact_164_disjoint__iff__not__equal,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ B )
=> ( X4 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_165_disjoint__iff__not__equal,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ B )
=> ( X4 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_166_disjoint__iff__not__equal,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ A2 @ B )
= bot_bo3357376287454694259od_a_a )
= ( ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A2 )
=> ! [Y4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y4 @ B )
=> ( X4 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_167_disjoint__iff__not__equal,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ B )
=> ( X4 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_168_Int__empty__right,axiom,
! [A2: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% Int_empty_right
thf(fact_169_Int__empty__right,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_170_Int__empty__right,axiom,
! [A2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A2 @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% Int_empty_right
thf(fact_171_Int__empty__right,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_172_Int__empty__left,axiom,
! [B: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ B )
= bot_bot_set_set_a ) ).
% Int_empty_left
thf(fact_173_Int__empty__left,axiom,
! [B: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_174_Int__empty__left,axiom,
! [B: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ B )
= bot_bo3357376287454694259od_a_a ) ).
% Int_empty_left
thf(fact_175_Int__empty__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_176_disjoint__iff,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ~ ( member_set_a @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_177_disjoint__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ~ ( member_a @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_178_disjoint__iff,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ A2 @ B )
= bot_bo3357376287454694259od_a_a )
= ( ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A2 )
=> ~ ( member1426531477525435216od_a_a @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_179_disjoint__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ~ ( member_nat @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_180_Int__emptyI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
=> ~ ( member_set_a @ X5 @ B ) )
=> ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_181_Int__emptyI,axiom,
! [A2: set_a,B: set_a] :
( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ~ ( member_a @ X5 @ B ) )
=> ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_182_Int__emptyI,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A2 )
=> ~ ( member1426531477525435216od_a_a @ X5 @ B ) )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ B )
= bot_bo3357376287454694259od_a_a ) ) ).
% Int_emptyI
thf(fact_183_Int__emptyI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ~ ( member_nat @ X5 @ B ) )
=> ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_184_Int__insert__right,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ A @ A2 )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ A @ B ) )
= ( insert4534936382041156343od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) ) ) )
& ( ~ ( member1426531477525435216od_a_a @ A @ A2 )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ A @ B ) )
= ( inf_in8905007599844390133od_a_a @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_185_Int__insert__right,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B ) ) ) )
& ( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_186_Int__insert__right,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_187_Int__insert__right,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B ) ) ) )
& ( ~ ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_188_Int__insert__left,axiom,
! [A: product_prod_a_a,C2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ A @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ B ) @ C2 )
= ( insert4534936382041156343od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ B @ C2 ) ) ) )
& ( ~ ( member1426531477525435216od_a_a @ A @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ B ) @ C2 )
= ( inf_in8905007599844390133od_a_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_189_Int__insert__left,axiom,
! [A: nat,C2: set_nat,B: set_nat] :
( ( ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B ) @ C2 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) ) ) )
& ( ~ ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_190_Int__insert__left,axiom,
! [A: a,C2: set_a,B: set_a] :
( ( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_191_Int__insert__left,axiom,
! [A: set_a,C2: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C2 )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_192_sgraph_Ono__loops,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_a] :
( ( undire6035205377725458044_set_a @ Vertices @ Edges )
=> ( ( member_set_a @ V @ Vertices )
=> ~ ( undire5774735625301615776_set_a @ Edges @ V ) ) ) ).
% sgraph.no_loops
thf(fact_193_sgraph_Ono__loops,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V: product_prod_a_a] :
( ( undire8797707285729112389od_a_a @ Vertices @ Edges )
=> ( ( member1426531477525435216od_a_a @ V @ Vertices )
=> ~ ( undire7777398424729533289od_a_a @ Edges @ V ) ) ) ).
% sgraph.no_loops
thf(fact_194_sgraph_Ono__loops,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V: nat] :
( ( undire7290660292559394354ph_nat @ Vertices @ Edges )
=> ( ( member_nat @ V @ Vertices )
=> ~ ( undire5005864372999571214op_nat @ Edges @ V ) ) ) ).
% sgraph.no_loops
thf(fact_195_sgraph_Ono__loops,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire3507641187627840796raph_a @ Vertices @ Edges )
=> ( ( member_a @ V @ Vertices )
=> ~ ( undire3617971648856834880loop_a @ Edges @ V ) ) ) ).
% sgraph.no_loops
thf(fact_196_mk__triangle__set_Ocases,axiom,
! [X: produc4044097585999906000od_a_a] :
~ ! [X5: a,Y5: a,Z3: a] :
( X
!= ( produc431845341423274048od_a_a @ X5 @ ( product_Pair_a_a @ Y5 @ Z3 ) ) ) ).
% mk_triangle_set.cases
thf(fact_197_mk__edge_Ocases,axiom,
! [X: product_prod_a_a] :
~ ! [U2: a,V3: a] :
( X
!= ( product_Pair_a_a @ U2 @ V3 ) ) ).
% mk_edge.cases
thf(fact_198_ex__in__conv,axiom,
! [A2: set_set_a] :
( ( ? [X4: set_a] : ( member_set_a @ X4 @ A2 ) )
= ( A2 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_199_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X4: a] : ( member_a @ X4 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_200_ex__in__conv,axiom,
! [A2: set_Product_prod_a_a] :
( ( ? [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ A2 ) )
= ( A2 != bot_bo3357376287454694259od_a_a ) ) ).
% ex_in_conv
thf(fact_201_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X4: nat] : ( member_nat @ X4 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_202_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y5: set_a] :
~ ( member_set_a @ Y5 @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_203_equals0I,axiom,
! [A2: set_a] :
( ! [Y5: a] :
~ ( member_a @ Y5 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_204_equals0I,axiom,
! [A2: set_Product_prod_a_a] :
( ! [Y5: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ Y5 @ A2 )
=> ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% equals0I
thf(fact_205_equals0I,axiom,
! [A2: set_nat] :
( ! [Y5: nat] :
~ ( member_nat @ Y5 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_206_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_207_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_208_equals0D,axiom,
! [A2: set_Product_prod_a_a,A: product_prod_a_a] :
( ( A2 = bot_bo3357376287454694259od_a_a )
=> ~ ( member1426531477525435216od_a_a @ A @ A2 ) ) ).
% equals0D
thf(fact_209_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_210_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_211_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_212_emptyE,axiom,
! [A: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ).
% emptyE
thf(fact_213_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_214_mk__disjoint__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ? [B4: set_set_a] :
( ( A2
= ( insert_set_a @ A @ B4 ) )
& ~ ( member_set_a @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_215_mk__disjoint__insert,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ A2 )
=> ? [B4: set_Product_prod_a_a] :
( ( A2
= ( insert4534936382041156343od_a_a @ A @ B4 ) )
& ~ ( member1426531477525435216od_a_a @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_216_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B4: set_a] :
( ( A2
= ( insert_a @ A @ B4 ) )
& ~ ( member_a @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_217_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B4: set_nat] :
( ( A2
= ( insert_nat @ A @ B4 ) )
& ~ ( member_nat @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_218_insert__commute,axiom,
! [X: a,Y: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ Y @ A2 ) )
= ( insert_a @ Y @ ( insert_a @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_219_insert__commute,axiom,
! [X: set_a,Y: set_a,A2: set_set_a] :
( ( insert_set_a @ X @ ( insert_set_a @ Y @ A2 ) )
= ( insert_set_a @ Y @ ( insert_set_a @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_220_insert__eq__iff,axiom,
! [A: set_a,A2: set_set_a,B2: set_a,B: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ~ ( member_set_a @ B2 @ B )
=> ( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_set_a] :
( ( A2
= ( insert_set_a @ B2 @ C3 ) )
& ~ ( member_set_a @ B2 @ C3 )
& ( B
= ( insert_set_a @ A @ C3 ) )
& ~ ( member_set_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_221_insert__eq__iff,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a,B2: product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A @ A2 )
=> ( ~ ( member1426531477525435216od_a_a @ B2 @ B )
=> ( ( ( insert4534936382041156343od_a_a @ A @ A2 )
= ( insert4534936382041156343od_a_a @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_Product_prod_a_a] :
( ( A2
= ( insert4534936382041156343od_a_a @ B2 @ C3 ) )
& ~ ( member1426531477525435216od_a_a @ B2 @ C3 )
& ( B
= ( insert4534936382041156343od_a_a @ A @ C3 ) )
& ~ ( member1426531477525435216od_a_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_222_insert__eq__iff,axiom,
! [A: a,A2: set_a,B2: a,B: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B2 @ B )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_a] :
( ( A2
= ( insert_a @ B2 @ C3 ) )
& ~ ( member_a @ B2 @ C3 )
& ( B
= ( insert_a @ A @ C3 ) )
& ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_223_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B2: nat,B: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B2 @ B )
=> ( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_nat] :
( ( A2
= ( insert_nat @ B2 @ C3 ) )
& ~ ( member_nat @ B2 @ C3 )
& ( B
= ( insert_nat @ A @ C3 ) )
& ~ ( member_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_224_insert__absorb,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( insert_set_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_225_insert__absorb,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ A2 )
=> ( ( insert4534936382041156343od_a_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_226_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_227_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_228_insert__ident,axiom,
! [X: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ~ ( member_set_a @ X @ B )
=> ( ( ( insert_set_a @ X @ A2 )
= ( insert_set_a @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_229_insert__ident,axiom,
! [X: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X @ A2 )
=> ( ~ ( member1426531477525435216od_a_a @ X @ B )
=> ( ( ( insert4534936382041156343od_a_a @ X @ A2 )
= ( insert4534936382041156343od_a_a @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_230_insert__ident,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_231_insert__ident,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ~ ( member_nat @ X @ B )
=> ( ( ( insert_nat @ X @ A2 )
= ( insert_nat @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_232_Set_Oset__insert,axiom,
! [X: set_a,A2: set_set_a] :
( ( member_set_a @ X @ A2 )
=> ~ ! [B4: set_set_a] :
( ( A2
= ( insert_set_a @ X @ B4 ) )
=> ( member_set_a @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_233_Set_Oset__insert,axiom,
! [X: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A2 )
=> ~ ! [B4: set_Product_prod_a_a] :
( ( A2
= ( insert4534936382041156343od_a_a @ X @ B4 ) )
=> ( member1426531477525435216od_a_a @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_234_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B4: set_a] :
( ( A2
= ( insert_a @ X @ B4 ) )
=> ( member_a @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_235_Set_Oset__insert,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat @ X @ A2 )
=> ~ ! [B4: set_nat] :
( ( A2
= ( insert_nat @ X @ B4 ) )
=> ( member_nat @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_236_insertI2,axiom,
! [A: set_a,B: set_set_a,B2: set_a] :
( ( member_set_a @ A @ B )
=> ( member_set_a @ A @ ( insert_set_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_237_insertI2,axiom,
! [A: product_prod_a_a,B: set_Product_prod_a_a,B2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ B )
=> ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_238_insertI2,axiom,
! [A: a,B: set_a,B2: a] :
( ( member_a @ A @ B )
=> ( member_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_239_insertI2,axiom,
! [A: nat,B: set_nat,B2: nat] :
( ( member_nat @ A @ B )
=> ( member_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_240_insertI1,axiom,
! [A: set_a,B: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B ) ) ).
% insertI1
thf(fact_241_insertI1,axiom,
! [A: product_prod_a_a,B: set_Product_prod_a_a] : ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ A @ B ) ) ).
% insertI1
thf(fact_242_insertI1,axiom,
! [A: a,B: set_a] : ( member_a @ A @ ( insert_a @ A @ B ) ) ).
% insertI1
thf(fact_243_insertI1,axiom,
! [A: nat,B: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B ) ) ).
% insertI1
thf(fact_244_insertE,axiom,
! [A: set_a,B2: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_set_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_245_insertE,axiom,
! [A: product_prod_a_a,B2: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member1426531477525435216od_a_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_246_insertE,axiom,
! [A: a,B2: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_247_insertE,axiom,
! [A: nat,B2: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_248_comp__sgraph_Ois__edge__between__def,axiom,
( undire2578756059399487229_set_a
= ( ^ [X3: set_set_a,Y3: set_set_a,E3: set_set_a] :
? [X4: set_a,Y4: set_a] :
( ( E3
= ( insert_set_a @ X4 @ ( insert_set_a @ Y4 @ bot_bot_set_set_a ) ) )
& ( member_set_a @ X4 @ X3 )
& ( member_set_a @ Y4 @ Y3 ) ) ) ) ).
% comp_sgraph.is_edge_between_def
thf(fact_249_comp__sgraph_Ois__edge__between__def,axiom,
( undire7011261089604658374od_a_a
= ( ^ [X3: set_Product_prod_a_a,Y3: set_Product_prod_a_a,E3: set_Product_prod_a_a] :
? [X4: product_prod_a_a,Y4: product_prod_a_a] :
( ( E3
= ( insert4534936382041156343od_a_a @ X4 @ ( insert4534936382041156343od_a_a @ Y4 @ bot_bo3357376287454694259od_a_a ) ) )
& ( member1426531477525435216od_a_a @ X4 @ X3 )
& ( member1426531477525435216od_a_a @ Y4 @ Y3 ) ) ) ) ).
% comp_sgraph.is_edge_between_def
thf(fact_250_comp__sgraph_Ois__edge__between__def,axiom,
( undire6814325412647357297en_nat
= ( ^ [X3: set_nat,Y3: set_nat,E3: set_nat] :
? [X4: nat,Y4: nat] :
( ( E3
= ( insert_nat @ X4 @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) )
& ( member_nat @ X4 @ X3 )
& ( member_nat @ Y4 @ Y3 ) ) ) ) ).
% comp_sgraph.is_edge_between_def
thf(fact_251_comp__sgraph_Ois__edge__between__def,axiom,
( undire8544646567961481629ween_a
= ( ^ [X3: set_a,Y3: set_a,E3: set_a] :
? [X4: a,Y4: a] :
( ( E3
= ( insert_a @ X4 @ ( insert_a @ Y4 @ bot_bot_set_a ) ) )
& ( member_a @ X4 @ X3 )
& ( member_a @ Y4 @ Y3 ) ) ) ) ).
% comp_sgraph.is_edge_between_def
thf(fact_252_ulgraph_Oall__edges__between_Ocong,axiom,
undire8383842906760478443ween_a = undire8383842906760478443ween_a ).
% ulgraph.all_edges_between.cong
thf(fact_253_ulgraph_Overt__adj_Ocong,axiom,
undire397441198561214472_adj_a = undire397441198561214472_adj_a ).
% ulgraph.vert_adj.cong
thf(fact_254_ulgraph_Oedge__density_Ocong,axiom,
undire297304480579013331sity_a = undire297304480579013331sity_a ).
% ulgraph.edge_density.cong
thf(fact_255_graph__system_Oedge__adj_Ocong,axiom,
undire4022703626023482010_adj_a = undire4022703626023482010_adj_a ).
% graph_system.edge_adj.cong
thf(fact_256_singleton__inject,axiom,
! [A: set_a,B2: set_a] :
( ( ( insert_set_a @ A @ bot_bot_set_set_a )
= ( insert_set_a @ B2 @ bot_bot_set_set_a ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_257_singleton__inject,axiom,
! [A: a,B2: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B2 @ bot_bot_set_a ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_258_singleton__inject,axiom,
! [A: product_prod_a_a,B2: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a )
= ( insert4534936382041156343od_a_a @ B2 @ bot_bo3357376287454694259od_a_a ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_259_singleton__inject,axiom,
! [A: nat,B2: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_260_insert__not__empty,axiom,
! [A: set_a,A2: set_set_a] :
( ( insert_set_a @ A @ A2 )
!= bot_bot_set_set_a ) ).
% insert_not_empty
thf(fact_261_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_262_insert__not__empty,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( insert4534936382041156343od_a_a @ A @ A2 )
!= bot_bo3357376287454694259od_a_a ) ).
% insert_not_empty
thf(fact_263_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_264_doubleton__eq__iff,axiom,
! [A: set_a,B2: set_a,C: set_a,D: set_a] :
( ( ( insert_set_a @ A @ ( insert_set_a @ B2 @ bot_bot_set_set_a ) )
= ( insert_set_a @ C @ ( insert_set_a @ D @ bot_bot_set_set_a ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_265_doubleton__eq__iff,axiom,
! [A: a,B2: a,C: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_266_doubleton__eq__iff,axiom,
! [A: product_prod_a_a,B2: product_prod_a_a,C: product_prod_a_a,D: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A @ ( insert4534936382041156343od_a_a @ B2 @ bot_bo3357376287454694259od_a_a ) )
= ( insert4534936382041156343od_a_a @ C @ ( insert4534936382041156343od_a_a @ D @ bot_bo3357376287454694259od_a_a ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_267_doubleton__eq__iff,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_268_singleton__iff,axiom,
! [B2: set_a,A: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_269_singleton__iff,axiom,
! [B2: a,A: a] :
( ( member_a @ B2 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_270_singleton__iff,axiom,
! [B2: product_prod_a_a,A: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B2 @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_271_singleton__iff,axiom,
! [B2: nat,A: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_272_singletonD,axiom,
! [B2: set_a,A: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_273_singletonD,axiom,
! [B2: a,A: a] :
( ( member_a @ B2 @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_274_singletonD,axiom,
! [B2: product_prod_a_a,A: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B2 @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_275_singletonD,axiom,
! [B2: nat,A: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_276_incident__loops__simp_I1_J,axiom,
! [V: a] :
( ( undire3617971648856834880loop_a @ edges @ V )
=> ( ( undire4753905205749729249oops_a @ edges @ V )
= ( insert_set_a @ ( insert_a @ V @ bot_bot_set_a ) @ bot_bot_set_set_a ) ) ) ).
% incident_loops_simp(1)
thf(fact_277_incident__loops__simp_I2_J,axiom,
! [V: a] :
( ~ ( undire3617971648856834880loop_a @ edges @ V )
=> ( ( undire4753905205749729249oops_a @ edges @ V )
= bot_bot_set_set_a ) ) ).
% incident_loops_simp(2)
thf(fact_278_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ X @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_279_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_280_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_281_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_282_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_283_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_284_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ X )
= bot_bo3357376287454694259od_a_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_285_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_286_inf__bot__right,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ X @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% inf_bot_right
thf(fact_287_inf__bot__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_288_inf__bot__right,axiom,
! [X: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% inf_bot_right
thf(fact_289_inf__bot__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_290_inf__bot__left,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X )
= bot_bot_set_set_a ) ).
% inf_bot_left
thf(fact_291_inf__bot__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_292_inf__bot__left,axiom,
! [X: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ X )
= bot_bo3357376287454694259od_a_a ) ).
% inf_bot_left
thf(fact_293_inf__bot__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_294_edge__density__eq0,axiom,
! [A2: set_a,B: set_a,X2: set_a,Y2: set_a] :
( ( ( undire8383842906760478443ween_a @ edges @ A2 @ B )
= bot_bo3357376287454694259od_a_a )
=> ( ( ord_less_eq_set_a @ X2 @ A2 )
=> ( ( ord_less_eq_set_a @ Y2 @ B )
=> ( ( undire297304480579013331sity_a @ edges @ X2 @ Y2 )
= zero_zero_real ) ) ) ) ).
% edge_density_eq0
thf(fact_295_edge__density__ge0,axiom,
! [X2: set_a,Y2: set_a] : ( ord_less_eq_real @ zero_zero_real @ ( undire297304480579013331sity_a @ edges @ X2 @ Y2 ) ) ).
% edge_density_ge0
thf(fact_296_mk__triangle__set_Osimps,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( undire4638465864238448455_set_a @ ( produc7299740244201487072_set_a @ X @ ( produc9088192753505129239_set_a @ Y @ Z ) ) )
= ( insert_set_a @ X @ ( insert_set_a @ Y @ ( insert_set_a @ Z @ bot_bot_set_set_a ) ) ) ) ).
% mk_triangle_set.simps
thf(fact_297_mk__triangle__set_Osimps,axiom,
! [X: a,Y: a,Z: a] :
( ( undire8536760333753235943_set_a @ ( produc431845341423274048od_a_a @ X @ ( product_Pair_a_a @ Y @ Z ) ) )
= ( insert_a @ X @ ( insert_a @ Y @ ( insert_a @ Z @ bot_bot_set_a ) ) ) ) ).
% mk_triangle_set.simps
thf(fact_298_mk__triangle__set_Osimps,axiom,
! [X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
( ( undire2459242765783757584od_a_a @ ( produc4925843558922497303od_a_a @ X @ ( produc7886510207707329367od_a_a @ Y @ Z ) ) )
= ( insert4534936382041156343od_a_a @ X @ ( insert4534936382041156343od_a_a @ Y @ ( insert4534936382041156343od_a_a @ Z @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% mk_triangle_set.simps
thf(fact_299_mk__triangle__set_Osimps,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( undire4970100481470743719et_nat @ ( produc487386426758144856at_nat @ X @ ( product_Pair_nat_nat @ Y @ Z ) ) )
= ( insert_nat @ X @ ( insert_nat @ Y @ ( insert_nat @ Z @ bot_bot_set_nat ) ) ) ) ).
% mk_triangle_set.simps
thf(fact_300_mk__triangle__set_Oelims,axiom,
! [X: produc3364680560414100336_set_a,Y: set_set_a] :
( ( ( undire4638465864238448455_set_a @ X )
= Y )
=> ~ ! [X5: set_a,Y5: set_a,Z3: set_a] :
( ( X
= ( produc7299740244201487072_set_a @ X5 @ ( produc9088192753505129239_set_a @ Y5 @ Z3 ) ) )
=> ( Y
!= ( insert_set_a @ X5 @ ( insert_set_a @ Y5 @ ( insert_set_a @ Z3 @ bot_bot_set_set_a ) ) ) ) ) ) ).
% mk_triangle_set.elims
thf(fact_301_mk__triangle__set_Oelims,axiom,
! [X: produc4044097585999906000od_a_a,Y: set_a] :
( ( ( undire8536760333753235943_set_a @ X )
= Y )
=> ~ ! [X5: a,Y5: a,Z3: a] :
( ( X
= ( produc431845341423274048od_a_a @ X5 @ ( product_Pair_a_a @ Y5 @ Z3 ) ) )
=> ( Y
!= ( insert_a @ X5 @ ( insert_a @ Y5 @ ( insert_a @ Z3 @ bot_bot_set_a ) ) ) ) ) ) ).
% mk_triangle_set.elims
thf(fact_302_mk__triangle__set_Oelims,axiom,
! [X: produc8857593507947890343od_a_a,Y: set_Product_prod_a_a] :
( ( ( undire2459242765783757584od_a_a @ X )
= Y )
=> ~ ! [X5: product_prod_a_a,Y5: product_prod_a_a,Z3: product_prod_a_a] :
( ( X
= ( produc4925843558922497303od_a_a @ X5 @ ( produc7886510207707329367od_a_a @ Y5 @ Z3 ) ) )
=> ( Y
!= ( insert4534936382041156343od_a_a @ X5 @ ( insert4534936382041156343od_a_a @ Y5 @ ( insert4534936382041156343od_a_a @ Z3 @ bot_bo3357376287454694259od_a_a ) ) ) ) ) ) ).
% mk_triangle_set.elims
thf(fact_303_mk__triangle__set_Oelims,axiom,
! [X: produc7248412053542808358at_nat,Y: set_nat] :
( ( ( undire4970100481470743719et_nat @ X )
= Y )
=> ~ ! [X5: nat,Y5: nat,Z3: nat] :
( ( X
= ( produc487386426758144856at_nat @ X5 @ ( product_Pair_nat_nat @ Y5 @ Z3 ) ) )
=> ( Y
!= ( insert_nat @ X5 @ ( insert_nat @ Y5 @ ( insert_nat @ Z3 @ bot_bot_set_nat ) ) ) ) ) ) ).
% mk_triangle_set.elims
thf(fact_304_subset__antisym,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_305_subset__antisym,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( ord_le746702958409616551od_a_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_306_subset__antisym,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_307_subsetI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( member_nat @ X5 @ B ) )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_308_subsetI,axiom,
! [A2: set_a,B: set_a] :
( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( member_a @ X5 @ B ) )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_309_subsetI,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A2 )
=> ( member1426531477525435216od_a_a @ X5 @ B ) )
=> ( ord_le746702958409616551od_a_a @ A2 @ B ) ) ).
% subsetI
thf(fact_310_subsetI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
=> ( member_set_a @ X5 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_311_inf_Oidem,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_312_inf_Oidem,axiom,
! [A: set_set_a] :
( ( inf_inf_set_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_313_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_314_inf__idem,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_315_inf_Oleft__idem,axiom,
! [A: set_a,B2: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B2 ) )
= ( inf_inf_set_a @ A @ B2 ) ) ).
% inf.left_idem
thf(fact_316_inf_Oleft__idem,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ A @ B2 ) )
= ( inf_inf_set_set_a @ A @ B2 ) ) ).
% inf.left_idem
thf(fact_317_inf__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_left_idem
thf(fact_318_inf__left__idem,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ X @ Y ) )
= ( inf_inf_set_set_a @ X @ Y ) ) ).
% inf_left_idem
thf(fact_319_inf_Oright__idem,axiom,
! [A: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B2 ) @ B2 )
= ( inf_inf_set_a @ A @ B2 ) ) ).
% inf.right_idem
thf(fact_320_inf_Oright__idem,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ B2 )
= ( inf_inf_set_set_a @ A @ B2 ) ) ).
% inf.right_idem
thf(fact_321_inf__right__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_322_inf__right__idem,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Y )
= ( inf_inf_set_set_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_323_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_324_le__inf__iff,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ ( inf_inf_real @ Y @ Z ) )
= ( ( ord_less_eq_real @ X @ Y )
& ( ord_less_eq_real @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_325_le__inf__iff,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ Y @ Z ) )
= ( ( ord_le746702958409616551od_a_a @ X @ Y )
& ( ord_le746702958409616551od_a_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_326_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_327_le__inf__iff,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) )
= ( ( ord_le3724670747650509150_set_a @ X @ Y )
& ( ord_le3724670747650509150_set_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_328_inf_Obounded__iff,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B2 @ C ) )
= ( ( ord_less_eq_set_a @ A @ B2 )
& ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_329_inf_Obounded__iff,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ ( inf_inf_real @ B2 @ C ) )
= ( ( ord_less_eq_real @ A @ B2 )
& ( ord_less_eq_real @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_330_inf_Obounded__iff,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ B2 @ C ) )
= ( ( ord_le746702958409616551od_a_a @ A @ B2 )
& ( ord_le746702958409616551od_a_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_331_inf_Obounded__iff,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B2 @ C ) )
= ( ( ord_less_eq_nat @ A @ B2 )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_332_inf_Obounded__iff,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( inf_inf_set_set_a @ B2 @ C ) )
= ( ( ord_le3724670747650509150_set_a @ A @ B2 )
& ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_333_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_334_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_335_subset__empty,axiom,
! [A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ bot_bo3357376287454694259od_a_a )
= ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% subset_empty
thf(fact_336_subset__empty,axiom,
! [A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a )
= ( A2 = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_337_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_338_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_339_empty__subsetI,axiom,
! [A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ bot_bo3357376287454694259od_a_a @ A2 ) ).
% empty_subsetI
thf(fact_340_empty__subsetI,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A2 ) ).
% empty_subsetI
thf(fact_341_insert__subset,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A2 ) @ B )
= ( ( member_nat @ X @ B )
& ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_342_insert__subset,axiom,
! [X: a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( ( member_a @ X @ B )
& ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_343_insert__subset,axiom,
! [X: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ X @ A2 ) @ B )
= ( ( member1426531477525435216od_a_a @ X @ B )
& ( ord_le746702958409616551od_a_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_344_insert__subset,axiom,
! [X: set_a,A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A2 ) @ B )
= ( ( member_set_a @ X @ B )
& ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_345_Int__subset__iff,axiom,
! [C2: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( ord_less_eq_set_a @ C2 @ A2 )
& ( ord_less_eq_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_346_Int__subset__iff,axiom,
! [C2: set_Product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C2 @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) )
= ( ( ord_le746702958409616551od_a_a @ C2 @ A2 )
& ( ord_le746702958409616551od_a_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_347_Int__subset__iff,axiom,
! [C2: set_set_a,A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) )
= ( ( ord_le3724670747650509150_set_a @ C2 @ A2 )
& ( ord_le3724670747650509150_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_348_singleton__insert__inj__eq,axiom,
! [B2: nat,A: nat,A2: set_nat] :
( ( ( insert_nat @ B2 @ bot_bot_set_nat )
= ( insert_nat @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_349_singleton__insert__inj__eq,axiom,
! [B2: a,A: a,A2: set_a] :
( ( ( insert_a @ B2 @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_350_singleton__insert__inj__eq,axiom,
! [B2: product_prod_a_a,A: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ B2 @ bot_bo3357376287454694259od_a_a )
= ( insert4534936382041156343od_a_a @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le746702958409616551od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B2 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_351_singleton__insert__inj__eq,axiom,
! [B2: set_a,A: set_a,A2: set_set_a] :
( ( ( insert_set_a @ B2 @ bot_bot_set_set_a )
= ( insert_set_a @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B2 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_352_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B2: nat] :
( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( ( A = B2 )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_353_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B2: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B2 @ bot_bot_set_a ) )
= ( ( A = B2 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_354_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a,B2: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A @ A2 )
= ( insert4534936382041156343od_a_a @ B2 @ bot_bo3357376287454694259od_a_a ) )
= ( ( A = B2 )
& ( ord_le746702958409616551od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B2 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_355_singleton__insert__inj__eq_H,axiom,
! [A: set_a,A2: set_set_a,B2: set_a] :
( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B2 @ bot_bot_set_set_a ) )
= ( ( A = B2 )
& ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B2 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_356_unique__triangles__def,axiom,
( ( graph_6144490306505338871gles_a @ edges )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ edges )
=> ? [Y4: set_a] :
( ? [Z4: a,Aa: a,Ab: a] :
( ( Y4
= ( insert_a @ Z4 @ ( insert_a @ Aa @ ( insert_a @ Ab @ bot_bot_set_a ) ) ) )
& ( graph_4582152751571636272raph_a @ edges @ Z4 @ Aa @ Ab )
& ( ord_less_eq_set_a @ X4 @ Y4 ) )
& ! [Z4: set_a] :
( ? [Aa: a,Ab: a,Ac: a] :
( ( Z4
= ( insert_a @ Aa @ ( insert_a @ Ab @ ( insert_a @ Ac @ bot_bot_set_a ) ) ) )
& ( graph_4582152751571636272raph_a @ edges @ Aa @ Ab @ Ac )
& ( ord_less_eq_set_a @ X4 @ Z4 ) )
=> ( Z4 = Y4 ) ) ) ) ) ) ).
% unique_triangles_def
thf(fact_357_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_358_Collect__mono__iff,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P ) @ ( collec3336397797384452498od_a_a @ Q ) )
= ( ! [X4: product_prod_a_a] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_359_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X4: set_a] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_360_set__eq__subset,axiom,
( ( ^ [Y6: set_a,Z5: set_a] : ( Y6 = Z5 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_361_set__eq__subset,axiom,
( ( ^ [Y6: set_Product_prod_a_a,Z5: set_Product_prod_a_a] : ( Y6 = Z5 ) )
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ B3 )
& ( ord_le746702958409616551od_a_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_362_set__eq__subset,axiom,
( ( ^ [Y6: set_set_a,Z5: set_set_a] : ( Y6 = Z5 ) )
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_363_subset__trans,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_364_subset__trans,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( ord_le746702958409616551od_a_a @ B @ C2 )
=> ( ord_le746702958409616551od_a_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_365_subset__trans,axiom,
! [A2: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_366_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X5: a] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_367_Collect__mono,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X5: product_prod_a_a] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P ) @ ( collec3336397797384452498od_a_a @ Q ) ) ) ).
% Collect_mono
thf(fact_368_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X5: set_a] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_369_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_370_subset__refl,axiom,
! [A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_371_subset__refl,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_372_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_373_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [T: a] :
( ( member_a @ T @ A3 )
=> ( member_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_374_subset__iff,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
! [T: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T @ A3 )
=> ( member1426531477525435216od_a_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_375_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A3 )
=> ( member_set_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_376_equalityD2,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_377_equalityD2,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A2 = B )
=> ( ord_le746702958409616551od_a_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_378_equalityD2,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( A2 = B )
=> ( ord_le3724670747650509150_set_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_379_equalityD1,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_380_equalityD1,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A2 = B )
=> ( ord_le746702958409616551od_a_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_381_equalityD1,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( A2 = B )
=> ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_382_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ( member_nat @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_383_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [X4: a] :
( ( member_a @ X4 @ A3 )
=> ( member_a @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_384_subset__eq,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A3 )
=> ( member1426531477525435216od_a_a @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_385_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
! [X4: set_a] :
( ( member_set_a @ X4 @ A3 )
=> ( member_set_a @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_386_equalityE,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B )
=> ~ ( ord_less_eq_set_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_387_equalityE,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A2 = B )
=> ~ ( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ~ ( ord_le746702958409616551od_a_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_388_equalityE,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( A2 = B )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ~ ( ord_le3724670747650509150_set_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_389_subsetD,axiom,
! [A2: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_390_subsetD,axiom,
! [A2: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_391_subsetD,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( member1426531477525435216od_a_a @ C @ A2 )
=> ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% subsetD
thf(fact_392_subsetD,axiom,
! [A2: set_set_a,B: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B ) ) ) ).
% subsetD
thf(fact_393_in__mono,axiom,
! [A2: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_394_in__mono,axiom,
! [A2: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_395_in__mono,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,X: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( member1426531477525435216od_a_a @ X @ A2 )
=> ( member1426531477525435216od_a_a @ X @ B ) ) ) ).
% in_mono
thf(fact_396_in__mono,axiom,
! [A2: set_set_a,B: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_a @ X @ A2 )
=> ( member_set_a @ X @ B ) ) ) ).
% in_mono
thf(fact_397_ulgraph_Oincident__loops_Ocong,axiom,
undire4753905205749729249oops_a = undire4753905205749729249oops_a ).
% ulgraph.incident_loops.cong
thf(fact_398_inf__sup__ord_I2_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_399_inf__sup__ord_I2_J,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_400_inf__sup__ord_I2_J,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_401_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_402_inf__sup__ord_I2_J,axiom,
! [X: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_403_inf__sup__ord_I1_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_404_inf__sup__ord_I1_J,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_405_inf__sup__ord_I1_J,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_406_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_407_inf__sup__ord_I1_J,axiom,
! [X: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_408_inf__le1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_409_inf__le1,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_410_inf__le1,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_411_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_412_inf__le1,axiom,
! [X: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_413_inf__le2,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_414_inf__le2,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_415_inf__le2,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_416_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_417_inf__le2,axiom,
! [X: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_418_le__infE,axiom,
! [X: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B2 ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A )
=> ~ ( ord_less_eq_set_a @ X @ B2 ) ) ) ).
% le_infE
thf(fact_419_le__infE,axiom,
! [X: real,A: real,B2: real] :
( ( ord_less_eq_real @ X @ ( inf_inf_real @ A @ B2 ) )
=> ~ ( ( ord_less_eq_real @ X @ A )
=> ~ ( ord_less_eq_real @ X @ B2 ) ) ) ).
% le_infE
thf(fact_420_le__infE,axiom,
! [X: set_Product_prod_a_a,A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ A @ B2 ) )
=> ~ ( ( ord_le746702958409616551od_a_a @ X @ A )
=> ~ ( ord_le746702958409616551od_a_a @ X @ B2 ) ) ) ).
% le_infE
thf(fact_421_le__infE,axiom,
! [X: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X @ A )
=> ~ ( ord_less_eq_nat @ X @ B2 ) ) ) ).
% le_infE
thf(fact_422_le__infE,axiom,
! [X: set_set_a,A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ A @ B2 ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ X @ A )
=> ~ ( ord_le3724670747650509150_set_a @ X @ B2 ) ) ) ).
% le_infE
thf(fact_423_le__infI,axiom,
! [X: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X @ B2 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_424_le__infI,axiom,
! [X: real,A: real,B2: real] :
( ( ord_less_eq_real @ X @ A )
=> ( ( ord_less_eq_real @ X @ B2 )
=> ( ord_less_eq_real @ X @ ( inf_inf_real @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_425_le__infI,axiom,
! [X: set_Product_prod_a_a,A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ A )
=> ( ( ord_le746702958409616551od_a_a @ X @ B2 )
=> ( ord_le746702958409616551od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_426_le__infI,axiom,
! [X: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_427_le__infI,axiom,
! [X: set_set_a,A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ A )
=> ( ( ord_le3724670747650509150_set_a @ X @ B2 )
=> ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_428_inf__mono,axiom,
! [A: set_a,C: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_429_inf__mono,axiom,
! [A: real,C: real,B2: real,D: real] :
( ( ord_less_eq_real @ A @ C )
=> ( ( ord_less_eq_real @ B2 @ D )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B2 ) @ ( inf_inf_real @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_430_inf__mono,axiom,
! [A: set_Product_prod_a_a,C: set_Product_prod_a_a,B2: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ C )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ D )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B2 ) @ ( inf_in8905007599844390133od_a_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_431_inf__mono,axiom,
! [A: nat,C: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_432_inf__mono,axiom,
! [A: set_set_a,C: set_set_a,B2: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ D )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ ( inf_inf_set_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_433_le__infI1,axiom,
! [A: set_a,X: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_434_le__infI1,axiom,
! [A: real,X: real,B2: real] :
( ( ord_less_eq_real @ A @ X )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_435_le__infI1,axiom,
! [A: set_Product_prod_a_a,X: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ X )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_436_le__infI1,axiom,
! [A: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_437_le__infI1,axiom,
! [A: set_set_a,X: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ X )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_438_le__infI2,axiom,
! [B2: set_a,X: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_439_le__infI2,axiom,
! [B2: real,X: real,A: real] :
( ( ord_less_eq_real @ B2 @ X )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_440_le__infI2,axiom,
! [B2: set_Product_prod_a_a,X: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B2 @ X )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_441_le__infI2,axiom,
! [B2: nat,X: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_442_le__infI2,axiom,
! [B2: set_set_a,X: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ X )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_443_inf_OorderE,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( A
= ( inf_inf_set_a @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_444_inf_OorderE,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( A
= ( inf_inf_real @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_445_inf_OorderE,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B2 )
=> ( A
= ( inf_in8905007599844390133od_a_a @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_446_inf_OorderE,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( A
= ( inf_inf_nat @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_447_inf_OorderE,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( A
= ( inf_inf_set_set_a @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_448_inf_OorderI,axiom,
! [A: set_a,B2: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B2 ) )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% inf.orderI
thf(fact_449_inf_OorderI,axiom,
! [A: real,B2: real] :
( ( A
= ( inf_inf_real @ A @ B2 ) )
=> ( ord_less_eq_real @ A @ B2 ) ) ).
% inf.orderI
thf(fact_450_inf_OorderI,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( A
= ( inf_in8905007599844390133od_a_a @ A @ B2 ) )
=> ( ord_le746702958409616551od_a_a @ A @ B2 ) ) ).
% inf.orderI
thf(fact_451_inf_OorderI,axiom,
! [A: nat,B2: nat] :
( ( A
= ( inf_inf_nat @ A @ B2 ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% inf.orderI
thf(fact_452_inf_OorderI,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( A
= ( inf_inf_set_set_a @ A @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A @ B2 ) ) ).
% inf.orderI
thf(fact_453_inf__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X5: set_a,Y5: set_a] : ( ord_less_eq_set_a @ ( F @ X5 @ Y5 ) @ X5 )
=> ( ! [X5: set_a,Y5: set_a] : ( ord_less_eq_set_a @ ( F @ X5 @ Y5 ) @ Y5 )
=> ( ! [X5: set_a,Y5: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ( ord_less_eq_set_a @ X5 @ Z3 )
=> ( ord_less_eq_set_a @ X5 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_454_inf__unique,axiom,
! [F: real > real > real,X: real,Y: real] :
( ! [X5: real,Y5: real] : ( ord_less_eq_real @ ( F @ X5 @ Y5 ) @ X5 )
=> ( ! [X5: real,Y5: real] : ( ord_less_eq_real @ ( F @ X5 @ Y5 ) @ Y5 )
=> ( ! [X5: real,Y5: real,Z3: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ( ord_less_eq_real @ X5 @ Z3 )
=> ( ord_less_eq_real @ X5 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_inf_real @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_455_inf__unique,axiom,
! [F: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a,X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ! [X5: set_Product_prod_a_a,Y5: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( F @ X5 @ Y5 ) @ X5 )
=> ( ! [X5: set_Product_prod_a_a,Y5: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( F @ X5 @ Y5 ) @ Y5 )
=> ( ! [X5: set_Product_prod_a_a,Y5: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X5 @ Y5 )
=> ( ( ord_le746702958409616551od_a_a @ X5 @ Z3 )
=> ( ord_le746702958409616551od_a_a @ X5 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_in8905007599844390133od_a_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_456_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X5: nat,Y5: nat] : ( ord_less_eq_nat @ ( F @ X5 @ Y5 ) @ X5 )
=> ( ! [X5: nat,Y5: nat] : ( ord_less_eq_nat @ ( F @ X5 @ Y5 ) @ Y5 )
=> ( ! [X5: nat,Y5: nat,Z3: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ( ord_less_eq_nat @ X5 @ Z3 )
=> ( ord_less_eq_nat @ X5 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_457_inf__unique,axiom,
! [F: set_set_a > set_set_a > set_set_a,X: set_set_a,Y: set_set_a] :
( ! [X5: set_set_a,Y5: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F @ X5 @ Y5 ) @ X5 )
=> ( ! [X5: set_set_a,Y5: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F @ X5 @ Y5 ) @ Y5 )
=> ( ! [X5: set_set_a,Y5: set_set_a,Z3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ Y5 )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ Z3 )
=> ( ord_le3724670747650509150_set_a @ X5 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_inf_set_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_458_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X4: set_a,Y4: set_a] :
( ( inf_inf_set_a @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_459_le__iff__inf,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y4: real] :
( ( inf_inf_real @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_460_le__iff__inf,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [X4: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_461_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( inf_inf_nat @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_462_le__iff__inf,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X4: set_set_a,Y4: set_set_a] :
( ( inf_inf_set_set_a @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_463_inf_Oabsorb1,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( inf_inf_set_a @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_464_inf_Oabsorb1,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( inf_inf_real @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_465_inf_Oabsorb1,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B2 )
=> ( ( inf_in8905007599844390133od_a_a @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_466_inf_Oabsorb1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( inf_inf_nat @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_467_inf_Oabsorb1,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( inf_inf_set_set_a @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_468_inf_Oabsorb2,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( inf_inf_set_a @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_469_inf_Oabsorb2,axiom,
! [B2: real,A: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( inf_inf_real @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_470_inf_Oabsorb2,axiom,
! [B2: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B2 @ A )
=> ( ( inf_in8905007599844390133od_a_a @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_471_inf_Oabsorb2,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( inf_inf_nat @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_472_inf_Oabsorb2,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( ( inf_inf_set_set_a @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_473_inf__absorb1,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( inf_inf_set_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_474_inf__absorb1,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( inf_inf_real @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_475_inf__absorb1,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ Y )
=> ( ( inf_in8905007599844390133od_a_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_476_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_477_inf__absorb1,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ( inf_inf_set_set_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_478_inf__absorb2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( inf_inf_set_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_479_inf__absorb2,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( inf_inf_real @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_480_inf__absorb2,axiom,
! [Y: set_Product_prod_a_a,X: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y @ X )
=> ( ( inf_in8905007599844390133od_a_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_481_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_482_inf__absorb2,axiom,
! [Y: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X )
=> ( ( inf_inf_set_set_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_483_inf_OboundedE,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B2 )
=> ~ ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_484_inf_OboundedE,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ ( inf_inf_real @ B2 @ C ) )
=> ~ ( ( ord_less_eq_real @ A @ B2 )
=> ~ ( ord_less_eq_real @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_485_inf_OboundedE,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ B2 @ C ) )
=> ~ ( ( ord_le746702958409616551od_a_a @ A @ B2 )
=> ~ ( ord_le746702958409616551od_a_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_486_inf_OboundedE,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B2 )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_487_inf_OboundedE,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( inf_inf_set_set_a @ B2 @ C ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ~ ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_488_inf_OboundedI,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_489_inf_OboundedI,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ A @ C )
=> ( ord_less_eq_real @ A @ ( inf_inf_real @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_490_inf_OboundedI,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B2 )
=> ( ( ord_le746702958409616551od_a_a @ A @ C )
=> ( ord_le746702958409616551od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_491_inf_OboundedI,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_492_inf_OboundedI,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A @ C )
=> ( ord_le3724670747650509150_set_a @ A @ ( inf_inf_set_set_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_493_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_494_inf__greatest,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Z )
=> ( ord_less_eq_real @ X @ ( inf_inf_real @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_495_inf__greatest,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ Y )
=> ( ( ord_le746702958409616551od_a_a @ X @ Z )
=> ( ord_le746702958409616551od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_496_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_497_inf__greatest,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ( ord_le3724670747650509150_set_a @ X @ Z )
=> ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_498_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B5: set_a] :
( A4
= ( inf_inf_set_a @ A4 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_499_inf_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B5: real] :
( A4
= ( inf_inf_real @ A4 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_500_inf_Oorder__iff,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A4: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( A4
= ( inf_in8905007599844390133od_a_a @ A4 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_501_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] :
( A4
= ( inf_inf_nat @ A4 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_502_inf_Oorder__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B5: set_set_a] :
( A4
= ( inf_inf_set_set_a @ A4 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_503_inf_Ocobounded1,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_504_inf_Ocobounded1,axiom,
! [A: real,B2: real] : ( ord_less_eq_real @ ( inf_inf_real @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_505_inf_Ocobounded1,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_506_inf_Ocobounded1,axiom,
! [A: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_507_inf_Ocobounded1,axiom,
! [A: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_508_inf_Ocobounded2,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_509_inf_Ocobounded2,axiom,
! [A: real,B2: real] : ( ord_less_eq_real @ ( inf_inf_real @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_510_inf_Ocobounded2,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_511_inf_Ocobounded2,axiom,
! [A: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_512_inf_Ocobounded2,axiom,
! [A: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_513_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B5: set_a] :
( ( inf_inf_set_a @ A4 @ B5 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_514_inf_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B5: real] :
( ( inf_inf_real @ A4 @ B5 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_515_inf_Oabsorb__iff1,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A4: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A4 @ B5 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_516_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] :
( ( inf_inf_nat @ A4 @ B5 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_517_inf_Oabsorb__iff1,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B5: set_set_a] :
( ( inf_inf_set_set_a @ A4 @ B5 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_518_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B5: set_a,A4: set_a] :
( ( inf_inf_set_a @ A4 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_519_inf_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [B5: real,A4: real] :
( ( inf_inf_real @ A4 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_520_inf_Oabsorb__iff2,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [B5: set_Product_prod_a_a,A4: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A4 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_521_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_522_inf_Oabsorb__iff2,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B5: set_set_a,A4: set_set_a] :
( ( inf_inf_set_set_a @ A4 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_523_inf_OcoboundedI1,axiom,
! [A: set_a,C: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_524_inf_OcoboundedI1,axiom,
! [A: real,C: real,B2: real] :
( ( ord_less_eq_real @ A @ C )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_525_inf_OcoboundedI1,axiom,
! [A: set_Product_prod_a_a,C: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ C )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_526_inf_OcoboundedI1,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_527_inf_OcoboundedI1,axiom,
! [A: set_set_a,C: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_528_inf_OcoboundedI2,axiom,
! [B2: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_529_inf_OcoboundedI2,axiom,
! [B2: real,C: real,A: real] :
( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_530_inf_OcoboundedI2,axiom,
! [B2: set_Product_prod_a_a,C: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B2 @ C )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_531_inf_OcoboundedI2,axiom,
! [B2: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_532_inf_OcoboundedI2,axiom,
! [B2: set_set_a,C: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_533_bot__set__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a @ bot_bot_set_a_o ) ) ).
% bot_set_def
thf(fact_534_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_535_bot__set__def,axiom,
( bot_bo3357376287454694259od_a_a
= ( collec3336397797384452498od_a_a @ bot_bo4160289986317612842_a_a_o ) ) ).
% bot_set_def
thf(fact_536_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_537_insert__mono,axiom,
! [C2: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C2 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C2 ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_538_insert__mono,axiom,
! [C2: set_Product_prod_a_a,D2: set_Product_prod_a_a,A: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C2 @ D2 )
=> ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ A @ C2 ) @ ( insert4534936382041156343od_a_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_539_insert__mono,axiom,
! [C2: set_set_a,D2: set_set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ A @ C2 ) @ ( insert_set_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_540_subset__insert,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B ) )
= ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_541_subset__insert,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_542_subset__insert,axiom,
! [X: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ X @ B ) )
= ( ord_le746702958409616551od_a_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_543_subset__insert,axiom,
! [X: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B ) )
= ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_544_subset__insertI,axiom,
! [B: set_a,A: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A @ B ) ) ).
% subset_insertI
thf(fact_545_subset__insertI,axiom,
! [B: set_Product_prod_a_a,A: product_prod_a_a] : ( ord_le746702958409616551od_a_a @ B @ ( insert4534936382041156343od_a_a @ A @ B ) ) ).
% subset_insertI
thf(fact_546_subset__insertI,axiom,
! [B: set_set_a,A: set_a] : ( ord_le3724670747650509150_set_a @ B @ ( insert_set_a @ A @ B ) ) ).
% subset_insertI
thf(fact_547_subset__insertI2,axiom,
! [A2: set_a,B: set_a,B2: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_548_subset__insertI2,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,B2: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ord_le746702958409616551od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_549_subset__insertI2,axiom,
! [A2: set_set_a,B: set_set_a,B2: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_550_Int__mono,axiom,
! [A2: set_a,C2: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_551_Int__mono,axiom,
! [A2: set_Product_prod_a_a,C2: set_Product_prod_a_a,B: set_Product_prod_a_a,D2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ C2 )
=> ( ( ord_le746702958409616551od_a_a @ B @ D2 )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) @ ( inf_in8905007599844390133od_a_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_552_Int__mono,axiom,
! [A2: set_set_a,C2: set_set_a,B: set_set_a,D2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ ( inf_inf_set_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_553_Int__lower1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_554_Int__lower1,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_555_Int__lower1,axiom,
! [A2: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_556_Int__lower2,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_557_Int__lower2,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_558_Int__lower2,axiom,
! [A2: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_559_Int__absorb1,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_560_Int__absorb1,axiom,
! [B: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B @ A2 )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_561_Int__absorb1,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_562_Int__absorb2,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( inf_inf_set_a @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_563_Int__absorb2,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_564_Int__absorb2,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( inf_inf_set_set_a @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_565_Int__greatest,axiom,
! [C2: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ B )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_566_Int__greatest,axiom,
! [C2: set_Product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C2 @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ C2 @ B )
=> ( ord_le746702958409616551od_a_a @ C2 @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_567_Int__greatest,axiom,
! [C2: set_set_a,A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ C2 @ B )
=> ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_568_Int__Collect__mono,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_569_Int__Collect__mono,axiom,
! [A2: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_570_Int__Collect__mono,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A2 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ ( collec3336397797384452498od_a_a @ P ) ) @ ( inf_in8905007599844390133od_a_a @ B @ ( collec3336397797384452498od_a_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_571_Int__Collect__mono,axiom,
! [A2: set_set_a,B: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_572_subset__singletonD,axiom,
! [A2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_573_subset__singletonD,axiom,
! [A2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A2 = bot_bot_set_a )
| ( A2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_574_subset__singletonD,axiom,
! [A2: set_Product_prod_a_a,X: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) )
=> ( ( A2 = bot_bo3357376287454694259od_a_a )
| ( A2
= ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% subset_singletonD
thf(fact_575_subset__singletonD,axiom,
! [A2: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
=> ( ( A2 = bot_bot_set_set_a )
| ( A2
= ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_576_subset__singleton__iff,axiom,
! [X2: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X2 = bot_bot_set_nat )
| ( X2
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_577_subset__singleton__iff,axiom,
! [X2: set_a,A: a] :
( ( ord_less_eq_set_a @ X2 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X2 = bot_bot_set_a )
| ( X2
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_578_subset__singleton__iff,axiom,
! [X2: set_Product_prod_a_a,A: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) )
= ( ( X2 = bot_bo3357376287454694259od_a_a )
| ( X2
= ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_579_subset__singleton__iff,axiom,
! [X2: set_set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( ( X2 = bot_bot_set_set_a )
| ( X2
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_580_inf__sup__aci_I4_J,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_581_inf__sup__aci_I4_J,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ X @ Y ) )
= ( inf_inf_set_set_a @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_582_inf__sup__aci_I3_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_583_inf__sup__aci_I3_J,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) )
= ( inf_inf_set_set_a @ Y @ ( inf_inf_set_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_584_inf__sup__aci_I2_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_585_inf__sup__aci_I2_J,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_586_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X4: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X4 ) ) ) ).
% inf_sup_aci(1)
thf(fact_587_inf__sup__aci_I1_J,axiom,
( inf_inf_set_set_a
= ( ^ [X4: set_set_a,Y4: set_set_a] : ( inf_inf_set_set_a @ Y4 @ X4 ) ) ) ).
% inf_sup_aci(1)
thf(fact_588_inf_Oassoc,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B2 ) @ C )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_589_inf_Oassoc,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ C )
= ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_590_inf__assoc,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_591_inf__assoc,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_592_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B5: set_a] : ( inf_inf_set_a @ B5 @ A4 ) ) ) ).
% inf.commute
thf(fact_593_inf_Ocommute,axiom,
( inf_inf_set_set_a
= ( ^ [A4: set_set_a,B5: set_set_a] : ( inf_inf_set_set_a @ B5 @ A4 ) ) ) ).
% inf.commute
thf(fact_594_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X4: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X4 ) ) ) ).
% inf_commute
thf(fact_595_inf__commute,axiom,
( inf_inf_set_set_a
= ( ^ [X4: set_set_a,Y4: set_set_a] : ( inf_inf_set_set_a @ Y4 @ X4 ) ) ) ).
% inf_commute
thf(fact_596_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_a,K: set_a,A: set_a,B2: set_a] :
( ( A2
= ( inf_inf_set_a @ K @ A ) )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_597_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_set_a,K: set_set_a,A: set_set_a,B2: set_set_a] :
( ( A2
= ( inf_inf_set_set_a @ K @ A ) )
=> ( ( inf_inf_set_set_a @ A2 @ B2 )
= ( inf_inf_set_set_a @ K @ ( inf_inf_set_set_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_598_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_a,K: set_a,B2: set_a,A: set_a] :
( ( B
= ( inf_inf_set_a @ K @ B2 ) )
=> ( ( inf_inf_set_a @ A @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_599_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_set_a,K: set_set_a,B2: set_set_a,A: set_set_a] :
( ( B
= ( inf_inf_set_set_a @ K @ B2 ) )
=> ( ( inf_inf_set_set_a @ A @ B )
= ( inf_inf_set_set_a @ K @ ( inf_inf_set_set_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_600_inf_Oleft__commute,axiom,
! [B2: set_a,A: set_a,C: set_a] :
( ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A @ C ) )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_601_inf_Oleft__commute,axiom,
! [B2: set_set_a,A: set_set_a,C: set_set_a] :
( ( inf_inf_set_set_a @ B2 @ ( inf_inf_set_set_a @ A @ C ) )
= ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_602_inf__left__commute,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_603_inf__left__commute,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) )
= ( inf_inf_set_set_a @ Y @ ( inf_inf_set_set_a @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_604_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_605_sgraph_Ounique__triangles__def,axiom,
! [Vertices: set_nat,Edges: set_set_nat] :
( ( undire7290660292559394354ph_nat @ Vertices @ Edges )
=> ( ( graph_4265997628758960791es_nat @ Edges )
= ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ Edges )
=> ? [Y4: set_nat] :
( ? [Z4: nat,Aa: nat,Ab: nat] :
( ( Y4
= ( insert_nat @ Z4 @ ( insert_nat @ Aa @ ( insert_nat @ Ab @ bot_bot_set_nat ) ) ) )
& ( graph_2911189250448956958ph_nat @ Edges @ Z4 @ Aa @ Ab )
& ( ord_less_eq_set_nat @ X4 @ Y4 ) )
& ! [Z4: set_nat] :
( ? [Aa: nat,Ab: nat,Ac: nat] :
( ( Z4
= ( insert_nat @ Aa @ ( insert_nat @ Ab @ ( insert_nat @ Ac @ bot_bot_set_nat ) ) ) )
& ( graph_2911189250448956958ph_nat @ Edges @ Aa @ Ab @ Ac )
& ( ord_less_eq_set_nat @ X4 @ Z4 ) )
=> ( Z4 = Y4 ) ) ) ) ) ) ) ).
% sgraph.unique_triangles_def
thf(fact_606_sgraph_Ounique__triangles__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a] :
( ( undire8797707285729112389od_a_a @ Vertices @ Edges )
=> ( ( graph_2552870840184084768od_a_a @ Edges )
= ( ! [X4: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X4 @ Edges )
=> ? [Y4: set_Product_prod_a_a] :
( ? [Z4: product_prod_a_a,Aa: product_prod_a_a,Ab: product_prod_a_a] :
( ( Y4
= ( insert4534936382041156343od_a_a @ Z4 @ ( insert4534936382041156343od_a_a @ Aa @ ( insert4534936382041156343od_a_a @ Ab @ bot_bo3357376287454694259od_a_a ) ) ) )
& ( graph_4803287029668059225od_a_a @ Edges @ Z4 @ Aa @ Ab )
& ( ord_le746702958409616551od_a_a @ X4 @ Y4 ) )
& ! [Z4: set_Product_prod_a_a] :
( ? [Aa: product_prod_a_a,Ab: product_prod_a_a,Ac: product_prod_a_a] :
( ( Z4
= ( insert4534936382041156343od_a_a @ Aa @ ( insert4534936382041156343od_a_a @ Ab @ ( insert4534936382041156343od_a_a @ Ac @ bot_bo3357376287454694259od_a_a ) ) ) )
& ( graph_4803287029668059225od_a_a @ Edges @ Aa @ Ab @ Ac )
& ( ord_le746702958409616551od_a_a @ X4 @ Z4 ) )
=> ( Z4 = Y4 ) ) ) ) ) ) ) ).
% sgraph.unique_triangles_def
thf(fact_607_sgraph_Ounique__triangles__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a] :
( ( undire6035205377725458044_set_a @ Vertices @ Edges )
=> ( ( graph_7001731429221021015_set_a @ Edges )
= ( ! [X4: set_set_a] :
( ( member_set_set_a @ X4 @ Edges )
=> ? [Y4: set_set_a] :
( ? [Z4: set_a,Aa: set_a,Ab: set_a] :
( ( Y4
= ( insert_set_a @ Z4 @ ( insert_set_a @ Aa @ ( insert_set_a @ Ab @ bot_bot_set_set_a ) ) ) )
& ( graph_3840782946058334608_set_a @ Edges @ Z4 @ Aa @ Ab )
& ( ord_le3724670747650509150_set_a @ X4 @ Y4 ) )
& ! [Z4: set_set_a] :
( ? [Aa: set_a,Ab: set_a,Ac: set_a] :
( ( Z4
= ( insert_set_a @ Aa @ ( insert_set_a @ Ab @ ( insert_set_a @ Ac @ bot_bot_set_set_a ) ) ) )
& ( graph_3840782946058334608_set_a @ Edges @ Aa @ Ab @ Ac )
& ( ord_le3724670747650509150_set_a @ X4 @ Z4 ) )
=> ( Z4 = Y4 ) ) ) ) ) ) ) ).
% sgraph.unique_triangles_def
thf(fact_608_sgraph_Ounique__triangles__def,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire3507641187627840796raph_a @ Vertices @ Edges )
=> ( ( graph_6144490306505338871gles_a @ Edges )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ Edges )
=> ? [Y4: set_a] :
( ? [Z4: a,Aa: a,Ab: a] :
( ( Y4
= ( insert_a @ Z4 @ ( insert_a @ Aa @ ( insert_a @ Ab @ bot_bot_set_a ) ) ) )
& ( graph_4582152751571636272raph_a @ Edges @ Z4 @ Aa @ Ab )
& ( ord_less_eq_set_a @ X4 @ Y4 ) )
& ! [Z4: set_a] :
( ? [Aa: a,Ab: a,Ac: a] :
( ( Z4
= ( insert_a @ Aa @ ( insert_a @ Ab @ ( insert_a @ Ac @ bot_bot_set_a ) ) ) )
& ( graph_4582152751571636272raph_a @ Edges @ Aa @ Ab @ Ac )
& ( ord_less_eq_set_a @ X4 @ Z4 ) )
=> ( Z4 = Y4 ) ) ) ) ) ) ) ).
% sgraph.unique_triangles_def
thf(fact_609_edge__density__le1,axiom,
! [X2: set_a,Y2: set_a] : ( ord_less_eq_real @ ( undire297304480579013331sity_a @ edges @ X2 @ Y2 ) @ one_one_real ) ).
% edge_density_le1
thf(fact_610_all__edges__between__mono2,axiom,
! [Y2: set_a,Z2: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y2 @ Z2 )
=> ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y2 ) @ ( undire8383842906760478443ween_a @ edges @ X2 @ Z2 ) ) ) ).
% all_edges_between_mono2
thf(fact_611_all__edges__between__mono1,axiom,
! [Y2: set_a,Z2: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y2 @ Z2 )
=> ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ edges @ Y2 @ X2 ) @ ( undire8383842906760478443ween_a @ edges @ Z2 @ X2 ) ) ) ).
% all_edges_between_mono1
thf(fact_612_finite__incident__loops,axiom,
! [V: a] : ( finite_finite_set_a @ ( undire4753905205749729249oops_a @ edges @ V ) ) ).
% finite_incident_loops
thf(fact_613_the__elem__eq,axiom,
! [X: set_a] :
( ( the_elem_set_a @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= X ) ).
% the_elem_eq
thf(fact_614_the__elem__eq,axiom,
! [X: a] :
( ( the_elem_a @ ( insert_a @ X @ bot_bot_set_a ) )
= X ) ).
% the_elem_eq
thf(fact_615_the__elem__eq,axiom,
! [X: product_prod_a_a] :
( ( the_el8589169208993665564od_a_a @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) )
= X ) ).
% the_elem_eq
thf(fact_616_the__elem__eq,axiom,
! [X: nat] :
( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% the_elem_eq
thf(fact_617_prod_Oinject,axiom,
! [X1: a,X22: a,Y1: a,Y22: a] :
( ( ( product_Pair_a_a @ X1 @ X22 )
= ( product_Pair_a_a @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_618_prod_Oinject,axiom,
! [X1: set_a,X22: set_set_a,Y1: set_a,Y22: set_set_a] :
( ( ( produc2116933609460601975_set_a @ X1 @ X22 )
= ( produc2116933609460601975_set_a @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_619_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_620_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_621_dual__order_Orefl,axiom,
! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A @ A ) ).
% dual_order.refl
thf(fact_622_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_623_dual__order_Orefl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_624_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_625_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_626_order__refl,axiom,
! [X: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ X @ X ) ).
% order_refl
thf(fact_627_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_628_order__refl,axiom,
! [X: set_set_a] : ( ord_le3724670747650509150_set_a @ X @ X ) ).
% order_refl
thf(fact_629_old_Oprod_Oinject,axiom,
! [A: a,B2: a,A5: a,B6: a] :
( ( ( product_Pair_a_a @ A @ B2 )
= ( product_Pair_a_a @ A5 @ B6 ) )
= ( ( A = A5 )
& ( B2 = B6 ) ) ) ).
% old.prod.inject
thf(fact_630_old_Oprod_Oinject,axiom,
! [A: set_a,B2: set_set_a,A5: set_a,B6: set_set_a] :
( ( ( produc2116933609460601975_set_a @ A @ B2 )
= ( produc2116933609460601975_set_a @ A5 @ B6 ) )
= ( ( A = A5 )
& ( B2 = B6 ) ) ) ).
% old.prod.inject
thf(fact_631_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_632_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_633_sgraph_Ounique__triangles_Ocong,axiom,
graph_6144490306505338871gles_a = graph_6144490306505338871gles_a ).
% sgraph.unique_triangles.cong
thf(fact_634_order__antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_635_order__antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_636_order__antisym__conv,axiom,
! [Y: set_Product_prod_a_a,X: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y @ X )
=> ( ( ord_le746702958409616551od_a_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_637_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_638_order__antisym__conv,axiom,
! [Y: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_639_linorder__le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_640_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_641_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_642_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_643_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_644_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_645_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > real,C: real] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_646_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_647_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > set_a,C: set_a] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_648_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_649_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_650_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > set_set_a,C: set_set_a] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_651_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_652_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_653_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_654_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_655_ord__eq__le__subst,axiom,
! [A: real,F: set_a > real,B2: set_a,C: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_656_ord__eq__le__subst,axiom,
! [A: nat,F: set_a > nat,B2: set_a,C: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_657_ord__eq__le__subst,axiom,
! [A: set_a,F: real > set_a,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_658_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_659_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_660_ord__eq__le__subst,axiom,
! [A: set_set_a,F: real > set_set_a,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_661_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_662_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_663_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_664_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_665_order__eq__refl,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( X = Y )
=> ( ord_le746702958409616551od_a_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_666_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_667_order__eq__refl,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( X = Y )
=> ( ord_le3724670747650509150_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_668_order__subst2,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_669_order__subst2,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_670_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_671_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_672_order__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > real,C: real] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_673_order__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_674_order__subst2,axiom,
! [A: real,B2: real,F: real > set_a,C: set_a] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_675_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_676_order__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_677_order__subst2,axiom,
! [A: real,B2: real,F: real > set_set_a,C: set_set_a] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B2 ) @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_678_order__subst1,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_679_order__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_680_order__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_681_order__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_682_order__subst1,axiom,
! [A: set_a,F: real > set_a,B2: real,C: real] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X5: real,Y5: real] :
( ( ord_less_eq_real @ X5 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_683_order__subst1,axiom,
! [A: set_a,F: nat > set_a,B2: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_684_order__subst1,axiom,
! [A: real,F: set_a > real,B2: set_a,C: set_a] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_685_order__subst1,axiom,
! [A: nat,F: set_a > nat,B2: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_686_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_687_order__subst1,axiom,
! [A: real,F: set_set_a > real,B2: set_set_a,C: set_set_a] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X5: set_set_a,Y5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_688_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_a,Z5: set_a] : ( Y6 = Z5 ) )
= ( ^ [A4: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_689_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: real,Z5: real] : ( Y6 = Z5 ) )
= ( ^ [A4: real,B5: real] :
( ( ord_less_eq_real @ A4 @ B5 )
& ( ord_less_eq_real @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_690_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_Product_prod_a_a,Z5: set_Product_prod_a_a] : ( Y6 = Z5 ) )
= ( ^ [A4: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A4 @ B5 )
& ( ord_le746702958409616551od_a_a @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_691_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
& ( ord_less_eq_nat @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_692_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_set_a,Z5: set_set_a] : ( Y6 = Z5 ) )
= ( ^ [A4: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_693_antisym,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_694_antisym,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_695_antisym,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_696_antisym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_697_antisym,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_698_dual__order_Otrans,axiom,
! [B2: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_699_dual__order_Otrans,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_700_dual__order_Otrans,axiom,
! [B2: set_Product_prod_a_a,A: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B2 @ A )
=> ( ( ord_le746702958409616551od_a_a @ C @ B2 )
=> ( ord_le746702958409616551od_a_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_701_dual__order_Otrans,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_702_dual__order_Otrans,axiom,
! [B2: set_set_a,A: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( ( ord_le3724670747650509150_set_a @ C @ B2 )
=> ( ord_le3724670747650509150_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_703_dual__order_Oantisym,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_704_dual__order_Oantisym,axiom,
! [B2: real,A: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_705_dual__order_Oantisym,axiom,
! [B2: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B2 @ A )
=> ( ( ord_le746702958409616551od_a_a @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_706_dual__order_Oantisym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_707_dual__order_Oantisym,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_708_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_a,Z5: set_a] : ( Y6 = Z5 ) )
= ( ^ [A4: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_709_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: real,Z5: real] : ( Y6 = Z5 ) )
= ( ^ [A4: real,B5: real] :
( ( ord_less_eq_real @ B5 @ A4 )
& ( ord_less_eq_real @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_710_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_Product_prod_a_a,Z5: set_Product_prod_a_a] : ( Y6 = Z5 ) )
= ( ^ [A4: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B5 @ A4 )
& ( ord_le746702958409616551od_a_a @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_711_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A4 )
& ( ord_less_eq_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_712_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_set_a,Z5: set_set_a] : ( Y6 = Z5 ) )
= ( ^ [A4: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ A4 )
& ( ord_le3724670747650509150_set_a @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_713_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B2: real] :
( ! [A6: real,B7: real] :
( ( ord_less_eq_real @ A6 @ B7 )
=> ( P @ A6 @ B7 ) )
=> ( ! [A6: real,B7: real] :
( ( P @ B7 @ A6 )
=> ( P @ A6 @ B7 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_714_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A6: nat,B7: nat] :
( ( ord_less_eq_nat @ A6 @ B7 )
=> ( P @ A6 @ B7 ) )
=> ( ! [A6: nat,B7: nat] :
( ( P @ B7 @ A6 )
=> ( P @ A6 @ B7 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_715_order__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_716_order__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_eq_real @ X @ Z ) ) ) ).
% order_trans
thf(fact_717_order__trans,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ Y )
=> ( ( ord_le746702958409616551od_a_a @ Y @ Z )
=> ( ord_le746702958409616551od_a_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_718_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_719_order__trans,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ Z )
=> ( ord_le3724670747650509150_set_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_720_order_Otrans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_721_order_Otrans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_722_order_Otrans,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ C )
=> ( ord_le746702958409616551od_a_a @ A @ C ) ) ) ).
% order.trans
thf(fact_723_order_Otrans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_724_order_Otrans,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_725_order__antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_726_order__antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_727_order__antisym,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ Y )
=> ( ( ord_le746702958409616551od_a_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_728_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_729_order__antisym,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_730_ord__le__eq__trans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_731_ord__le__eq__trans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_732_ord__le__eq__trans,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le746702958409616551od_a_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_733_ord__le__eq__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_734_ord__le__eq__trans,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_735_ord__eq__le__trans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( A = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_736_ord__eq__le__trans,axiom,
! [A: real,B2: real,C: real] :
( ( A = B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_737_ord__eq__le__trans,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( A = B2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ C )
=> ( ord_le746702958409616551od_a_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_738_ord__eq__le__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_739_ord__eq__le__trans,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( A = B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_740_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_a,Z5: set_a] : ( Y6 = Z5 ) )
= ( ^ [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_741_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: real,Z5: real] : ( Y6 = Z5 ) )
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_742_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_Product_prod_a_a,Z5: set_Product_prod_a_a] : ( Y6 = Z5 ) )
= ( ^ [X4: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X4 @ Y4 )
& ( ord_le746702958409616551od_a_a @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_743_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_744_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_set_a,Z5: set_set_a] : ( Y6 = Z5 ) )
= ( ^ [X4: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y4 )
& ( ord_le3724670747650509150_set_a @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_745_le__cases3,axiom,
! [X: real,Y: real,Z: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z ) )
=> ( ( ( ord_less_eq_real @ X @ Z )
=> ~ ( ord_less_eq_real @ Z @ Y ) )
=> ( ( ( ord_less_eq_real @ Z @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z )
=> ~ ( ord_less_eq_real @ Z @ X ) )
=> ~ ( ( ord_less_eq_real @ Z @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_746_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_747_nle__le,axiom,
! [A: real,B2: real] :
( ( ~ ( ord_less_eq_real @ A @ B2 ) )
= ( ( ord_less_eq_real @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_748_nle__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_749_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_750_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_751_Pair__inject,axiom,
! [A: a,B2: a,A5: a,B6: a] :
( ( ( product_Pair_a_a @ A @ B2 )
= ( product_Pair_a_a @ A5 @ B6 ) )
=> ~ ( ( A = A5 )
=> ( B2 != B6 ) ) ) ).
% Pair_inject
thf(fact_752_Pair__inject,axiom,
! [A: set_a,B2: set_set_a,A5: set_a,B6: set_set_a] :
( ( ( produc2116933609460601975_set_a @ A @ B2 )
= ( produc2116933609460601975_set_a @ A5 @ B6 ) )
=> ~ ( ( A = A5 )
=> ( B2 != B6 ) ) ) ).
% Pair_inject
thf(fact_753_prod__cases,axiom,
! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
( ! [A6: a,B7: a] : ( P @ ( product_Pair_a_a @ A6 @ B7 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_754_prod__cases,axiom,
! [P: produc7943277765024757383_set_a > $o,P2: produc7943277765024757383_set_a] :
( ! [A6: set_a,B7: set_set_a] : ( P @ ( produc2116933609460601975_set_a @ A6 @ B7 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_755_surj__pair,axiom,
! [P2: product_prod_a_a] :
? [X5: a,Y5: a] :
( P2
= ( product_Pair_a_a @ X5 @ Y5 ) ) ).
% surj_pair
thf(fact_756_surj__pair,axiom,
! [P2: produc7943277765024757383_set_a] :
? [X5: set_a,Y5: set_set_a] :
( P2
= ( produc2116933609460601975_set_a @ X5 @ Y5 ) ) ).
% surj_pair
thf(fact_757_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_a_a] :
~ ! [A6: a,B7: a] :
( Y
!= ( product_Pair_a_a @ A6 @ B7 ) ) ).
% old.prod.exhaust
thf(fact_758_old_Oprod_Oexhaust,axiom,
! [Y: produc7943277765024757383_set_a] :
~ ! [A6: set_a,B7: set_set_a] :
( Y
!= ( produc2116933609460601975_set_a @ A6 @ B7 ) ) ).
% old.prod.exhaust
thf(fact_759_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_760_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_761_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_762_bot_Oextremum__uniqueI,axiom,
! [A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ bot_bo3357376287454694259od_a_a )
=> ( A = bot_bo3357376287454694259od_a_a ) ) ).
% bot.extremum_uniqueI
thf(fact_763_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_764_bot_Oextremum__uniqueI,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
=> ( A = bot_bot_set_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_765_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_766_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_767_bot_Oextremum__unique,axiom,
! [A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ bot_bo3357376287454694259od_a_a )
= ( A = bot_bo3357376287454694259od_a_a ) ) ).
% bot.extremum_unique
thf(fact_768_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_769_bot_Oextremum__unique,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% bot.extremum_unique
thf(fact_770_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_771_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_772_bot_Oextremum,axiom,
! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ bot_bo3357376287454694259od_a_a @ A ) ).
% bot.extremum
thf(fact_773_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_774_bot_Oextremum,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% bot.extremum
thf(fact_775_card__0__eq,axiom,
! [A2: set_Pr5530083903271594800od_a_a] :
( ( finite5848958031409366265od_a_a @ A2 )
=> ( ( ( finite6893194910719049976od_a_a @ A2 )
= zero_zero_nat )
= ( A2 = bot_bo4436838304982128028od_a_a ) ) ) ).
% card_0_eq
thf(fact_776_card__0__eq,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( ( finite_card_set_a @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_set_a ) ) ) ).
% card_0_eq
thf(fact_777_card__0__eq,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ( finite_card_a @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_a ) ) ) ).
% card_0_eq
thf(fact_778_card__0__eq,axiom,
! [A2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A2 )
=> ( ( ( finite4795055649997197647od_a_a @ A2 )
= zero_zero_nat )
= ( A2 = bot_bo3357376287454694259od_a_a ) ) ) ).
% card_0_eq
thf(fact_779_card__0__eq,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_780_finite__inc__sedges,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( finite_finite_set_a @ ( undire1270416042309875431dges_a @ edges @ V ) ) ) ).
% finite_inc_sedges
thf(fact_781_card_Oinfinite,axiom,
! [A2: set_Pr5530083903271594800od_a_a] :
( ~ ( finite5848958031409366265od_a_a @ A2 )
=> ( ( finite6893194910719049976od_a_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_782_card_Oinfinite,axiom,
! [A2: set_set_a] :
( ~ ( finite_finite_set_a @ A2 )
=> ( ( finite_card_set_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_783_card_Oinfinite,axiom,
! [A2: set_a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_card_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_784_card_Oinfinite,axiom,
! [A2: set_Product_prod_a_a] :
( ~ ( finite6544458595007987280od_a_a @ A2 )
=> ( ( finite4795055649997197647od_a_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_785_card_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_786_finite__Int,axiom,
! [F2: set_Product_prod_a_a,G: set_Product_prod_a_a] :
( ( ( finite6544458595007987280od_a_a @ F2 )
| ( finite6544458595007987280od_a_a @ G ) )
=> ( finite6544458595007987280od_a_a @ ( inf_in8905007599844390133od_a_a @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_787_finite__Int,axiom,
! [F2: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_788_finite__Int,axiom,
! [F2: set_a,G: set_a] :
( ( ( finite_finite_a @ F2 )
| ( finite_finite_a @ G ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_789_finite__Int,axiom,
! [F2: set_set_a,G: set_set_a] :
( ( ( finite_finite_set_a @ F2 )
| ( finite_finite_set_a @ G ) )
=> ( finite_finite_set_a @ ( inf_inf_set_set_a @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_790_card_Oempty,axiom,
( ( finite6893194910719049976od_a_a @ bot_bo4436838304982128028od_a_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_791_card_Oempty,axiom,
( ( finite_card_set_a @ bot_bot_set_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_792_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_793_card_Oempty,axiom,
( ( finite4795055649997197647od_a_a @ bot_bo3357376287454694259od_a_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_794_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_795_finite__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( finite_finite_set_a @ ( insert_set_a @ A @ A2 ) )
= ( finite_finite_set_a @ A2 ) ) ).
% finite_insert
thf(fact_796_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_797_finite__insert,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ ( insert4534936382041156343od_a_a @ A @ A2 ) )
= ( finite6544458595007987280od_a_a @ A2 ) ) ).
% finite_insert
thf(fact_798_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_799_finite__subset__induct_H,axiom,
! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A6 @ A2 )
=> ( ( ord_less_eq_set_nat @ F3 @ A2 )
=> ( ~ ( member_nat @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A6 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_800_finite__subset__induct_H,axiom,
! [F2: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A6: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A6 @ A2 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ~ ( member_a @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A6 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_801_finite__subset__induct_H,axiom,
! [F2: set_Product_prod_a_a,A2: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ F2 )
=> ( ( ord_le746702958409616551od_a_a @ F2 @ A2 )
=> ( ( P @ bot_bo3357376287454694259od_a_a )
=> ( ! [A6: product_prod_a_a,F3: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F3 )
=> ( ( member1426531477525435216od_a_a @ A6 @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ F3 @ A2 )
=> ( ~ ( member1426531477525435216od_a_a @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert4534936382041156343od_a_a @ A6 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_802_finite__subset__induct_H,axiom,
! [F2: set_set_a,A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A6: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A6 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ F3 @ A2 )
=> ( ~ ( member_set_a @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ A6 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_803_is__loop__def,axiom,
( undire2905028936066782638loop_a
= ( ^ [E3: set_a] :
( ( finite_card_a @ E3 )
= one_one_nat ) ) ) ).
% is_loop_def
thf(fact_804_finite__all__edges__between,axiom,
! [X2: set_a,Y2: set_a] :
( ( finite_finite_a @ X2 )
=> ( ( finite_finite_a @ Y2 )
=> ( finite6544458595007987280od_a_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y2 ) ) ) ) ).
% finite_all_edges_between
thf(fact_805_card__all__edges__between__commute,axiom,
! [X2: set_a,Y2: set_a] :
( ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y2 ) )
= ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ Y2 @ X2 ) ) ) ).
% card_all_edges_between_commute
thf(fact_806_incident__loops__card,axiom,
! [V: a] : ( ord_less_eq_nat @ ( finite_card_set_a @ ( undire4753905205749729249oops_a @ edges @ V ) ) @ one_one_nat ) ).
% incident_loops_card
thf(fact_807_card1__incident__imp__vert,axiom,
! [V: a,E: set_a] :
( ( ( undire1521409233611534436dent_a @ V @ E )
& ( ( finite_card_a @ E )
= one_one_nat ) )
=> ( E
= ( insert_a @ V @ bot_bot_set_a ) ) ) ).
% card1_incident_imp_vert
thf(fact_808_ulgraph_Oincident__sedges_Ocong,axiom,
undire1270416042309875431dges_a = undire1270416042309875431dges_a ).
% ulgraph.incident_sedges.cong
thf(fact_809_card__1__singletonE,axiom,
! [A2: set_Pr5530083903271594800od_a_a] :
( ( ( finite6893194910719049976od_a_a @ A2 )
= one_one_nat )
=> ~ ! [X5: produc4044097585999906000od_a_a] :
( A2
!= ( insert5959526376311583392od_a_a @ X5 @ bot_bo4436838304982128028od_a_a ) ) ) ).
% card_1_singletonE
thf(fact_810_card__1__singletonE,axiom,
! [A2: set_set_a] :
( ( ( finite_card_set_a @ A2 )
= one_one_nat )
=> ~ ! [X5: set_a] :
( A2
!= ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) ) ).
% card_1_singletonE
thf(fact_811_card__1__singletonE,axiom,
! [A2: set_a] :
( ( ( finite_card_a @ A2 )
= one_one_nat )
=> ~ ! [X5: a] :
( A2
!= ( insert_a @ X5 @ bot_bot_set_a ) ) ) ).
% card_1_singletonE
thf(fact_812_card__1__singletonE,axiom,
! [A2: set_Product_prod_a_a] :
( ( ( finite4795055649997197647od_a_a @ A2 )
= one_one_nat )
=> ~ ! [X5: product_prod_a_a] :
( A2
!= ( insert4534936382041156343od_a_a @ X5 @ bot_bo3357376287454694259od_a_a ) ) ) ).
% card_1_singletonE
thf(fact_813_card__1__singletonE,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= one_one_nat )
=> ~ ! [X5: nat] :
( A2
!= ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ).
% card_1_singletonE
thf(fact_814_comp__sgraph_Ois__loop__def,axiom,
( undire2959850631191844736od_a_a
= ( ^ [E3: set_Pr5530083903271594800od_a_a] :
( ( finite6893194910719049976od_a_a @ E3 )
= one_one_nat ) ) ) ).
% comp_sgraph.is_loop_def
thf(fact_815_comp__sgraph_Ois__loop__def,axiom,
( undire3428022325429088215od_a_a
= ( ^ [E3: set_Product_prod_a_a] :
( ( finite4795055649997197647od_a_a @ E3 )
= one_one_nat ) ) ) ).
% comp_sgraph.is_loop_def
thf(fact_816_comp__sgraph_Ois__loop__def,axiom,
( undire3618949687197220622_set_a
= ( ^ [E3: set_set_a] :
( ( finite_card_set_a @ E3 )
= one_one_nat ) ) ) ).
% comp_sgraph.is_loop_def
thf(fact_817_comp__sgraph_Ois__loop__def,axiom,
( undire2905028936066782638loop_a
= ( ^ [E3: set_a] :
( ( finite_card_a @ E3 )
= one_one_nat ) ) ) ).
% comp_sgraph.is_loop_def
thf(fact_818_comp__sgraph_Ocard1__incident__imp__vert,axiom,
! [V: produc4044097585999906000od_a_a,E: set_Pr5530083903271594800od_a_a] :
( ( ( undire8732585234338801206od_a_a @ V @ E )
& ( ( finite6893194910719049976od_a_a @ E )
= one_one_nat ) )
=> ( E
= ( insert5959526376311583392od_a_a @ V @ bot_bo4436838304982128028od_a_a ) ) ) ).
% comp_sgraph.card1_incident_imp_vert
thf(fact_819_comp__sgraph_Ocard1__incident__imp__vert,axiom,
! [V: set_a,E: set_set_a] :
( ( ( undire2320338297334612420_set_a @ V @ E )
& ( ( finite_card_set_a @ E )
= one_one_nat ) )
=> ( E
= ( insert_set_a @ V @ bot_bot_set_set_a ) ) ) ).
% comp_sgraph.card1_incident_imp_vert
thf(fact_820_comp__sgraph_Ocard1__incident__imp__vert,axiom,
! [V: product_prod_a_a,E: set_Product_prod_a_a] :
( ( ( undire3369688177417741453od_a_a @ V @ E )
& ( ( finite4795055649997197647od_a_a @ E )
= one_one_nat ) )
=> ( E
= ( insert4534936382041156343od_a_a @ V @ bot_bo3357376287454694259od_a_a ) ) ) ).
% comp_sgraph.card1_incident_imp_vert
thf(fact_821_comp__sgraph_Ocard1__incident__imp__vert,axiom,
! [V: nat,E: set_nat] :
( ( ( undire7858122600432113898nt_nat @ V @ E )
& ( ( finite_card_nat @ E )
= one_one_nat ) )
=> ( E
= ( insert_nat @ V @ bot_bot_set_nat ) ) ) ).
% comp_sgraph.card1_incident_imp_vert
thf(fact_822_comp__sgraph_Ocard1__incident__imp__vert,axiom,
! [V: a,E: set_a] :
( ( ( undire1521409233611534436dent_a @ V @ E )
& ( ( finite_card_a @ E )
= one_one_nat ) )
=> ( E
= ( insert_a @ V @ bot_bot_set_a ) ) ) ).
% comp_sgraph.card1_incident_imp_vert
thf(fact_823_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
& ( ord_less_eq_set_a @ X5 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_824_finite__has__minimal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X5: real] :
( ( member_real @ X5 @ A2 )
& ( ord_less_eq_real @ X5 @ A )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_825_finite__has__minimal2,axiom,
! [A2: set_se5735800977113168103od_a_a,A: set_Product_prod_a_a] :
( ( finite8717734299975451184od_a_a @ A2 )
=> ( ( member1816616512716248880od_a_a @ A @ A2 )
=> ? [X5: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X5 @ A2 )
& ( ord_le746702958409616551od_a_a @ X5 @ A )
& ! [Xa: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ Xa @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_826_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ord_less_eq_nat @ X5 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_827_finite__has__minimal2,axiom,
! [A2: set_set_set_a,A: set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( member_set_set_a @ A @ A2 )
=> ? [X5: set_set_a] :
( ( member_set_set_a @ X5 @ A2 )
& ( ord_le3724670747650509150_set_a @ X5 @ A )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_828_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
& ( ord_less_eq_set_a @ A @ X5 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_829_finite__has__maximal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X5: real] :
( ( member_real @ X5 @ A2 )
& ( ord_less_eq_real @ A @ X5 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_830_finite__has__maximal2,axiom,
! [A2: set_se5735800977113168103od_a_a,A: set_Product_prod_a_a] :
( ( finite8717734299975451184od_a_a @ A2 )
=> ( ( member1816616512716248880od_a_a @ A @ A2 )
=> ? [X5: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X5 @ A2 )
& ( ord_le746702958409616551od_a_a @ A @ X5 )
& ! [Xa: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ Xa @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_831_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ord_less_eq_nat @ A @ X5 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_832_finite__has__maximal2,axiom,
! [A2: set_set_set_a,A: set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( member_set_set_a @ A @ A2 )
=> ? [X5: set_set_a] :
( ( member_set_set_a @ X5 @ A2 )
& ( ord_le3724670747650509150_set_a @ A @ X5 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_833_infinite__imp__nonempty,axiom,
! [S: set_set_a] :
( ~ ( finite_finite_set_a @ S )
=> ( S != bot_bot_set_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_834_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_835_infinite__imp__nonempty,axiom,
! [S: set_Product_prod_a_a] :
( ~ ( finite6544458595007987280od_a_a @ S )
=> ( S != bot_bo3357376287454694259od_a_a ) ) ).
% infinite_imp_nonempty
thf(fact_836_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_837_finite_OemptyI,axiom,
finite_finite_set_a @ bot_bot_set_set_a ).
% finite.emptyI
thf(fact_838_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_839_finite_OemptyI,axiom,
finite6544458595007987280od_a_a @ bot_bo3357376287454694259od_a_a ).
% finite.emptyI
thf(fact_840_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_841_finite__subset,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_842_finite__subset,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_843_finite__subset,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( finite6544458595007987280od_a_a @ B )
=> ( finite6544458595007987280od_a_a @ A2 ) ) ) ).
% finite_subset
thf(fact_844_finite__subset,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( finite_finite_set_a @ B )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% finite_subset
thf(fact_845_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_846_infinite__super,axiom,
! [S: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_847_infinite__super,axiom,
! [S: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ S @ T2 )
=> ( ~ ( finite6544458595007987280od_a_a @ S )
=> ~ ( finite6544458595007987280od_a_a @ T2 ) ) ) ).
% infinite_super
thf(fact_848_infinite__super,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ S @ T2 )
=> ( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ T2 ) ) ) ).
% infinite_super
thf(fact_849_rev__finite__subset,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_850_rev__finite__subset,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_851_rev__finite__subset,axiom,
! [B: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ B )
=> ( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( finite6544458595007987280od_a_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_852_rev__finite__subset,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_853_finite_OinsertI,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( finite_finite_set_a @ ( insert_set_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_854_finite_OinsertI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_855_finite_OinsertI,axiom,
! [A2: set_Product_prod_a_a,A: product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A2 )
=> ( finite6544458595007987280od_a_a @ ( insert4534936382041156343od_a_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_856_finite_OinsertI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_857_card__insert__le,axiom,
! [A2: set_Pr5530083903271594800od_a_a,X: produc4044097585999906000od_a_a] : ( ord_less_eq_nat @ ( finite6893194910719049976od_a_a @ A2 ) @ ( finite6893194910719049976od_a_a @ ( insert5959526376311583392od_a_a @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_858_card__insert__le,axiom,
! [A2: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ ( insert_a @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_859_card__insert__le,axiom,
! [A2: set_Product_prod_a_a,X: product_prod_a_a] : ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ A2 ) @ ( finite4795055649997197647od_a_a @ ( insert4534936382041156343od_a_a @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_860_card__insert__le,axiom,
! [A2: set_set_a,X: set_a] : ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ ( insert_set_a @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_861_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_862_finite__has__maximal,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ? [X5: real] :
( ( member_real @ X5 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_863_finite__has__maximal,axiom,
! [A2: set_se5735800977113168103od_a_a] :
( ( finite8717734299975451184od_a_a @ A2 )
=> ( ( A2 != bot_bo777872063958040403od_a_a )
=> ? [X5: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X5 @ A2 )
& ! [Xa: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ Xa @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_864_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_865_finite__has__maximal,axiom,
! [A2: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( A2 != bot_bo3380559777022489994_set_a )
=> ? [X5: set_set_a] :
( ( member_set_set_a @ X5 @ A2 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ Xa )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_866_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_867_finite__has__minimal,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ? [X5: real] :
( ( member_real @ X5 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_868_finite__has__minimal,axiom,
! [A2: set_se5735800977113168103od_a_a] :
( ( finite8717734299975451184od_a_a @ A2 )
=> ( ( A2 != bot_bo777872063958040403od_a_a )
=> ? [X5: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X5 @ A2 )
& ! [Xa: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ Xa @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_869_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_870_finite__has__minimal,axiom,
! [A2: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( A2 != bot_bo3380559777022489994_set_a )
=> ? [X5: set_set_a] :
( ( member_set_set_a @ X5 @ A2 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ Xa @ X5 )
=> ( X5 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_871_finite_Ocases,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ~ ! [A7: set_set_a] :
( ? [A6: set_a] :
( A
= ( insert_set_a @ A6 @ A7 ) )
=> ~ ( finite_finite_set_a @ A7 ) ) ) ) ).
% finite.cases
thf(fact_872_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A7: set_a] :
( ? [A6: a] :
( A
= ( insert_a @ A6 @ A7 ) )
=> ~ ( finite_finite_a @ A7 ) ) ) ) ).
% finite.cases
thf(fact_873_finite_Ocases,axiom,
! [A: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A )
=> ( ( A != bot_bo3357376287454694259od_a_a )
=> ~ ! [A7: set_Product_prod_a_a] :
( ? [A6: product_prod_a_a] :
( A
= ( insert4534936382041156343od_a_a @ A6 @ A7 ) )
=> ~ ( finite6544458595007987280od_a_a @ A7 ) ) ) ) ).
% finite.cases
thf(fact_874_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A7: set_nat] :
( ? [A6: nat] :
( A
= ( insert_nat @ A6 @ A7 ) )
=> ~ ( finite_finite_nat @ A7 ) ) ) ) ).
% finite.cases
thf(fact_875_finite_Osimps,axiom,
( finite_finite_set_a
= ( ^ [A4: set_set_a] :
( ( A4 = bot_bot_set_set_a )
| ? [A3: set_set_a,B5: set_a] :
( ( A4
= ( insert_set_a @ B5 @ A3 ) )
& ( finite_finite_set_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_876_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A4: set_a] :
( ( A4 = bot_bot_set_a )
| ? [A3: set_a,B5: a] :
( ( A4
= ( insert_a @ B5 @ A3 ) )
& ( finite_finite_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_877_finite_Osimps,axiom,
( finite6544458595007987280od_a_a
= ( ^ [A4: set_Product_prod_a_a] :
( ( A4 = bot_bo3357376287454694259od_a_a )
| ? [A3: set_Product_prod_a_a,B5: product_prod_a_a] :
( ( A4
= ( insert4534936382041156343od_a_a @ B5 @ A3 ) )
& ( finite6544458595007987280od_a_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_878_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A4: set_nat] :
( ( A4 = bot_bot_set_nat )
| ? [A3: set_nat,B5: nat] :
( ( A4
= ( insert_nat @ B5 @ A3 ) )
& ( finite_finite_nat @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_879_finite__induct,axiom,
! [F2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X5: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X5 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_880_finite__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X5: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X5 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_881_finite__induct,axiom,
! [F2: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ F2 )
=> ( ( P @ bot_bo3357376287454694259od_a_a )
=> ( ! [X5: product_prod_a_a,F3: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F3 )
=> ( ~ ( member1426531477525435216od_a_a @ X5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert4534936382041156343od_a_a @ X5 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_882_finite__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X5: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X5 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_883_finite__ne__induct,axiom,
! [F2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( F2 != bot_bot_set_set_a )
=> ( ! [X5: set_a] : ( P @ ( insert_set_a @ X5 @ bot_bot_set_set_a ) )
=> ( ! [X5: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( F3 != bot_bot_set_set_a )
=> ( ~ ( member_set_a @ X5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_884_finite__ne__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ! [X5: a] : ( P @ ( insert_a @ X5 @ bot_bot_set_a ) )
=> ( ! [X5: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_885_finite__ne__induct,axiom,
! [F2: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ F2 )
=> ( ( F2 != bot_bo3357376287454694259od_a_a )
=> ( ! [X5: product_prod_a_a] : ( P @ ( insert4534936382041156343od_a_a @ X5 @ bot_bo3357376287454694259od_a_a ) )
=> ( ! [X5: product_prod_a_a,F3: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F3 )
=> ( ( F3 != bot_bo3357376287454694259od_a_a )
=> ( ~ ( member1426531477525435216od_a_a @ X5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert4534936382041156343od_a_a @ X5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_886_finite__ne__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X5: nat] : ( P @ ( insert_nat @ X5 @ bot_bot_set_nat ) )
=> ( ! [X5: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_887_infinite__finite__induct,axiom,
! [P: set_set_a > $o,A2: set_set_a] :
( ! [A7: set_set_a] :
( ~ ( finite_finite_set_a @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X5: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X5 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_888_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A7: set_a] :
( ~ ( finite_finite_a @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X5: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X5 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_889_infinite__finite__induct,axiom,
! [P: set_Product_prod_a_a > $o,A2: set_Product_prod_a_a] :
( ! [A7: set_Product_prod_a_a] :
( ~ ( finite6544458595007987280od_a_a @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bo3357376287454694259od_a_a )
=> ( ! [X5: product_prod_a_a,F3: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F3 )
=> ( ~ ( member1426531477525435216od_a_a @ X5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert4534936382041156343od_a_a @ X5 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_890_infinite__finite__induct,axiom,
! [P: set_nat > $o,A2: set_nat] :
( ! [A7: set_nat] :
( ~ ( finite_finite_nat @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X5: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X5 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_891_card__eq__0__iff,axiom,
! [A2: set_Pr5530083903271594800od_a_a] :
( ( ( finite6893194910719049976od_a_a @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bo4436838304982128028od_a_a )
| ~ ( finite5848958031409366265od_a_a @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_892_card__eq__0__iff,axiom,
! [A2: set_set_a] :
( ( ( finite_card_set_a @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_set_a )
| ~ ( finite_finite_set_a @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_893_card__eq__0__iff,axiom,
! [A2: set_a] :
( ( ( finite_card_a @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_a )
| ~ ( finite_finite_a @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_894_card__eq__0__iff,axiom,
! [A2: set_Product_prod_a_a] :
( ( ( finite4795055649997197647od_a_a @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bo3357376287454694259od_a_a )
| ~ ( finite6544458595007987280od_a_a @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_895_card__eq__0__iff,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_896_card__subset__eq,axiom,
! [B: set_Pr5530083903271594800od_a_a,A2: set_Pr5530083903271594800od_a_a] :
( ( finite5848958031409366265od_a_a @ B )
=> ( ( ord_le114883831454073552od_a_a @ A2 @ B )
=> ( ( ( finite6893194910719049976od_a_a @ A2 )
= ( finite6893194910719049976od_a_a @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_897_card__subset__eq,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_898_card__subset__eq,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ( finite_card_a @ A2 )
= ( finite_card_a @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_899_card__subset__eq,axiom,
! [B: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ B )
=> ( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( ( finite4795055649997197647od_a_a @ A2 )
= ( finite4795055649997197647od_a_a @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_900_card__subset__eq,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( ( finite_card_set_a @ A2 )
= ( finite_card_set_a @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_901_infinite__arbitrarily__large,axiom,
! [A2: set_Pr5530083903271594800od_a_a,N: nat] :
( ~ ( finite5848958031409366265od_a_a @ A2 )
=> ? [B4: set_Pr5530083903271594800od_a_a] :
( ( finite5848958031409366265od_a_a @ B4 )
& ( ( finite6893194910719049976od_a_a @ B4 )
= N )
& ( ord_le114883831454073552od_a_a @ B4 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_902_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ( finite_card_nat @ B4 )
= N )
& ( ord_less_eq_set_nat @ B4 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_903_infinite__arbitrarily__large,axiom,
! [A2: set_a,N: nat] :
( ~ ( finite_finite_a @ A2 )
=> ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ( finite_card_a @ B4 )
= N )
& ( ord_less_eq_set_a @ B4 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_904_infinite__arbitrarily__large,axiom,
! [A2: set_Product_prod_a_a,N: nat] :
( ~ ( finite6544458595007987280od_a_a @ A2 )
=> ? [B4: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ B4 )
& ( ( finite4795055649997197647od_a_a @ B4 )
= N )
& ( ord_le746702958409616551od_a_a @ B4 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_905_infinite__arbitrarily__large,axiom,
! [A2: set_set_a,N: nat] :
( ~ ( finite_finite_set_a @ A2 )
=> ? [B4: set_set_a] :
( ( finite_finite_set_a @ B4 )
& ( ( finite_card_set_a @ B4 )
= N )
& ( ord_le3724670747650509150_set_a @ B4 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_906_card__mono,axiom,
! [B: set_Pr5530083903271594800od_a_a,A2: set_Pr5530083903271594800od_a_a] :
( ( finite5848958031409366265od_a_a @ B )
=> ( ( ord_le114883831454073552od_a_a @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite6893194910719049976od_a_a @ A2 ) @ ( finite6893194910719049976od_a_a @ B ) ) ) ) ).
% card_mono
thf(fact_907_card__mono,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ).
% card_mono
thf(fact_908_card__mono,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) ) ) ).
% card_mono
thf(fact_909_card__mono,axiom,
! [B: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ B )
=> ( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ A2 ) @ ( finite4795055649997197647od_a_a @ B ) ) ) ) ).
% card_mono
thf(fact_910_card__mono,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B ) ) ) ) ).
% card_mono
thf(fact_911_card__seteq,axiom,
! [B: set_Pr5530083903271594800od_a_a,A2: set_Pr5530083903271594800od_a_a] :
( ( finite5848958031409366265od_a_a @ B )
=> ( ( ord_le114883831454073552od_a_a @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite6893194910719049976od_a_a @ B ) @ ( finite6893194910719049976od_a_a @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_912_card__seteq,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_913_card__seteq,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_914_card__seteq,axiom,
! [B: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ B )
=> ( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ B ) @ ( finite4795055649997197647od_a_a @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_915_card__seteq,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ B ) @ ( finite_card_set_a @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_916_exists__subset__between,axiom,
! [A2: set_Pr5530083903271594800od_a_a,N: nat,C2: set_Pr5530083903271594800od_a_a] :
( ( ord_less_eq_nat @ ( finite6893194910719049976od_a_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite6893194910719049976od_a_a @ C2 ) )
=> ( ( ord_le114883831454073552od_a_a @ A2 @ C2 )
=> ( ( finite5848958031409366265od_a_a @ C2 )
=> ? [B4: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ A2 @ B4 )
& ( ord_le114883831454073552od_a_a @ B4 @ C2 )
& ( ( finite6893194910719049976od_a_a @ B4 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_917_exists__subset__between,axiom,
! [A2: set_nat,N: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ C2 )
& ( ( finite_card_nat @ B4 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_918_exists__subset__between,axiom,
! [A2: set_a,N: nat,C2: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_a @ C2 ) )
=> ( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( finite_finite_a @ C2 )
=> ? [B4: set_a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
& ( ord_less_eq_set_a @ B4 @ C2 )
& ( ( finite_card_a @ B4 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_919_exists__subset__between,axiom,
! [A2: set_Product_prod_a_a,N: nat,C2: set_Product_prod_a_a] :
( ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite4795055649997197647od_a_a @ C2 ) )
=> ( ( ord_le746702958409616551od_a_a @ A2 @ C2 )
=> ( ( finite6544458595007987280od_a_a @ C2 )
=> ? [B4: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B4 )
& ( ord_le746702958409616551od_a_a @ B4 @ C2 )
& ( ( finite4795055649997197647od_a_a @ B4 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_920_exists__subset__between,axiom,
! [A2: set_set_a,N: nat,C2: set_set_a] :
( ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_set_a @ C2 ) )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ C2 )
=> ( ( finite_finite_set_a @ C2 )
=> ? [B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ C2 )
& ( ( finite_card_set_a @ B4 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_921_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_Pr5530083903271594800od_a_a] :
( ( ord_less_eq_nat @ N @ ( finite6893194910719049976od_a_a @ S ) )
=> ~ ! [T3: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ T3 @ S )
=> ( ( ( finite6893194910719049976od_a_a @ T3 )
= N )
=> ~ ( finite5848958031409366265od_a_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_922_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S )
=> ( ( ( finite_card_nat @ T3 )
= N )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_923_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ S ) )
=> ~ ! [T3: set_a] :
( ( ord_less_eq_set_a @ T3 @ S )
=> ( ( ( finite_card_a @ T3 )
= N )
=> ~ ( finite_finite_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_924_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_Product_prod_a_a] :
( ( ord_less_eq_nat @ N @ ( finite4795055649997197647od_a_a @ S ) )
=> ~ ! [T3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ T3 @ S )
=> ( ( ( finite4795055649997197647od_a_a @ T3 )
= N )
=> ~ ( finite6544458595007987280od_a_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_925_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_set_a @ S ) )
=> ~ ! [T3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ T3 @ S )
=> ( ( ( finite_card_set_a @ T3 )
= N )
=> ~ ( finite_finite_set_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_926_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_Pr5530083903271594800od_a_a,C2: nat] :
( ! [G2: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ G2 @ F2 )
=> ( ( finite5848958031409366265od_a_a @ G2 )
=> ( ord_less_eq_nat @ ( finite6893194910719049976od_a_a @ G2 ) @ C2 ) ) )
=> ( ( finite5848958031409366265od_a_a @ F2 )
& ( ord_less_eq_nat @ ( finite6893194910719049976od_a_a @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_927_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C2: nat] :
( ! [G2: set_nat] :
( ( ord_less_eq_set_nat @ G2 @ F2 )
=> ( ( finite_finite_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_928_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_a,C2: nat] :
( ! [G2: set_a] :
( ( ord_less_eq_set_a @ G2 @ F2 )
=> ( ( finite_finite_a @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ G2 ) @ C2 ) ) )
=> ( ( finite_finite_a @ F2 )
& ( ord_less_eq_nat @ ( finite_card_a @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_929_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_Product_prod_a_a,C2: nat] :
( ! [G2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ G2 @ F2 )
=> ( ( finite6544458595007987280od_a_a @ G2 )
=> ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ G2 ) @ C2 ) ) )
=> ( ( finite6544458595007987280od_a_a @ F2 )
& ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_930_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_set_a,C2: nat] :
( ! [G2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ G2 @ F2 )
=> ( ( finite_finite_set_a @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ G2 ) @ C2 ) ) )
=> ( ( finite_finite_set_a @ F2 )
& ( ord_less_eq_nat @ ( finite_card_set_a @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_931_finite__subset__induct,axiom,
! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A6 @ A2 )
=> ( ~ ( member_nat @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A6 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_932_finite__subset__induct,axiom,
! [F2: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A6: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A6 @ A2 )
=> ( ~ ( member_a @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A6 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_933_finite__subset__induct,axiom,
! [F2: set_Product_prod_a_a,A2: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ F2 )
=> ( ( ord_le746702958409616551od_a_a @ F2 @ A2 )
=> ( ( P @ bot_bo3357376287454694259od_a_a )
=> ( ! [A6: product_prod_a_a,F3: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F3 )
=> ( ( member1426531477525435216od_a_a @ A6 @ A2 )
=> ( ~ ( member1426531477525435216od_a_a @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert4534936382041156343od_a_a @ A6 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_934_finite__subset__induct,axiom,
! [F2: set_set_a,A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A6: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A6 @ A2 )
=> ( ~ ( member_set_a @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ A6 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_935_finite__ranking__induct,axiom,
! [S: set_set_a,P: set_set_a > $o,F: set_a > real] :
( ( finite_finite_set_a @ S )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X5: set_a,S2: set_set_a] :
( ( finite_finite_set_a @ S2 )
=> ( ! [Y7: set_a] :
( ( member_set_a @ Y7 @ S2 )
=> ( ord_less_eq_real @ ( F @ Y7 ) @ ( F @ X5 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_set_a @ X5 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_936_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F: a > real] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X5: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y7: a] :
( ( member_a @ Y7 @ S2 )
=> ( ord_less_eq_real @ ( F @ Y7 ) @ ( F @ X5 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X5 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_937_finite__ranking__induct,axiom,
! [S: set_Product_prod_a_a,P: set_Product_prod_a_a > $o,F: product_prod_a_a > real] :
( ( finite6544458595007987280od_a_a @ S )
=> ( ( P @ bot_bo3357376287454694259od_a_a )
=> ( ! [X5: product_prod_a_a,S2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ S2 )
=> ( ! [Y7: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y7 @ S2 )
=> ( ord_less_eq_real @ ( F @ Y7 ) @ ( F @ X5 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert4534936382041156343od_a_a @ X5 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_938_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X5: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y7: nat] :
( ( member_nat @ Y7 @ S2 )
=> ( ord_less_eq_real @ ( F @ Y7 ) @ ( F @ X5 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat @ X5 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_939_finite__ranking__induct,axiom,
! [S: set_set_a,P: set_set_a > $o,F: set_a > nat] :
( ( finite_finite_set_a @ S )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X5: set_a,S2: set_set_a] :
( ( finite_finite_set_a @ S2 )
=> ( ! [Y7: set_a] :
( ( member_set_a @ Y7 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y7 ) @ ( F @ X5 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_set_a @ X5 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_940_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X5: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y7: a] :
( ( member_a @ Y7 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y7 ) @ ( F @ X5 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X5 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_941_finite__ranking__induct,axiom,
! [S: set_Product_prod_a_a,P: set_Product_prod_a_a > $o,F: product_prod_a_a > nat] :
( ( finite6544458595007987280od_a_a @ S )
=> ( ( P @ bot_bo3357376287454694259od_a_a )
=> ( ! [X5: product_prod_a_a,S2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ S2 )
=> ( ! [Y7: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y7 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y7 ) @ ( F @ X5 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert4534936382041156343od_a_a @ X5 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_942_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X5: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y7: nat] :
( ( member_nat @ Y7 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y7 ) @ ( F @ X5 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat @ X5 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_943_finite__incident__edges,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( finite_finite_set_a @ ( undire3231912044278729248dges_a @ edges @ V ) ) ) ).
% finite_incident_edges
thf(fact_944_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_945_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_946_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_947_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_948_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_949_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_950_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_951_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_952_incident__edges__sedges,axiom,
! [V: a] :
( ~ ( undire3617971648856834880loop_a @ edges @ V )
=> ( ( undire3231912044278729248dges_a @ edges @ V )
= ( undire1270416042309875431dges_a @ edges @ V ) ) ) ).
% incident_edges_sedges
thf(fact_953_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_954_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_955_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_956_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_957_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_958_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ B2 ) )
=> ? [X5: nat] :
( ( P @ X5 )
& ! [Y7: nat] :
( ( P @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ X5 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_959_graph__system_Oincident__edges_Ocong,axiom,
undire3231912044278729248dges_a = undire3231912044278729248dges_a ).
% graph_system.incident_edges.cong
thf(fact_960_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_961_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_962_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_963_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_964_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_965_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_966_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_967_degree0__inc__edges__empt__iff,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( ( ( undire8867928226783802224gree_a @ edges @ V )
= zero_zero_nat )
= ( ( undire3231912044278729248dges_a @ edges @ V )
= bot_bot_set_set_a ) ) ) ).
% degree0_inc_edges_empt_iff
thf(fact_968_max__all__edges__between,axiom,
! [X2: set_a,Y2: set_a] :
( ( finite_finite_a @ X2 )
=> ( ( finite_finite_a @ Y2 )
=> ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y2 ) ) @ ( times_times_nat @ ( finite_card_a @ X2 ) @ ( finite_card_a @ Y2 ) ) ) ) ) ).
% max_all_edges_between
thf(fact_969_incident__edges__union,axiom,
! [V: a] :
( ( undire3231912044278729248dges_a @ edges @ V )
= ( sup_sup_set_set_a @ ( undire1270416042309875431dges_a @ edges @ V ) @ ( undire4753905205749729249oops_a @ edges @ V ) ) ) ).
% incident_edges_union
thf(fact_970_card__Ex__subset,axiom,
! [K: nat,M2: set_Pr5530083903271594800od_a_a] :
( ( ord_less_eq_nat @ K @ ( finite6893194910719049976od_a_a @ M2 ) )
=> ? [N2: set_Pr5530083903271594800od_a_a] :
( ( ord_le114883831454073552od_a_a @ N2 @ M2 )
& ( ( finite6893194910719049976od_a_a @ N2 )
= K ) ) ) ).
% card_Ex_subset
thf(fact_971_card__Ex__subset,axiom,
! [K: nat,M2: set_a] :
( ( ord_less_eq_nat @ K @ ( finite_card_a @ M2 ) )
=> ? [N2: set_a] :
( ( ord_less_eq_set_a @ N2 @ M2 )
& ( ( finite_card_a @ N2 )
= K ) ) ) ).
% card_Ex_subset
thf(fact_972_card__Ex__subset,axiom,
! [K: nat,M2: set_Product_prod_a_a] :
( ( ord_less_eq_nat @ K @ ( finite4795055649997197647od_a_a @ M2 ) )
=> ? [N2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ N2 @ M2 )
& ( ( finite4795055649997197647od_a_a @ N2 )
= K ) ) ) ).
% card_Ex_subset
thf(fact_973_card__Ex__subset,axiom,
! [K: nat,M2: set_set_a] :
( ( ord_less_eq_nat @ K @ ( finite_card_set_a @ M2 ) )
=> ? [N2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ N2 @ M2 )
& ( ( finite_card_set_a @ N2 )
= K ) ) ) ).
% card_Ex_subset
thf(fact_974_card__le__if__inj__on__rel,axiom,
! [B: set_a,A2: set_nat,R: nat > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A2 )
=> ? [B8: a] :
( ( member_a @ B8 @ B )
& ( R @ A6 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B7: a] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_a @ B7 @ B )
=> ( ( R @ A1 @ B7 )
=> ( ( R @ A22 @ B7 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_975_card__le__if__inj__on__rel,axiom,
! [B: set_a,A2: set_a,R: a > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A6: a] :
( ( member_a @ A6 @ A2 )
=> ? [B8: a] :
( ( member_a @ B8 @ B )
& ( R @ A6 @ B8 ) ) )
=> ( ! [A1: a,A22: a,B7: a] :
( ( member_a @ A1 @ A2 )
=> ( ( member_a @ A22 @ A2 )
=> ( ( member_a @ B7 @ B )
=> ( ( R @ A1 @ B7 )
=> ( ( R @ A22 @ B7 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_976_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A2: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A2 )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B )
& ( R @ A6 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B7: nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_nat @ B7 @ B )
=> ( ( R @ A1 @ B7 )
=> ( ( R @ A22 @ B7 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_977_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A2: set_a,R: a > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A6: a] :
( ( member_a @ A6 @ A2 )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B )
& ( R @ A6 @ B8 ) ) )
=> ( ! [A1: a,A22: a,B7: nat] :
( ( member_a @ A1 @ A2 )
=> ( ( member_a @ A22 @ A2 )
=> ( ( member_nat @ B7 @ B )
=> ( ( R @ A1 @ B7 )
=> ( ( R @ A22 @ B7 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_978_card__le__if__inj__on__rel,axiom,
! [B: set_set_a,A2: set_nat,R: nat > set_a > $o] :
( ( finite_finite_set_a @ B )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A2 )
=> ? [B8: set_a] :
( ( member_set_a @ B8 @ B )
& ( R @ A6 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B7: set_a] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_set_a @ B7 @ B )
=> ( ( R @ A1 @ B7 )
=> ( ( R @ A22 @ B7 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_set_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_979_card__le__if__inj__on__rel,axiom,
! [B: set_set_a,A2: set_a,R: a > set_a > $o] :
( ( finite_finite_set_a @ B )
=> ( ! [A6: a] :
( ( member_a @ A6 @ A2 )
=> ? [B8: set_a] :
( ( member_set_a @ B8 @ B )
& ( R @ A6 @ B8 ) ) )
=> ( ! [A1: a,A22: a,B7: set_a] :
( ( member_a @ A1 @ A2 )
=> ( ( member_a @ A22 @ A2 )
=> ( ( member_set_a @ B7 @ B )
=> ( ( R @ A1 @ B7 )
=> ( ( R @ A22 @ B7 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_set_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_980_card__le__if__inj__on__rel,axiom,
! [B: set_a,A2: set_set_a,R: set_a > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A6: set_a] :
( ( member_set_a @ A6 @ A2 )
=> ? [B8: a] :
( ( member_a @ B8 @ B )
& ( R @ A6 @ B8 ) ) )
=> ( ! [A1: set_a,A22: set_a,B7: a] :
( ( member_set_a @ A1 @ A2 )
=> ( ( member_set_a @ A22 @ A2 )
=> ( ( member_a @ B7 @ B )
=> ( ( R @ A1 @ B7 )
=> ( ( R @ A22 @ B7 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_981_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A2: set_set_a,R: set_a > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A6: set_a] :
( ( member_set_a @ A6 @ A2 )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B )
& ( R @ A6 @ B8 ) ) )
=> ( ! [A1: set_a,A22: set_a,B7: nat] :
( ( member_set_a @ A1 @ A2 )
=> ( ( member_set_a @ A22 @ A2 )
=> ( ( member_nat @ B7 @ B )
=> ( ( R @ A1 @ B7 )
=> ( ( R @ A22 @ B7 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_982_card__le__if__inj__on__rel,axiom,
! [B: set_set_a,A2: set_set_a,R: set_a > set_a > $o] :
( ( finite_finite_set_a @ B )
=> ( ! [A6: set_a] :
( ( member_set_a @ A6 @ A2 )
=> ? [B8: set_a] :
( ( member_set_a @ B8 @ B )
& ( R @ A6 @ B8 ) ) )
=> ( ! [A1: set_a,A22: set_a,B7: set_a] :
( ( member_set_a @ A1 @ A2 )
=> ( ( member_set_a @ A22 @ A2 )
=> ( ( member_set_a @ B7 @ B )
=> ( ( R @ A1 @ B7 )
=> ( ( R @ A22 @ B7 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_983_card__le__if__inj__on__rel,axiom,
! [B: set_a,A2: set_Product_prod_a_a,R: product_prod_a_a > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A6: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A6 @ A2 )
=> ? [B8: a] :
( ( member_a @ B8 @ B )
& ( R @ A6 @ B8 ) ) )
=> ( ! [A1: product_prod_a_a,A22: product_prod_a_a,B7: a] :
( ( member1426531477525435216od_a_a @ A1 @ A2 )
=> ( ( member1426531477525435216od_a_a @ A22 @ A2 )
=> ( ( member_a @ B7 @ B )
=> ( ( R @ A1 @ B7 )
=> ( ( R @ A22 @ B7 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ A2 ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_984_degree__no__loops,axiom,
! [V: a] :
( ~ ( undire3617971648856834880loop_a @ edges @ V )
=> ( ( undire8867928226783802224gree_a @ edges @ V )
= ( finite_card_set_a @ ( undire3231912044278729248dges_a @ edges @ V ) ) ) ) ).
% degree_no_loops
thf(fact_985_sup_Oright__idem,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A @ B2 ) @ B2 )
= ( sup_sup_set_set_a @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_986_sup_Oright__idem,axiom,
! [A: set_a,B2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B2 ) @ B2 )
= ( sup_sup_set_a @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_987_sup_Oright__idem,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ A @ B2 ) @ B2 )
= ( sup_su3048258781599657691od_a_a @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_988_sup__left__idem,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( sup_sup_set_set_a @ X @ ( sup_sup_set_set_a @ X @ Y ) )
= ( sup_sup_set_set_a @ X @ Y ) ) ).
% sup_left_idem
thf(fact_989_sup__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% sup_left_idem
thf(fact_990_sup__left__idem,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ X @ Y ) )
= ( sup_su3048258781599657691od_a_a @ X @ Y ) ) ).
% sup_left_idem
thf(fact_991_sup_Oleft__idem,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( sup_sup_set_set_a @ A @ ( sup_sup_set_set_a @ A @ B2 ) )
= ( sup_sup_set_set_a @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_992_sup_Oleft__idem,axiom,
! [A: set_a,B2: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B2 ) )
= ( sup_sup_set_a @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_993_sup_Oleft__idem,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ A @ B2 ) )
= ( sup_su3048258781599657691od_a_a @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_994_sup__idem,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_995_sup__idem,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_996_sup__idem,axiom,
! [X: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_997_sup_Oidem,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_998_sup_Oidem,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_999_sup_Oidem,axiom,
! [A: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_1000_Un__iff,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) )
= ( ( member_nat @ C @ A2 )
| ( member_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_1001_Un__iff,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B ) )
= ( ( member_set_a @ C @ A2 )
| ( member_set_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_1002_Un__iff,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) )
= ( ( member_a @ C @ A2 )
| ( member_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_1003_Un__iff,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) )
= ( ( member1426531477525435216od_a_a @ C @ A2 )
| ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_1004_UnCI,axiom,
! [C: nat,B: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ A2 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_1005_UnCI,axiom,
! [C: set_a,B: set_set_a,A2: set_set_a] :
( ( ~ ( member_set_a @ C @ B )
=> ( member_set_a @ C @ A2 ) )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B ) ) ) ).
% UnCI
thf(fact_1006_UnCI,axiom,
! [C: a,B: set_a,A2: set_a] :
( ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ A2 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnCI
thf(fact_1007_UnCI,axiom,
! [C: product_prod_a_a,B: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ~ ( member1426531477525435216od_a_a @ C @ B )
=> ( member1426531477525435216od_a_a @ C @ A2 ) )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ) ).
% UnCI
thf(fact_1008_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_1009_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1010_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_1011_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1012_mult__eq__0__iff,axiom,
! [A: real,B2: real] :
( ( ( times_times_real @ A @ B2 )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B2 = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_1013_mult__eq__0__iff,axiom,
! [A: nat,B2: nat] :
( ( ( times_times_nat @ A @ B2 )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_1014_mult__cancel__left,axiom,
! [C: real,A: real,B2: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B2 ) )
= ( ( C = zero_zero_real )
| ( A = B2 ) ) ) ).
% mult_cancel_left
thf(fact_1015_mult__cancel__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B2 ) )
= ( ( C = zero_zero_nat )
| ( A = B2 ) ) ) ).
% mult_cancel_left
thf(fact_1016_mult__cancel__right,axiom,
! [A: real,C: real,B2: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B2 @ C ) )
= ( ( C = zero_zero_real )
| ( A = B2 ) ) ) ).
% mult_cancel_right
thf(fact_1017_mult__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B2 @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B2 ) ) ) ).
% mult_cancel_right
thf(fact_1018_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_1019_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_1020_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_1021_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_1022_le__sup__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( ( ord_less_eq_set_a @ X @ Z )
& ( ord_less_eq_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_1023_le__sup__iff,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ X @ Y ) @ Z )
= ( ( ord_less_eq_real @ X @ Z )
& ( ord_less_eq_real @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_1024_le__sup__iff,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ X @ Y ) @ Z )
= ( ( ord_le746702958409616551od_a_a @ X @ Z )
& ( ord_le746702958409616551od_a_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_1025_le__sup__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_1026_le__sup__iff,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ X @ Y ) @ Z )
= ( ( ord_le3724670747650509150_set_a @ X @ Z )
& ( ord_le3724670747650509150_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_1027_sup_Obounded__iff,axiom,
! [B2: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A )
= ( ( ord_less_eq_set_a @ B2 @ A )
& ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_1028_sup_Obounded__iff,axiom,
! [B2: real,C: real,A: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ B2 @ C ) @ A )
= ( ( ord_less_eq_real @ B2 @ A )
& ( ord_less_eq_real @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_1029_sup_Obounded__iff,axiom,
! [B2: set_Product_prod_a_a,C: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ B2 @ C ) @ A )
= ( ( ord_le746702958409616551od_a_a @ B2 @ A )
& ( ord_le746702958409616551od_a_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_1030_sup_Obounded__iff,axiom,
! [B2: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_1031_sup_Obounded__iff,axiom,
! [B2: set_set_a,C: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B2 @ C ) @ A )
= ( ( ord_le3724670747650509150_set_a @ B2 @ A )
& ( ord_le3724670747650509150_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_1032_sup__bot__left,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_1033_sup__bot__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_1034_sup__bot__left,axiom,
! [X: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ bot_bo3357376287454694259od_a_a @ X )
= X ) ).
% sup_bot_left
thf(fact_1035_sup__bot__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_1036_sup__bot__right,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ X @ bot_bot_set_set_a )
= X ) ).
% sup_bot_right
thf(fact_1037_sup__bot__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% sup_bot_right
thf(fact_1038_sup__bot__right,axiom,
! [X: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ bot_bo3357376287454694259od_a_a )
= X ) ).
% sup_bot_right
thf(fact_1039_sup__bot__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% sup_bot_right
thf(fact_1040_bot__eq__sup__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( bot_bot_set_set_a
= ( sup_sup_set_set_a @ X @ Y ) )
= ( ( X = bot_bot_set_set_a )
& ( Y = bot_bot_set_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_1041_bot__eq__sup__iff,axiom,
! [X: set_a,Y: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X @ Y ) )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_1042_bot__eq__sup__iff,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( sup_su3048258781599657691od_a_a @ X @ Y ) )
= ( ( X = bot_bo3357376287454694259od_a_a )
& ( Y = bot_bo3357376287454694259od_a_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_1043_bot__eq__sup__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X @ Y ) )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_1044_sup__eq__bot__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ( sup_sup_set_set_a @ X @ Y )
= bot_bot_set_set_a )
= ( ( X = bot_bot_set_set_a )
& ( Y = bot_bot_set_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_1045_sup__eq__bot__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X @ Y )
= bot_bot_set_a )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_1046_sup__eq__bot__iff,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ( sup_su3048258781599657691od_a_a @ X @ Y )
= bot_bo3357376287454694259od_a_a )
= ( ( X = bot_bo3357376287454694259od_a_a )
& ( Y = bot_bo3357376287454694259od_a_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_1047_sup__eq__bot__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_1048_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ( sup_sup_set_set_a @ A @ B2 )
= bot_bot_set_set_a )
= ( ( A = bot_bot_set_set_a )
& ( B2 = bot_bot_set_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1049_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A @ B2 )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1050_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ( sup_su3048258781599657691od_a_a @ A @ B2 )
= bot_bo3357376287454694259od_a_a )
= ( ( A = bot_bo3357376287454694259od_a_a )
& ( B2 = bot_bo3357376287454694259od_a_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1051_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A @ B2 )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1052_sup__bot_Oleft__neutral,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_1053_sup__bot_Oleft__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_1054_sup__bot_Oleft__neutral,axiom,
! [A: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ bot_bo3357376287454694259od_a_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_1055_sup__bot_Oleft__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_1056_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( bot_bot_set_set_a
= ( sup_sup_set_set_a @ A @ B2 ) )
= ( ( A = bot_bot_set_set_a )
& ( B2 = bot_bot_set_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1057_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A @ B2 ) )
= ( ( A = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1058_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( sup_su3048258781599657691od_a_a @ A @ B2 ) )
= ( ( A = bot_bo3357376287454694259od_a_a )
& ( B2 = bot_bo3357376287454694259od_a_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1059_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A @ B2 ) )
= ( ( A = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1060_sup__bot_Oright__neutral,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ A @ bot_bot_set_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_1061_sup__bot_Oright__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_1062_sup__bot_Oright__neutral,axiom,
! [A: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A @ bot_bo3357376287454694259od_a_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_1063_sup__bot_Oright__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_1064_inf__sup__absorb,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( sup_sup_set_set_a @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_1065_inf__sup__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_1066_inf__sup__absorb,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_1067_sup__inf__absorb,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( sup_sup_set_set_a @ X @ ( inf_inf_set_set_a @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_1068_sup__inf__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_1069_sup__inf__absorb,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_1070_Un__empty,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ( sup_sup_set_set_a @ A2 @ B )
= bot_bot_set_set_a )
= ( ( A2 = bot_bot_set_set_a )
& ( B = bot_bot_set_set_a ) ) ) ).
% Un_empty
thf(fact_1071_Un__empty,axiom,
! [A2: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_1072_Un__empty,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( sup_su3048258781599657691od_a_a @ A2 @ B )
= bot_bo3357376287454694259od_a_a )
= ( ( A2 = bot_bo3357376287454694259od_a_a )
& ( B = bot_bo3357376287454694259od_a_a ) ) ) ).
% Un_empty
thf(fact_1073_Un__empty,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_1074_finite__Un,axiom,
! [F2: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) )
= ( ( finite_finite_nat @ F2 )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_1075_finite__Un,axiom,
! [F2: set_set_a,G: set_set_a] :
( ( finite_finite_set_a @ ( sup_sup_set_set_a @ F2 @ G ) )
= ( ( finite_finite_set_a @ F2 )
& ( finite_finite_set_a @ G ) ) ) ).
% finite_Un
thf(fact_1076_finite__Un,axiom,
! [F2: set_a,G: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G ) )
= ( ( finite_finite_a @ F2 )
& ( finite_finite_a @ G ) ) ) ).
% finite_Un
thf(fact_1077_finite__Un,axiom,
! [F2: set_Product_prod_a_a,G: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ ( sup_su3048258781599657691od_a_a @ F2 @ G ) )
= ( ( finite6544458595007987280od_a_a @ F2 )
& ( finite6544458595007987280od_a_a @ G ) ) ) ).
% finite_Un
thf(fact_1078_Un__subset__iff,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C2 )
= ( ( ord_less_eq_set_a @ A2 @ C2 )
& ( ord_less_eq_set_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_1079_Un__subset__iff,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) @ C2 )
= ( ( ord_le746702958409616551od_a_a @ A2 @ C2 )
& ( ord_le746702958409616551od_a_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_1080_Un__subset__iff,axiom,
! [A2: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B ) @ C2 )
= ( ( ord_le3724670747650509150_set_a @ A2 @ C2 )
& ( ord_le3724670747650509150_set_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_1081_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1082_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1083_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1084_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1085_Un__insert__left,axiom,
! [A: set_a,B: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ ( insert_set_a @ A @ B ) @ C2 )
= ( insert_set_a @ A @ ( sup_sup_set_set_a @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_1086_Un__insert__left,axiom,
! [A: a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A @ B ) @ C2 )
= ( insert_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_1087_Un__insert__left,axiom,
! [A: product_prod_a_a,B: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( insert4534936382041156343od_a_a @ A @ B ) @ C2 )
= ( insert4534936382041156343od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_1088_Un__insert__right,axiom,
! [A2: set_set_a,A: set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( insert_set_a @ A @ ( sup_sup_set_set_a @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_1089_Un__insert__right,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_1090_Un__insert__right,axiom,
! [A2: set_Product_prod_a_a,A: product_prod_a_a,B: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ A @ B ) )
= ( insert4534936382041156343od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_1091_Int__Un__eq_I4_J,axiom,
! [T2: set_set_a,S: set_set_a] :
( ( sup_sup_set_set_a @ T2 @ ( inf_inf_set_set_a @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_1092_Int__Un__eq_I4_J,axiom,
! [T2: set_a,S: set_a] :
( ( sup_sup_set_a @ T2 @ ( inf_inf_set_a @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_1093_Int__Un__eq_I4_J,axiom,
! [T2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ T2 @ ( inf_in8905007599844390133od_a_a @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_1094_Int__Un__eq_I3_J,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( sup_sup_set_set_a @ S @ ( inf_inf_set_set_a @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_1095_Int__Un__eq_I3_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_1096_Int__Un__eq_I3_J,axiom,
! [S: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ S @ ( inf_in8905007599844390133od_a_a @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_1097_Int__Un__eq_I2_J,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_1098_Int__Un__eq_I2_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_1099_Int__Un__eq_I2_J,axiom,
! [S: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_1100_Int__Un__eq_I1_J,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_1101_Int__Un__eq_I1_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_1102_Int__Un__eq_I1_J,axiom,
! [S: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_1103_Un__Int__eq_I4_J,axiom,
! [T2: set_set_a,S: set_set_a] :
( ( inf_inf_set_set_a @ T2 @ ( sup_sup_set_set_a @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_1104_Un__Int__eq_I4_J,axiom,
! [T2: set_a,S: set_a] :
( ( inf_inf_set_a @ T2 @ ( sup_sup_set_a @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_1105_Un__Int__eq_I4_J,axiom,
! [T2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ T2 @ ( sup_su3048258781599657691od_a_a @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_1106_Un__Int__eq_I3_J,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( inf_inf_set_set_a @ S @ ( sup_sup_set_set_a @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_1107_Un__Int__eq_I3_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_1108_Un__Int__eq_I3_J,axiom,
! [S: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ S @ ( sup_su3048258781599657691od_a_a @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_1109_Un__Int__eq_I2_J,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_1110_Un__Int__eq_I2_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_1111_Un__Int__eq_I2_J,axiom,
! [S: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_1112_Un__Int__eq_I1_J,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_1113_Un__Int__eq_I1_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_1114_Un__Int__eq_I1_J,axiom,
! [S: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_1115_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1116_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1117_mult__cancel__right2,axiom,
! [A: real,C: real] :
( ( ( times_times_real @ A @ C )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_1118_mult__cancel__right1,axiom,
! [C: real,B2: real] :
( ( C
= ( times_times_real @ B2 @ C ) )
= ( ( C = zero_zero_real )
| ( B2 = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_1119_mult__cancel__left2,axiom,
! [C: real,A: real] :
( ( ( times_times_real @ C @ A )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_1120_mult__cancel__left1,axiom,
! [C: real,B2: real] :
( ( C
= ( times_times_real @ C @ B2 ) )
= ( ( C = zero_zero_real )
| ( B2 = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_1121_alt__degree__def,axiom,
! [V: a] :
( ( undire8867928226783802224gree_a @ edges @ V )
= ( finite_card_set_a @ ( undire3231912044278729248dges_a @ edges @ V ) ) ) ).
% alt_degree_def
thf(fact_1122_mult_Oleft__commute,axiom,
! [B2: nat,A: nat,C: nat] :
( ( times_times_nat @ B2 @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% mult.left_commute
thf(fact_1123_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B5: nat] : ( times_times_nat @ B5 @ A4 ) ) ) ).
% mult.commute
thf(fact_1124_mult_Oassoc,axiom,
! [A: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B2 ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% mult.assoc
thf(fact_1125_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B2 ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1126_Un__left__commute,axiom,
! [A2: set_set_a,B: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ ( sup_sup_set_set_a @ B @ C2 ) )
= ( sup_sup_set_set_a @ B @ ( sup_sup_set_set_a @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_1127_Un__left__commute,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B @ C2 ) )
= ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_1128_Un__left__commute,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A2 @ ( sup_su3048258781599657691od_a_a @ B @ C2 ) )
= ( sup_su3048258781599657691od_a_a @ B @ ( sup_su3048258781599657691od_a_a @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_1129_Un__left__absorb,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ ( sup_sup_set_set_a @ A2 @ B ) )
= ( sup_sup_set_set_a @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_1130_Un__left__absorb,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_1131_Un__left__absorb,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A2 @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) )
= ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_1132_Un__commute,axiom,
( sup_sup_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] : ( sup_sup_set_set_a @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_1133_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_1134_Un__commute,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_1135_Un__absorb,axiom,
! [A2: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_1136_Un__absorb,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_1137_Un__absorb,axiom,
! [A2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_1138_Un__assoc,axiom,
! [A2: set_set_a,B: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A2 @ B ) @ C2 )
= ( sup_sup_set_set_a @ A2 @ ( sup_sup_set_set_a @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_1139_Un__assoc,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C2 )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_1140_Un__assoc,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) @ C2 )
= ( sup_su3048258781599657691od_a_a @ A2 @ ( sup_su3048258781599657691od_a_a @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_1141_ball__Un,axiom,
! [A2: set_set_a,B: set_set_a,P: set_a > $o] :
( ( ! [X4: set_a] :
( ( member_set_a @ X4 @ ( sup_sup_set_set_a @ A2 @ B ) )
=> ( P @ X4 ) ) )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ( P @ X4 ) )
& ! [X4: set_a] :
( ( member_set_a @ X4 @ B )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_1142_ball__Un,axiom,
! [A2: set_a,B: set_a,P: a > $o] :
( ( ! [X4: a] :
( ( member_a @ X4 @ ( sup_sup_set_a @ A2 @ B ) )
=> ( P @ X4 ) ) )
= ( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( P @ X4 ) )
& ! [X4: a] :
( ( member_a @ X4 @ B )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_1143_ball__Un,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ( ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) )
=> ( P @ X4 ) ) )
= ( ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A2 )
=> ( P @ X4 ) )
& ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ B )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_1144_bex__Un,axiom,
! [A2: set_set_a,B: set_set_a,P: set_a > $o] :
( ( ? [X4: set_a] :
( ( member_set_a @ X4 @ ( sup_sup_set_set_a @ A2 @ B ) )
& ( P @ X4 ) ) )
= ( ? [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
& ( P @ X4 ) )
| ? [X4: set_a] :
( ( member_set_a @ X4 @ B )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_1145_bex__Un,axiom,
! [A2: set_a,B: set_a,P: a > $o] :
( ( ? [X4: a] :
( ( member_a @ X4 @ ( sup_sup_set_a @ A2 @ B ) )
& ( P @ X4 ) ) )
= ( ? [X4: a] :
( ( member_a @ X4 @ A2 )
& ( P @ X4 ) )
| ? [X4: a] :
( ( member_a @ X4 @ B )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_1146_bex__Un,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ( ? [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) )
& ( P @ X4 ) ) )
= ( ? [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A2 )
& ( P @ X4 ) )
| ? [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ B )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_1147_UnI2,axiom,
! [C: nat,B: set_nat,A2: set_nat] :
( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_1148_UnI2,axiom,
! [C: set_a,B: set_set_a,A2: set_set_a] :
( ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B ) ) ) ).
% UnI2
thf(fact_1149_UnI2,axiom,
! [C: a,B: set_a,A2: set_a] :
( ( member_a @ C @ B )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnI2
thf(fact_1150_UnI2,axiom,
! [C: product_prod_a_a,B: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ B )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ) ).
% UnI2
thf(fact_1151_UnI1,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_1152_UnI1,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B ) ) ) ).
% UnI1
thf(fact_1153_UnI1,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ A2 )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnI1
thf(fact_1154_UnI1,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A2 )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ) ).
% UnI1
thf(fact_1155_UnE,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) )
=> ( ~ ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B ) ) ) ).
% UnE
thf(fact_1156_UnE,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B ) )
=> ( ~ ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B ) ) ) ).
% UnE
thf(fact_1157_UnE,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) )
=> ( ~ ( member_a @ C @ A2 )
=> ( member_a @ C @ B ) ) ) ).
% UnE
thf(fact_1158_UnE,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) )
=> ( ~ ( member1426531477525435216od_a_a @ C @ A2 )
=> ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% UnE
thf(fact_1159_ulgraph_Odegree_Ocong,axiom,
undire8867928226783802224gree_a = undire8867928226783802224gree_a ).
% ulgraph.degree.cong
thf(fact_1160_sup__left__commute,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ X @ ( sup_sup_set_set_a @ Y @ Z ) )
= ( sup_sup_set_set_a @ Y @ ( sup_sup_set_set_a @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_1161_sup__left__commute,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_1162_sup__left__commute,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ Y @ Z ) )
= ( sup_su3048258781599657691od_a_a @ Y @ ( sup_su3048258781599657691od_a_a @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_1163_sup_Oleft__commute,axiom,
! [B2: set_set_a,A: set_set_a,C: set_set_a] :
( ( sup_sup_set_set_a @ B2 @ ( sup_sup_set_set_a @ A @ C ) )
= ( sup_sup_set_set_a @ A @ ( sup_sup_set_set_a @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_1164_sup_Oleft__commute,axiom,
! [B2: set_a,A: set_a,C: set_a] :
( ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A @ C ) )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_1165_sup_Oleft__commute,axiom,
! [B2: set_Product_prod_a_a,A: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ B2 @ ( sup_su3048258781599657691od_a_a @ A @ C ) )
= ( sup_su3048258781599657691od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_1166_boolean__algebra__cancel_Osup2,axiom,
! [B: set_set_a,K: set_set_a,B2: set_set_a,A: set_set_a] :
( ( B
= ( sup_sup_set_set_a @ K @ B2 ) )
=> ( ( sup_sup_set_set_a @ A @ B )
= ( sup_sup_set_set_a @ K @ ( sup_sup_set_set_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_1167_boolean__algebra__cancel_Osup2,axiom,
! [B: set_a,K: set_a,B2: set_a,A: set_a] :
( ( B
= ( sup_sup_set_a @ K @ B2 ) )
=> ( ( sup_sup_set_a @ A @ B )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_1168_boolean__algebra__cancel_Osup2,axiom,
! [B: set_Product_prod_a_a,K: set_Product_prod_a_a,B2: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( B
= ( sup_su3048258781599657691od_a_a @ K @ B2 ) )
=> ( ( sup_su3048258781599657691od_a_a @ A @ B )
= ( sup_su3048258781599657691od_a_a @ K @ ( sup_su3048258781599657691od_a_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_1169_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_set_a,K: set_set_a,A: set_set_a,B2: set_set_a] :
( ( A2
= ( sup_sup_set_set_a @ K @ A ) )
=> ( ( sup_sup_set_set_a @ A2 @ B2 )
= ( sup_sup_set_set_a @ K @ ( sup_sup_set_set_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_1170_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_a,K: set_a,A: set_a,B2: set_a] :
( ( A2
= ( sup_sup_set_a @ K @ A ) )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_1171_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_Product_prod_a_a,K: set_Product_prod_a_a,A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( A2
= ( sup_su3048258781599657691od_a_a @ K @ A ) )
=> ( ( sup_su3048258781599657691od_a_a @ A2 @ B2 )
= ( sup_su3048258781599657691od_a_a @ K @ ( sup_su3048258781599657691od_a_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_1172_sup__commute,axiom,
( sup_sup_set_set_a
= ( ^ [X4: set_set_a,Y4: set_set_a] : ( sup_sup_set_set_a @ Y4 @ X4 ) ) ) ).
% sup_commute
thf(fact_1173_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X4: set_a,Y4: set_a] : ( sup_sup_set_a @ Y4 @ X4 ) ) ) ).
% sup_commute
thf(fact_1174_sup__commute,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [X4: set_Product_prod_a_a,Y4: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ Y4 @ X4 ) ) ) ).
% sup_commute
thf(fact_1175_sup_Ocommute,axiom,
( sup_sup_set_set_a
= ( ^ [A4: set_set_a,B5: set_set_a] : ( sup_sup_set_set_a @ B5 @ A4 ) ) ) ).
% sup.commute
thf(fact_1176_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B5: set_a] : ( sup_sup_set_a @ B5 @ A4 ) ) ) ).
% sup.commute
thf(fact_1177_sup_Ocommute,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [A4: set_Product_prod_a_a,B5: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ B5 @ A4 ) ) ) ).
% sup.commute
thf(fact_1178_sup__assoc,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ X @ Y ) @ Z )
= ( sup_sup_set_set_a @ X @ ( sup_sup_set_set_a @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_1179_sup__assoc,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_1180_sup__assoc,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ X @ Y ) @ Z )
= ( sup_su3048258781599657691od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_1181_sup_Oassoc,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A @ B2 ) @ C )
= ( sup_sup_set_set_a @ A @ ( sup_sup_set_set_a @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_1182_sup_Oassoc,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B2 ) @ C )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_1183_sup_Oassoc,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ A @ B2 ) @ C )
= ( sup_su3048258781599657691od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_1184_inf__sup__aci_I5_J,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [X4: set_Product_prod_a_a,Y4: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ Y4 @ X4 ) ) ) ).
% inf_sup_aci(5)
thf(fact_1185_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1186_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1187_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1188_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1189_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1190_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1191_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1192_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1193_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1194_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M3: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N3 )
=> ( ord_less_eq_nat @ X4 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1195_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M2: nat] :
( ( P @ X )
=> ( ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M2 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1196_all__edges__between__Un1,axiom,
! [X2: set_a,Y2: set_a,Z2: set_a] :
( ( undire8383842906760478443ween_a @ edges @ ( sup_sup_set_a @ X2 @ Y2 ) @ Z2 )
= ( sup_su3048258781599657691od_a_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Z2 ) @ ( undire8383842906760478443ween_a @ edges @ Y2 @ Z2 ) ) ) ).
% all_edges_between_Un1
thf(fact_1197_all__edges__between__Un2,axiom,
! [X2: set_a,Y2: set_a,Z2: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X2 @ ( sup_sup_set_a @ Y2 @ Z2 ) )
= ( sup_su3048258781599657691od_a_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y2 ) @ ( undire8383842906760478443ween_a @ edges @ X2 @ Z2 ) ) ) ).
% all_edges_between_Un2
thf(fact_1198_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1199_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M3: nat] :
? [N4: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
& ( member_nat @ N4 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_1200_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1201_sgraph__axioms,axiom,
undire3507641187627840796raph_a @ vertices @ edges ).
% sgraph_axioms
thf(fact_1202_wellformed,axiom,
! [E: set_a] :
( ( member_set_a @ E @ edges )
=> ( ord_less_eq_set_a @ E @ vertices ) ) ).
% wellformed
thf(fact_1203_vert__adj__imp__inV,axiom,
! [V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
=> ( ( member_a @ V1 @ vertices )
& ( member_a @ V2 @ vertices ) ) ) ).
% vert_adj_imp_inV
thf(fact_1204_incident__edge__in__wf,axiom,
! [E: set_a,V: a] :
( ( member_set_a @ E @ edges )
=> ( ( undire1521409233611534436dent_a @ V @ E )
=> ( member_a @ V @ vertices ) ) ) ).
% incident_edge_in_wf
thf(fact_1205_has__loop__in__verts,axiom,
! [V: a] :
( ( undire3617971648856834880loop_a @ edges @ V )
=> ( member_a @ V @ vertices ) ) ).
% has_loop_in_verts
thf(fact_1206_no__loops,axiom,
! [V: a] :
( ( member_a @ V @ vertices )
=> ~ ( undire3617971648856834880loop_a @ edges @ V ) ) ).
% no_loops
thf(fact_1207_edge__adjacent__alt__def,axiom,
! [E1: set_a,E2: set_a] :
( ( member_set_a @ E1 @ edges )
=> ( ( member_set_a @ E2 @ edges )
=> ( ? [X6: a] :
( ( member_a @ X6 @ vertices )
& ( member_a @ X6 @ E1 )
& ( member_a @ X6 @ E2 ) )
=> ( undire4022703626023482010_adj_a @ edges @ E1 @ E2 ) ) ) ) ).
% edge_adjacent_alt_def
thf(fact_1208_wellformed__alt__fst,axiom,
! [X: a,Y: a] :
( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges )
=> ( member_a @ X @ vertices ) ) ).
% wellformed_alt_fst
thf(fact_1209_wellformed__alt__snd,axiom,
! [X: a,Y: a] :
( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges )
=> ( member_a @ Y @ vertices ) ) ).
% wellformed_alt_snd
thf(fact_1210_all__edges__between__rem__wf,axiom,
! [X2: set_a,Y2: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X2 @ Y2 )
= ( undire8383842906760478443ween_a @ edges @ ( inf_inf_set_a @ X2 @ vertices ) @ ( inf_inf_set_a @ Y2 @ vertices ) ) ) ).
% all_edges_between_rem_wf
thf(fact_1211_incident__edges__empty,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire3231912044278729248dges_a @ edges @ V )
= bot_bot_set_set_a ) ) ).
% incident_edges_empty
thf(fact_1212_degree__none,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire8867928226783802224gree_a @ edges @ V )
= zero_zero_nat ) ) ).
% degree_none
thf(fact_1213_incident__sedges__empty,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire1270416042309875431dges_a @ edges @ V )
= bot_bot_set_set_a ) ) ).
% incident_sedges_empty
thf(fact_1214_is__isolated__vertex__degree0,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
=> ( ( undire8867928226783802224gree_a @ edges @ V )
= zero_zero_nat ) ) ).
% is_isolated_vertex_degree0
thf(fact_1215_is__isolated__vertex__no__loop,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
=> ~ ( undire3617971648856834880loop_a @ edges @ V ) ) ).
% is_isolated_vertex_no_loop
thf(fact_1216_is__isolated__vertex__edge,axiom,
! [V: a,E: set_a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
=> ( ( member_set_a @ E @ edges )
=> ~ ( undire1521409233611534436dent_a @ V @ E ) ) ) ).
% is_isolated_vertex_edge
thf(fact_1217_is__isolated__vertex__def,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
= ( ( member_a @ V @ vertices )
& ! [X4: a] :
( ( member_a @ X4 @ vertices )
=> ~ ( undire397441198561214472_adj_a @ edges @ X4 @ V ) ) ) ) ).
% is_isolated_vertex_def
thf(fact_1218_degree0__neighborhood__empt__iff,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( ( ( undire8867928226783802224gree_a @ edges @ V )
= zero_zero_nat )
= ( ( undire8504279938402040014hood_a @ vertices @ edges @ V )
= bot_bot_set_a ) ) ) ).
% degree0_neighborhood_empt_iff
thf(fact_1219_card__incident__sedges__neighborhood,axiom,
! [V: a] :
( ( finite_card_set_a @ ( undire3231912044278729248dges_a @ edges @ V ) )
= ( finite_card_a @ ( undire8504279938402040014hood_a @ vertices @ edges @ V ) ) ) ).
% card_incident_sedges_neighborhood
thf(fact_1220_neighborhood__incident,axiom,
! [U: a,V: a] :
( ( member_a @ U @ ( undire8504279938402040014hood_a @ vertices @ edges @ V ) )
= ( member_set_a @ ( insert_a @ U @ ( insert_a @ V @ bot_bot_set_a ) ) @ ( undire3231912044278729248dges_a @ edges @ V ) ) ) ).
% neighborhood_incident
thf(fact_1221_alt__deg__neighborhood,axiom,
! [V: a] :
( ( undire8867928226783802224gree_a @ edges @ V )
= ( finite_card_a @ ( undire8504279938402040014hood_a @ vertices @ edges @ V ) ) ) ).
% alt_deg_neighborhood
thf(fact_1222_iso__vertex__empty__neighborhood,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
=> ( ( undire8504279938402040014hood_a @ vertices @ edges @ V )
= bot_bot_set_a ) ) ).
% iso_vertex_empty_neighborhood
thf(fact_1223_induced__edges__ss,axiom,
! [V4: set_a] :
( ( ord_less_eq_set_a @ V4 @ vertices )
=> ( ord_le3724670747650509150_set_a @ ( undire7777452895879145676dges_a @ edges @ V4 ) @ edges ) ) ).
% induced_edges_ss
thf(fact_1224_induced__edges__self,axiom,
( ( undire7777452895879145676dges_a @ edges @ vertices )
= edges ) ).
% induced_edges_self
thf(fact_1225_subgraph__refl,axiom,
undire7103218114511261257raph_a @ vertices @ edges @ vertices @ edges ).
% subgraph_refl
thf(fact_1226_induced__is__subgraph,axiom,
! [V4: set_a] :
( ( ord_less_eq_set_a @ V4 @ vertices )
=> ( undire7103218114511261257raph_a @ V4 @ ( undire7777452895879145676dges_a @ edges @ V4 ) @ vertices @ edges ) ) ).
% induced_is_subgraph
thf(fact_1227_induced__edges__union,axiom,
! [VH1: set_a,S: set_a,VH2: set_a,T2: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T2 )
=> ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
=> ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
=> ( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( undire7777452895879145676dges_a @ edges @ ( sup_sup_set_a @ S @ T2 ) ) )
=> ( ord_le3724670747650509150_set_a @ EH1 @ ( undire7777452895879145676dges_a @ edges @ S ) ) ) ) ) ) ) ).
% induced_edges_union
thf(fact_1228_induced__edges__union__subgraph__single,axiom,
! [VH1: set_a,S: set_a,VH2: set_a,T2: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T2 )
=> ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
=> ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
=> ( ( undire7103218114511261257raph_a @ ( sup_sup_set_a @ VH1 @ VH2 ) @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( sup_sup_set_a @ S @ T2 ) @ ( undire7777452895879145676dges_a @ edges @ ( sup_sup_set_a @ S @ T2 ) ) )
=> ( undire7103218114511261257raph_a @ VH1 @ EH1 @ S @ ( undire7777452895879145676dges_a @ edges @ S ) ) ) ) ) ) ) ).
% induced_edges_union_subgraph_single
thf(fact_1229_induced__union__subgraph,axiom,
! [VH1: set_a,S: set_a,VH2: set_a,T2: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T2 )
=> ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
=> ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
=> ( ( ( undire7103218114511261257raph_a @ VH1 @ EH1 @ S @ ( undire7777452895879145676dges_a @ edges @ S ) )
& ( undire7103218114511261257raph_a @ VH2 @ EH2 @ T2 @ ( undire7777452895879145676dges_a @ edges @ T2 ) ) )
= ( undire7103218114511261257raph_a @ ( sup_sup_set_a @ VH1 @ VH2 ) @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( sup_sup_set_a @ S @ T2 ) @ ( undire7777452895879145676dges_a @ edges @ ( sup_sup_set_a @ S @ T2 ) ) ) ) ) ) ) ) ).
% induced_union_subgraph
thf(fact_1230_induced__is__graph__sys,axiom,
! [V4: set_a] : ( undire2554140024507503526stem_a @ V4 @ ( undire7777452895879145676dges_a @ edges @ V4 ) ) ).
% induced_is_graph_sys
thf(fact_1231_graph__system__axioms,axiom,
undire2554140024507503526stem_a @ vertices @ edges ).
% graph_system_axioms
thf(fact_1232_wellformed__all__edges,axiom,
ord_le3724670747650509150_set_a @ edges @ ( undire2918257014606996450dges_a @ vertices ) ).
% wellformed_all_edges
thf(fact_1233_subgraph__complete,axiom,
undire7103218114511261257raph_a @ vertices @ edges @ vertices @ ( undire2918257014606996450dges_a @ vertices ) ).
% subgraph_complete
thf(fact_1234_e__in__all__edges__ss,axiom,
! [E: set_a,V4: set_a] :
( ( member_set_a @ E @ edges )
=> ( ( ord_less_eq_set_a @ E @ V4 )
=> ( ( ord_less_eq_set_a @ V4 @ vertices )
=> ( member_set_a @ E @ ( undire2918257014606996450dges_a @ V4 ) ) ) ) ) ).
% e_in_all_edges_ss
thf(fact_1235_induced__edges__alt,axiom,
! [V4: set_a] :
( ( undire7777452895879145676dges_a @ edges @ V4 )
= ( inf_inf_set_set_a @ edges @ ( undire2918257014606996450dges_a @ V4 ) ) ) ).
% induced_edges_alt
thf(fact_1236_e__in__all__edges,axiom,
! [E: set_a] :
( ( member_set_a @ E @ edges )
=> ( member_set_a @ E @ ( undire2918257014606996450dges_a @ vertices ) ) ) ).
% e_in_all_edges
thf(fact_1237_is__complete__n__graph__def,axiom,
! [N: nat] :
( ( undire6087271738840788937raph_a @ vertices @ edges @ N )
= ( ( ( finite_card_a @ vertices )
= N )
& ( edges
= ( undire2918257014606996450dges_a @ vertices ) ) ) ) ).
% is_complete_n_graph_def
thf(fact_1238_ulgraph__axioms,axiom,
undire7251896706689453996raph_a @ vertices @ edges ).
% ulgraph_axioms
thf(fact_1239_is__complement__edges,axiom,
! [V4: set_a,E4: set_set_a] :
( ( undire8013100667316154652ment_a @ vertices @ edges @ ( produc2116933609460601975_set_a @ V4 @ E4 ) )
= ( ( vertices = V4 )
& ( ( undire4625228487420481630dges_a @ vertices @ edges )
= E4 ) ) ) ).
% is_complement_edges
thf(fact_1240_complement__edges__def,axiom,
( ( undire4625228487420481630dges_a @ vertices @ edges )
= ( minus_5736297505244876581_set_a @ ( undire2918257014606996450dges_a @ vertices ) @ edges ) ) ).
% complement_edges_def
thf(fact_1241_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1242_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1243_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1244_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1245_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1246_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1247_le__diff__iff_H,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1248_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1249_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1250_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1251_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1252_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1253_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1254_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1255_is__complement__def,axiom,
! [G: produc7943277765024757383_set_a] :
( ( undire8013100667316154652ment_a @ vertices @ edges @ G )
= ( ( vertices
= ( produc7697427544025156195_set_a @ G ) )
& ( ( produc1930393432416776613_set_a @ G )
= ( minus_5736297505244876581_set_a @ ( undire2918257014606996450dges_a @ vertices ) @ edges ) ) ) ) ).
% is_complement_def
thf(fact_1256_edge__density__def,axiom,
! [X2: set_a,Y2: set_a] :
( ( undire297304480579013331sity_a @ edges @ X2 @ Y2 )
= ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y2 ) ) ) @ ( semiri5074537144036343181t_real @ ( times_times_nat @ ( finite_card_a @ X2 ) @ ( finite_card_a @ Y2 ) ) ) ) ) ).
% edge_density_def
thf(fact_1257_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1258_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1259_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1260_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1261_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1262_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1263_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1264_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1265_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1266_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1267_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1268_diff__less__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1269_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1270_bounded__nat__set__is__finite,axiom,
! [N5: set_nat,N: nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ N5 )
=> ( ord_less_nat @ X5 @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1271_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M3: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N3 )
=> ( ord_less_nat @ X4 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1272_unbounded__k__infinite,axiom,
! [K: nat,S: set_nat] :
( ! [M4: nat] :
( ( ord_less_nat @ K @ M4 )
=> ? [N6: nat] :
( ( ord_less_nat @ M4 @ N6 )
& ( member_nat @ N6 @ S ) ) )
=> ~ ( finite_finite_nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_1273_infinite__nat__iff__unbounded,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M3: nat] :
? [N4: nat] :
( ( ord_less_nat @ M3 @ N4 )
& ( member_nat @ N4 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
% Conjectures (1)
thf(conj_0,conjecture,
graph_4582152751571636272raph_a @ edges @ x2 @ y2 @ z2 ).
%------------------------------------------------------------------------------