TPTP Problem File: SLH0134^1.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/0016_Undirected_Graph_Walks/prob_00572_021776__13273840_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1482 ( 590 unt; 209 typ; 0 def)
% Number of atoms : 3747 (1355 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 10586 ( 386 ~; 72 |; 378 &;8185 @)
% ( 0 <=>;1565 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 503 ( 503 >; 0 *; 0 +; 0 <<)
% Number of symbols : 195 ( 194 usr; 18 con; 0-4 aty)
% Number of variables : 3340 ( 199 ^;2982 !; 159 ?;3340 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:33:38.250
%------------------------------------------------------------------------------
% Could-be-implicit typings (15)
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thf(ty_n_t__Set__Oset_It__Nat__Onat_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 (194)
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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_Undirected__Graph__Basics_Ograph__system_001tf__a,type,
undire2554140024507503526stem_a: set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oedge__adj_001t__Nat__Onat,type,
undire1664191744716346676dj_nat: set_set_nat > set_nat > set_nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oedge__adj_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire9186443406341554371od_a_a: set_se5735800977113168103od_a_a > set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oedge__adj_001t__Set__Oset_Itf__a_J,type,
undire3485422320110889978_set_a: set_set_set_a > set_set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oedge__adj_001tf__a,type,
undire4022703626023482010_adj_a: set_set_a > set_a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident_001t__Nat__Onat,type,
undire7858122600432113898nt_nat: nat > set_nat > $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_001t__Nat__Onat,type,
undire4176300566717384750es_nat: set_set_nat > nat > set_set_nat ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident__edges_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire8905369280470868553od_a_a: set_se5735800977113168103od_a_a > product_prod_a_a > set_se5735800977113168103od_a_a ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oincident__edges_001t__Set__Oset_Itf__a_J,type,
undire4631953023069350784_set_a: set_set_set_a > set_a > set_set_set_a ).
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_001t__List__Olist_Itf__a_J,type,
undire8521487854958249554list_a: set_set_list_a > set_list_a > set_set_list_a ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oinduced__edges_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire5906991851038061813od_a_a: set_se5735800977113168103od_a_a > set_Product_prod_a_a > set_se5735800977113168103od_a_a ).
thf(sy_c_Undirected__Graph__Basics_Ograph__system_Oinduced__edges_001t__Set__Oset_Itf__a_J,type,
undire7854589003810675244_set_a: set_set_set_a > set_set_a > set_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__edge_001tf__a,type,
undire6670514144573423676edge_a: product_prod_a_a > set_a ).
thf(sy_c_Undirected__Graph__Basics_Osubgraph_001t__List__Olist_Itf__a_J,type,
undire761398192061991247list_a: set_list_a > set_set_list_a > set_list_a > set_set_list_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Osubgraph_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire398746457437328754od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > set_Product_prod_a_a > set_se5735800977113168103od_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Osubgraph_001t__Set__Oset_Itf__a_J,type,
undire1186139521737116585_set_a: set_set_a > set_set_set_a > set_set_a > set_set_set_a > $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_001t__Nat__Onat,type,
undire3269267262472140706ph_nat: set_nat > set_set_nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire4585262585102564309od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_001t__Set__Oset_Itf__a_J,type,
undire6886684016831807756_set_a: set_set_a > set_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_001t__Nat__Onat,type,
undire6581030323043281630ee_nat: set_set_nat > nat > nat ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Odegree_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire1436394852029823897od_a_a: set_se5735800977113168103od_a_a > product_prod_a_a > nat ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Odegree_001t__Set__Oset_Itf__a_J,type,
undire8939077443744732368_set_a: set_set_set_a > set_a > nat ).
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_001t__Nat__Onat,type,
undire8640779321340989627ty_nat: set_set_nat > set_nat > set_nat > real ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oedge__density_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire8410861505230878716od_a_a: set_se5735800977113168103od_a_a > set_Product_prod_a_a > set_Product_prod_a_a > real ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oedge__density_001t__Set__Oset_Itf__a_J,type,
undire8927637694342045747_set_a: set_set_set_a > set_set_a > set_set_a > real ).
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_001t__Nat__Onat,type,
undire1050940535076293677ps_nat: set_set_nat > nat > set_set_nat ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oincident__loops_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire3049230956220217098od_a_a: set_se5735800977113168103od_a_a > product_prod_a_a > set_se5735800977113168103od_a_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oincident__loops_001t__Set__Oset_Itf__a_J,type,
undire7215034953758041409_set_a: set_set_set_a > set_a > set_set_set_a ).
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_001t__Nat__Onat,type,
undire996053960663353255es_nat: set_set_nat > nat > set_set_nat ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oincident__sedges_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire1583524423955984400od_a_a: set_se5735800977113168103od_a_a > product_prod_a_a > set_se5735800977113168103od_a_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oincident__sedges_001t__Set__Oset_Itf__a_J,type,
undire5844230293943614535_set_a: set_set_set_a > set_a > set_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_001t__Nat__Onat,type,
undire5609513041723151865ex_nat: set_nat > set_set_nat > nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__isolated__vertex_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire3207556238582723646od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > product_prod_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__isolated__vertex_001t__Set__Oset_Itf__a_J,type,
undire6879241558604981877_set_a: set_set_a > set_set_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__Nat__Onat,type,
undire643512044667278624op_nat: set_nat > $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_001t__Nat__Onat,type,
undire8190396521545869824od_nat: set_nat > set_set_nat > nat > set_nat ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oneighborhood_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire7963753511165915895od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oneighborhood_001t__Set__Oset_Itf__a_J,type,
undire2074812191327625774_set_a: set_set_a > set_set_set_a > set_a > set_set_a ).
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_Oneighbors__ss_001tf__a,type,
undire401937927514038589s_ss_a: set_set_a > a > set_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_Undirected__Graph__Walks_Oulgraph_Ocycles_001tf__a,type,
undire2685670433559799090cles_a: set_a > set_set_a > set_list_a ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ogen__paths_001tf__a,type,
undire6235733737954427521aths_a: set_a > set_set_a > set_list_a ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__cycle_001tf__a,type,
undire2407311113669455967ycle_a: set_a > set_set_a > list_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__gen__path_001tf__a,type,
undire3562951555376170320path_a: set_a > set_set_a > list_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__path_001tf__a,type,
undire427332500224447920path_a: set_a > set_set_a > list_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Ois__walk_001tf__a,type,
undire6133010728901294956walk_a: set_a > set_set_a > list_a > $o ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Opaths_001tf__a,type,
undire1387732426225024653aths_a: set_a > set_set_a > set_list_a ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Owalk__edges_001tf__a,type,
undire7337870655677353998dges_a: list_a > list_set_a ).
thf(sy_c_Undirected__Graph__Walks_Oulgraph_Owalks_001tf__a,type,
undire3736599831911450577alks_a: set_a > set_set_a > set_list_a ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_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__Set__Oset_It__List__Olist_Itf__a_J_J,type,
member_set_list_a: set_list_a > set_set_list_a > $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_edges,type,
edges: set_set_a ).
thf(sy_v_vertices,type,
vertices: set_a ).
% Relevant facts (1272)
thf(fact_0_fin__ulgraph__axioms,axiom,
undire7599193295422474955raph_a @ vertices @ edges ).
% fin_ulgraph_axioms
thf(fact_1_wellformed,axiom,
! [E: set_a] :
( ( member_set_a @ E @ edges )
=> ( ord_less_eq_set_a @ E @ vertices ) ) ).
% wellformed
thf(fact_2_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_3_edge__adjacent__alt__def,axiom,
! [E1: set_a,E2: set_a] :
( ( member_set_a @ E1 @ edges )
=> ( ( member_set_a @ E2 @ edges )
=> ( ? [X: a] :
( ( member_a @ X @ vertices )
& ( member_a @ X @ E1 )
& ( member_a @ X @ E2 ) )
=> ( undire4022703626023482010_adj_a @ edges @ E1 @ E2 ) ) ) ) ).
% edge_adjacent_alt_def
thf(fact_4_ulgraph_Ocycles_Ocong,axiom,
undire2685670433559799090cles_a = undire2685670433559799090cles_a ).
% ulgraph.cycles.cong
thf(fact_5_finite__paths,axiom,
finite_finite_list_a @ ( undire1387732426225024653aths_a @ vertices @ edges ) ).
% finite_paths
thf(fact_6_fin__graph__system__axioms,axiom,
undire945497512398942277stem_a @ vertices @ edges ).
% fin_graph_system_axioms
thf(fact_7_cycles__ss__gen__paths,axiom,
ord_le8861187494160871172list_a @ ( undire2685670433559799090cles_a @ vertices @ edges ) @ ( undire6235733737954427521aths_a @ vertices @ edges ) ).
% cycles_ss_gen_paths
thf(fact_8_has__loop__in__verts,axiom,
! [V: a] :
( ( undire3617971648856834880loop_a @ edges @ V )
=> ( member_a @ V @ vertices ) ) ).
% has_loop_in_verts
thf(fact_9_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_10_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_11_subgraph__refl,axiom,
undire7103218114511261257raph_a @ vertices @ edges @ vertices @ edges ).
% subgraph_refl
thf(fact_12_graph__system__axioms,axiom,
undire2554140024507503526stem_a @ vertices @ edges ).
% graph_system_axioms
thf(fact_13_edge__density__commute,axiom,
! [X2: set_a,Y: set_a] :
( ( undire297304480579013331sity_a @ edges @ X2 @ Y )
= ( undire297304480579013331sity_a @ edges @ Y @ X2 ) ) ).
% edge_density_commute
thf(fact_14_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_15_ulgraph__axioms,axiom,
undire7251896706689453996raph_a @ vertices @ edges ).
% ulgraph_axioms
thf(fact_16_incident__def,axiom,
undire1521409233611534436dent_a = member_a ).
% incident_def
thf(fact_17_fin__edges,axiom,
finite_finite_set_a @ edges ).
% fin_edges
thf(fact_18_finV,axiom,
finite_finite_a @ vertices ).
% finV
thf(fact_19_vert__adj__edge__iff2,axiom,
! [V1: a,V2: a] :
( ( V1 != V2 )
=> ( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
= ( ? [X3: set_a] :
( ( member_set_a @ X3 @ edges )
& ( undire1521409233611534436dent_a @ V1 @ X3 )
& ( undire1521409233611534436dent_a @ V2 @ X3 ) ) ) ) ) ).
% vert_adj_edge_iff2
thf(fact_20_gen__paths__ss__walks,axiom,
ord_le8861187494160871172list_a @ ( undire6235733737954427521aths_a @ vertices @ edges ) @ ( undire3736599831911450577alks_a @ vertices @ edges ) ).
% gen_paths_ss_walks
thf(fact_21_paths__ss__walk,axiom,
ord_le8861187494160871172list_a @ ( undire1387732426225024653aths_a @ vertices @ edges ) @ ( undire3736599831911450577alks_a @ vertices @ edges ) ).
% paths_ss_walk
thf(fact_22_ulgraph_Ogen__paths_Ocong,axiom,
undire6235733737954427521aths_a = undire6235733737954427521aths_a ).
% ulgraph.gen_paths.cong
thf(fact_23_ulgraph_Opaths_Ocong,axiom,
undire1387732426225024653aths_a = undire1387732426225024653aths_a ).
% ulgraph.paths.cong
thf(fact_24_ulgraph_Ocycles__ss__gen__paths,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ord_le8861187494160871172list_a @ ( undire2685670433559799090cles_a @ Vertices @ Edges ) @ ( undire6235733737954427521aths_a @ Vertices @ Edges ) ) ) ).
% ulgraph.cycles_ss_gen_paths
thf(fact_25_fin__ulgraph_Ofinite__paths,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire7599193295422474955raph_a @ Vertices @ Edges )
=> ( finite_finite_list_a @ ( undire1387732426225024653aths_a @ Vertices @ Edges ) ) ) ).
% fin_ulgraph.finite_paths
thf(fact_26_finite__has__minimal2,axiom,
! [A: set_se5735800977113168103od_a_a,A2: set_Product_prod_a_a] :
( ( finite8717734299975451184od_a_a @ A )
=> ( ( member1816616512716248880od_a_a @ A2 @ A )
=> ? [X4: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X4 @ A )
& ( ord_le746702958409616551od_a_a @ X4 @ A2 )
& ! [Xa: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ Xa @ A )
=> ( ( ord_le746702958409616551od_a_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_27_finite__has__minimal2,axiom,
! [A: set_set_list_a,A2: set_list_a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( member_set_list_a @ A2 @ A )
=> ? [X4: set_list_a] :
( ( member_set_list_a @ X4 @ A )
& ( ord_le8861187494160871172list_a @ X4 @ A2 )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A )
=> ( ( ord_le8861187494160871172list_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_28_finite__has__minimal2,axiom,
! [A: set_set_set_a,A2: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( member_set_set_a @ A2 @ A )
=> ? [X4: set_set_a] :
( ( member_set_set_a @ X4 @ A )
& ( ord_le3724670747650509150_set_a @ X4 @ A2 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A )
=> ( ( ord_le3724670747650509150_set_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_29_finite__has__minimal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A )
& ( ord_less_eq_set_a @ X4 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_30_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_nat @ X4 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_31_finite__has__maximal2,axiom,
! [A: set_se5735800977113168103od_a_a,A2: set_Product_prod_a_a] :
( ( finite8717734299975451184od_a_a @ A )
=> ( ( member1816616512716248880od_a_a @ A2 @ A )
=> ? [X4: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X4 @ A )
& ( ord_le746702958409616551od_a_a @ A2 @ X4 )
& ! [Xa: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ Xa @ A )
=> ( ( ord_le746702958409616551od_a_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_32_finite__has__maximal2,axiom,
! [A: set_set_list_a,A2: set_list_a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( member_set_list_a @ A2 @ A )
=> ? [X4: set_list_a] :
( ( member_set_list_a @ X4 @ A )
& ( ord_le8861187494160871172list_a @ A2 @ X4 )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A )
=> ( ( ord_le8861187494160871172list_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_33_finite__has__maximal2,axiom,
! [A: set_set_set_a,A2: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( member_set_set_a @ A2 @ A )
=> ? [X4: set_set_a] :
( ( member_set_set_a @ X4 @ A )
& ( ord_le3724670747650509150_set_a @ A2 @ X4 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A )
=> ( ( ord_le3724670747650509150_set_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_34_finite__has__maximal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A )
& ( ord_less_eq_set_a @ A2 @ X4 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_35_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_nat @ A2 @ X4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_36_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_37_rev__finite__subset,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ B )
=> ( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( finite6544458595007987280od_a_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_38_rev__finite__subset,axiom,
! [B: set_list_a,A: set_list_a] :
( ( finite_finite_list_a @ B )
=> ( ( ord_le8861187494160871172list_a @ A @ B )
=> ( finite_finite_list_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_39_rev__finite__subset,axiom,
! [B: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( finite_finite_set_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_40_rev__finite__subset,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( finite_finite_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_41_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_42_infinite__super,axiom,
! [S: set_Product_prod_a_a,T: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ S @ T )
=> ( ~ ( finite6544458595007987280od_a_a @ S )
=> ~ ( finite6544458595007987280od_a_a @ T ) ) ) ).
% infinite_super
thf(fact_43_infinite__super,axiom,
! [S: set_list_a,T: set_list_a] :
( ( ord_le8861187494160871172list_a @ S @ T )
=> ( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ T ) ) ) ).
% infinite_super
thf(fact_44_infinite__super,axiom,
! [S: set_set_a,T: set_set_a] :
( ( ord_le3724670747650509150_set_a @ S @ T )
=> ( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ T ) ) ) ).
% infinite_super
thf(fact_45_infinite__super,axiom,
! [S: set_a,T: set_a] :
( ( ord_less_eq_set_a @ S @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T ) ) ) ).
% infinite_super
thf(fact_46_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_47_finite__subset,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( finite6544458595007987280od_a_a @ B )
=> ( finite6544458595007987280od_a_a @ A ) ) ) ).
% finite_subset
thf(fact_48_finite__subset,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( finite_finite_list_a @ B )
=> ( finite_finite_list_a @ A ) ) ) ).
% finite_subset
thf(fact_49_finite__subset,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( finite_finite_set_a @ B )
=> ( finite_finite_set_a @ A ) ) ) ).
% finite_subset
thf(fact_50_finite__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A ) ) ) ).
% finite_subset
thf(fact_51_is__isolated__vertex__no__loop,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
=> ~ ( undire3617971648856834880loop_a @ edges @ V ) ) ).
% is_isolated_vertex_no_loop
thf(fact_52_is__isolated__vertex__def,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
= ( ( member_a @ V @ vertices )
& ! [X3: a] :
( ( member_a @ X3 @ vertices )
=> ~ ( undire397441198561214472_adj_a @ edges @ X3 @ V ) ) ) ) ).
% is_isolated_vertex_def
thf(fact_53_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_54_is__subgraphI,axiom,
! [V3: set_Product_prod_a_a,V4: set_Product_prod_a_a,E3: set_se5735800977113168103od_a_a,E4: set_se5735800977113168103od_a_a] :
( ( ord_le746702958409616551od_a_a @ V3 @ V4 )
=> ( ( ord_le1995061765932249223od_a_a @ E3 @ E4 )
=> ( ( undire1860116983885411791od_a_a @ V3 @ E3 )
=> ( ( undire1860116983885411791od_a_a @ V4 @ E4 )
=> ( undire398746457437328754od_a_a @ V3 @ E3 @ V4 @ E4 ) ) ) ) ) ).
% is_subgraphI
thf(fact_55_is__subgraphI,axiom,
! [V3: set_list_a,V4: set_list_a,E3: set_set_list_a,E4: set_set_list_a] :
( ( ord_le8861187494160871172list_a @ V3 @ V4 )
=> ( ( ord_le8877086941679407844list_a @ E3 @ E4 )
=> ( ( undire5959234994740280364list_a @ V3 @ E3 )
=> ( ( undire5959234994740280364list_a @ V4 @ E4 )
=> ( undire761398192061991247list_a @ V3 @ E3 @ V4 @ E4 ) ) ) ) ) ).
% is_subgraphI
thf(fact_56_is__subgraphI,axiom,
! [V3: set_set_a,V4: set_set_a,E3: set_set_set_a,E4: set_set_set_a] :
( ( ord_le3724670747650509150_set_a @ V3 @ V4 )
=> ( ( ord_le5722252365846178494_set_a @ E3 @ E4 )
=> ( ( undire7159349782766787846_set_a @ V3 @ E3 )
=> ( ( undire7159349782766787846_set_a @ V4 @ E4 )
=> ( undire1186139521737116585_set_a @ V3 @ E3 @ V4 @ E4 ) ) ) ) ) ).
% is_subgraphI
thf(fact_57_is__subgraphI,axiom,
! [V3: set_a,V4: set_a,E3: set_set_a,E4: set_set_a] :
( ( ord_less_eq_set_a @ V3 @ V4 )
=> ( ( ord_le3724670747650509150_set_a @ E3 @ E4 )
=> ( ( undire2554140024507503526stem_a @ V3 @ E3 )
=> ( ( undire2554140024507503526stem_a @ V4 @ E4 )
=> ( undire7103218114511261257raph_a @ V3 @ E3 @ V4 @ E4 ) ) ) ) ) ).
% is_subgraphI
thf(fact_58_induced__is__subgraph,axiom,
! [V3: set_a] :
( ( ord_less_eq_set_a @ V3 @ vertices )
=> ( undire7103218114511261257raph_a @ V3 @ ( undire7777452895879145676dges_a @ edges @ V3 ) @ vertices @ edges ) ) ).
% induced_is_subgraph
thf(fact_59_fin__ulgraph_Ointro,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire945497512398942277stem_a @ Vertices @ Edges )
=> ( undire7599193295422474955raph_a @ Vertices @ Edges ) ) ) ).
% fin_ulgraph.intro
thf(fact_60_fin__ulgraph__def,axiom,
( undire7599193295422474955raph_a
= ( ^ [Vertices2: set_a,Edges2: set_set_a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
& ( undire945497512398942277stem_a @ Vertices2 @ Edges2 ) ) ) ) ).
% fin_ulgraph_def
thf(fact_61_ulgraph_Overt__adj__edge__iff2,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( V1 != V2 )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V2 )
= ( ? [X3: set_a] :
( ( member_set_a @ X3 @ Edges )
& ( undire1521409233611534436dent_a @ V1 @ X3 )
& ( undire1521409233611534436dent_a @ V2 @ X3 ) ) ) ) ) ) ).
% ulgraph.vert_adj_edge_iff2
thf(fact_62_finite__inc__sedges,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( finite_finite_set_a @ ( undire1270416042309875431dges_a @ edges @ V ) ) ) ).
% finite_inc_sedges
thf(fact_63_induced__is__graph__sys,axiom,
! [V3: set_a] : ( undire2554140024507503526stem_a @ V3 @ ( undire7777452895879145676dges_a @ edges @ V3 ) ) ).
% induced_is_graph_sys
thf(fact_64_fin__ulgraph_Oaxioms_I2_J,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire7599193295422474955raph_a @ Vertices @ Edges )
=> ( undire945497512398942277stem_a @ Vertices @ Edges ) ) ).
% fin_ulgraph.axioms(2)
thf(fact_65_mem__Collect__eq,axiom,
! [A2: list_a,P: list_a > $o] :
( ( member_list_a @ A2 @ ( collect_list_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_66_mem__Collect__eq,axiom,
! [A2: set_a,P: set_a > $o] :
( ( member_set_a @ A2 @ ( collect_set_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_67_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_68_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_69_mem__Collect__eq,axiom,
! [A2: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A2 @ ( collec3336397797384452498od_a_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_70_Collect__mem__eq,axiom,
! [A: set_list_a] :
( ( collect_list_a
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_71_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_72_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_73_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_74_Collect__mem__eq,axiom,
! [A: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_75_Collect__cong,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ! [X4: list_a] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_list_a @ P )
= ( collect_list_a @ Q ) ) ) ).
% Collect_cong
thf(fact_76_Collect__cong,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X4: set_a] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_set_a @ P )
= ( collect_set_a @ Q ) ) ) ).
% Collect_cong
thf(fact_77_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_78_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_79_Collect__cong,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X4: product_prod_a_a] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collec3336397797384452498od_a_a @ P )
= ( collec3336397797384452498od_a_a @ Q ) ) ) ).
% Collect_cong
thf(fact_80_finite__incident__loops,axiom,
! [V: a] : ( finite_finite_set_a @ ( undire4753905205749729249oops_a @ edges @ V ) ) ).
% finite_incident_loops
thf(fact_81_induced__edges__ss,axiom,
! [V3: set_a] :
( ( ord_less_eq_set_a @ V3 @ vertices )
=> ( ord_le3724670747650509150_set_a @ ( undire7777452895879145676dges_a @ edges @ V3 ) @ edges ) ) ).
% induced_edges_ss
thf(fact_82_ulgraph_Ois__isolated__vertex_Ocong,axiom,
undire8931668460104145173rtex_a = undire8931668460104145173rtex_a ).
% ulgraph.is_isolated_vertex.cong
thf(fact_83_graph__system_Oinduced__edges_Ocong,axiom,
undire7777452895879145676dges_a = undire7777452895879145676dges_a ).
% graph_system.induced_edges.cong
thf(fact_84_ulgraph_Oincident__sedges_Ocong,axiom,
undire1270416042309875431dges_a = undire1270416042309875431dges_a ).
% ulgraph.incident_sedges.cong
thf(fact_85_ulgraph_Oincident__loops_Ocong,axiom,
undire4753905205749729249oops_a = undire4753905205749729249oops_a ).
% ulgraph.incident_loops.cong
thf(fact_86_ulgraph_Owalks_Ocong,axiom,
undire3736599831911450577alks_a = undire3736599831911450577alks_a ).
% ulgraph.walks.cong
thf(fact_87_subgraph_Oedges__ss,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ord_le3724670747650509150_set_a @ E_H @ E_G ) ) ).
% subgraph.edges_ss
thf(fact_88_graph__system_Oinduced__is__graph__sys,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( undire2554140024507503526stem_a @ V3 @ ( undire7777452895879145676dges_a @ Edges @ V3 ) ) ) ).
% graph_system.induced_is_graph_sys
thf(fact_89_ulgraph_Ofinite__incident__loops,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( finite_finite_set_a @ ( undire4753905205749729249oops_a @ Edges @ V ) ) ) ).
% ulgraph.finite_incident_loops
thf(fact_90_ulgraph_Ofinite__inc__sedges,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( finite_finite_set_a @ Edges )
=> ( finite_finite_set_a @ ( undire1270416042309875431dges_a @ Edges @ V ) ) ) ) ).
% ulgraph.finite_inc_sedges
thf(fact_91_graph__system_Oinduced__edges__ss,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V3: set_Product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( ord_le746702958409616551od_a_a @ V3 @ Vertices )
=> ( ord_le1995061765932249223od_a_a @ ( undire5906991851038061813od_a_a @ Edges @ V3 ) @ Edges ) ) ) ).
% graph_system.induced_edges_ss
thf(fact_92_graph__system_Oinduced__edges__ss,axiom,
! [Vertices: set_list_a,Edges: set_set_list_a,V3: set_list_a] :
( ( undire5959234994740280364list_a @ Vertices @ Edges )
=> ( ( ord_le8861187494160871172list_a @ V3 @ Vertices )
=> ( ord_le8877086941679407844list_a @ ( undire8521487854958249554list_a @ Edges @ V3 ) @ Edges ) ) ) ).
% graph_system.induced_edges_ss
thf(fact_93_graph__system_Oinduced__edges__ss,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V3: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( ord_le3724670747650509150_set_a @ V3 @ Vertices )
=> ( ord_le5722252365846178494_set_a @ ( undire7854589003810675244_set_a @ Edges @ V3 ) @ Edges ) ) ) ).
% graph_system.induced_edges_ss
thf(fact_94_graph__system_Oinduced__edges__ss,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( ord_less_eq_set_a @ V3 @ Vertices )
=> ( ord_le3724670747650509150_set_a @ ( undire7777452895879145676dges_a @ Edges @ V3 ) @ Edges ) ) ) ).
% graph_system.induced_edges_ss
thf(fact_95_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire6879241558604981877_set_a @ Vertices @ Edges @ V )
= ( ( member_set_a @ V @ Vertices )
& ! [X3: set_a] :
( ( member_set_a @ X3 @ Vertices )
=> ~ ( undire3510646817838285160_set_a @ Edges @ X3 @ V ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_96_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire5609513041723151865ex_nat @ Vertices @ Edges @ V )
= ( ( member_nat @ V @ Vertices )
& ! [X3: nat] :
( ( member_nat @ X3 @ Vertices )
=> ~ ( undire1083030068171319366dj_nat @ Edges @ X3 @ V ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_97_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire3207556238582723646od_a_a @ Vertices @ Edges @ V )
= ( ( member1426531477525435216od_a_a @ V @ Vertices )
& ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ Vertices )
=> ~ ( undire6135774327024169009od_a_a @ Edges @ X3 @ V ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_98_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V )
= ( ( member_a @ V @ Vertices )
& ! [X3: a] :
( ( member_a @ X3 @ Vertices )
=> ~ ( undire397441198561214472_adj_a @ Edges @ X3 @ V ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_99_ulgraph_Ois__isolated__vertex__edge,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a,E: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V )
=> ( ( member_set_a @ E @ Edges )
=> ~ ( undire1521409233611534436dent_a @ V @ E ) ) ) ) ).
% ulgraph.is_isolated_vertex_edge
thf(fact_100_ulgraph_Ois__isolated__vertex__no__loop,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V )
=> ~ ( undire3617971648856834880loop_a @ Edges @ V ) ) ) ).
% ulgraph.is_isolated_vertex_no_loop
thf(fact_101_graph__system_Oinduced__is__subgraph,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V3: set_Product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( ord_le746702958409616551od_a_a @ V3 @ Vertices )
=> ( undire398746457437328754od_a_a @ V3 @ ( undire5906991851038061813od_a_a @ Edges @ V3 ) @ Vertices @ Edges ) ) ) ).
% graph_system.induced_is_subgraph
thf(fact_102_graph__system_Oinduced__is__subgraph,axiom,
! [Vertices: set_list_a,Edges: set_set_list_a,V3: set_list_a] :
( ( undire5959234994740280364list_a @ Vertices @ Edges )
=> ( ( ord_le8861187494160871172list_a @ V3 @ Vertices )
=> ( undire761398192061991247list_a @ V3 @ ( undire8521487854958249554list_a @ Edges @ V3 ) @ Vertices @ Edges ) ) ) ).
% graph_system.induced_is_subgraph
thf(fact_103_graph__system_Oinduced__is__subgraph,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V3: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( ord_le3724670747650509150_set_a @ V3 @ Vertices )
=> ( undire1186139521737116585_set_a @ V3 @ ( undire7854589003810675244_set_a @ Edges @ V3 ) @ Vertices @ Edges ) ) ) ).
% graph_system.induced_is_subgraph
thf(fact_104_graph__system_Oinduced__is__subgraph,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( ord_less_eq_set_a @ V3 @ Vertices )
=> ( undire7103218114511261257raph_a @ V3 @ ( undire7777452895879145676dges_a @ Edges @ V3 ) @ Vertices @ Edges ) ) ) ).
% graph_system.induced_is_subgraph
thf(fact_105_subgraph_Osubgraph__antisym,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a,V3: set_a,E3: set_set_a,V4: set_a,E4: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ( undire7103218114511261257raph_a @ V3 @ E3 @ V4 @ E4 )
=> ( ( undire7103218114511261257raph_a @ V4 @ E4 @ V3 @ E3 )
=> ( ( V4 = V3 )
& ( E4 = E3 ) ) ) ) ) ).
% subgraph.subgraph_antisym
thf(fact_106_ulgraph_Overt__adj_Ocong,axiom,
undire397441198561214472_adj_a = undire397441198561214472_adj_a ).
% ulgraph.vert_adj.cong
thf(fact_107_comp__sgraph_Oincident__def,axiom,
undire2320338297334612420_set_a = member_set_a ).
% comp_sgraph.incident_def
thf(fact_108_comp__sgraph_Oincident__def,axiom,
undire7858122600432113898nt_nat = member_nat ).
% comp_sgraph.incident_def
thf(fact_109_comp__sgraph_Oincident__def,axiom,
undire3369688177417741453od_a_a = member1426531477525435216od_a_a ).
% comp_sgraph.incident_def
thf(fact_110_comp__sgraph_Oincident__def,axiom,
undire1521409233611534436dent_a = member_a ).
% comp_sgraph.incident_def
thf(fact_111_ulgraph_Oedge__density_Ocong,axiom,
undire297304480579013331sity_a = undire297304480579013331sity_a ).
% ulgraph.edge_density.cong
thf(fact_112_ulgraph_Ohas__loop_Ocong,axiom,
undire3617971648856834880loop_a = undire3617971648856834880loop_a ).
% ulgraph.has_loop.cong
thf(fact_113_fin__graph__system_Ofin__edges,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire945497512398942277stem_a @ Vertices @ Edges )
=> ( finite_finite_set_a @ Edges ) ) ).
% fin_graph_system.fin_edges
thf(fact_114_graph__system_Oedge__adj_Ocong,axiom,
undire4022703626023482010_adj_a = undire4022703626023482010_adj_a ).
% graph_system.edge_adj.cong
thf(fact_115_graph__system__def,axiom,
( undire1860116983885411791od_a_a
= ( ^ [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a] :
! [E5: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ E5 @ Edges2 )
=> ( ord_le746702958409616551od_a_a @ E5 @ Vertices2 ) ) ) ) ).
% graph_system_def
thf(fact_116_graph__system__def,axiom,
( undire5959234994740280364list_a
= ( ^ [Vertices2: set_list_a,Edges2: set_set_list_a] :
! [E5: set_list_a] :
( ( member_set_list_a @ E5 @ Edges2 )
=> ( ord_le8861187494160871172list_a @ E5 @ Vertices2 ) ) ) ) ).
% graph_system_def
thf(fact_117_graph__system__def,axiom,
( undire7159349782766787846_set_a
= ( ^ [Vertices2: set_set_a,Edges2: set_set_set_a] :
! [E5: set_set_a] :
( ( member_set_set_a @ E5 @ Edges2 )
=> ( ord_le3724670747650509150_set_a @ E5 @ Vertices2 ) ) ) ) ).
% graph_system_def
thf(fact_118_graph__system__def,axiom,
( undire2554140024507503526stem_a
= ( ^ [Vertices2: set_a,Edges2: set_set_a] :
! [E5: set_a] :
( ( member_set_a @ E5 @ Edges2 )
=> ( ord_less_eq_set_a @ E5 @ Vertices2 ) ) ) ) ).
% graph_system_def
thf(fact_119_graph__system_Owellformed,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,E: set_Product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( member1816616512716248880od_a_a @ E @ Edges )
=> ( ord_le746702958409616551od_a_a @ E @ Vertices ) ) ) ).
% graph_system.wellformed
thf(fact_120_graph__system_Owellformed,axiom,
! [Vertices: set_list_a,Edges: set_set_list_a,E: set_list_a] :
( ( undire5959234994740280364list_a @ Vertices @ Edges )
=> ( ( member_set_list_a @ E @ Edges )
=> ( ord_le8861187494160871172list_a @ E @ Vertices ) ) ) ).
% graph_system.wellformed
thf(fact_121_graph__system_Owellformed,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,E: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( member_set_set_a @ E @ Edges )
=> ( ord_le3724670747650509150_set_a @ E @ Vertices ) ) ) ).
% graph_system.wellformed
thf(fact_122_graph__system_Owellformed,axiom,
! [Vertices: set_a,Edges: set_set_a,E: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( member_set_a @ E @ Edges )
=> ( ord_less_eq_set_a @ E @ Vertices ) ) ) ).
% graph_system.wellformed
thf(fact_123_graph__system_Ointro,axiom,
! [Edges: set_se5735800977113168103od_a_a,Vertices: set_Product_prod_a_a] :
( ! [E6: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ E6 @ Edges )
=> ( ord_le746702958409616551od_a_a @ E6 @ Vertices ) )
=> ( undire1860116983885411791od_a_a @ Vertices @ Edges ) ) ).
% graph_system.intro
thf(fact_124_graph__system_Ointro,axiom,
! [Edges: set_set_list_a,Vertices: set_list_a] :
( ! [E6: set_list_a] :
( ( member_set_list_a @ E6 @ Edges )
=> ( ord_le8861187494160871172list_a @ E6 @ Vertices ) )
=> ( undire5959234994740280364list_a @ Vertices @ Edges ) ) ).
% graph_system.intro
thf(fact_125_graph__system_Ointro,axiom,
! [Edges: set_set_set_a,Vertices: set_set_a] :
( ! [E6: set_set_a] :
( ( member_set_set_a @ E6 @ Edges )
=> ( ord_le3724670747650509150_set_a @ E6 @ Vertices ) )
=> ( undire7159349782766787846_set_a @ Vertices @ Edges ) ) ).
% graph_system.intro
thf(fact_126_graph__system_Ointro,axiom,
! [Edges: set_set_a,Vertices: set_a] :
( ! [E6: set_a] :
( ( member_set_a @ E6 @ Edges )
=> ( ord_less_eq_set_a @ E6 @ Vertices ) )
=> ( undire2554140024507503526stem_a @ Vertices @ Edges ) ) ).
% graph_system.intro
thf(fact_127_ulgraph_Ogen__paths__ss__walks,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ord_le8861187494160871172list_a @ ( undire6235733737954427521aths_a @ Vertices @ Edges ) @ ( undire3736599831911450577alks_a @ Vertices @ Edges ) ) ) ).
% ulgraph.gen_paths_ss_walks
thf(fact_128_subgraph_Overts__ss,axiom,
! [V_H: set_Product_prod_a_a,E_H: set_se5735800977113168103od_a_a,V_G: set_Product_prod_a_a,E_G: set_se5735800977113168103od_a_a] :
( ( undire398746457437328754od_a_a @ V_H @ E_H @ V_G @ E_G )
=> ( ord_le746702958409616551od_a_a @ V_H @ V_G ) ) ).
% subgraph.verts_ss
thf(fact_129_subgraph_Overts__ss,axiom,
! [V_H: set_list_a,E_H: set_set_list_a,V_G: set_list_a,E_G: set_set_list_a] :
( ( undire761398192061991247list_a @ V_H @ E_H @ V_G @ E_G )
=> ( ord_le8861187494160871172list_a @ V_H @ V_G ) ) ).
% subgraph.verts_ss
thf(fact_130_subgraph_Overts__ss,axiom,
! [V_H: set_set_a,E_H: set_set_set_a,V_G: set_set_a,E_G: set_set_set_a] :
( ( undire1186139521737116585_set_a @ V_H @ E_H @ V_G @ E_G )
=> ( ord_le3724670747650509150_set_a @ V_H @ V_G ) ) ).
% subgraph.verts_ss
thf(fact_131_subgraph_Overts__ss,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ord_less_eq_set_a @ V_H @ V_G ) ) ).
% subgraph.verts_ss
thf(fact_132_ulgraph_Opaths__ss__walk,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ord_le8861187494160871172list_a @ ( undire1387732426225024653aths_a @ Vertices @ Edges ) @ ( undire3736599831911450577alks_a @ Vertices @ Edges ) ) ) ).
% ulgraph.paths_ss_walk
thf(fact_133_ulgraph_Oaxioms_I1_J,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( undire2554140024507503526stem_a @ Vertices @ Edges ) ) ).
% ulgraph.axioms(1)
thf(fact_134_subgraph_Ois__subgraph__ulgraph,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ( undire7251896706689453996raph_a @ V_G @ E_G )
=> ( undire7251896706689453996raph_a @ V_H @ E_H ) ) ) ).
% subgraph.is_subgraph_ulgraph
thf(fact_135_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V1: set_a,V2: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3510646817838285160_set_a @ Edges @ V1 @ V2 )
=> ( ( member_set_a @ V1 @ Vertices )
& ( member_set_a @ V2 @ Vertices ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_136_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V1: nat,V2: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire1083030068171319366dj_nat @ Edges @ V1 @ V2 )
=> ( ( member_nat @ V1 @ Vertices )
& ( member_nat @ V2 @ Vertices ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_137_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V1: product_prod_a_a,V2: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire6135774327024169009od_a_a @ Edges @ V1 @ V2 )
=> ( ( member1426531477525435216od_a_a @ V1 @ Vertices )
& ( member1426531477525435216od_a_a @ V2 @ Vertices ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_138_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V2 )
=> ( ( member_a @ V1 @ Vertices )
& ( member_a @ V2 @ Vertices ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_139_ulgraph_Overt__adj__sym,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V2 )
= ( undire397441198561214472_adj_a @ Edges @ V2 @ V1 ) ) ) ).
% ulgraph.vert_adj_sym
thf(fact_140_ulgraph_Oedge__density__commute,axiom,
! [Vertices: set_a,Edges: set_set_a,X2: set_a,Y: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire297304480579013331sity_a @ Edges @ X2 @ Y )
= ( undire297304480579013331sity_a @ Edges @ Y @ X2 ) ) ) ).
% ulgraph.edge_density_commute
thf(fact_141_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire5774735625301615776_set_a @ Edges @ V )
=> ( member_set_a @ V @ Vertices ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_142_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire5005864372999571214op_nat @ Edges @ V )
=> ( member_nat @ V @ Vertices ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_143_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire7777398424729533289od_a_a @ Edges @ V )
=> ( member1426531477525435216od_a_a @ V @ Vertices ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_144_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3617971648856834880loop_a @ Edges @ V )
=> ( member_a @ V @ Vertices ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_145_fin__graph__system_OfinV,axiom,
! [Vertices: set_list_a,Edges: set_set_list_a] :
( ( undire8457811296688094539list_a @ Vertices @ Edges )
=> ( finite_finite_list_a @ Vertices ) ) ).
% fin_graph_system.finV
thf(fact_146_fin__graph__system_OfinV,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a] :
( ( undire1228946095791996325_set_a @ Vertices @ Edges )
=> ( finite_finite_set_a @ Vertices ) ) ).
% fin_graph_system.finV
thf(fact_147_fin__graph__system_OfinV,axiom,
! [Vertices: set_nat,Edges: set_set_nat] :
( ( undire6162778924909679305em_nat @ Vertices @ Edges )
=> ( finite_finite_nat @ Vertices ) ) ).
% fin_graph_system.finV
thf(fact_148_fin__graph__system_OfinV,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a] :
( ( undire8735137233620540270od_a_a @ Vertices @ Edges )
=> ( finite6544458595007987280od_a_a @ Vertices ) ) ).
% fin_graph_system.finV
thf(fact_149_fin__graph__system_OfinV,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire945497512398942277stem_a @ Vertices @ Edges )
=> ( finite_finite_a @ Vertices ) ) ).
% fin_graph_system.finV
thf(fact_150_graph__system_Osubgraph__refl,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( undire7103218114511261257raph_a @ Vertices @ Edges @ Vertices @ Edges ) ) ).
% graph_system.subgraph_refl
thf(fact_151_subgraph_Osubgraph__trans,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a,V4: set_a,E4: set_set_a,V3: set_a,E3: set_set_a,V5: set_a,E7: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ( undire2554140024507503526stem_a @ V4 @ E4 )
=> ( ( undire2554140024507503526stem_a @ V3 @ E3 )
=> ( ( undire2554140024507503526stem_a @ V5 @ E7 )
=> ( ( undire7103218114511261257raph_a @ V5 @ E7 @ V3 @ E3 )
=> ( ( undire7103218114511261257raph_a @ V3 @ E3 @ V4 @ E4 )
=> ( undire7103218114511261257raph_a @ V5 @ E7 @ V4 @ E4 ) ) ) ) ) ) ) ).
% subgraph.subgraph_trans
thf(fact_152_subgraph_Oaxioms_I1_J,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( undire2554140024507503526stem_a @ V_H @ E_H ) ) ).
% subgraph.axioms(1)
thf(fact_153_subgraph_Oaxioms_I2_J,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( undire2554140024507503526stem_a @ V_G @ E_G ) ) ).
% subgraph.axioms(2)
thf(fact_154_graph__system_Oincident__edge__in__wf,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,E: set_set_a,V: set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( member_set_set_a @ E @ Edges )
=> ( ( undire2320338297334612420_set_a @ V @ E )
=> ( member_set_a @ V @ Vertices ) ) ) ) ).
% graph_system.incident_edge_in_wf
thf(fact_155_graph__system_Oincident__edge__in__wf,axiom,
! [Vertices: set_nat,Edges: set_set_nat,E: set_nat,V: nat] :
( ( undire7481384412329822504em_nat @ Vertices @ Edges )
=> ( ( member_set_nat @ E @ Edges )
=> ( ( undire7858122600432113898nt_nat @ V @ E )
=> ( member_nat @ V @ Vertices ) ) ) ) ).
% graph_system.incident_edge_in_wf
thf(fact_156_graph__system_Oincident__edge__in__wf,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,E: set_Product_prod_a_a,V: product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( member1816616512716248880od_a_a @ E @ Edges )
=> ( ( undire3369688177417741453od_a_a @ V @ E )
=> ( member1426531477525435216od_a_a @ V @ Vertices ) ) ) ) ).
% graph_system.incident_edge_in_wf
thf(fact_157_graph__system_Oincident__edge__in__wf,axiom,
! [Vertices: set_a,Edges: set_set_a,E: set_a,V: a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( member_set_a @ E @ Edges )
=> ( ( undire1521409233611534436dent_a @ V @ E )
=> ( member_a @ V @ Vertices ) ) ) ) ).
% graph_system.incident_edge_in_wf
thf(fact_158_graph__system_Oincident__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_a,E: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( undire2320338297334612420_set_a @ V @ E )
= ( member_set_a @ V @ E ) ) ) ).
% graph_system.incident_def
thf(fact_159_graph__system_Oincident__def,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V: nat,E: set_nat] :
( ( undire7481384412329822504em_nat @ Vertices @ Edges )
=> ( ( undire7858122600432113898nt_nat @ V @ E )
= ( member_nat @ V @ E ) ) ) ).
% graph_system.incident_def
thf(fact_160_graph__system_Oincident__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V: product_prod_a_a,E: set_Product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( undire3369688177417741453od_a_a @ V @ E )
= ( member1426531477525435216od_a_a @ V @ E ) ) ) ).
% graph_system.incident_def
thf(fact_161_graph__system_Oincident__def,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a,E: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( undire1521409233611534436dent_a @ V @ E )
= ( member_a @ V @ E ) ) ) ).
% graph_system.incident_def
thf(fact_162_fin__graph__system_Oaxioms_I1_J,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire945497512398942277stem_a @ Vertices @ Edges )
=> ( undire2554140024507503526stem_a @ Vertices @ Edges ) ) ).
% fin_graph_system.axioms(1)
thf(fact_163_subgraph_Ois__finite__subgraph,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ( undire945497512398942277stem_a @ V_G @ E_G )
=> ( undire945497512398942277stem_a @ V_H @ E_H ) ) ) ).
% subgraph.is_finite_subgraph
thf(fact_164_fin__ulgraph_Oaxioms_I1_J,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire7599193295422474955raph_a @ Vertices @ Edges )
=> ( undire7251896706689453996raph_a @ Vertices @ Edges ) ) ).
% fin_ulgraph.axioms(1)
thf(fact_165_graph__system_Oedge__adjacent__alt__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,E1: set_set_a,E2: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( member_set_set_a @ E1 @ Edges )
=> ( ( member_set_set_a @ E2 @ Edges )
=> ( ? [X: set_a] :
( ( member_set_a @ X @ Vertices )
& ( member_set_a @ X @ E1 )
& ( member_set_a @ X @ E2 ) )
=> ( undire3485422320110889978_set_a @ Edges @ E1 @ E2 ) ) ) ) ) ).
% graph_system.edge_adjacent_alt_def
thf(fact_166_graph__system_Oedge__adjacent__alt__def,axiom,
! [Vertices: set_nat,Edges: set_set_nat,E1: set_nat,E2: set_nat] :
( ( undire7481384412329822504em_nat @ Vertices @ Edges )
=> ( ( member_set_nat @ E1 @ Edges )
=> ( ( member_set_nat @ E2 @ Edges )
=> ( ? [X: nat] :
( ( member_nat @ X @ Vertices )
& ( member_nat @ X @ E1 )
& ( member_nat @ X @ E2 ) )
=> ( undire1664191744716346676dj_nat @ Edges @ E1 @ E2 ) ) ) ) ) ).
% graph_system.edge_adjacent_alt_def
thf(fact_167_graph__system_Oedge__adjacent__alt__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,E1: set_Product_prod_a_a,E2: set_Product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( member1816616512716248880od_a_a @ E1 @ Edges )
=> ( ( member1816616512716248880od_a_a @ E2 @ Edges )
=> ( ? [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ Vertices )
& ( member1426531477525435216od_a_a @ X @ E1 )
& ( member1426531477525435216od_a_a @ X @ E2 ) )
=> ( undire9186443406341554371od_a_a @ Edges @ E1 @ E2 ) ) ) ) ) ).
% graph_system.edge_adjacent_alt_def
thf(fact_168_graph__system_Oedge__adjacent__alt__def,axiom,
! [Vertices: set_a,Edges: set_set_a,E1: set_a,E2: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( member_set_a @ E1 @ Edges )
=> ( ( member_set_a @ E2 @ Edges )
=> ( ? [X: a] :
( ( member_a @ X @ Vertices )
& ( member_a @ X @ E1 )
& ( member_a @ X @ E2 ) )
=> ( undire4022703626023482010_adj_a @ Edges @ E1 @ E2 ) ) ) ) ) ).
% graph_system.edge_adjacent_alt_def
thf(fact_169_graph__system_Oedge__adj__inE,axiom,
! [Vertices: set_a,Edges: set_set_a,E1: set_a,E2: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( undire4022703626023482010_adj_a @ Edges @ E1 @ E2 )
=> ( ( member_set_a @ E1 @ Edges )
& ( member_set_a @ E2 @ Edges ) ) ) ) ).
% graph_system.edge_adj_inE
thf(fact_170_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_171_finite__incident__edges,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( finite_finite_set_a @ ( undire3231912044278729248dges_a @ edges @ V ) ) ) ).
% finite_incident_edges
thf(fact_172_edge__density__ge0,axiom,
! [X2: set_a,Y: set_a] : ( ord_less_eq_real @ zero_zero_real @ ( undire297304480579013331sity_a @ edges @ X2 @ Y ) ) ).
% edge_density_ge0
thf(fact_173_edge__density__le1,axiom,
! [X2: set_a,Y: set_a] : ( ord_less_eq_real @ ( undire297304480579013331sity_a @ edges @ X2 @ Y ) @ one_one_real ) ).
% edge_density_le1
thf(fact_174_card__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).
% card_mono
thf(fact_175_card__mono,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ B )
=> ( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ A ) @ ( finite4795055649997197647od_a_a @ B ) ) ) ) ).
% card_mono
thf(fact_176_card__mono,axiom,
! [B: set_list_a,A: set_list_a] :
( ( finite_finite_list_a @ B )
=> ( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_list_a @ A ) @ ( finite_card_list_a @ B ) ) ) ) ).
% card_mono
thf(fact_177_card__mono,axiom,
! [B: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ A ) @ ( finite_card_set_a @ B ) ) ) ) ).
% card_mono
thf(fact_178_card__mono,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ).
% card_mono
thf(fact_179_card__seteq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_180_card__seteq,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ B )
=> ( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ B ) @ ( finite4795055649997197647od_a_a @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_181_card__seteq,axiom,
! [B: set_list_a,A: set_list_a] :
( ( finite_finite_list_a @ B )
=> ( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_list_a @ B ) @ ( finite_card_list_a @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_182_card__seteq,axiom,
! [B: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ B ) @ ( finite_card_set_a @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_183_card__seteq,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_184_exists__subset__between,axiom,
! [A: set_nat,N: nat,C: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C ) )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ( finite_finite_nat @ C )
=> ? [B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
& ( ord_less_eq_set_nat @ B2 @ C )
& ( ( finite_card_nat @ B2 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_185_exists__subset__between,axiom,
! [A: set_Product_prod_a_a,N: nat,C: set_Product_prod_a_a] :
( ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite4795055649997197647od_a_a @ C ) )
=> ( ( ord_le746702958409616551od_a_a @ A @ C )
=> ( ( finite6544458595007987280od_a_a @ C )
=> ? [B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B2 )
& ( ord_le746702958409616551od_a_a @ B2 @ C )
& ( ( finite4795055649997197647od_a_a @ B2 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_186_exists__subset__between,axiom,
! [A: set_list_a,N: nat,C: set_list_a] :
( ( ord_less_eq_nat @ ( finite_card_list_a @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_list_a @ C ) )
=> ( ( ord_le8861187494160871172list_a @ A @ C )
=> ( ( finite_finite_list_a @ C )
=> ? [B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
& ( ord_le8861187494160871172list_a @ B2 @ C )
& ( ( finite_card_list_a @ B2 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_187_exists__subset__between,axiom,
! [A: set_set_a,N: nat,C: set_set_a] :
( ( ord_less_eq_nat @ ( finite_card_set_a @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_set_a @ C ) )
=> ( ( ord_le3724670747650509150_set_a @ A @ C )
=> ( ( finite_finite_set_a @ C )
=> ? [B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
& ( ord_le3724670747650509150_set_a @ B2 @ C )
& ( ( finite_card_set_a @ B2 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_188_exists__subset__between,axiom,
! [A: set_a,N: nat,C: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_a @ C ) )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ( finite_finite_a @ C )
=> ? [B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
& ( ord_less_eq_set_a @ B2 @ C )
& ( ( finite_card_a @ B2 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_189_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T2: set_nat] :
( ( ord_less_eq_set_nat @ T2 @ S )
=> ( ( ( finite_card_nat @ T2 )
= N )
=> ~ ( finite_finite_nat @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_190_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_Product_prod_a_a] :
( ( ord_less_eq_nat @ N @ ( finite4795055649997197647od_a_a @ S ) )
=> ~ ! [T2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ T2 @ S )
=> ( ( ( finite4795055649997197647od_a_a @ T2 )
= N )
=> ~ ( finite6544458595007987280od_a_a @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_191_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_list_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_list_a @ S ) )
=> ~ ! [T2: set_list_a] :
( ( ord_le8861187494160871172list_a @ T2 @ S )
=> ( ( ( finite_card_list_a @ T2 )
= N )
=> ~ ( finite_finite_list_a @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_192_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_set_a @ S ) )
=> ~ ! [T2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ T2 @ S )
=> ( ( ( finite_card_set_a @ T2 )
= N )
=> ~ ( finite_finite_set_a @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_193_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ S ) )
=> ~ ! [T2: set_a] :
( ( ord_less_eq_set_a @ T2 @ S )
=> ( ( ( finite_card_a @ T2 )
= N )
=> ~ ( finite_finite_a @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_194_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_nat,C: nat] :
( ! [G: set_nat] :
( ( ord_less_eq_set_nat @ G @ F )
=> ( ( finite_finite_nat @ G )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C ) ) )
=> ( ( finite_finite_nat @ F )
& ( ord_less_eq_nat @ ( finite_card_nat @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_195_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_Product_prod_a_a,C: nat] :
( ! [G: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ G @ F )
=> ( ( finite6544458595007987280od_a_a @ G )
=> ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ G ) @ C ) ) )
=> ( ( finite6544458595007987280od_a_a @ F )
& ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_196_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_list_a,C: nat] :
( ! [G: set_list_a] :
( ( ord_le8861187494160871172list_a @ G @ F )
=> ( ( finite_finite_list_a @ G )
=> ( ord_less_eq_nat @ ( finite_card_list_a @ G ) @ C ) ) )
=> ( ( finite_finite_list_a @ F )
& ( ord_less_eq_nat @ ( finite_card_list_a @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_197_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_set_a,C: nat] :
( ! [G: set_set_a] :
( ( ord_le3724670747650509150_set_a @ G @ F )
=> ( ( finite_finite_set_a @ G )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ G ) @ C ) ) )
=> ( ( finite_finite_set_a @ F )
& ( ord_less_eq_nat @ ( finite_card_set_a @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_198_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_a,C: nat] :
( ! [G: set_a] :
( ( ord_less_eq_set_a @ G @ F )
=> ( ( finite_finite_a @ G )
=> ( ord_less_eq_nat @ ( finite_card_a @ G ) @ C ) ) )
=> ( ( finite_finite_a @ F )
& ( ord_less_eq_nat @ ( finite_card_a @ F ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_199_card__subset__eq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_200_card__subset__eq,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ B )
=> ( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( ( finite4795055649997197647od_a_a @ A )
= ( finite4795055649997197647od_a_a @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_201_card__subset__eq,axiom,
! [B: set_list_a,A: set_list_a] :
( ( finite_finite_list_a @ B )
=> ( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ( finite_card_list_a @ A )
= ( finite_card_list_a @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_202_card__subset__eq,axiom,
! [B: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ( finite_card_set_a @ A )
= ( finite_card_set_a @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_203_card__subset__eq,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( finite_card_a @ A )
= ( finite_card_a @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_204_infinite__arbitrarily__large,axiom,
! [A: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ( finite_card_nat @ B2 )
= N )
& ( ord_less_eq_set_nat @ B2 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_205_infinite__arbitrarily__large,axiom,
! [A: set_Product_prod_a_a,N: nat] :
( ~ ( finite6544458595007987280od_a_a @ A )
=> ? [B2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ B2 )
& ( ( finite4795055649997197647od_a_a @ B2 )
= N )
& ( ord_le746702958409616551od_a_a @ B2 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_206_infinite__arbitrarily__large,axiom,
! [A: set_list_a,N: nat] :
( ~ ( finite_finite_list_a @ A )
=> ? [B2: set_list_a] :
( ( finite_finite_list_a @ B2 )
& ( ( finite_card_list_a @ B2 )
= N )
& ( ord_le8861187494160871172list_a @ B2 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_207_infinite__arbitrarily__large,axiom,
! [A: set_set_a,N: nat] :
( ~ ( finite_finite_set_a @ A )
=> ? [B2: set_set_a] :
( ( finite_finite_set_a @ B2 )
& ( ( finite_card_set_a @ B2 )
= N )
& ( ord_le3724670747650509150_set_a @ B2 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_208_infinite__arbitrarily__large,axiom,
! [A: set_a,N: nat] :
( ~ ( finite_finite_a @ A )
=> ? [B2: set_a] :
( ( finite_finite_a @ B2 )
& ( ( finite_card_a @ B2 )
= N )
& ( ord_less_eq_set_a @ B2 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_209_incident__edges__empty,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire3231912044278729248dges_a @ edges @ V )
= bot_bot_set_set_a ) ) ).
% incident_edges_empty
thf(fact_210_incident__edges__sedges,axiom,
! [V: a] :
( ~ ( undire3617971648856834880loop_a @ edges @ V )
=> ( ( undire3231912044278729248dges_a @ edges @ V )
= ( undire1270416042309875431dges_a @ edges @ V ) ) ) ).
% incident_edges_sedges
thf(fact_211_incident__sedges__empty,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire1270416042309875431dges_a @ edges @ V )
= bot_bot_set_set_a ) ) ).
% incident_sedges_empty
thf(fact_212_graph__system_Oincident__edges_Ocong,axiom,
undire3231912044278729248dges_a = undire3231912044278729248dges_a ).
% graph_system.incident_edges.cong
thf(fact_213_graph__system_Oincident__edges__empty,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ~ ( member_set_a @ V @ Vertices )
=> ( ( undire4631953023069350784_set_a @ Edges @ V )
= bot_bo3380559777022489994_set_a ) ) ) ).
% graph_system.incident_edges_empty
thf(fact_214_graph__system_Oincident__edges__empty,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V: nat] :
( ( undire7481384412329822504em_nat @ Vertices @ Edges )
=> ( ~ ( member_nat @ V @ Vertices )
=> ( ( undire4176300566717384750es_nat @ Edges @ V )
= bot_bot_set_set_nat ) ) ) ).
% graph_system.incident_edges_empty
thf(fact_215_graph__system_Oincident__edges__empty,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V: product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ~ ( member1426531477525435216od_a_a @ V @ Vertices )
=> ( ( undire8905369280470868553od_a_a @ Edges @ V )
= bot_bo777872063958040403od_a_a ) ) ) ).
% graph_system.incident_edges_empty
thf(fact_216_graph__system_Oincident__edges__empty,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ~ ( member_a @ V @ Vertices )
=> ( ( undire3231912044278729248dges_a @ Edges @ V )
= bot_bot_set_set_a ) ) ) ).
% graph_system.incident_edges_empty
thf(fact_217_ulgraph_Oedge__density__zero,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Y: set_set_a,X2: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( Y = bot_bot_set_set_a )
=> ( ( undire8927637694342045747_set_a @ Edges @ X2 @ Y )
= zero_zero_real ) ) ) ).
% ulgraph.edge_density_zero
thf(fact_218_ulgraph_Oedge__density__zero,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,Y: set_Product_prod_a_a,X2: set_Product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( Y = bot_bo3357376287454694259od_a_a )
=> ( ( undire8410861505230878716od_a_a @ Edges @ X2 @ Y )
= zero_zero_real ) ) ) ).
% ulgraph.edge_density_zero
thf(fact_219_ulgraph_Oedge__density__zero,axiom,
! [Vertices: set_nat,Edges: set_set_nat,Y: set_nat,X2: set_nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( Y = bot_bot_set_nat )
=> ( ( undire8640779321340989627ty_nat @ Edges @ X2 @ Y )
= zero_zero_real ) ) ) ).
% ulgraph.edge_density_zero
thf(fact_220_ulgraph_Oedge__density__zero,axiom,
! [Vertices: set_a,Edges: set_set_a,Y: set_a,X2: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( Y = bot_bot_set_a )
=> ( ( undire297304480579013331sity_a @ Edges @ X2 @ Y )
= zero_zero_real ) ) ) ).
% ulgraph.edge_density_zero
thf(fact_221_infinite__imp__nonempty,axiom,
! [S: set_list_a] :
( ~ ( finite_finite_list_a @ S )
=> ( S != bot_bot_set_list_a ) ) ).
% infinite_imp_nonempty
thf(fact_222_infinite__imp__nonempty,axiom,
! [S: set_set_a] :
( ~ ( finite_finite_set_a @ S )
=> ( S != bot_bot_set_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_223_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_224_infinite__imp__nonempty,axiom,
! [S: set_Product_prod_a_a] :
( ~ ( finite6544458595007987280od_a_a @ S )
=> ( S != bot_bo3357376287454694259od_a_a ) ) ).
% infinite_imp_nonempty
thf(fact_225_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_226_finite_OemptyI,axiom,
finite_finite_list_a @ bot_bot_set_list_a ).
% finite.emptyI
thf(fact_227_finite_OemptyI,axiom,
finite_finite_set_a @ bot_bot_set_set_a ).
% finite.emptyI
thf(fact_228_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_229_finite_OemptyI,axiom,
finite6544458595007987280od_a_a @ bot_bo3357376287454694259od_a_a ).
% finite.emptyI
thf(fact_230_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_231_ulgraph_Oempty__not__edge,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ~ ( member_set_set_a @ bot_bot_set_set_a @ Edges ) ) ).
% ulgraph.empty_not_edge
thf(fact_232_ulgraph_Oempty__not__edge,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ~ ( member1816616512716248880od_a_a @ bot_bo3357376287454694259od_a_a @ Edges ) ) ).
% ulgraph.empty_not_edge
thf(fact_233_ulgraph_Oempty__not__edge,axiom,
! [Vertices: set_nat,Edges: set_set_nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ~ ( member_set_nat @ bot_bot_set_nat @ Edges ) ) ).
% ulgraph.empty_not_edge
thf(fact_234_ulgraph_Oempty__not__edge,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ~ ( member_set_a @ bot_bot_set_a @ Edges ) ) ).
% ulgraph.empty_not_edge
thf(fact_235_finite__has__minimal,axiom,
! [A: set_se5735800977113168103od_a_a] :
( ( finite8717734299975451184od_a_a @ A )
=> ( ( A != bot_bo777872063958040403od_a_a )
=> ? [X4: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X4 @ A )
& ! [Xa: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ Xa @ A )
=> ( ( ord_le746702958409616551od_a_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_236_finite__has__minimal,axiom,
! [A: set_set_list_a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( A != bot_bo3186585308812441520list_a )
=> ? [X4: set_list_a] :
( ( member_set_list_a @ X4 @ A )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A )
=> ( ( ord_le8861187494160871172list_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_237_finite__has__minimal,axiom,
! [A: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ? [X4: set_set_a] :
( ( member_set_set_a @ X4 @ A )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A )
=> ( ( ord_le3724670747650509150_set_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_238_finite__has__minimal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_239_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_240_finite__has__maximal,axiom,
! [A: set_se5735800977113168103od_a_a] :
( ( finite8717734299975451184od_a_a @ A )
=> ( ( A != bot_bo777872063958040403od_a_a )
=> ? [X4: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X4 @ A )
& ! [Xa: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ Xa @ A )
=> ( ( ord_le746702958409616551od_a_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_241_finite__has__maximal,axiom,
! [A: set_set_list_a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( A != bot_bo3186585308812441520list_a )
=> ? [X4: set_list_a] :
( ( member_set_list_a @ X4 @ A )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A )
=> ( ( ord_le8861187494160871172list_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_242_finite__has__maximal,axiom,
! [A: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ? [X4: set_set_a] :
( ( member_set_set_a @ X4 @ A )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A )
=> ( ( ord_le3724670747650509150_set_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_243_finite__has__maximal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_244_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_245_ulgraph_Oedge__density__le1,axiom,
! [Vertices: set_a,Edges: set_set_a,X2: set_a,Y: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ord_less_eq_real @ ( undire297304480579013331sity_a @ Edges @ X2 @ Y ) @ one_one_real ) ) ).
% ulgraph.edge_density_le1
thf(fact_246_ulgraph_Oincident__sedges__empty,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ~ ( member_set_a @ V @ Vertices )
=> ( ( undire5844230293943614535_set_a @ Edges @ V )
= bot_bo3380559777022489994_set_a ) ) ) ).
% ulgraph.incident_sedges_empty
thf(fact_247_ulgraph_Oincident__sedges__empty,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ~ ( member_nat @ V @ Vertices )
=> ( ( undire996053960663353255es_nat @ Edges @ V )
= bot_bot_set_set_nat ) ) ) ).
% ulgraph.incident_sedges_empty
thf(fact_248_ulgraph_Oincident__sedges__empty,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ~ ( member1426531477525435216od_a_a @ V @ Vertices )
=> ( ( undire1583524423955984400od_a_a @ Edges @ V )
= bot_bo777872063958040403od_a_a ) ) ) ).
% ulgraph.incident_sedges_empty
thf(fact_249_ulgraph_Oincident__sedges__empty,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ~ ( member_a @ V @ Vertices )
=> ( ( undire1270416042309875431dges_a @ Edges @ V )
= bot_bot_set_set_a ) ) ) ).
% ulgraph.incident_sedges_empty
thf(fact_250_ulgraph_Oedge__density__ge0,axiom,
! [Vertices: set_a,Edges: set_set_a,X2: set_a,Y: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ord_less_eq_real @ zero_zero_real @ ( undire297304480579013331sity_a @ Edges @ X2 @ Y ) ) ) ).
% ulgraph.edge_density_ge0
thf(fact_251_graph__system_Ofinite__incident__edges,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( finite_finite_set_a @ Edges )
=> ( finite_finite_set_a @ ( undire3231912044278729248dges_a @ Edges @ V ) ) ) ) ).
% graph_system.finite_incident_edges
thf(fact_252_ulgraph_Oincident__loops__simp_I2_J,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ~ ( undire3617971648856834880loop_a @ Edges @ V )
=> ( ( undire4753905205749729249oops_a @ Edges @ V )
= bot_bot_set_set_a ) ) ) ).
% ulgraph.incident_loops_simp(2)
thf(fact_253_ulgraph_Oincident__edges__sedges,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ~ ( undire3617971648856834880loop_a @ Edges @ V )
=> ( ( undire3231912044278729248dges_a @ Edges @ V )
= ( undire1270416042309875431dges_a @ Edges @ V ) ) ) ) ).
% ulgraph.incident_edges_sedges
thf(fact_254_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_255_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_256_subset__empty,axiom,
! [A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ bot_bo3357376287454694259od_a_a )
= ( A = bot_bo3357376287454694259od_a_a ) ) ).
% subset_empty
thf(fact_257_subset__empty,axiom,
! [A: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ bot_bot_set_list_a )
= ( A = bot_bot_set_list_a ) ) ).
% subset_empty
thf(fact_258_subset__empty,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_259_subset__empty,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_260_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_261_empty__subsetI,axiom,
! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ bot_bo3357376287454694259od_a_a @ A ) ).
% empty_subsetI
thf(fact_262_empty__subsetI,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A ) ).
% empty_subsetI
thf(fact_263_empty__subsetI,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% empty_subsetI
thf(fact_264_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_265_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_266_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_267_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_268_edge__density__zero,axiom,
! [Y: set_a,X2: set_a] :
( ( Y = bot_bot_set_a )
=> ( ( undire297304480579013331sity_a @ edges @ X2 @ Y )
= zero_zero_real ) ) ).
% edge_density_zero
thf(fact_269_card__Ex__subset,axiom,
! [K: nat,M: set_nat] :
( ( ord_less_eq_nat @ K @ ( finite_card_nat @ M ) )
=> ? [N2: set_nat] :
( ( ord_less_eq_set_nat @ N2 @ M )
& ( ( finite_card_nat @ N2 )
= K ) ) ) ).
% card_Ex_subset
thf(fact_270_card__Ex__subset,axiom,
! [K: nat,M: set_Product_prod_a_a] :
( ( ord_less_eq_nat @ K @ ( finite4795055649997197647od_a_a @ M ) )
=> ? [N2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ N2 @ M )
& ( ( finite4795055649997197647od_a_a @ N2 )
= K ) ) ) ).
% card_Ex_subset
thf(fact_271_card__Ex__subset,axiom,
! [K: nat,M: set_list_a] :
( ( ord_less_eq_nat @ K @ ( finite_card_list_a @ M ) )
=> ? [N2: set_list_a] :
( ( ord_le8861187494160871172list_a @ N2 @ M )
& ( ( finite_card_list_a @ N2 )
= K ) ) ) ).
% card_Ex_subset
thf(fact_272_card__Ex__subset,axiom,
! [K: nat,M: set_set_a] :
( ( ord_less_eq_nat @ K @ ( finite_card_set_a @ M ) )
=> ? [N2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ N2 @ M )
& ( ( finite_card_set_a @ N2 )
= K ) ) ) ).
% card_Ex_subset
thf(fact_273_card__Ex__subset,axiom,
! [K: nat,M: set_a] :
( ( ord_less_eq_nat @ K @ ( finite_card_a @ M ) )
=> ? [N2: set_a] :
( ( ord_less_eq_set_a @ N2 @ M )
& ( ( finite_card_a @ N2 )
= K ) ) ) ).
% card_Ex_subset
thf(fact_274_card__le__if__inj__on__rel,axiom,
! [B: set_a,A: set_a,R: a > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A )
=> ? [B3: a] :
( ( member_a @ B3 @ B )
& ( R @ A3 @ B3 ) ) )
=> ( ! [A1: a,A22: a,B4: a] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_a @ B4 @ B )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_275_card__le__if__inj__on__rel,axiom,
! [B: set_a,A: set_nat,R: nat > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ? [B3: a] :
( ( member_a @ B3 @ B )
& ( R @ A3 @ B3 ) ) )
=> ( ! [A1: nat,A22: nat,B4: a] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_a @ B4 @ B )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_276_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_a,R: a > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A )
=> ? [B3: nat] :
( ( member_nat @ B3 @ B )
& ( R @ A3 @ B3 ) ) )
=> ( ! [A1: a,A22: a,B4: nat] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_nat @ B4 @ B )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_277_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ? [B3: nat] :
( ( member_nat @ B3 @ B )
& ( R @ A3 @ B3 ) ) )
=> ( ! [A1: nat,A22: nat,B4: nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_nat @ B4 @ B )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_278_card__le__if__inj__on__rel,axiom,
! [B: set_list_a,A: set_a,R: a > list_a > $o] :
( ( finite_finite_list_a @ B )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A )
=> ? [B3: list_a] :
( ( member_list_a @ B3 @ B )
& ( R @ A3 @ B3 ) ) )
=> ( ! [A1: a,A22: a,B4: list_a] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_list_a @ B4 @ B )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_list_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_279_card__le__if__inj__on__rel,axiom,
! [B: set_list_a,A: set_nat,R: nat > list_a > $o] :
( ( finite_finite_list_a @ B )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ? [B3: list_a] :
( ( member_list_a @ B3 @ B )
& ( R @ A3 @ B3 ) ) )
=> ( ! [A1: nat,A22: nat,B4: list_a] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_list_a @ B4 @ B )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_list_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_280_card__le__if__inj__on__rel,axiom,
! [B: set_set_a,A: set_a,R: a > set_a > $o] :
( ( finite_finite_set_a @ B )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A )
=> ? [B3: set_a] :
( ( member_set_a @ B3 @ B )
& ( R @ A3 @ B3 ) ) )
=> ( ! [A1: a,A22: a,B4: set_a] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_set_a @ B4 @ B )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_set_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_281_card__le__if__inj__on__rel,axiom,
! [B: set_set_a,A: set_nat,R: nat > set_a > $o] :
( ( finite_finite_set_a @ B )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ? [B3: set_a] :
( ( member_set_a @ B3 @ B )
& ( R @ A3 @ B3 ) ) )
=> ( ! [A1: nat,A22: nat,B4: set_a] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_set_a @ B4 @ B )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_set_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_282_card__le__if__inj__on__rel,axiom,
! [B: set_a,A: set_set_a,R: set_a > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A3: set_a] :
( ( member_set_a @ A3 @ A )
=> ? [B3: a] :
( ( member_a @ B3 @ B )
& ( R @ A3 @ B3 ) ) )
=> ( ! [A1: set_a,A22: set_a,B4: a] :
( ( member_set_a @ A1 @ A )
=> ( ( member_set_a @ A22 @ A )
=> ( ( member_a @ B4 @ B )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_283_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_set_a,R: set_a > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A3: set_a] :
( ( member_set_a @ A3 @ A )
=> ? [B3: nat] :
( ( member_nat @ B3 @ B )
& ( R @ A3 @ B3 ) ) )
=> ( ! [A1: set_a,A22: set_a,B4: nat] :
( ( member_set_a @ A1 @ A )
=> ( ( member_set_a @ A22 @ A )
=> ( ( member_nat @ B4 @ B )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_284_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_285_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_286_empty__not__edge,axiom,
~ ( member_set_a @ bot_bot_set_a @ edges ) ).
% empty_not_edge
thf(fact_287_empty__Collect__eq,axiom,
! [P: list_a > $o] :
( ( bot_bot_set_list_a
= ( collect_list_a @ P ) )
= ( ! [X3: list_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_288_empty__Collect__eq,axiom,
! [P: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P ) )
= ( ! [X3: set_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_289_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_290_empty__Collect__eq,axiom,
! [P: product_prod_a_a > $o] :
( ( bot_bo3357376287454694259od_a_a
= ( collec3336397797384452498od_a_a @ P ) )
= ( ! [X3: product_prod_a_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_291_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_292_Collect__empty__eq,axiom,
! [P: list_a > $o] :
( ( ( collect_list_a @ P )
= bot_bot_set_list_a )
= ( ! [X3: list_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_293_Collect__empty__eq,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_294_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_295_Collect__empty__eq,axiom,
! [P: product_prod_a_a > $o] :
( ( ( collec3336397797384452498od_a_a @ P )
= bot_bo3357376287454694259od_a_a )
= ( ! [X3: product_prod_a_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_296_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_297_all__not__in__conv,axiom,
! [A: set_set_a] :
( ( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A ) )
= ( A = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_298_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_299_all__not__in__conv,axiom,
! [A: set_Product_prod_a_a] :
( ( ! [X3: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X3 @ A ) )
= ( A = bot_bo3357376287454694259od_a_a ) ) ).
% all_not_in_conv
thf(fact_300_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_301_empty__iff,axiom,
! [C2: set_a] :
~ ( member_set_a @ C2 @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_302_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_303_empty__iff,axiom,
! [C2: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ C2 @ bot_bo3357376287454694259od_a_a ) ).
% empty_iff
thf(fact_304_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_305_subset__antisym,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( ord_le746702958409616551od_a_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_306_subset__antisym,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_307_subset__antisym,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_308_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_309_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ X4 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_310_subsetI,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A )
=> ( member1426531477525435216od_a_a @ X4 @ B ) )
=> ( ord_le746702958409616551od_a_a @ A @ B ) ) ).
% subsetI
thf(fact_311_subsetI,axiom,
! [A: set_list_a,B: set_list_a] :
( ! [X4: list_a] :
( ( member_list_a @ X4 @ A )
=> ( member_list_a @ X4 @ B ) )
=> ( ord_le8861187494160871172list_a @ A @ B ) ) ).
% subsetI
thf(fact_312_subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ( member_set_a @ X4 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% subsetI
thf(fact_313_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( member_a @ X4 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_314_Un__iff,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat @ C2 @ A )
| ( member_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_315_Un__iff,axiom,
! [C2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A @ B ) )
= ( ( member_set_a @ C2 @ A )
| ( member_set_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_316_Un__iff,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
= ( ( member_a @ C2 @ A )
| ( member_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_317_Un__iff,axiom,
! [C2: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C2 @ ( sup_su3048258781599657691od_a_a @ A @ B ) )
= ( ( member1426531477525435216od_a_a @ C2 @ A )
| ( member1426531477525435216od_a_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_318_Un__iff,axiom,
! [C2: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a @ C2 @ ( sup_sup_set_list_a @ A @ B ) )
= ( ( member_list_a @ C2 @ A )
| ( member_list_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_319_UnCI,axiom,
! [C2: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ A ) )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_320_UnCI,axiom,
! [C2: set_a,B: set_set_a,A: set_set_a] :
( ( ~ ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ A ) )
=> ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_321_UnCI,axiom,
! [C2: a,B: set_a,A: set_a] :
( ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ A ) )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_322_UnCI,axiom,
! [C2: product_prod_a_a,B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ~ ( member1426531477525435216od_a_a @ C2 @ B )
=> ( member1426531477525435216od_a_a @ C2 @ A ) )
=> ( member1426531477525435216od_a_a @ C2 @ ( sup_su3048258781599657691od_a_a @ A @ B ) ) ) ).
% UnCI
thf(fact_323_UnCI,axiom,
! [C2: list_a,B: set_list_a,A: set_list_a] :
( ( ~ ( member_list_a @ C2 @ B )
=> ( member_list_a @ C2 @ A ) )
=> ( member_list_a @ C2 @ ( sup_sup_set_list_a @ A @ B ) ) ) ).
% UnCI
thf(fact_324_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_325_card_Oempty,axiom,
( ( finite_card_set_a @ bot_bot_set_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_326_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_327_card_Oempty,axiom,
( ( finite4795055649997197647od_a_a @ bot_bo3357376287454694259od_a_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_328_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_329_Un__empty,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ( sup_sup_set_list_a @ A @ B )
= bot_bot_set_list_a )
= ( ( A = bot_bot_set_list_a )
& ( B = bot_bot_set_list_a ) ) ) ).
% Un_empty
thf(fact_330_Un__empty,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( sup_sup_set_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ( A = bot_bot_set_set_a )
& ( B = bot_bot_set_set_a ) ) ) ).
% Un_empty
thf(fact_331_Un__empty,axiom,
! [A: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A @ B )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_332_Un__empty,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( sup_su3048258781599657691od_a_a @ A @ B )
= bot_bo3357376287454694259od_a_a )
= ( ( A = bot_bo3357376287454694259od_a_a )
& ( B = bot_bo3357376287454694259od_a_a ) ) ) ).
% Un_empty
thf(fact_333_Un__empty,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_334_card_Oinfinite,axiom,
! [A: set_list_a] :
( ~ ( finite_finite_list_a @ A )
=> ( ( finite_card_list_a @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_335_card_Oinfinite,axiom,
! [A: set_set_a] :
( ~ ( finite_finite_set_a @ A )
=> ( ( finite_card_set_a @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_336_card_Oinfinite,axiom,
! [A: set_a] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_card_a @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_337_card_Oinfinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_card_nat @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_338_card_Oinfinite,axiom,
! [A: set_Product_prod_a_a] :
( ~ ( finite6544458595007987280od_a_a @ A )
=> ( ( finite4795055649997197647od_a_a @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_339_finite__Un,axiom,
! [F: set_nat,G2: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F @ G2 ) )
= ( ( finite_finite_nat @ F )
& ( finite_finite_nat @ G2 ) ) ) ).
% finite_Un
thf(fact_340_finite__Un,axiom,
! [F: set_set_a,G2: set_set_a] :
( ( finite_finite_set_a @ ( sup_sup_set_set_a @ F @ G2 ) )
= ( ( finite_finite_set_a @ F )
& ( finite_finite_set_a @ G2 ) ) ) ).
% finite_Un
thf(fact_341_finite__Un,axiom,
! [F: set_a,G2: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F @ G2 ) )
= ( ( finite_finite_a @ F )
& ( finite_finite_a @ G2 ) ) ) ).
% finite_Un
thf(fact_342_finite__Un,axiom,
! [F: set_Product_prod_a_a,G2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ ( sup_su3048258781599657691od_a_a @ F @ G2 ) )
= ( ( finite6544458595007987280od_a_a @ F )
& ( finite6544458595007987280od_a_a @ G2 ) ) ) ).
% finite_Un
thf(fact_343_finite__Un,axiom,
! [F: set_list_a,G2: set_list_a] :
( ( finite_finite_list_a @ ( sup_sup_set_list_a @ F @ G2 ) )
= ( ( finite_finite_list_a @ F )
& ( finite_finite_list_a @ G2 ) ) ) ).
% finite_Un
thf(fact_344_Un__subset__iff,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ A @ B ) @ C )
= ( ( ord_le746702958409616551od_a_a @ A @ C )
& ( ord_le746702958409616551od_a_a @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_345_Un__subset__iff,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A @ B ) @ C )
= ( ( ord_le8861187494160871172list_a @ A @ C )
& ( ord_le8861187494160871172list_a @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_346_Un__subset__iff,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A @ B ) @ C )
= ( ( ord_le3724670747650509150_set_a @ A @ C )
& ( ord_le3724670747650509150_set_a @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_347_Un__subset__iff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C )
= ( ( ord_less_eq_set_a @ A @ C )
& ( ord_less_eq_set_a @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_348_card__0__eq,axiom,
! [A: set_list_a] :
( ( finite_finite_list_a @ A )
=> ( ( ( finite_card_list_a @ A )
= zero_zero_nat )
= ( A = bot_bot_set_list_a ) ) ) ).
% card_0_eq
thf(fact_349_card__0__eq,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( ( finite_card_set_a @ A )
= zero_zero_nat )
= ( A = bot_bot_set_set_a ) ) ) ).
% card_0_eq
thf(fact_350_card__0__eq,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( ( finite_card_a @ A )
= zero_zero_nat )
= ( A = bot_bot_set_a ) ) ) ).
% card_0_eq
thf(fact_351_card__0__eq,axiom,
! [A: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A )
=> ( ( ( finite4795055649997197647od_a_a @ A )
= zero_zero_nat )
= ( A = bot_bo3357376287454694259od_a_a ) ) ) ).
% card_0_eq
thf(fact_352_card__0__eq,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ( finite_card_nat @ A )
= zero_zero_nat )
= ( A = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_353_Un__left__commute,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( sup_sup_set_set_a @ A @ ( sup_sup_set_set_a @ B @ C ) )
= ( sup_sup_set_set_a @ B @ ( sup_sup_set_set_a @ A @ C ) ) ) ).
% Un_left_commute
thf(fact_354_Un__left__commute,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C ) )
= ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A @ C ) ) ) ).
% Un_left_commute
thf(fact_355_Un__left__commute,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ B @ C ) )
= ( sup_su3048258781599657691od_a_a @ B @ ( sup_su3048258781599657691od_a_a @ A @ C ) ) ) ).
% Un_left_commute
thf(fact_356_Un__left__commute,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( sup_sup_set_list_a @ A @ ( sup_sup_set_list_a @ B @ C ) )
= ( sup_sup_set_list_a @ B @ ( sup_sup_set_list_a @ A @ C ) ) ) ).
% Un_left_commute
thf(fact_357_Un__left__absorb,axiom,
! [A: set_set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ A @ ( sup_sup_set_set_a @ A @ B ) )
= ( sup_sup_set_set_a @ A @ B ) ) ).
% Un_left_absorb
thf(fact_358_Un__left__absorb,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_a @ A @ B ) ) ).
% Un_left_absorb
thf(fact_359_Un__left__absorb,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ A @ B ) )
= ( sup_su3048258781599657691od_a_a @ A @ B ) ) ).
% Un_left_absorb
thf(fact_360_Un__left__absorb,axiom,
! [A: set_list_a,B: set_list_a] :
( ( sup_sup_set_list_a @ A @ ( sup_sup_set_list_a @ A @ B ) )
= ( sup_sup_set_list_a @ A @ B ) ) ).
% Un_left_absorb
thf(fact_361_Un__commute,axiom,
( sup_sup_set_set_a
= ( ^ [A4: set_set_a,B5: set_set_a] : ( sup_sup_set_set_a @ B5 @ A4 ) ) ) ).
% Un_commute
thf(fact_362_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B5: set_a] : ( sup_sup_set_a @ B5 @ A4 ) ) ) ).
% Un_commute
thf(fact_363_Un__commute,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [A4: set_Product_prod_a_a,B5: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ B5 @ A4 ) ) ) ).
% Un_commute
thf(fact_364_Un__commute,axiom,
( sup_sup_set_list_a
= ( ^ [A4: set_list_a,B5: set_list_a] : ( sup_sup_set_list_a @ B5 @ A4 ) ) ) ).
% Un_commute
thf(fact_365_Un__absorb,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_366_Un__absorb,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_367_Un__absorb,axiom,
! [A: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_368_Un__absorb,axiom,
! [A: set_list_a] :
( ( sup_sup_set_list_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_369_Un__assoc,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A @ B ) @ C )
= ( sup_sup_set_set_a @ A @ ( sup_sup_set_set_a @ B @ C ) ) ) ).
% Un_assoc
thf(fact_370_Un__assoc,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ C )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Un_assoc
thf(fact_371_Un__assoc,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ A @ B ) @ C )
= ( sup_su3048258781599657691od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ B @ C ) ) ) ).
% Un_assoc
thf(fact_372_Un__assoc,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( sup_sup_set_list_a @ ( sup_sup_set_list_a @ A @ B ) @ C )
= ( sup_sup_set_list_a @ A @ ( sup_sup_set_list_a @ B @ C ) ) ) ).
% Un_assoc
thf(fact_373_ball__Un,axiom,
! [A: set_set_a,B: set_set_a,P: set_a > $o] :
( ( ! [X3: set_a] :
( ( member_set_a @ X3 @ ( sup_sup_set_set_a @ A @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( P @ X3 ) )
& ! [X3: set_a] :
( ( member_set_a @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_374_ball__Un,axiom,
! [A: set_a,B: set_a,P: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( P @ X3 ) )
& ! [X3: a] :
( ( member_a @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_375_ball__Un,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( sup_su3048258781599657691od_a_a @ A @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A )
=> ( P @ X3 ) )
& ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_376_ball__Un,axiom,
! [A: set_list_a,B: set_list_a,P: list_a > $o] :
( ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( sup_sup_set_list_a @ A @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A )
=> ( P @ X3 ) )
& ! [X3: list_a] :
( ( member_list_a @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_377_bex__Un,axiom,
! [A: set_set_a,B: set_set_a,P: set_a > $o] :
( ( ? [X3: set_a] :
( ( member_set_a @ X3 @ ( sup_sup_set_set_a @ A @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: set_a] :
( ( member_set_a @ X3 @ A )
& ( P @ X3 ) )
| ? [X3: set_a] :
( ( member_set_a @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_378_bex__Un,axiom,
! [A: set_a,B: set_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ X3 ) )
| ? [X3: a] :
( ( member_a @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_379_bex__Un,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ( ? [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( sup_su3048258781599657691od_a_a @ A @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A )
& ( P @ X3 ) )
| ? [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_380_bex__Un,axiom,
! [A: set_list_a,B: set_list_a,P: list_a > $o] :
( ( ? [X3: list_a] :
( ( member_list_a @ X3 @ ( sup_sup_set_list_a @ A @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: list_a] :
( ( member_list_a @ X3 @ A )
& ( P @ X3 ) )
| ? [X3: list_a] :
( ( member_list_a @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_381_UnI2,axiom,
! [C2: nat,B: set_nat,A: set_nat] :
( ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_382_UnI2,axiom,
! [C2: set_a,B: set_set_a,A: set_set_a] :
( ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_383_UnI2,axiom,
! [C2: a,B: set_a,A: set_a] :
( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_384_UnI2,axiom,
! [C2: product_prod_a_a,B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C2 @ B )
=> ( member1426531477525435216od_a_a @ C2 @ ( sup_su3048258781599657691od_a_a @ A @ B ) ) ) ).
% UnI2
thf(fact_385_UnI2,axiom,
! [C2: list_a,B: set_list_a,A: set_list_a] :
( ( member_list_a @ C2 @ B )
=> ( member_list_a @ C2 @ ( sup_sup_set_list_a @ A @ B ) ) ) ).
% UnI2
thf(fact_386_UnI1,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_387_UnI1,axiom,
! [C2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ A )
=> ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_388_UnI1,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_389_UnI1,axiom,
! [C2: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C2 @ A )
=> ( member1426531477525435216od_a_a @ C2 @ ( sup_su3048258781599657691od_a_a @ A @ B ) ) ) ).
% UnI1
thf(fact_390_UnI1,axiom,
! [C2: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a @ C2 @ A )
=> ( member_list_a @ C2 @ ( sup_sup_set_list_a @ A @ B ) ) ) ).
% UnI1
thf(fact_391_UnE,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) )
=> ( ~ ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_392_UnE,axiom,
! [C2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( sup_sup_set_set_a @ A @ B ) )
=> ( ~ ( member_set_a @ C2 @ A )
=> ( member_set_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_393_UnE,axiom,
! [C2: a,A: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
=> ( ~ ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_394_UnE,axiom,
! [C2: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C2 @ ( sup_su3048258781599657691od_a_a @ A @ B ) )
=> ( ~ ( member1426531477525435216od_a_a @ C2 @ A )
=> ( member1426531477525435216od_a_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_395_UnE,axiom,
! [C2: list_a,A: set_list_a,B: set_list_a] :
( ( member_list_a @ C2 @ ( sup_sup_set_list_a @ A @ B ) )
=> ( ~ ( member_list_a @ C2 @ A )
=> ( member_list_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_396_ulgraph_Oneighborhood_Ocong,axiom,
undire8504279938402040014hood_a = undire8504279938402040014hood_a ).
% ulgraph.neighborhood.cong
thf(fact_397_Un__empty__right,axiom,
! [A: set_list_a] :
( ( sup_sup_set_list_a @ A @ bot_bot_set_list_a )
= A ) ).
% Un_empty_right
thf(fact_398_Un__empty__right,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ A @ bot_bot_set_set_a )
= A ) ).
% Un_empty_right
thf(fact_399_Un__empty__right,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% Un_empty_right
thf(fact_400_Un__empty__right,axiom,
! [A: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A @ bot_bo3357376287454694259od_a_a )
= A ) ).
% Un_empty_right
thf(fact_401_Un__empty__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Un_empty_right
thf(fact_402_Un__empty__left,axiom,
! [B: set_list_a] :
( ( sup_sup_set_list_a @ bot_bot_set_list_a @ B )
= B ) ).
% Un_empty_left
thf(fact_403_Un__empty__left,axiom,
! [B: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_404_Un__empty__left,axiom,
! [B: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_405_Un__empty__left,axiom,
! [B: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ bot_bo3357376287454694259od_a_a @ B )
= B ) ).
% Un_empty_left
thf(fact_406_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_407_subset__Un__eq,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A4: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A4 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_408_subset__Un__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A4: set_list_a,B5: set_list_a] :
( ( sup_sup_set_list_a @ A4 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_409_subset__Un__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B5: set_set_a] :
( ( sup_sup_set_set_a @ A4 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_410_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B5: set_a] :
( ( sup_sup_set_a @ A4 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_411_subset__UnE,axiom,
! [C: set_Product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B ) )
=> ~ ! [A5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A5 @ A )
=> ! [B6: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B6 @ B )
=> ( C
!= ( sup_su3048258781599657691od_a_a @ A5 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_412_subset__UnE,axiom,
! [C: set_list_a,A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ C @ ( sup_sup_set_list_a @ A @ B ) )
=> ~ ! [A5: set_list_a] :
( ( ord_le8861187494160871172list_a @ A5 @ A )
=> ! [B6: set_list_a] :
( ( ord_le8861187494160871172list_a @ B6 @ B )
=> ( C
!= ( sup_sup_set_list_a @ A5 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_413_subset__UnE,axiom,
! [C: set_set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) )
=> ~ ! [A5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ A )
=> ! [B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B6 @ B )
=> ( C
!= ( sup_sup_set_set_a @ A5 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_414_subset__UnE,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B ) )
=> ~ ! [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ A )
=> ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ B )
=> ( C
!= ( sup_sup_set_a @ A5 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_415_Un__absorb2,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B @ A )
=> ( ( sup_su3048258781599657691od_a_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_416_Un__absorb2,axiom,
! [B: set_list_a,A: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ A )
=> ( ( sup_sup_set_list_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_417_Un__absorb2,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( sup_sup_set_set_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_418_Un__absorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_419_Un__absorb1,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( sup_su3048258781599657691od_a_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_420_Un__absorb1,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( sup_sup_set_list_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_421_Un__absorb1,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( sup_sup_set_set_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_422_Un__absorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_423_Un__upper2,axiom,
! [B: set_Product_prod_a_a,A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ B @ ( sup_su3048258781599657691od_a_a @ A @ B ) ) ).
% Un_upper2
thf(fact_424_Un__upper2,axiom,
! [B: set_list_a,A: set_list_a] : ( ord_le8861187494160871172list_a @ B @ ( sup_sup_set_list_a @ A @ B ) ) ).
% Un_upper2
thf(fact_425_Un__upper2,axiom,
! [B: set_set_a,A: set_set_a] : ( ord_le3724670747650509150_set_a @ B @ ( sup_sup_set_set_a @ A @ B ) ) ).
% Un_upper2
thf(fact_426_Un__upper2,axiom,
! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper2
thf(fact_427_Un__upper1,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ A @ B ) ) ).
% Un_upper1
thf(fact_428_Un__upper1,axiom,
! [A: set_list_a,B: set_list_a] : ( ord_le8861187494160871172list_a @ A @ ( sup_sup_set_list_a @ A @ B ) ) ).
% Un_upper1
thf(fact_429_Un__upper1,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ ( sup_sup_set_set_a @ A @ B ) ) ).
% Un_upper1
thf(fact_430_Un__upper1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).
% Un_upper1
thf(fact_431_Un__least,axiom,
! [A: set_Product_prod_a_a,C: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ C )
=> ( ( ord_le746702958409616551od_a_a @ B @ C )
=> ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ A @ B ) @ C ) ) ) ).
% Un_least
thf(fact_432_Un__least,axiom,
! [A: set_list_a,C: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ C )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A @ B ) @ C ) ) ) ).
% Un_least
thf(fact_433_Un__least,axiom,
! [A: set_set_a,C: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A @ B ) @ C ) ) ) ).
% Un_least
thf(fact_434_Un__least,axiom,
! [A: set_a,C: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C ) ) ) ).
% Un_least
thf(fact_435_Un__mono,axiom,
! [A: set_Product_prod_a_a,C: set_Product_prod_a_a,B: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ C )
=> ( ( ord_le746702958409616551od_a_a @ B @ D )
=> ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ A @ B ) @ ( sup_su3048258781599657691od_a_a @ C @ D ) ) ) ) ).
% Un_mono
thf(fact_436_Un__mono,axiom,
! [A: set_list_a,C: set_list_a,B: set_list_a,D: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ C )
=> ( ( ord_le8861187494160871172list_a @ B @ D )
=> ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A @ B ) @ ( sup_sup_set_list_a @ C @ D ) ) ) ) ).
% Un_mono
thf(fact_437_Un__mono,axiom,
! [A: set_set_a,C: set_set_a,B: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C )
=> ( ( ord_le3724670747650509150_set_a @ B @ D )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A @ B ) @ ( sup_sup_set_set_a @ C @ D ) ) ) ) ).
% Un_mono
thf(fact_438_Un__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C @ D ) ) ) ) ).
% Un_mono
thf(fact_439_finite__UnI,axiom,
! [F: set_nat,G2: set_nat] :
( ( finite_finite_nat @ F )
=> ( ( finite_finite_nat @ G2 )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_440_finite__UnI,axiom,
! [F: set_set_a,G2: set_set_a] :
( ( finite_finite_set_a @ F )
=> ( ( finite_finite_set_a @ G2 )
=> ( finite_finite_set_a @ ( sup_sup_set_set_a @ F @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_441_finite__UnI,axiom,
! [F: set_a,G2: set_a] :
( ( finite_finite_a @ F )
=> ( ( finite_finite_a @ G2 )
=> ( finite_finite_a @ ( sup_sup_set_a @ F @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_442_finite__UnI,axiom,
! [F: set_Product_prod_a_a,G2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F )
=> ( ( finite6544458595007987280od_a_a @ G2 )
=> ( finite6544458595007987280od_a_a @ ( sup_su3048258781599657691od_a_a @ F @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_443_finite__UnI,axiom,
! [F: set_list_a,G2: set_list_a] :
( ( finite_finite_list_a @ F )
=> ( ( finite_finite_list_a @ G2 )
=> ( finite_finite_list_a @ ( sup_sup_set_list_a @ F @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_444_Un__infinite,axiom,
! [S: set_nat,T: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_445_Un__infinite,axiom,
! [S: set_set_a,T: set_set_a] :
( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ ( sup_sup_set_set_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_446_Un__infinite,axiom,
! [S: set_a,T: set_a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_447_Un__infinite,axiom,
! [S: set_Product_prod_a_a,T: set_Product_prod_a_a] :
( ~ ( finite6544458595007987280od_a_a @ S )
=> ~ ( finite6544458595007987280od_a_a @ ( sup_su3048258781599657691od_a_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_448_Un__infinite,axiom,
! [S: set_list_a,T: set_list_a] :
( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ ( sup_sup_set_list_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_449_infinite__Un,axiom,
! [S: set_nat,T: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_Un
thf(fact_450_infinite__Un,axiom,
! [S: set_set_a,T: set_set_a] :
( ( ~ ( finite_finite_set_a @ ( sup_sup_set_set_a @ S @ T ) ) )
= ( ~ ( finite_finite_set_a @ S )
| ~ ( finite_finite_set_a @ T ) ) ) ).
% infinite_Un
thf(fact_451_infinite__Un,axiom,
! [S: set_a,T: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) )
= ( ~ ( finite_finite_a @ S )
| ~ ( finite_finite_a @ T ) ) ) ).
% infinite_Un
thf(fact_452_infinite__Un,axiom,
! [S: set_Product_prod_a_a,T: set_Product_prod_a_a] :
( ( ~ ( finite6544458595007987280od_a_a @ ( sup_su3048258781599657691od_a_a @ S @ T ) ) )
= ( ~ ( finite6544458595007987280od_a_a @ S )
| ~ ( finite6544458595007987280od_a_a @ T ) ) ) ).
% infinite_Un
thf(fact_453_infinite__Un,axiom,
! [S: set_list_a,T: set_list_a] :
( ( ~ ( finite_finite_list_a @ ( sup_sup_set_list_a @ S @ T ) ) )
= ( ~ ( finite_finite_list_a @ S )
| ~ ( finite_finite_list_a @ T ) ) ) ).
% infinite_Un
thf(fact_454_graph__system_Oinduced__edges__union,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,VH1: set_Product_prod_a_a,S: set_Product_prod_a_a,VH2: set_Product_prod_a_a,T: set_Product_prod_a_a,EH1: set_se5735800977113168103od_a_a,EH2: set_se5735800977113168103od_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( ord_le746702958409616551od_a_a @ VH1 @ S )
=> ( ( ord_le746702958409616551od_a_a @ VH2 @ T )
=> ( ( undire1860116983885411791od_a_a @ VH1 @ EH1 )
=> ( ( undire1860116983885411791od_a_a @ VH2 @ EH2 )
=> ( ( ord_le1995061765932249223od_a_a @ ( sup_su6839823885543768763od_a_a @ EH1 @ EH2 ) @ ( undire5906991851038061813od_a_a @ Edges @ ( sup_su3048258781599657691od_a_a @ S @ T ) ) )
=> ( ord_le1995061765932249223od_a_a @ EH1 @ ( undire5906991851038061813od_a_a @ Edges @ S ) ) ) ) ) ) ) ) ).
% graph_system.induced_edges_union
thf(fact_455_graph__system_Oinduced__edges__union,axiom,
! [Vertices: set_list_a,Edges: set_set_list_a,VH1: set_list_a,S: set_list_a,VH2: set_list_a,T: set_list_a,EH1: set_set_list_a,EH2: set_set_list_a] :
( ( undire5959234994740280364list_a @ Vertices @ Edges )
=> ( ( ord_le8861187494160871172list_a @ VH1 @ S )
=> ( ( ord_le8861187494160871172list_a @ VH2 @ T )
=> ( ( undire5959234994740280364list_a @ VH1 @ EH1 )
=> ( ( undire5959234994740280364list_a @ VH2 @ EH2 )
=> ( ( ord_le8877086941679407844list_a @ ( sup_su4537662296134749976list_a @ EH1 @ EH2 ) @ ( undire8521487854958249554list_a @ Edges @ ( sup_sup_set_list_a @ S @ T ) ) )
=> ( ord_le8877086941679407844list_a @ EH1 @ ( undire8521487854958249554list_a @ Edges @ S ) ) ) ) ) ) ) ) ).
% graph_system.induced_edges_union
thf(fact_456_graph__system_Oinduced__edges__union,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,VH1: set_set_a,S: set_set_a,VH2: set_set_a,T: set_set_a,EH1: set_set_set_a,EH2: set_set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( ord_le3724670747650509150_set_a @ VH1 @ S )
=> ( ( ord_le3724670747650509150_set_a @ VH2 @ T )
=> ( ( undire7159349782766787846_set_a @ VH1 @ EH1 )
=> ( ( undire7159349782766787846_set_a @ VH2 @ EH2 )
=> ( ( ord_le5722252365846178494_set_a @ ( sup_su2076012971530813682_set_a @ EH1 @ EH2 ) @ ( undire7854589003810675244_set_a @ Edges @ ( sup_sup_set_set_a @ S @ T ) ) )
=> ( ord_le5722252365846178494_set_a @ EH1 @ ( undire7854589003810675244_set_a @ Edges @ S ) ) ) ) ) ) ) ) ).
% graph_system.induced_edges_union
thf(fact_457_graph__system_Oinduced__edges__union,axiom,
! [Vertices: set_a,Edges: set_set_a,VH1: set_a,S: set_a,VH2: set_a,T: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T )
=> ( ( 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 @ T ) ) )
=> ( ord_le3724670747650509150_set_a @ EH1 @ ( undire7777452895879145676dges_a @ Edges @ S ) ) ) ) ) ) ) ) ).
% graph_system.induced_edges_union
thf(fact_458_graph__system_Oinduced__edges__union__subgraph__single,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,VH1: set_Product_prod_a_a,S: set_Product_prod_a_a,VH2: set_Product_prod_a_a,T: set_Product_prod_a_a,EH1: set_se5735800977113168103od_a_a,EH2: set_se5735800977113168103od_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( ord_le746702958409616551od_a_a @ VH1 @ S )
=> ( ( ord_le746702958409616551od_a_a @ VH2 @ T )
=> ( ( undire1860116983885411791od_a_a @ VH1 @ EH1 )
=> ( ( undire1860116983885411791od_a_a @ VH2 @ EH2 )
=> ( ( undire398746457437328754od_a_a @ ( sup_su3048258781599657691od_a_a @ VH1 @ VH2 ) @ ( sup_su6839823885543768763od_a_a @ EH1 @ EH2 ) @ ( sup_su3048258781599657691od_a_a @ S @ T ) @ ( undire5906991851038061813od_a_a @ Edges @ ( sup_su3048258781599657691od_a_a @ S @ T ) ) )
=> ( undire398746457437328754od_a_a @ VH1 @ EH1 @ S @ ( undire5906991851038061813od_a_a @ Edges @ S ) ) ) ) ) ) ) ) ).
% graph_system.induced_edges_union_subgraph_single
thf(fact_459_graph__system_Oinduced__edges__union__subgraph__single,axiom,
! [Vertices: set_list_a,Edges: set_set_list_a,VH1: set_list_a,S: set_list_a,VH2: set_list_a,T: set_list_a,EH1: set_set_list_a,EH2: set_set_list_a] :
( ( undire5959234994740280364list_a @ Vertices @ Edges )
=> ( ( ord_le8861187494160871172list_a @ VH1 @ S )
=> ( ( ord_le8861187494160871172list_a @ VH2 @ T )
=> ( ( undire5959234994740280364list_a @ VH1 @ EH1 )
=> ( ( undire5959234994740280364list_a @ VH2 @ EH2 )
=> ( ( undire761398192061991247list_a @ ( sup_sup_set_list_a @ VH1 @ VH2 ) @ ( sup_su4537662296134749976list_a @ EH1 @ EH2 ) @ ( sup_sup_set_list_a @ S @ T ) @ ( undire8521487854958249554list_a @ Edges @ ( sup_sup_set_list_a @ S @ T ) ) )
=> ( undire761398192061991247list_a @ VH1 @ EH1 @ S @ ( undire8521487854958249554list_a @ Edges @ S ) ) ) ) ) ) ) ) ).
% graph_system.induced_edges_union_subgraph_single
thf(fact_460_graph__system_Oinduced__edges__union__subgraph__single,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,VH1: set_set_a,S: set_set_a,VH2: set_set_a,T: set_set_a,EH1: set_set_set_a,EH2: set_set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( ord_le3724670747650509150_set_a @ VH1 @ S )
=> ( ( ord_le3724670747650509150_set_a @ VH2 @ T )
=> ( ( undire7159349782766787846_set_a @ VH1 @ EH1 )
=> ( ( undire7159349782766787846_set_a @ VH2 @ EH2 )
=> ( ( undire1186139521737116585_set_a @ ( sup_sup_set_set_a @ VH1 @ VH2 ) @ ( sup_su2076012971530813682_set_a @ EH1 @ EH2 ) @ ( sup_sup_set_set_a @ S @ T ) @ ( undire7854589003810675244_set_a @ Edges @ ( sup_sup_set_set_a @ S @ T ) ) )
=> ( undire1186139521737116585_set_a @ VH1 @ EH1 @ S @ ( undire7854589003810675244_set_a @ Edges @ S ) ) ) ) ) ) ) ) ).
% graph_system.induced_edges_union_subgraph_single
thf(fact_461_graph__system_Oinduced__edges__union__subgraph__single,axiom,
! [Vertices: set_a,Edges: set_set_a,VH1: set_a,S: set_a,VH2: set_a,T: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T )
=> ( ( 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 @ T ) @ ( undire7777452895879145676dges_a @ Edges @ ( sup_sup_set_a @ S @ T ) ) )
=> ( undire7103218114511261257raph_a @ VH1 @ EH1 @ S @ ( undire7777452895879145676dges_a @ Edges @ S ) ) ) ) ) ) ) ) ).
% graph_system.induced_edges_union_subgraph_single
thf(fact_462_graph__system_Oinduced__union__subgraph,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,VH1: set_Product_prod_a_a,S: set_Product_prod_a_a,VH2: set_Product_prod_a_a,T: set_Product_prod_a_a,EH1: set_se5735800977113168103od_a_a,EH2: set_se5735800977113168103od_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( ord_le746702958409616551od_a_a @ VH1 @ S )
=> ( ( ord_le746702958409616551od_a_a @ VH2 @ T )
=> ( ( undire1860116983885411791od_a_a @ VH1 @ EH1 )
=> ( ( undire1860116983885411791od_a_a @ VH2 @ EH2 )
=> ( ( ( undire398746457437328754od_a_a @ VH1 @ EH1 @ S @ ( undire5906991851038061813od_a_a @ Edges @ S ) )
& ( undire398746457437328754od_a_a @ VH2 @ EH2 @ T @ ( undire5906991851038061813od_a_a @ Edges @ T ) ) )
= ( undire398746457437328754od_a_a @ ( sup_su3048258781599657691od_a_a @ VH1 @ VH2 ) @ ( sup_su6839823885543768763od_a_a @ EH1 @ EH2 ) @ ( sup_su3048258781599657691od_a_a @ S @ T ) @ ( undire5906991851038061813od_a_a @ Edges @ ( sup_su3048258781599657691od_a_a @ S @ T ) ) ) ) ) ) ) ) ) ).
% graph_system.induced_union_subgraph
thf(fact_463_graph__system_Oinduced__union__subgraph,axiom,
! [Vertices: set_list_a,Edges: set_set_list_a,VH1: set_list_a,S: set_list_a,VH2: set_list_a,T: set_list_a,EH1: set_set_list_a,EH2: set_set_list_a] :
( ( undire5959234994740280364list_a @ Vertices @ Edges )
=> ( ( ord_le8861187494160871172list_a @ VH1 @ S )
=> ( ( ord_le8861187494160871172list_a @ VH2 @ T )
=> ( ( undire5959234994740280364list_a @ VH1 @ EH1 )
=> ( ( undire5959234994740280364list_a @ VH2 @ EH2 )
=> ( ( ( undire761398192061991247list_a @ VH1 @ EH1 @ S @ ( undire8521487854958249554list_a @ Edges @ S ) )
& ( undire761398192061991247list_a @ VH2 @ EH2 @ T @ ( undire8521487854958249554list_a @ Edges @ T ) ) )
= ( undire761398192061991247list_a @ ( sup_sup_set_list_a @ VH1 @ VH2 ) @ ( sup_su4537662296134749976list_a @ EH1 @ EH2 ) @ ( sup_sup_set_list_a @ S @ T ) @ ( undire8521487854958249554list_a @ Edges @ ( sup_sup_set_list_a @ S @ T ) ) ) ) ) ) ) ) ) ).
% graph_system.induced_union_subgraph
thf(fact_464_graph__system_Oinduced__union__subgraph,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,VH1: set_set_a,S: set_set_a,VH2: set_set_a,T: set_set_a,EH1: set_set_set_a,EH2: set_set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( ord_le3724670747650509150_set_a @ VH1 @ S )
=> ( ( ord_le3724670747650509150_set_a @ VH2 @ T )
=> ( ( undire7159349782766787846_set_a @ VH1 @ EH1 )
=> ( ( undire7159349782766787846_set_a @ VH2 @ EH2 )
=> ( ( ( undire1186139521737116585_set_a @ VH1 @ EH1 @ S @ ( undire7854589003810675244_set_a @ Edges @ S ) )
& ( undire1186139521737116585_set_a @ VH2 @ EH2 @ T @ ( undire7854589003810675244_set_a @ Edges @ T ) ) )
= ( undire1186139521737116585_set_a @ ( sup_sup_set_set_a @ VH1 @ VH2 ) @ ( sup_su2076012971530813682_set_a @ EH1 @ EH2 ) @ ( sup_sup_set_set_a @ S @ T ) @ ( undire7854589003810675244_set_a @ Edges @ ( sup_sup_set_set_a @ S @ T ) ) ) ) ) ) ) ) ) ).
% graph_system.induced_union_subgraph
thf(fact_465_graph__system_Oinduced__union__subgraph,axiom,
! [Vertices: set_a,Edges: set_set_a,VH1: set_a,S: set_a,VH2: set_a,T: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T )
=> ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
=> ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
=> ( ( ( undire7103218114511261257raph_a @ VH1 @ EH1 @ S @ ( undire7777452895879145676dges_a @ Edges @ S ) )
& ( undire7103218114511261257raph_a @ VH2 @ EH2 @ T @ ( undire7777452895879145676dges_a @ Edges @ T ) ) )
= ( undire7103218114511261257raph_a @ ( sup_sup_set_a @ VH1 @ VH2 ) @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( sup_sup_set_a @ S @ T ) @ ( undire7777452895879145676dges_a @ Edges @ ( sup_sup_set_a @ S @ T ) ) ) ) ) ) ) ) ) ).
% graph_system.induced_union_subgraph
thf(fact_466_zero__reorient,axiom,
! [X5: real] :
( ( zero_zero_real = X5 )
= ( X5 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_467_zero__reorient,axiom,
! [X5: nat] :
( ( zero_zero_nat = X5 )
= ( X5 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_468_one__reorient,axiom,
! [X5: real] :
( ( one_one_real = X5 )
= ( X5 = one_one_real ) ) ).
% one_reorient
thf(fact_469_one__reorient,axiom,
! [X5: nat] :
( ( one_one_nat = X5 )
= ( X5 = one_one_nat ) ) ).
% one_reorient
thf(fact_470_ex__in__conv,axiom,
! [A: set_set_a] :
( ( ? [X3: set_a] : ( member_set_a @ X3 @ A ) )
= ( A != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_471_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_472_ex__in__conv,axiom,
! [A: set_Product_prod_a_a] :
( ( ? [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ A ) )
= ( A != bot_bo3357376287454694259od_a_a ) ) ).
% ex_in_conv
thf(fact_473_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_474_equals0I,axiom,
! [A: set_set_a] :
( ! [Y2: set_a] :
~ ( member_set_a @ Y2 @ A )
=> ( A = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_475_equals0I,axiom,
! [A: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_476_equals0I,axiom,
! [A: set_Product_prod_a_a] :
( ! [Y2: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ Y2 @ A )
=> ( A = bot_bo3357376287454694259od_a_a ) ) ).
% equals0I
thf(fact_477_equals0I,axiom,
! [A: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_478_equals0D,axiom,
! [A: set_set_a,A2: set_a] :
( ( A = bot_bot_set_set_a )
=> ~ ( member_set_a @ A2 @ A ) ) ).
% equals0D
thf(fact_479_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_480_equals0D,axiom,
! [A: set_Product_prod_a_a,A2: product_prod_a_a] :
( ( A = bot_bo3357376287454694259od_a_a )
=> ~ ( member1426531477525435216od_a_a @ A2 @ A ) ) ).
% equals0D
thf(fact_481_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_482_emptyE,axiom,
! [A2: set_a] :
~ ( member_set_a @ A2 @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_483_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_484_emptyE,axiom,
! [A2: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ).
% emptyE
thf(fact_485_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_486_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_487_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 ) )
= ( ! [X3: product_prod_a_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_488_Collect__mono__iff,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) )
= ( ! [X3: list_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_489_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X3: set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_490_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_491_set__eq__subset,axiom,
( ( ^ [Y3: set_Product_prod_a_a,Z: set_Product_prod_a_a] : ( Y3 = Z ) )
= ( ^ [A4: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A4 @ B5 )
& ( ord_le746702958409616551od_a_a @ B5 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_492_set__eq__subset,axiom,
( ( ^ [Y3: set_list_a,Z: set_list_a] : ( Y3 = Z ) )
= ( ^ [A4: set_list_a,B5: set_list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ B5 )
& ( ord_le8861187494160871172list_a @ B5 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_493_set__eq__subset,axiom,
( ( ^ [Y3: set_set_a,Z: set_set_a] : ( Y3 = Z ) )
= ( ^ [A4: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_494_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A4: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_495_subset__trans,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( ord_le746702958409616551od_a_a @ B @ C )
=> ( ord_le746702958409616551od_a_a @ A @ C ) ) ) ).
% subset_trans
thf(fact_496_subset__trans,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ord_le8861187494160871172list_a @ A @ C ) ) ) ).
% subset_trans
thf(fact_497_subset__trans,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% subset_trans
thf(fact_498_subset__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% subset_trans
thf(fact_499_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_500_Collect__mono,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X4: product_prod_a_a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P ) @ ( collec3336397797384452498od_a_a @ Q ) ) ) ).
% Collect_mono
thf(fact_501_Collect__mono,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ! [X4: list_a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) ) ) ).
% Collect_mono
thf(fact_502_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X4: set_a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_503_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_504_subset__refl,axiom,
! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A @ A ) ).
% subset_refl
thf(fact_505_subset__refl,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ A @ A ) ).
% subset_refl
thf(fact_506_subset__refl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% subset_refl
thf(fact_507_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_508_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A4 )
=> ( member_nat @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_509_subset__iff,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A4: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
! [T3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T3 @ A4 )
=> ( member1426531477525435216od_a_a @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_510_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A4: set_list_a,B5: set_list_a] :
! [T3: list_a] :
( ( member_list_a @ T3 @ A4 )
=> ( member_list_a @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_511_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B5: set_set_a] :
! [T3: set_a] :
( ( member_set_a @ T3 @ A4 )
=> ( member_set_a @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_512_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B5: set_a] :
! [T3: a] :
( ( member_a @ T3 @ A4 )
=> ( member_a @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_513_equalityD2,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A = B )
=> ( ord_le746702958409616551od_a_a @ B @ A ) ) ).
% equalityD2
thf(fact_514_equalityD2,axiom,
! [A: set_list_a,B: set_list_a] :
( ( A = B )
=> ( ord_le8861187494160871172list_a @ B @ A ) ) ).
% equalityD2
thf(fact_515_equalityD2,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le3724670747650509150_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_516_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_517_equalityD1,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A = B )
=> ( ord_le746702958409616551od_a_a @ A @ B ) ) ).
% equalityD1
thf(fact_518_equalityD1,axiom,
! [A: set_list_a,B: set_list_a] :
( ( A = B )
=> ( ord_le8861187494160871172list_a @ A @ B ) ) ).
% equalityD1
thf(fact_519_equalityD1,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_520_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_521_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( member_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_522_subset__eq,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A4: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A4 )
=> ( member1426531477525435216od_a_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_523_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A4: set_list_a,B5: set_list_a] :
! [X3: list_a] :
( ( member_list_a @ X3 @ A4 )
=> ( member_list_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_524_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B5: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A4 )
=> ( member_set_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_525_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B5: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( member_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_526_equalityE,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A = B )
=> ~ ( ( ord_le746702958409616551od_a_a @ A @ B )
=> ~ ( ord_le746702958409616551od_a_a @ B @ A ) ) ) ).
% equalityE
thf(fact_527_equalityE,axiom,
! [A: set_list_a,B: set_list_a] :
( ( A = B )
=> ~ ( ( ord_le8861187494160871172list_a @ A @ B )
=> ~ ( ord_le8861187494160871172list_a @ B @ A ) ) ) ).
% equalityE
thf(fact_528_equalityE,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ~ ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ~ ( ord_le3724670747650509150_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_529_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_530_subsetD,axiom,
! [A: set_nat,B: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_531_subsetD,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( member1426531477525435216od_a_a @ C2 @ A )
=> ( member1426531477525435216od_a_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_532_subsetD,axiom,
! [A: set_list_a,B: set_list_a,C2: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( member_list_a @ C2 @ A )
=> ( member_list_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_533_subsetD,axiom,
! [A: set_set_a,B: set_set_a,C2: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ C2 @ A )
=> ( member_set_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_534_subsetD,axiom,
! [A: set_a,B: set_a,C2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C2 @ A )
=> ( member_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_535_in__mono,axiom,
! [A: set_nat,B: set_nat,X5: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X5 @ A )
=> ( member_nat @ X5 @ B ) ) ) ).
% in_mono
thf(fact_536_in__mono,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,X5: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( member1426531477525435216od_a_a @ X5 @ A )
=> ( member1426531477525435216od_a_a @ X5 @ B ) ) ) ).
% in_mono
thf(fact_537_in__mono,axiom,
! [A: set_list_a,B: set_list_a,X5: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( member_list_a @ X5 @ A )
=> ( member_list_a @ X5 @ B ) ) ) ).
% in_mono
thf(fact_538_in__mono,axiom,
! [A: set_set_a,B: set_set_a,X5: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ X5 @ A )
=> ( member_set_a @ X5 @ B ) ) ) ).
% in_mono
thf(fact_539_in__mono,axiom,
! [A: set_a,B: set_a,X5: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X5 @ A )
=> ( member_a @ X5 @ B ) ) ) ).
% in_mono
thf(fact_540_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M2: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N3 )
=> ( ord_less_eq_nat @ X3 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_541_bounded__Max__nat,axiom,
! [P: nat > $o,X5: nat,M: nat] :
( ( P @ X5 )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_542_card__eq__0__iff,axiom,
! [A: set_list_a] :
( ( ( finite_card_list_a @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_list_a )
| ~ ( finite_finite_list_a @ A ) ) ) ).
% card_eq_0_iff
thf(fact_543_card__eq__0__iff,axiom,
! [A: set_set_a] :
( ( ( finite_card_set_a @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_set_a )
| ~ ( finite_finite_set_a @ A ) ) ) ).
% card_eq_0_iff
thf(fact_544_card__eq__0__iff,axiom,
! [A: set_a] :
( ( ( finite_card_a @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_a )
| ~ ( finite_finite_a @ A ) ) ) ).
% card_eq_0_iff
thf(fact_545_card__eq__0__iff,axiom,
! [A: set_Product_prod_a_a] :
( ( ( finite4795055649997197647od_a_a @ A )
= zero_zero_nat )
= ( ( A = bot_bo3357376287454694259od_a_a )
| ~ ( finite6544458595007987280od_a_a @ A ) ) ) ).
% card_eq_0_iff
thf(fact_546_card__eq__0__iff,axiom,
! [A: set_nat] :
( ( ( finite_card_nat @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A ) ) ) ).
% card_eq_0_iff
thf(fact_547_ulgraph_Oincident__loops__card,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ ( undire4753905205749729249oops_a @ Edges @ V ) ) @ one_one_nat ) ) ).
% ulgraph.incident_loops_card
thf(fact_548_ulgraph_Oincident__edges__union,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3231912044278729248dges_a @ Edges @ V )
= ( sup_sup_set_set_a @ ( undire1270416042309875431dges_a @ Edges @ V ) @ ( undire4753905205749729249oops_a @ Edges @ V ) ) ) ) ).
% ulgraph.incident_edges_union
thf(fact_549_ulgraph_Ocard__incident__sedges__neighborhood,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( finite6524359278146944486_set_a @ ( undire4631953023069350784_set_a @ Edges @ V ) )
= ( finite_card_set_a @ ( undire2074812191327625774_set_a @ Vertices @ Edges @ V ) ) ) ) ).
% ulgraph.card_incident_sedges_neighborhood
thf(fact_550_ulgraph_Ocard__incident__sedges__neighborhood,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( finite145373428556175663od_a_a @ ( undire8905369280470868553od_a_a @ Edges @ V ) )
= ( finite4795055649997197647od_a_a @ ( undire7963753511165915895od_a_a @ Vertices @ Edges @ V ) ) ) ) ).
% ulgraph.card_incident_sedges_neighborhood
thf(fact_551_ulgraph_Ocard__incident__sedges__neighborhood,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( finite_card_set_nat @ ( undire4176300566717384750es_nat @ Edges @ V ) )
= ( finite_card_nat @ ( undire8190396521545869824od_nat @ Vertices @ Edges @ V ) ) ) ) ).
% ulgraph.card_incident_sedges_neighborhood
thf(fact_552_ulgraph_Ocard__incident__sedges__neighborhood,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( finite_card_set_a @ ( undire3231912044278729248dges_a @ Edges @ V ) )
= ( finite_card_a @ ( undire8504279938402040014hood_a @ Vertices @ Edges @ V ) ) ) ) ).
% ulgraph.card_incident_sedges_neighborhood
thf(fact_553_ulgraph_Oiso__vertex__empty__neighborhood,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire6879241558604981877_set_a @ Vertices @ Edges @ V )
=> ( ( undire2074812191327625774_set_a @ Vertices @ Edges @ V )
= bot_bot_set_set_a ) ) ) ).
% ulgraph.iso_vertex_empty_neighborhood
thf(fact_554_ulgraph_Oiso__vertex__empty__neighborhood,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire3207556238582723646od_a_a @ Vertices @ Edges @ V )
=> ( ( undire7963753511165915895od_a_a @ Vertices @ Edges @ V )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% ulgraph.iso_vertex_empty_neighborhood
thf(fact_555_ulgraph_Oiso__vertex__empty__neighborhood,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire5609513041723151865ex_nat @ Vertices @ Edges @ V )
=> ( ( undire8190396521545869824od_nat @ Vertices @ Edges @ V )
= bot_bot_set_nat ) ) ) ).
% ulgraph.iso_vertex_empty_neighborhood
thf(fact_556_ulgraph_Oiso__vertex__empty__neighborhood,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V )
=> ( ( undire8504279938402040014hood_a @ Vertices @ Edges @ V )
= bot_bot_set_a ) ) ) ).
% ulgraph.iso_vertex_empty_neighborhood
thf(fact_557_zero__le,axiom,
! [X5: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X5 ) ).
% zero_le
thf(fact_558_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_559_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_560_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_561_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_562_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_563_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_564_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_565_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_566_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_567_induced__union__subgraph,axiom,
! [VH1: set_a,S: set_a,VH2: set_a,T: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T )
=> ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
=> ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
=> ( ( ( undire7103218114511261257raph_a @ VH1 @ EH1 @ S @ ( undire7777452895879145676dges_a @ edges @ S ) )
& ( undire7103218114511261257raph_a @ VH2 @ EH2 @ T @ ( undire7777452895879145676dges_a @ edges @ T ) ) )
= ( undire7103218114511261257raph_a @ ( sup_sup_set_a @ VH1 @ VH2 ) @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( sup_sup_set_a @ S @ T ) @ ( undire7777452895879145676dges_a @ edges @ ( sup_sup_set_a @ S @ T ) ) ) ) ) ) ) ) ).
% induced_union_subgraph
thf(fact_568_induced__edges__union__subgraph__single,axiom,
! [VH1: set_a,S: set_a,VH2: set_a,T: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T )
=> ( ( 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 @ T ) @ ( undire7777452895879145676dges_a @ edges @ ( sup_sup_set_a @ S @ T ) ) )
=> ( undire7103218114511261257raph_a @ VH1 @ EH1 @ S @ ( undire7777452895879145676dges_a @ edges @ S ) ) ) ) ) ) ) ).
% induced_edges_union_subgraph_single
thf(fact_569_induced__edges__union,axiom,
! [VH1: set_a,S: set_a,VH2: set_a,T: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T )
=> ( ( 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 @ T ) ) )
=> ( ord_le3724670747650509150_set_a @ EH1 @ ( undire7777452895879145676dges_a @ edges @ S ) ) ) ) ) ) ) ).
% induced_edges_union
thf(fact_570_is__loop__def,axiom,
( undire2905028936066782638loop_a
= ( ^ [E5: set_a] :
( ( finite_card_a @ E5 )
= one_one_nat ) ) ) ).
% is_loop_def
thf(fact_571_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_572_sup__bot_Oright__neutral,axiom,
! [A2: set_list_a] :
( ( sup_sup_set_list_a @ A2 @ bot_bot_set_list_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_573_sup__bot_Oright__neutral,axiom,
! [A2: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ bot_bot_set_set_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_574_sup__bot_Oright__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_575_sup__bot_Oright__neutral,axiom,
! [A2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A2 @ bot_bo3357376287454694259od_a_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_576_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_577_insert__absorb2,axiom,
! [X5: a,A: set_a] :
( ( insert_a @ X5 @ ( insert_a @ X5 @ A ) )
= ( insert_a @ X5 @ A ) ) ).
% insert_absorb2
thf(fact_578_insert__absorb2,axiom,
! [X5: set_a,A: set_set_a] :
( ( insert_set_a @ X5 @ ( insert_set_a @ X5 @ A ) )
= ( insert_set_a @ X5 @ A ) ) ).
% insert_absorb2
thf(fact_579_insert__iff,axiom,
! [A2: set_a,B7: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ ( insert_set_a @ B7 @ A ) )
= ( ( A2 = B7 )
| ( member_set_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_580_insert__iff,axiom,
! [A2: a,B7: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B7 @ A ) )
= ( ( A2 = B7 )
| ( member_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_581_insert__iff,axiom,
! [A2: nat,B7: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B7 @ A ) )
= ( ( A2 = B7 )
| ( member_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_582_insert__iff,axiom,
! [A2: product_prod_a_a,B7: product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B7 @ A ) )
= ( ( A2 = B7 )
| ( member1426531477525435216od_a_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_583_insertCI,axiom,
! [A2: set_a,B: set_set_a,B7: set_a] :
( ( ~ ( member_set_a @ A2 @ B )
=> ( A2 = B7 ) )
=> ( member_set_a @ A2 @ ( insert_set_a @ B7 @ B ) ) ) ).
% insertCI
thf(fact_584_insertCI,axiom,
! [A2: a,B: set_a,B7: a] :
( ( ~ ( member_a @ A2 @ B )
=> ( A2 = B7 ) )
=> ( member_a @ A2 @ ( insert_a @ B7 @ B ) ) ) ).
% insertCI
thf(fact_585_insertCI,axiom,
! [A2: nat,B: set_nat,B7: nat] :
( ( ~ ( member_nat @ A2 @ B )
=> ( A2 = B7 ) )
=> ( member_nat @ A2 @ ( insert_nat @ B7 @ B ) ) ) ).
% insertCI
thf(fact_586_insertCI,axiom,
! [A2: product_prod_a_a,B: set_Product_prod_a_a,B7: product_prod_a_a] :
( ( ~ ( member1426531477525435216od_a_a @ A2 @ B )
=> ( A2 = B7 ) )
=> ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B7 @ B ) ) ) ).
% insertCI
thf(fact_587_sup_Oidem,axiom,
! [A2: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_588_sup_Oidem,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_589_sup_Oidem,axiom,
! [A2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_590_sup_Oidem,axiom,
! [A2: set_list_a] :
( ( sup_sup_set_list_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_591_sup__idem,axiom,
! [X5: set_set_a] :
( ( sup_sup_set_set_a @ X5 @ X5 )
= X5 ) ).
% sup_idem
thf(fact_592_sup__idem,axiom,
! [X5: set_a] :
( ( sup_sup_set_a @ X5 @ X5 )
= X5 ) ).
% sup_idem
thf(fact_593_sup__idem,axiom,
! [X5: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X5 @ X5 )
= X5 ) ).
% sup_idem
thf(fact_594_sup__idem,axiom,
! [X5: set_list_a] :
( ( sup_sup_set_list_a @ X5 @ X5 )
= X5 ) ).
% sup_idem
thf(fact_595_sup_Oleft__idem,axiom,
! [A2: set_set_a,B7: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ ( sup_sup_set_set_a @ A2 @ B7 ) )
= ( sup_sup_set_set_a @ A2 @ B7 ) ) ).
% sup.left_idem
thf(fact_596_sup_Oleft__idem,axiom,
! [A2: set_a,B7: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B7 ) )
= ( sup_sup_set_a @ A2 @ B7 ) ) ).
% sup.left_idem
thf(fact_597_sup_Oleft__idem,axiom,
! [A2: set_Product_prod_a_a,B7: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A2 @ ( sup_su3048258781599657691od_a_a @ A2 @ B7 ) )
= ( sup_su3048258781599657691od_a_a @ A2 @ B7 ) ) ).
% sup.left_idem
thf(fact_598_sup_Oleft__idem,axiom,
! [A2: set_list_a,B7: set_list_a] :
( ( sup_sup_set_list_a @ A2 @ ( sup_sup_set_list_a @ A2 @ B7 ) )
= ( sup_sup_set_list_a @ A2 @ B7 ) ) ).
% sup.left_idem
thf(fact_599_sup__left__idem,axiom,
! [X5: set_set_a,Y4: set_set_a] :
( ( sup_sup_set_set_a @ X5 @ ( sup_sup_set_set_a @ X5 @ Y4 ) )
= ( sup_sup_set_set_a @ X5 @ Y4 ) ) ).
% sup_left_idem
thf(fact_600_sup__left__idem,axiom,
! [X5: set_a,Y4: set_a] :
( ( sup_sup_set_a @ X5 @ ( sup_sup_set_a @ X5 @ Y4 ) )
= ( sup_sup_set_a @ X5 @ Y4 ) ) ).
% sup_left_idem
thf(fact_601_sup__left__idem,axiom,
! [X5: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X5 @ ( sup_su3048258781599657691od_a_a @ X5 @ Y4 ) )
= ( sup_su3048258781599657691od_a_a @ X5 @ Y4 ) ) ).
% sup_left_idem
thf(fact_602_sup__left__idem,axiom,
! [X5: set_list_a,Y4: set_list_a] :
( ( sup_sup_set_list_a @ X5 @ ( sup_sup_set_list_a @ X5 @ Y4 ) )
= ( sup_sup_set_list_a @ X5 @ Y4 ) ) ).
% sup_left_idem
thf(fact_603_sup_Oright__idem,axiom,
! [A2: set_set_a,B7: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A2 @ B7 ) @ B7 )
= ( sup_sup_set_set_a @ A2 @ B7 ) ) ).
% sup.right_idem
thf(fact_604_sup_Oright__idem,axiom,
! [A2: set_a,B7: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B7 ) @ B7 )
= ( sup_sup_set_a @ A2 @ B7 ) ) ).
% sup.right_idem
thf(fact_605_sup_Oright__idem,axiom,
! [A2: set_Product_prod_a_a,B7: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ A2 @ B7 ) @ B7 )
= ( sup_su3048258781599657691od_a_a @ A2 @ B7 ) ) ).
% sup.right_idem
thf(fact_606_sup_Oright__idem,axiom,
! [A2: set_list_a,B7: set_list_a] :
( ( sup_sup_set_list_a @ ( sup_sup_set_list_a @ A2 @ B7 ) @ B7 )
= ( sup_sup_set_list_a @ A2 @ B7 ) ) ).
% sup.right_idem
thf(fact_607_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_608_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_609_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_610_wellformed__alt__snd,axiom,
! [X5: a,Y4: a] :
( ( member_set_a @ ( insert_a @ X5 @ ( insert_a @ Y4 @ bot_bot_set_a ) ) @ edges )
=> ( member_a @ Y4 @ vertices ) ) ).
% wellformed_alt_snd
thf(fact_611_wellformed__alt__fst,axiom,
! [X5: a,Y4: a] :
( ( member_set_a @ ( insert_a @ X5 @ ( insert_a @ Y4 @ bot_bot_set_a ) ) @ edges )
=> ( member_a @ X5 @ vertices ) ) ).
% wellformed_alt_fst
thf(fact_612_is__edge__between__def,axiom,
( undire8544646567961481629ween_a
= ( ^ [X6: set_a,Y5: set_a,E5: set_a] :
? [X3: a,Y6: a] :
( ( E5
= ( insert_a @ X3 @ ( insert_a @ Y6 @ bot_bot_set_a ) ) )
& ( member_a @ X3 @ X6 )
& ( member_a @ Y6 @ Y5 ) ) ) ) ).
% is_edge_between_def
thf(fact_613_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_614_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_615_le__sup__iff,axiom,
! [X5: set_Product_prod_a_a,Y4: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ X5 @ Y4 ) @ Z2 )
= ( ( ord_le746702958409616551od_a_a @ X5 @ Z2 )
& ( ord_le746702958409616551od_a_a @ Y4 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_616_le__sup__iff,axiom,
! [X5: set_list_a,Y4: set_list_a,Z2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ X5 @ Y4 ) @ Z2 )
= ( ( ord_le8861187494160871172list_a @ X5 @ Z2 )
& ( ord_le8861187494160871172list_a @ Y4 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_617_le__sup__iff,axiom,
! [X5: set_set_a,Y4: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ X5 @ Y4 ) @ Z2 )
= ( ( ord_le3724670747650509150_set_a @ X5 @ Z2 )
& ( ord_le3724670747650509150_set_a @ Y4 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_618_le__sup__iff,axiom,
! [X5: set_a,Y4: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X5 @ Y4 ) @ Z2 )
= ( ( ord_less_eq_set_a @ X5 @ Z2 )
& ( ord_less_eq_set_a @ Y4 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_619_le__sup__iff,axiom,
! [X5: nat,Y4: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X5 @ Y4 ) @ Z2 )
= ( ( ord_less_eq_nat @ X5 @ Z2 )
& ( ord_less_eq_nat @ Y4 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_620_sup_Obounded__iff,axiom,
! [B7: set_Product_prod_a_a,C2: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ B7 @ C2 ) @ A2 )
= ( ( ord_le746702958409616551od_a_a @ B7 @ A2 )
& ( ord_le746702958409616551od_a_a @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_621_sup_Obounded__iff,axiom,
! [B7: set_list_a,C2: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ B7 @ C2 ) @ A2 )
= ( ( ord_le8861187494160871172list_a @ B7 @ A2 )
& ( ord_le8861187494160871172list_a @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_622_sup_Obounded__iff,axiom,
! [B7: set_set_a,C2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B7 @ C2 ) @ A2 )
= ( ( ord_le3724670747650509150_set_a @ B7 @ A2 )
& ( ord_le3724670747650509150_set_a @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_623_sup_Obounded__iff,axiom,
! [B7: set_a,C2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B7 @ C2 ) @ A2 )
= ( ( ord_less_eq_set_a @ B7 @ A2 )
& ( ord_less_eq_set_a @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_624_sup_Obounded__iff,axiom,
! [B7: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B7 @ C2 ) @ A2 )
= ( ( ord_less_eq_nat @ B7 @ A2 )
& ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_625_singletonI,axiom,
! [A2: set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_626_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_627_singletonI,axiom,
! [A2: product_prod_a_a] : ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) ).
% singletonI
thf(fact_628_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_629_finite__insert,axiom,
! [A2: list_a,A: set_list_a] :
( ( finite_finite_list_a @ ( insert_list_a @ A2 @ A ) )
= ( finite_finite_list_a @ A ) ) ).
% finite_insert
thf(fact_630_finite__insert,axiom,
! [A2: set_a,A: set_set_a] :
( ( finite_finite_set_a @ ( insert_set_a @ A2 @ A ) )
= ( finite_finite_set_a @ A ) ) ).
% finite_insert
thf(fact_631_finite__insert,axiom,
! [A2: a,A: set_a] :
( ( finite_finite_a @ ( insert_a @ A2 @ A ) )
= ( finite_finite_a @ A ) ) ).
% finite_insert
thf(fact_632_finite__insert,axiom,
! [A2: nat,A: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A2 @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_insert
thf(fact_633_finite__insert,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ A ) )
= ( finite6544458595007987280od_a_a @ A ) ) ).
% finite_insert
thf(fact_634_sup__bot__left,axiom,
! [X5: set_list_a] :
( ( sup_sup_set_list_a @ bot_bot_set_list_a @ X5 )
= X5 ) ).
% sup_bot_left
thf(fact_635_sup__bot__left,axiom,
! [X5: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ X5 )
= X5 ) ).
% sup_bot_left
thf(fact_636_sup__bot__left,axiom,
! [X5: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X5 )
= X5 ) ).
% sup_bot_left
thf(fact_637_sup__bot__left,axiom,
! [X5: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ bot_bo3357376287454694259od_a_a @ X5 )
= X5 ) ).
% sup_bot_left
thf(fact_638_sup__bot__left,axiom,
! [X5: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X5 )
= X5 ) ).
% sup_bot_left
thf(fact_639_sup__bot__right,axiom,
! [X5: set_list_a] :
( ( sup_sup_set_list_a @ X5 @ bot_bot_set_list_a )
= X5 ) ).
% sup_bot_right
thf(fact_640_sup__bot__right,axiom,
! [X5: set_set_a] :
( ( sup_sup_set_set_a @ X5 @ bot_bot_set_set_a )
= X5 ) ).
% sup_bot_right
thf(fact_641_sup__bot__right,axiom,
! [X5: set_a] :
( ( sup_sup_set_a @ X5 @ bot_bot_set_a )
= X5 ) ).
% sup_bot_right
thf(fact_642_sup__bot__right,axiom,
! [X5: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X5 @ bot_bo3357376287454694259od_a_a )
= X5 ) ).
% sup_bot_right
thf(fact_643_sup__bot__right,axiom,
! [X5: set_nat] :
( ( sup_sup_set_nat @ X5 @ bot_bot_set_nat )
= X5 ) ).
% sup_bot_right
thf(fact_644_bot__eq__sup__iff,axiom,
! [X5: set_list_a,Y4: set_list_a] :
( ( bot_bot_set_list_a
= ( sup_sup_set_list_a @ X5 @ Y4 ) )
= ( ( X5 = bot_bot_set_list_a )
& ( Y4 = bot_bot_set_list_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_645_bot__eq__sup__iff,axiom,
! [X5: set_set_a,Y4: set_set_a] :
( ( bot_bot_set_set_a
= ( sup_sup_set_set_a @ X5 @ Y4 ) )
= ( ( X5 = bot_bot_set_set_a )
& ( Y4 = bot_bot_set_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_646_bot__eq__sup__iff,axiom,
! [X5: set_a,Y4: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X5 @ Y4 ) )
= ( ( X5 = bot_bot_set_a )
& ( Y4 = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_647_bot__eq__sup__iff,axiom,
! [X5: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( sup_su3048258781599657691od_a_a @ X5 @ Y4 ) )
= ( ( X5 = bot_bo3357376287454694259od_a_a )
& ( Y4 = bot_bo3357376287454694259od_a_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_648_bot__eq__sup__iff,axiom,
! [X5: set_nat,Y4: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X5 @ Y4 ) )
= ( ( X5 = bot_bot_set_nat )
& ( Y4 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_649_sup__eq__bot__iff,axiom,
! [X5: set_list_a,Y4: set_list_a] :
( ( ( sup_sup_set_list_a @ X5 @ Y4 )
= bot_bot_set_list_a )
= ( ( X5 = bot_bot_set_list_a )
& ( Y4 = bot_bot_set_list_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_650_sup__eq__bot__iff,axiom,
! [X5: set_set_a,Y4: set_set_a] :
( ( ( sup_sup_set_set_a @ X5 @ Y4 )
= bot_bot_set_set_a )
= ( ( X5 = bot_bot_set_set_a )
& ( Y4 = bot_bot_set_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_651_sup__eq__bot__iff,axiom,
! [X5: set_a,Y4: set_a] :
( ( ( sup_sup_set_a @ X5 @ Y4 )
= bot_bot_set_a )
= ( ( X5 = bot_bot_set_a )
& ( Y4 = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_652_sup__eq__bot__iff,axiom,
! [X5: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( ( sup_su3048258781599657691od_a_a @ X5 @ Y4 )
= bot_bo3357376287454694259od_a_a )
= ( ( X5 = bot_bo3357376287454694259od_a_a )
& ( Y4 = bot_bo3357376287454694259od_a_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_653_sup__eq__bot__iff,axiom,
! [X5: set_nat,Y4: set_nat] :
( ( ( sup_sup_set_nat @ X5 @ Y4 )
= bot_bot_set_nat )
= ( ( X5 = bot_bot_set_nat )
& ( Y4 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_654_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_list_a,B7: set_list_a] :
( ( ( sup_sup_set_list_a @ A2 @ B7 )
= bot_bot_set_list_a )
= ( ( A2 = bot_bot_set_list_a )
& ( B7 = bot_bot_set_list_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_655_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_set_a,B7: set_set_a] :
( ( ( sup_sup_set_set_a @ A2 @ B7 )
= bot_bot_set_set_a )
= ( ( A2 = bot_bot_set_set_a )
& ( B7 = bot_bot_set_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_656_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_a,B7: set_a] :
( ( ( sup_sup_set_a @ A2 @ B7 )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B7 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_657_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_Product_prod_a_a,B7: set_Product_prod_a_a] :
( ( ( sup_su3048258781599657691od_a_a @ A2 @ B7 )
= bot_bo3357376287454694259od_a_a )
= ( ( A2 = bot_bo3357376287454694259od_a_a )
& ( B7 = bot_bo3357376287454694259od_a_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_658_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat,B7: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B7 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B7 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_659_sup__bot_Oleft__neutral,axiom,
! [A2: set_list_a] :
( ( sup_sup_set_list_a @ bot_bot_set_list_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_660_sup__bot_Oleft__neutral,axiom,
! [A2: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_661_sup__bot_Oleft__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_662_sup__bot_Oleft__neutral,axiom,
! [A2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ bot_bo3357376287454694259od_a_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_663_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_664_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_list_a,B7: set_list_a] :
( ( bot_bot_set_list_a
= ( sup_sup_set_list_a @ A2 @ B7 ) )
= ( ( A2 = bot_bot_set_list_a )
& ( B7 = bot_bot_set_list_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_665_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_set_a,B7: set_set_a] :
( ( bot_bot_set_set_a
= ( sup_sup_set_set_a @ A2 @ B7 ) )
= ( ( A2 = bot_bot_set_set_a )
& ( B7 = bot_bot_set_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_666_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_a,B7: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A2 @ B7 ) )
= ( ( A2 = bot_bot_set_a )
& ( B7 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_667_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_Product_prod_a_a,B7: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( sup_su3048258781599657691od_a_a @ A2 @ B7 ) )
= ( ( A2 = bot_bo3357376287454694259od_a_a )
& ( B7 = bot_bo3357376287454694259od_a_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_668_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat,B7: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A2 @ B7 ) )
= ( ( A2 = bot_bot_set_nat )
& ( B7 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_669_insert__subset,axiom,
! [X5: nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X5 @ A ) @ B )
= ( ( member_nat @ X5 @ B )
& ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_670_insert__subset,axiom,
! [X5: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ X5 @ A ) @ B )
= ( ( member1426531477525435216od_a_a @ X5 @ B )
& ( ord_le746702958409616551od_a_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_671_insert__subset,axiom,
! [X5: list_a,A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( insert_list_a @ X5 @ A ) @ B )
= ( ( member_list_a @ X5 @ B )
& ( ord_le8861187494160871172list_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_672_insert__subset,axiom,
! [X5: set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X5 @ A ) @ B )
= ( ( member_set_a @ X5 @ B )
& ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_673_insert__subset,axiom,
! [X5: a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X5 @ A ) @ B )
= ( ( member_a @ X5 @ B )
& ( ord_less_eq_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_674_Un__insert__left,axiom,
! [A2: set_a,B: set_set_a,C: set_set_a] :
( ( sup_sup_set_set_a @ ( insert_set_a @ A2 @ B ) @ C )
= ( insert_set_a @ A2 @ ( sup_sup_set_set_a @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_675_Un__insert__left,axiom,
! [A2: a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A2 @ B ) @ C )
= ( insert_a @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_676_Un__insert__left,axiom,
! [A2: product_prod_a_a,B: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ B ) @ C )
= ( insert4534936382041156343od_a_a @ A2 @ ( sup_su3048258781599657691od_a_a @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_677_Un__insert__left,axiom,
! [A2: list_a,B: set_list_a,C: set_list_a] :
( ( sup_sup_set_list_a @ ( insert_list_a @ A2 @ B ) @ C )
= ( insert_list_a @ A2 @ ( sup_sup_set_list_a @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_678_Un__insert__right,axiom,
! [A: set_set_a,A2: set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( insert_set_a @ A2 @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_679_Un__insert__right,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( sup_sup_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_680_Un__insert__right,axiom,
! [A: set_Product_prod_a_a,A2: product_prod_a_a,B: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A @ ( insert4534936382041156343od_a_a @ A2 @ B ) )
= ( insert4534936382041156343od_a_a @ A2 @ ( sup_su3048258781599657691od_a_a @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_681_Un__insert__right,axiom,
! [A: set_list_a,A2: list_a,B: set_list_a] :
( ( sup_sup_set_list_a @ A @ ( insert_list_a @ A2 @ B ) )
= ( insert_list_a @ A2 @ ( sup_sup_set_list_a @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_682_singleton__insert__inj__eq,axiom,
! [B7: nat,A2: nat,A: set_nat] :
( ( ( insert_nat @ B7 @ bot_bot_set_nat )
= ( insert_nat @ A2 @ A ) )
= ( ( A2 = B7 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B7 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_683_singleton__insert__inj__eq,axiom,
! [B7: product_prod_a_a,A2: product_prod_a_a,A: set_Product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ B7 @ bot_bo3357376287454694259od_a_a )
= ( insert4534936382041156343od_a_a @ A2 @ A ) )
= ( ( A2 = B7 )
& ( ord_le746702958409616551od_a_a @ A @ ( insert4534936382041156343od_a_a @ B7 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_684_singleton__insert__inj__eq,axiom,
! [B7: list_a,A2: list_a,A: set_list_a] :
( ( ( insert_list_a @ B7 @ bot_bot_set_list_a )
= ( insert_list_a @ A2 @ A ) )
= ( ( A2 = B7 )
& ( ord_le8861187494160871172list_a @ A @ ( insert_list_a @ B7 @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_685_singleton__insert__inj__eq,axiom,
! [B7: set_a,A2: set_a,A: set_set_a] :
( ( ( insert_set_a @ B7 @ bot_bot_set_set_a )
= ( insert_set_a @ A2 @ A ) )
= ( ( A2 = B7 )
& ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ B7 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_686_singleton__insert__inj__eq,axiom,
! [B7: a,A2: a,A: set_a] :
( ( ( insert_a @ B7 @ bot_bot_set_a )
= ( insert_a @ A2 @ A ) )
= ( ( A2 = B7 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B7 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_687_singleton__insert__inj__eq_H,axiom,
! [A2: nat,A: set_nat,B7: nat] :
( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B7 @ bot_bot_set_nat ) )
= ( ( A2 = B7 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B7 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_688_singleton__insert__inj__eq_H,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a,B7: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A2 @ A )
= ( insert4534936382041156343od_a_a @ B7 @ bot_bo3357376287454694259od_a_a ) )
= ( ( A2 = B7 )
& ( ord_le746702958409616551od_a_a @ A @ ( insert4534936382041156343od_a_a @ B7 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_689_singleton__insert__inj__eq_H,axiom,
! [A2: list_a,A: set_list_a,B7: list_a] :
( ( ( insert_list_a @ A2 @ A )
= ( insert_list_a @ B7 @ bot_bot_set_list_a ) )
= ( ( A2 = B7 )
& ( ord_le8861187494160871172list_a @ A @ ( insert_list_a @ B7 @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_690_singleton__insert__inj__eq_H,axiom,
! [A2: set_a,A: set_set_a,B7: set_a] :
( ( ( insert_set_a @ A2 @ A )
= ( insert_set_a @ B7 @ bot_bot_set_set_a ) )
= ( ( A2 = B7 )
& ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ B7 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_691_singleton__insert__inj__eq_H,axiom,
! [A2: a,A: set_a,B7: a] :
( ( ( insert_a @ A2 @ A )
= ( insert_a @ B7 @ bot_bot_set_a ) )
= ( ( A2 = B7 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B7 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_692_degree__none,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire8867928226783802224gree_a @ edges @ V )
= zero_zero_nat ) ) ).
% degree_none
thf(fact_693_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_694_mk__disjoint__insert,axiom,
! [A2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ? [B2: set_set_a] :
( ( A
= ( insert_set_a @ A2 @ B2 ) )
& ~ ( member_set_a @ A2 @ B2 ) ) ) ).
% mk_disjoint_insert
thf(fact_695_mk__disjoint__insert,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ? [B2: set_a] :
( ( A
= ( insert_a @ A2 @ B2 ) )
& ~ ( member_a @ A2 @ B2 ) ) ) ).
% mk_disjoint_insert
thf(fact_696_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ? [B2: set_nat] :
( ( A
= ( insert_nat @ A2 @ B2 ) )
& ~ ( member_nat @ A2 @ B2 ) ) ) ).
% mk_disjoint_insert
thf(fact_697_mk__disjoint__insert,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ A )
=> ? [B2: set_Product_prod_a_a] :
( ( A
= ( insert4534936382041156343od_a_a @ A2 @ B2 ) )
& ~ ( member1426531477525435216od_a_a @ A2 @ B2 ) ) ) ).
% mk_disjoint_insert
thf(fact_698_insert__commute,axiom,
! [X5: a,Y4: a,A: set_a] :
( ( insert_a @ X5 @ ( insert_a @ Y4 @ A ) )
= ( insert_a @ Y4 @ ( insert_a @ X5 @ A ) ) ) ).
% insert_commute
thf(fact_699_insert__commute,axiom,
! [X5: set_a,Y4: set_a,A: set_set_a] :
( ( insert_set_a @ X5 @ ( insert_set_a @ Y4 @ A ) )
= ( insert_set_a @ Y4 @ ( insert_set_a @ X5 @ A ) ) ) ).
% insert_commute
thf(fact_700_insert__eq__iff,axiom,
! [A2: set_a,A: set_set_a,B7: set_a,B: set_set_a] :
( ~ ( member_set_a @ A2 @ A )
=> ( ~ ( member_set_a @ B7 @ B )
=> ( ( ( insert_set_a @ A2 @ A )
= ( insert_set_a @ B7 @ B ) )
= ( ( ( A2 = B7 )
=> ( A = B ) )
& ( ( A2 != B7 )
=> ? [C3: set_set_a] :
( ( A
= ( insert_set_a @ B7 @ C3 ) )
& ~ ( member_set_a @ B7 @ C3 )
& ( B
= ( insert_set_a @ A2 @ C3 ) )
& ~ ( member_set_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_701_insert__eq__iff,axiom,
! [A2: a,A: set_a,B7: a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ~ ( member_a @ B7 @ B )
=> ( ( ( insert_a @ A2 @ A )
= ( insert_a @ B7 @ B ) )
= ( ( ( A2 = B7 )
=> ( A = B ) )
& ( ( A2 != B7 )
=> ? [C3: set_a] :
( ( A
= ( insert_a @ B7 @ C3 ) )
& ~ ( member_a @ B7 @ C3 )
& ( B
= ( insert_a @ A2 @ C3 ) )
& ~ ( member_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_702_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B7: nat,B: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ B7 @ B )
=> ( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B7 @ B ) )
= ( ( ( A2 = B7 )
=> ( A = B ) )
& ( ( A2 != B7 )
=> ? [C3: set_nat] :
( ( A
= ( insert_nat @ B7 @ C3 ) )
& ~ ( member_nat @ B7 @ C3 )
& ( B
= ( insert_nat @ A2 @ C3 ) )
& ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_703_insert__eq__iff,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a,B7: product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A2 @ A )
=> ( ~ ( member1426531477525435216od_a_a @ B7 @ B )
=> ( ( ( insert4534936382041156343od_a_a @ A2 @ A )
= ( insert4534936382041156343od_a_a @ B7 @ B ) )
= ( ( ( A2 = B7 )
=> ( A = B ) )
& ( ( A2 != B7 )
=> ? [C3: set_Product_prod_a_a] :
( ( A
= ( insert4534936382041156343od_a_a @ B7 @ C3 ) )
& ~ ( member1426531477525435216od_a_a @ B7 @ C3 )
& ( B
= ( insert4534936382041156343od_a_a @ A2 @ C3 ) )
& ~ ( member1426531477525435216od_a_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_704_insert__absorb,axiom,
! [A2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ( ( insert_set_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_705_insert__absorb,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_706_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_707_insert__absorb,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ A )
=> ( ( insert4534936382041156343od_a_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_708_insert__ident,axiom,
! [X5: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X5 @ A )
=> ( ~ ( member_set_a @ X5 @ B )
=> ( ( ( insert_set_a @ X5 @ A )
= ( insert_set_a @ X5 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_709_insert__ident,axiom,
! [X5: a,A: set_a,B: set_a] :
( ~ ( member_a @ X5 @ A )
=> ( ~ ( member_a @ X5 @ B )
=> ( ( ( insert_a @ X5 @ A )
= ( insert_a @ X5 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_710_insert__ident,axiom,
! [X5: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X5 @ A )
=> ( ~ ( member_nat @ X5 @ B )
=> ( ( ( insert_nat @ X5 @ A )
= ( insert_nat @ X5 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_711_insert__ident,axiom,
! [X5: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X5 @ A )
=> ( ~ ( member1426531477525435216od_a_a @ X5 @ B )
=> ( ( ( insert4534936382041156343od_a_a @ X5 @ A )
= ( insert4534936382041156343od_a_a @ X5 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_712_Set_Oset__insert,axiom,
! [X5: set_a,A: set_set_a] :
( ( member_set_a @ X5 @ A )
=> ~ ! [B2: set_set_a] :
( ( A
= ( insert_set_a @ X5 @ B2 ) )
=> ( member_set_a @ X5 @ B2 ) ) ) ).
% Set.set_insert
thf(fact_713_Set_Oset__insert,axiom,
! [X5: a,A: set_a] :
( ( member_a @ X5 @ A )
=> ~ ! [B2: set_a] :
( ( A
= ( insert_a @ X5 @ B2 ) )
=> ( member_a @ X5 @ B2 ) ) ) ).
% Set.set_insert
thf(fact_714_Set_Oset__insert,axiom,
! [X5: nat,A: set_nat] :
( ( member_nat @ X5 @ A )
=> ~ ! [B2: set_nat] :
( ( A
= ( insert_nat @ X5 @ B2 ) )
=> ( member_nat @ X5 @ B2 ) ) ) ).
% Set.set_insert
thf(fact_715_Set_Oset__insert,axiom,
! [X5: product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A )
=> ~ ! [B2: set_Product_prod_a_a] :
( ( A
= ( insert4534936382041156343od_a_a @ X5 @ B2 ) )
=> ( member1426531477525435216od_a_a @ X5 @ B2 ) ) ) ).
% Set.set_insert
thf(fact_716_insertI2,axiom,
! [A2: set_a,B: set_set_a,B7: set_a] :
( ( member_set_a @ A2 @ B )
=> ( member_set_a @ A2 @ ( insert_set_a @ B7 @ B ) ) ) ).
% insertI2
thf(fact_717_insertI2,axiom,
! [A2: a,B: set_a,B7: a] :
( ( member_a @ A2 @ B )
=> ( member_a @ A2 @ ( insert_a @ B7 @ B ) ) ) ).
% insertI2
thf(fact_718_insertI2,axiom,
! [A2: nat,B: set_nat,B7: nat] :
( ( member_nat @ A2 @ B )
=> ( member_nat @ A2 @ ( insert_nat @ B7 @ B ) ) ) ).
% insertI2
thf(fact_719_insertI2,axiom,
! [A2: product_prod_a_a,B: set_Product_prod_a_a,B7: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ B )
=> ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B7 @ B ) ) ) ).
% insertI2
thf(fact_720_insertI1,axiom,
! [A2: set_a,B: set_set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ B ) ) ).
% insertI1
thf(fact_721_insertI1,axiom,
! [A2: a,B: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B ) ) ).
% insertI1
thf(fact_722_insertI1,axiom,
! [A2: nat,B: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_723_insertI1,axiom,
! [A2: product_prod_a_a,B: set_Product_prod_a_a] : ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ A2 @ B ) ) ).
% insertI1
thf(fact_724_insertE,axiom,
! [A2: set_a,B7: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ ( insert_set_a @ B7 @ A ) )
=> ( ( A2 != B7 )
=> ( member_set_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_725_insertE,axiom,
! [A2: a,B7: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B7 @ A ) )
=> ( ( A2 != B7 )
=> ( member_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_726_insertE,axiom,
! [A2: nat,B7: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B7 @ A ) )
=> ( ( A2 != B7 )
=> ( member_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_727_insertE,axiom,
! [A2: product_prod_a_a,B7: product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B7 @ A ) )
=> ( ( A2 != B7 )
=> ( member1426531477525435216od_a_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_728_ulgraph_Odegree_Ocong,axiom,
undire8867928226783802224gree_a = undire8867928226783802224gree_a ).
% ulgraph.degree.cong
thf(fact_729_bot__set__def,axiom,
( bot_bot_set_list_a
= ( collect_list_a @ bot_bot_list_a_o ) ) ).
% bot_set_def
thf(fact_730_bot__set__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a @ bot_bot_set_a_o ) ) ).
% bot_set_def
thf(fact_731_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_732_bot__set__def,axiom,
( bot_bo3357376287454694259od_a_a
= ( collec3336397797384452498od_a_a @ bot_bo4160289986317612842_a_a_o ) ) ).
% bot_set_def
thf(fact_733_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_734_singletonD,axiom,
! [B7: set_a,A2: set_a] :
( ( member_set_a @ B7 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
=> ( B7 = A2 ) ) ).
% singletonD
thf(fact_735_singletonD,axiom,
! [B7: a,A2: a] :
( ( member_a @ B7 @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B7 = A2 ) ) ).
% singletonD
thf(fact_736_singletonD,axiom,
! [B7: product_prod_a_a,A2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B7 @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) )
=> ( B7 = A2 ) ) ).
% singletonD
thf(fact_737_singletonD,axiom,
! [B7: nat,A2: nat] :
( ( member_nat @ B7 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B7 = A2 ) ) ).
% singletonD
thf(fact_738_singleton__iff,axiom,
! [B7: set_a,A2: set_a] :
( ( member_set_a @ B7 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
= ( B7 = A2 ) ) ).
% singleton_iff
thf(fact_739_singleton__iff,axiom,
! [B7: a,A2: a] :
( ( member_a @ B7 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B7 = A2 ) ) ).
% singleton_iff
thf(fact_740_singleton__iff,axiom,
! [B7: product_prod_a_a,A2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B7 @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) )
= ( B7 = A2 ) ) ).
% singleton_iff
thf(fact_741_singleton__iff,axiom,
! [B7: nat,A2: nat] :
( ( member_nat @ B7 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B7 = A2 ) ) ).
% singleton_iff
thf(fact_742_doubleton__eq__iff,axiom,
! [A2: set_a,B7: set_a,C2: set_a,D2: set_a] :
( ( ( insert_set_a @ A2 @ ( insert_set_a @ B7 @ bot_bot_set_set_a ) )
= ( insert_set_a @ C2 @ ( insert_set_a @ D2 @ bot_bot_set_set_a ) ) )
= ( ( ( A2 = C2 )
& ( B7 = D2 ) )
| ( ( A2 = D2 )
& ( B7 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_743_doubleton__eq__iff,axiom,
! [A2: a,B7: a,C2: a,D2: a] :
( ( ( insert_a @ A2 @ ( insert_a @ B7 @ bot_bot_set_a ) )
= ( insert_a @ C2 @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
= ( ( ( A2 = C2 )
& ( B7 = D2 ) )
| ( ( A2 = D2 )
& ( B7 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_744_doubleton__eq__iff,axiom,
! [A2: product_prod_a_a,B7: product_prod_a_a,C2: product_prod_a_a,D2: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B7 @ bot_bo3357376287454694259od_a_a ) )
= ( insert4534936382041156343od_a_a @ C2 @ ( insert4534936382041156343od_a_a @ D2 @ bot_bo3357376287454694259od_a_a ) ) )
= ( ( ( A2 = C2 )
& ( B7 = D2 ) )
| ( ( A2 = D2 )
& ( B7 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_745_doubleton__eq__iff,axiom,
! [A2: nat,B7: nat,C2: nat,D2: nat] :
( ( ( insert_nat @ A2 @ ( insert_nat @ B7 @ bot_bot_set_nat ) )
= ( insert_nat @ C2 @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A2 = C2 )
& ( B7 = D2 ) )
| ( ( A2 = D2 )
& ( B7 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_746_insert__not__empty,axiom,
! [A2: set_a,A: set_set_a] :
( ( insert_set_a @ A2 @ A )
!= bot_bot_set_set_a ) ).
% insert_not_empty
thf(fact_747_insert__not__empty,axiom,
! [A2: a,A: set_a] :
( ( insert_a @ A2 @ A )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_748_insert__not__empty,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a] :
( ( insert4534936382041156343od_a_a @ A2 @ A )
!= bot_bo3357376287454694259od_a_a ) ).
% insert_not_empty
thf(fact_749_insert__not__empty,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ A )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_750_singleton__inject,axiom,
! [A2: set_a,B7: set_a] :
( ( ( insert_set_a @ A2 @ bot_bot_set_set_a )
= ( insert_set_a @ B7 @ bot_bot_set_set_a ) )
=> ( A2 = B7 ) ) ).
% singleton_inject
thf(fact_751_singleton__inject,axiom,
! [A2: a,B7: a] :
( ( ( insert_a @ A2 @ bot_bot_set_a )
= ( insert_a @ B7 @ bot_bot_set_a ) )
=> ( A2 = B7 ) ) ).
% singleton_inject
thf(fact_752_singleton__inject,axiom,
! [A2: product_prod_a_a,B7: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a )
= ( insert4534936382041156343od_a_a @ B7 @ bot_bo3357376287454694259od_a_a ) )
=> ( A2 = B7 ) ) ).
% singleton_inject
thf(fact_753_singleton__inject,axiom,
! [A2: nat,B7: nat] :
( ( ( insert_nat @ A2 @ bot_bot_set_nat )
= ( insert_nat @ B7 @ bot_bot_set_nat ) )
=> ( A2 = B7 ) ) ).
% singleton_inject
thf(fact_754_finite_OinsertI,axiom,
! [A: set_list_a,A2: list_a] :
( ( finite_finite_list_a @ A )
=> ( finite_finite_list_a @ ( insert_list_a @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_755_finite_OinsertI,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( finite_finite_set_a @ ( insert_set_a @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_756_finite_OinsertI,axiom,
! [A: set_a,A2: a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( insert_a @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_757_finite_OinsertI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( insert_nat @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_758_finite_OinsertI,axiom,
! [A: set_Product_prod_a_a,A2: product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A )
=> ( finite6544458595007987280od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_759_insert__mono,axiom,
! [C: set_Product_prod_a_a,D: set_Product_prod_a_a,A2: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C @ D )
=> ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ C ) @ ( insert4534936382041156343od_a_a @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_760_insert__mono,axiom,
! [C: set_list_a,D: set_list_a,A2: list_a] :
( ( ord_le8861187494160871172list_a @ C @ D )
=> ( ord_le8861187494160871172list_a @ ( insert_list_a @ A2 @ C ) @ ( insert_list_a @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_761_insert__mono,axiom,
! [C: set_set_a,D: set_set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ C @ D )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ A2 @ C ) @ ( insert_set_a @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_762_insert__mono,axiom,
! [C: set_a,D: set_a,A2: a] :
( ( ord_less_eq_set_a @ C @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A2 @ C ) @ ( insert_a @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_763_subset__insert,axiom,
! [X5: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X5 @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X5 @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_764_subset__insert,axiom,
! [X5: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X5 @ A )
=> ( ( ord_le746702958409616551od_a_a @ A @ ( insert4534936382041156343od_a_a @ X5 @ B ) )
= ( ord_le746702958409616551od_a_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_765_subset__insert,axiom,
! [X5: list_a,A: set_list_a,B: set_list_a] :
( ~ ( member_list_a @ X5 @ A )
=> ( ( ord_le8861187494160871172list_a @ A @ ( insert_list_a @ X5 @ B ) )
= ( ord_le8861187494160871172list_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_766_subset__insert,axiom,
! [X5: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X5 @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X5 @ B ) )
= ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_767_subset__insert,axiom,
! [X5: a,A: set_a,B: set_a] :
( ~ ( member_a @ X5 @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a @ X5 @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_768_subset__insertI,axiom,
! [B: set_Product_prod_a_a,A2: product_prod_a_a] : ( ord_le746702958409616551od_a_a @ B @ ( insert4534936382041156343od_a_a @ A2 @ B ) ) ).
% subset_insertI
thf(fact_769_subset__insertI,axiom,
! [B: set_list_a,A2: list_a] : ( ord_le8861187494160871172list_a @ B @ ( insert_list_a @ A2 @ B ) ) ).
% subset_insertI
thf(fact_770_subset__insertI,axiom,
! [B: set_set_a,A2: set_a] : ( ord_le3724670747650509150_set_a @ B @ ( insert_set_a @ A2 @ B ) ) ).
% subset_insertI
thf(fact_771_subset__insertI,axiom,
! [B: set_a,A2: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A2 @ B ) ) ).
% subset_insertI
thf(fact_772_subset__insertI2,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,B7: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ord_le746702958409616551od_a_a @ A @ ( insert4534936382041156343od_a_a @ B7 @ B ) ) ) ).
% subset_insertI2
thf(fact_773_subset__insertI2,axiom,
! [A: set_list_a,B: set_list_a,B7: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ord_le8861187494160871172list_a @ A @ ( insert_list_a @ B7 @ B ) ) ) ).
% subset_insertI2
thf(fact_774_subset__insertI2,axiom,
! [A: set_set_a,B: set_set_a,B7: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ B7 @ B ) ) ) ).
% subset_insertI2
thf(fact_775_subset__insertI2,axiom,
! [A: set_a,B: set_a,B7: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ B7 @ B ) ) ) ).
% subset_insertI2
thf(fact_776_ulgraph_Odegree__none,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ~ ( member_set_a @ V @ Vertices )
=> ( ( undire8939077443744732368_set_a @ Edges @ V )
= zero_zero_nat ) ) ) ).
% ulgraph.degree_none
thf(fact_777_ulgraph_Odegree__none,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ~ ( member_nat @ V @ Vertices )
=> ( ( undire6581030323043281630ee_nat @ Edges @ V )
= zero_zero_nat ) ) ) ).
% ulgraph.degree_none
thf(fact_778_ulgraph_Odegree__none,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ~ ( member1426531477525435216od_a_a @ V @ Vertices )
=> ( ( undire1436394852029823897od_a_a @ Edges @ V )
= zero_zero_nat ) ) ) ).
% ulgraph.degree_none
thf(fact_779_ulgraph_Odegree__none,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ~ ( member_a @ V @ Vertices )
=> ( ( undire8867928226783802224gree_a @ Edges @ V )
= zero_zero_nat ) ) ) ).
% ulgraph.degree_none
thf(fact_780_finite_Ocases,axiom,
! [A2: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( ( A2 != bot_bot_set_list_a )
=> ~ ! [A6: set_list_a] :
( ? [A3: list_a] :
( A2
= ( insert_list_a @ A3 @ A6 ) )
=> ~ ( finite_finite_list_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_781_finite_Ocases,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ~ ! [A6: set_set_a] :
( ? [A3: set_a] :
( A2
= ( insert_set_a @ A3 @ A6 ) )
=> ~ ( finite_finite_set_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_782_finite_Ocases,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( A2 != bot_bot_set_a )
=> ~ ! [A6: set_a] :
( ? [A3: a] :
( A2
= ( insert_a @ A3 @ A6 ) )
=> ~ ( finite_finite_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_783_finite_Ocases,axiom,
! [A2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A2 )
=> ( ( A2 != bot_bo3357376287454694259od_a_a )
=> ~ ! [A6: set_Product_prod_a_a] :
( ? [A3: product_prod_a_a] :
( A2
= ( insert4534936382041156343od_a_a @ A3 @ A6 ) )
=> ~ ( finite6544458595007987280od_a_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_784_finite_Ocases,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ~ ! [A6: set_nat] :
( ? [A3: nat] :
( A2
= ( insert_nat @ A3 @ A6 ) )
=> ~ ( finite_finite_nat @ A6 ) ) ) ) ).
% finite.cases
thf(fact_785_finite_Osimps,axiom,
( finite_finite_list_a
= ( ^ [A7: set_list_a] :
( ( A7 = bot_bot_set_list_a )
| ? [A4: set_list_a,B8: list_a] :
( ( A7
= ( insert_list_a @ B8 @ A4 ) )
& ( finite_finite_list_a @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_786_finite_Osimps,axiom,
( finite_finite_set_a
= ( ^ [A7: set_set_a] :
( ( A7 = bot_bot_set_set_a )
| ? [A4: set_set_a,B8: set_a] :
( ( A7
= ( insert_set_a @ B8 @ A4 ) )
& ( finite_finite_set_a @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_787_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A7: set_a] :
( ( A7 = bot_bot_set_a )
| ? [A4: set_a,B8: a] :
( ( A7
= ( insert_a @ B8 @ A4 ) )
& ( finite_finite_a @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_788_finite_Osimps,axiom,
( finite6544458595007987280od_a_a
= ( ^ [A7: set_Product_prod_a_a] :
( ( A7 = bot_bo3357376287454694259od_a_a )
| ? [A4: set_Product_prod_a_a,B8: product_prod_a_a] :
( ( A7
= ( insert4534936382041156343od_a_a @ B8 @ A4 ) )
& ( finite6544458595007987280od_a_a @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_789_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A7: set_nat] :
( ( A7 = bot_bot_set_nat )
| ? [A4: set_nat,B8: nat] :
( ( A7
= ( insert_nat @ B8 @ A4 ) )
& ( finite_finite_nat @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_790_finite__induct,axiom,
! [F: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [X4: list_a,F2: set_list_a] :
( ( finite_finite_list_a @ F2 )
=> ( ~ ( member_list_a @ X4 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_list_a @ X4 @ F2 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_791_finite__induct,axiom,
! [F: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X4: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ~ ( member_set_a @ X4 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_a @ X4 @ F2 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_792_finite__induct,axiom,
! [F: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X4: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ X4 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X4 @ F2 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_793_finite__induct,axiom,
! [F: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ F )
=> ( ( P @ bot_bo3357376287454694259od_a_a )
=> ( ! [X4: product_prod_a_a,F2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F2 )
=> ( ~ ( member1426531477525435216od_a_a @ X4 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert4534936382041156343od_a_a @ X4 @ F2 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_794_finite__induct,axiom,
! [F: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X4: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ~ ( member_nat @ X4 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ X4 @ F2 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_795_finite__ne__induct,axiom,
! [F: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F )
=> ( ( F != bot_bot_set_list_a )
=> ( ! [X4: list_a] : ( P @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) )
=> ( ! [X4: list_a,F2: set_list_a] :
( ( finite_finite_list_a @ F2 )
=> ( ( F2 != bot_bot_set_list_a )
=> ( ~ ( member_list_a @ X4 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_list_a @ X4 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_796_finite__ne__induct,axiom,
! [F: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F )
=> ( ( F != bot_bot_set_set_a )
=> ( ! [X4: set_a] : ( P @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) )
=> ( ! [X4: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ( F2 != bot_bot_set_set_a )
=> ( ~ ( member_set_a @ X4 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_a @ X4 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_797_finite__ne__induct,axiom,
! [F: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( F != bot_bot_set_a )
=> ( ! [X4: a] : ( P @ ( insert_a @ X4 @ bot_bot_set_a ) )
=> ( ! [X4: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ~ ( member_a @ X4 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X4 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_798_finite__ne__induct,axiom,
! [F: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ F )
=> ( ( F != bot_bo3357376287454694259od_a_a )
=> ( ! [X4: product_prod_a_a] : ( P @ ( insert4534936382041156343od_a_a @ X4 @ bot_bo3357376287454694259od_a_a ) )
=> ( ! [X4: product_prod_a_a,F2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F2 )
=> ( ( F2 != bot_bo3357376287454694259od_a_a )
=> ( ~ ( member1426531477525435216od_a_a @ X4 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert4534936382041156343od_a_a @ X4 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_799_finite__ne__induct,axiom,
! [F: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( F != bot_bot_set_nat )
=> ( ! [X4: nat] : ( P @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
=> ( ! [X4: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X4 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ X4 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_800_infinite__finite__induct,axiom,
! [P: set_list_a > $o,A: set_list_a] :
( ! [A6: set_list_a] :
( ~ ( finite_finite_list_a @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [X4: list_a,F2: set_list_a] :
( ( finite_finite_list_a @ F2 )
=> ( ~ ( member_list_a @ X4 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_list_a @ X4 @ F2 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_801_infinite__finite__induct,axiom,
! [P: set_set_a > $o,A: set_set_a] :
( ! [A6: set_set_a] :
( ~ ( finite_finite_set_a @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X4: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ~ ( member_set_a @ X4 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_a @ X4 @ F2 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_802_infinite__finite__induct,axiom,
! [P: set_a > $o,A: set_a] :
( ! [A6: set_a] :
( ~ ( finite_finite_a @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X4: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ X4 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X4 @ F2 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_803_infinite__finite__induct,axiom,
! [P: set_Product_prod_a_a > $o,A: set_Product_prod_a_a] :
( ! [A6: set_Product_prod_a_a] :
( ~ ( finite6544458595007987280od_a_a @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bo3357376287454694259od_a_a )
=> ( ! [X4: product_prod_a_a,F2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F2 )
=> ( ~ ( member1426531477525435216od_a_a @ X4 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert4534936382041156343od_a_a @ X4 @ F2 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_804_infinite__finite__induct,axiom,
! [P: set_nat > $o,A: set_nat] :
( ! [A6: set_nat] :
( ~ ( finite_finite_nat @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X4: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ~ ( member_nat @ X4 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ X4 @ F2 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_805_subset__singletonD,axiom,
! [A: set_nat,X5: nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X5 @ bot_bot_set_nat ) )
=> ( ( A = bot_bot_set_nat )
| ( A
= ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_806_subset__singletonD,axiom,
! [A: set_Product_prod_a_a,X5: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ ( insert4534936382041156343od_a_a @ X5 @ bot_bo3357376287454694259od_a_a ) )
=> ( ( A = bot_bo3357376287454694259od_a_a )
| ( A
= ( insert4534936382041156343od_a_a @ X5 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% subset_singletonD
thf(fact_807_subset__singletonD,axiom,
! [A: set_list_a,X5: list_a] :
( ( ord_le8861187494160871172list_a @ A @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) )
=> ( ( A = bot_bot_set_list_a )
| ( A
= ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) ) ) ).
% subset_singletonD
thf(fact_808_subset__singletonD,axiom,
! [A: set_set_a,X5: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X5 @ bot_bot_set_set_a ) )
=> ( ( A = bot_bot_set_set_a )
| ( A
= ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_809_subset__singletonD,axiom,
! [A: set_a,X5: a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X5 @ bot_bot_set_a ) )
=> ( ( A = bot_bot_set_a )
| ( A
= ( insert_a @ X5 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_810_subset__singleton__iff,axiom,
! [X2: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ X2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( ( X2 = bot_bot_set_nat )
| ( X2
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_811_subset__singleton__iff,axiom,
! [X2: set_Product_prod_a_a,A2: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X2 @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) )
= ( ( X2 = bot_bo3357376287454694259od_a_a )
| ( X2
= ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_812_subset__singleton__iff,axiom,
! [X2: set_list_a,A2: list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) )
= ( ( X2 = bot_bot_set_list_a )
| ( X2
= ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_813_subset__singleton__iff,axiom,
! [X2: set_set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
= ( ( X2 = bot_bot_set_set_a )
| ( X2
= ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_814_subset__singleton__iff,axiom,
! [X2: set_a,A2: a] :
( ( ord_less_eq_set_a @ X2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( ( X2 = bot_bot_set_a )
| ( X2
= ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_815_singleton__Un__iff,axiom,
! [X5: list_a,A: set_list_a,B: set_list_a] :
( ( ( insert_list_a @ X5 @ bot_bot_set_list_a )
= ( sup_sup_set_list_a @ A @ B ) )
= ( ( ( A = bot_bot_set_list_a )
& ( B
= ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) )
| ( ( A
= ( insert_list_a @ X5 @ bot_bot_set_list_a ) )
& ( B = bot_bot_set_list_a ) )
| ( ( A
= ( insert_list_a @ X5 @ bot_bot_set_list_a ) )
& ( B
= ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_816_singleton__Un__iff,axiom,
! [X5: set_a,A: set_set_a,B: set_set_a] :
( ( ( insert_set_a @ X5 @ bot_bot_set_set_a )
= ( sup_sup_set_set_a @ A @ B ) )
= ( ( ( A = bot_bot_set_set_a )
& ( B
= ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) )
| ( ( A
= ( insert_set_a @ X5 @ bot_bot_set_set_a ) )
& ( B = bot_bot_set_set_a ) )
| ( ( A
= ( insert_set_a @ X5 @ bot_bot_set_set_a ) )
& ( B
= ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_817_singleton__Un__iff,axiom,
! [X5: a,A: set_a,B: set_a] :
( ( ( insert_a @ X5 @ bot_bot_set_a )
= ( sup_sup_set_a @ A @ B ) )
= ( ( ( A = bot_bot_set_a )
& ( B
= ( insert_a @ X5 @ bot_bot_set_a ) ) )
| ( ( A
= ( insert_a @ X5 @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A
= ( insert_a @ X5 @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X5 @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_818_singleton__Un__iff,axiom,
! [X5: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ X5 @ bot_bo3357376287454694259od_a_a )
= ( sup_su3048258781599657691od_a_a @ A @ B ) )
= ( ( ( A = bot_bo3357376287454694259od_a_a )
& ( B
= ( insert4534936382041156343od_a_a @ X5 @ bot_bo3357376287454694259od_a_a ) ) )
| ( ( A
= ( insert4534936382041156343od_a_a @ X5 @ bot_bo3357376287454694259od_a_a ) )
& ( B = bot_bo3357376287454694259od_a_a ) )
| ( ( A
= ( insert4534936382041156343od_a_a @ X5 @ bot_bo3357376287454694259od_a_a ) )
& ( B
= ( insert4534936382041156343od_a_a @ X5 @ bot_bo3357376287454694259od_a_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_819_singleton__Un__iff,axiom,
! [X5: nat,A: set_nat,B: set_nat] :
( ( ( insert_nat @ X5 @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A @ B ) )
= ( ( ( A = bot_bot_set_nat )
& ( B
= ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat @ X5 @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat @ X5 @ bot_bot_set_nat ) )
& ( B
= ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_820_Un__singleton__iff,axiom,
! [A: set_list_a,B: set_list_a,X5: list_a] :
( ( ( sup_sup_set_list_a @ A @ B )
= ( insert_list_a @ X5 @ bot_bot_set_list_a ) )
= ( ( ( A = bot_bot_set_list_a )
& ( B
= ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) )
| ( ( A
= ( insert_list_a @ X5 @ bot_bot_set_list_a ) )
& ( B = bot_bot_set_list_a ) )
| ( ( A
= ( insert_list_a @ X5 @ bot_bot_set_list_a ) )
& ( B
= ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_821_Un__singleton__iff,axiom,
! [A: set_set_a,B: set_set_a,X5: set_a] :
( ( ( sup_sup_set_set_a @ A @ B )
= ( insert_set_a @ X5 @ bot_bot_set_set_a ) )
= ( ( ( A = bot_bot_set_set_a )
& ( B
= ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) )
| ( ( A
= ( insert_set_a @ X5 @ bot_bot_set_set_a ) )
& ( B = bot_bot_set_set_a ) )
| ( ( A
= ( insert_set_a @ X5 @ bot_bot_set_set_a ) )
& ( B
= ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_822_Un__singleton__iff,axiom,
! [A: set_a,B: set_a,X5: a] :
( ( ( sup_sup_set_a @ A @ B )
= ( insert_a @ X5 @ bot_bot_set_a ) )
= ( ( ( A = bot_bot_set_a )
& ( B
= ( insert_a @ X5 @ bot_bot_set_a ) ) )
| ( ( A
= ( insert_a @ X5 @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A
= ( insert_a @ X5 @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X5 @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_823_Un__singleton__iff,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,X5: product_prod_a_a] :
( ( ( sup_su3048258781599657691od_a_a @ A @ B )
= ( insert4534936382041156343od_a_a @ X5 @ bot_bo3357376287454694259od_a_a ) )
= ( ( ( A = bot_bo3357376287454694259od_a_a )
& ( B
= ( insert4534936382041156343od_a_a @ X5 @ bot_bo3357376287454694259od_a_a ) ) )
| ( ( A
= ( insert4534936382041156343od_a_a @ X5 @ bot_bo3357376287454694259od_a_a ) )
& ( B = bot_bo3357376287454694259od_a_a ) )
| ( ( A
= ( insert4534936382041156343od_a_a @ X5 @ bot_bo3357376287454694259od_a_a ) )
& ( B
= ( insert4534936382041156343od_a_a @ X5 @ bot_bo3357376287454694259od_a_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_824_Un__singleton__iff,axiom,
! [A: set_nat,B: set_nat,X5: nat] :
( ( ( sup_sup_set_nat @ A @ B )
= ( insert_nat @ X5 @ bot_bot_set_nat ) )
= ( ( ( A = bot_bot_set_nat )
& ( B
= ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat @ X5 @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat @ X5 @ bot_bot_set_nat ) )
& ( B
= ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_825_insert__is__Un,axiom,
( insert_list_a
= ( ^ [A7: list_a] : ( sup_sup_set_list_a @ ( insert_list_a @ A7 @ bot_bot_set_list_a ) ) ) ) ).
% insert_is_Un
thf(fact_826_insert__is__Un,axiom,
( insert_set_a
= ( ^ [A7: set_a] : ( sup_sup_set_set_a @ ( insert_set_a @ A7 @ bot_bot_set_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_827_insert__is__Un,axiom,
( insert_a
= ( ^ [A7: a] : ( sup_sup_set_a @ ( insert_a @ A7 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_828_insert__is__Un,axiom,
( insert4534936382041156343od_a_a
= ( ^ [A7: product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ ( insert4534936382041156343od_a_a @ A7 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% insert_is_Un
thf(fact_829_insert__is__Un,axiom,
( insert_nat
= ( ^ [A7: nat] : ( sup_sup_set_nat @ ( insert_nat @ A7 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_830_card__insert__le,axiom,
! [A: set_set_a,X5: set_a] : ( ord_less_eq_nat @ ( finite_card_set_a @ A ) @ ( finite_card_set_a @ ( insert_set_a @ X5 @ A ) ) ) ).
% card_insert_le
thf(fact_831_card__insert__le,axiom,
! [A: set_a,X5: a] : ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ ( insert_a @ X5 @ A ) ) ) ).
% card_insert_le
thf(fact_832_card__insert__le,axiom,
! [A: set_Product_prod_a_a,X5: product_prod_a_a] : ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ A ) @ ( finite4795055649997197647od_a_a @ ( insert4534936382041156343od_a_a @ X5 @ A ) ) ) ).
% card_insert_le
thf(fact_833_card__insert__le,axiom,
! [A: set_nat,X5: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ ( insert_nat @ X5 @ A ) ) ) ).
% card_insert_le
thf(fact_834_graph__system_Owellformed__alt__snd,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X5: set_a,Y4: set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( member_set_set_a @ ( insert_set_a @ X5 @ ( insert_set_a @ Y4 @ bot_bot_set_set_a ) ) @ Edges )
=> ( member_set_a @ Y4 @ Vertices ) ) ) ).
% graph_system.wellformed_alt_snd
thf(fact_835_graph__system_Owellformed__alt__snd,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,X5: product_prod_a_a,Y4: product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ X5 @ ( insert4534936382041156343od_a_a @ Y4 @ bot_bo3357376287454694259od_a_a ) ) @ Edges )
=> ( member1426531477525435216od_a_a @ Y4 @ Vertices ) ) ) ).
% graph_system.wellformed_alt_snd
thf(fact_836_graph__system_Owellformed__alt__snd,axiom,
! [Vertices: set_nat,Edges: set_set_nat,X5: nat,Y4: nat] :
( ( undire7481384412329822504em_nat @ Vertices @ Edges )
=> ( ( member_set_nat @ ( insert_nat @ X5 @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) @ Edges )
=> ( member_nat @ Y4 @ Vertices ) ) ) ).
% graph_system.wellformed_alt_snd
thf(fact_837_graph__system_Owellformed__alt__snd,axiom,
! [Vertices: set_a,Edges: set_set_a,X5: a,Y4: a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( member_set_a @ ( insert_a @ X5 @ ( insert_a @ Y4 @ bot_bot_set_a ) ) @ Edges )
=> ( member_a @ Y4 @ Vertices ) ) ) ).
% graph_system.wellformed_alt_snd
thf(fact_838_graph__system_Owellformed__alt__fst,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X5: set_a,Y4: set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( member_set_set_a @ ( insert_set_a @ X5 @ ( insert_set_a @ Y4 @ bot_bot_set_set_a ) ) @ Edges )
=> ( member_set_a @ X5 @ Vertices ) ) ) ).
% graph_system.wellformed_alt_fst
thf(fact_839_graph__system_Owellformed__alt__fst,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,X5: product_prod_a_a,Y4: product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices @ Edges )
=> ( ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ X5 @ ( insert4534936382041156343od_a_a @ Y4 @ bot_bo3357376287454694259od_a_a ) ) @ Edges )
=> ( member1426531477525435216od_a_a @ X5 @ Vertices ) ) ) ).
% graph_system.wellformed_alt_fst
thf(fact_840_graph__system_Owellformed__alt__fst,axiom,
! [Vertices: set_nat,Edges: set_set_nat,X5: nat,Y4: nat] :
( ( undire7481384412329822504em_nat @ Vertices @ Edges )
=> ( ( member_set_nat @ ( insert_nat @ X5 @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) @ Edges )
=> ( member_nat @ X5 @ Vertices ) ) ) ).
% graph_system.wellformed_alt_fst
thf(fact_841_graph__system_Owellformed__alt__fst,axiom,
! [Vertices: set_a,Edges: set_set_a,X5: a,Y4: a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( member_set_a @ ( insert_a @ X5 @ ( insert_a @ Y4 @ bot_bot_set_a ) ) @ Edges )
=> ( member_a @ X5 @ Vertices ) ) ) ).
% graph_system.wellformed_alt_fst
thf(fact_842_ulgraph_Oincident__loops__simp_I1_J,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire5774735625301615776_set_a @ Edges @ V )
=> ( ( undire7215034953758041409_set_a @ Edges @ V )
= ( insert_set_set_a @ ( insert_set_a @ V @ bot_bot_set_set_a ) @ bot_bo3380559777022489994_set_a ) ) ) ) ).
% ulgraph.incident_loops_simp(1)
thf(fact_843_ulgraph_Oincident__loops__simp_I1_J,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire7777398424729533289od_a_a @ Edges @ V )
=> ( ( undire3049230956220217098od_a_a @ Edges @ V )
= ( insert914553114930139863od_a_a @ ( insert4534936382041156343od_a_a @ V @ bot_bo3357376287454694259od_a_a ) @ bot_bo777872063958040403od_a_a ) ) ) ) ).
% ulgraph.incident_loops_simp(1)
thf(fact_844_ulgraph_Oincident__loops__simp_I1_J,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire5005864372999571214op_nat @ Edges @ V )
=> ( ( undire1050940535076293677ps_nat @ Edges @ V )
= ( insert_set_nat @ ( insert_nat @ V @ bot_bot_set_nat ) @ bot_bot_set_set_nat ) ) ) ) ).
% ulgraph.incident_loops_simp(1)
thf(fact_845_ulgraph_Oincident__loops__simp_I1_J,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3617971648856834880loop_a @ Edges @ V )
=> ( ( undire4753905205749729249oops_a @ Edges @ V )
= ( insert_set_a @ ( insert_a @ V @ bot_bot_set_a ) @ bot_bot_set_set_a ) ) ) ) ).
% ulgraph.incident_loops_simp(1)
thf(fact_846_finite__subset__induct,axiom,
! [F: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( ord_less_eq_set_nat @ F @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( member_nat @ A3 @ A )
=> ( ~ ( member_nat @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ A3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_847_finite__subset__induct,axiom,
! [F: set_Product_prod_a_a,A: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ F )
=> ( ( ord_le746702958409616551od_a_a @ F @ A )
=> ( ( P @ bot_bo3357376287454694259od_a_a )
=> ( ! [A3: product_prod_a_a,F2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F2 )
=> ( ( member1426531477525435216od_a_a @ A3 @ A )
=> ( ~ ( member1426531477525435216od_a_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert4534936382041156343od_a_a @ A3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_848_finite__subset__induct,axiom,
! [F: set_list_a,A: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F )
=> ( ( ord_le8861187494160871172list_a @ F @ A )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A3: list_a,F2: set_list_a] :
( ( finite_finite_list_a @ F2 )
=> ( ( member_list_a @ A3 @ A )
=> ( ~ ( member_list_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_list_a @ A3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_849_finite__subset__induct,axiom,
! [F: set_set_a,A: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F )
=> ( ( ord_le3724670747650509150_set_a @ F @ A )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A3: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ( member_set_a @ A3 @ A )
=> ( ~ ( member_set_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_a @ A3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_850_finite__subset__induct,axiom,
! [F: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( ord_less_eq_set_a @ F @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( member_a @ A3 @ A )
=> ( ~ ( member_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ A3 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_851_finite__subset__induct_H,axiom,
! [F: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( ord_less_eq_set_nat @ F @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( member_nat @ A3 @ A )
=> ( ( ord_less_eq_set_nat @ F2 @ A )
=> ( ~ ( member_nat @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_nat @ A3 @ F2 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_852_finite__subset__induct_H,axiom,
! [F: set_Product_prod_a_a,A: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ F )
=> ( ( ord_le746702958409616551od_a_a @ F @ A )
=> ( ( P @ bot_bo3357376287454694259od_a_a )
=> ( ! [A3: product_prod_a_a,F2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F2 )
=> ( ( member1426531477525435216od_a_a @ A3 @ A )
=> ( ( ord_le746702958409616551od_a_a @ F2 @ A )
=> ( ~ ( member1426531477525435216od_a_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert4534936382041156343od_a_a @ A3 @ F2 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_853_finite__subset__induct_H,axiom,
! [F: set_list_a,A: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F )
=> ( ( ord_le8861187494160871172list_a @ F @ A )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A3: list_a,F2: set_list_a] :
( ( finite_finite_list_a @ F2 )
=> ( ( member_list_a @ A3 @ A )
=> ( ( ord_le8861187494160871172list_a @ F2 @ A )
=> ( ~ ( member_list_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_list_a @ A3 @ F2 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_854_finite__subset__induct_H,axiom,
! [F: set_set_a,A: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F )
=> ( ( ord_le3724670747650509150_set_a @ F @ A )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A3: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ( member_set_a @ A3 @ A )
=> ( ( ord_le3724670747650509150_set_a @ F2 @ A )
=> ( ~ ( member_set_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_a @ A3 @ F2 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_855_finite__subset__induct_H,axiom,
! [F: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( ord_less_eq_set_a @ F @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( member_a @ A3 @ A )
=> ( ( ord_less_eq_set_a @ F2 @ A )
=> ( ~ ( member_a @ A3 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ A3 @ F2 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_856_card__1__singletonE,axiom,
! [A: set_set_a] :
( ( ( finite_card_set_a @ A )
= one_one_nat )
=> ~ ! [X4: set_a] :
( A
!= ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) ) ).
% card_1_singletonE
thf(fact_857_card__1__singletonE,axiom,
! [A: set_a] :
( ( ( finite_card_a @ A )
= one_one_nat )
=> ~ ! [X4: a] :
( A
!= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ).
% card_1_singletonE
thf(fact_858_card__1__singletonE,axiom,
! [A: set_Product_prod_a_a] :
( ( ( finite4795055649997197647od_a_a @ A )
= one_one_nat )
=> ~ ! [X4: product_prod_a_a] :
( A
!= ( insert4534936382041156343od_a_a @ X4 @ bot_bo3357376287454694259od_a_a ) ) ) ).
% card_1_singletonE
thf(fact_859_card__1__singletonE,axiom,
! [A: set_nat] :
( ( ( finite_card_nat @ A )
= one_one_nat )
=> ~ ! [X4: nat] :
( A
!= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ).
% card_1_singletonE
thf(fact_860_inf__sup__aci_I8_J,axiom,
! [X5: set_set_a,Y4: set_set_a] :
( ( sup_sup_set_set_a @ X5 @ ( sup_sup_set_set_a @ X5 @ Y4 ) )
= ( sup_sup_set_set_a @ X5 @ Y4 ) ) ).
% inf_sup_aci(8)
thf(fact_861_inf__sup__aci_I8_J,axiom,
! [X5: set_a,Y4: set_a] :
( ( sup_sup_set_a @ X5 @ ( sup_sup_set_a @ X5 @ Y4 ) )
= ( sup_sup_set_a @ X5 @ Y4 ) ) ).
% inf_sup_aci(8)
thf(fact_862_inf__sup__aci_I8_J,axiom,
! [X5: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X5 @ ( sup_su3048258781599657691od_a_a @ X5 @ Y4 ) )
= ( sup_su3048258781599657691od_a_a @ X5 @ Y4 ) ) ).
% inf_sup_aci(8)
thf(fact_863_inf__sup__aci_I8_J,axiom,
! [X5: set_list_a,Y4: set_list_a] :
( ( sup_sup_set_list_a @ X5 @ ( sup_sup_set_list_a @ X5 @ Y4 ) )
= ( sup_sup_set_list_a @ X5 @ Y4 ) ) ).
% inf_sup_aci(8)
thf(fact_864_inf__sup__aci_I7_J,axiom,
! [X5: set_set_a,Y4: set_set_a,Z2: set_set_a] :
( ( sup_sup_set_set_a @ X5 @ ( sup_sup_set_set_a @ Y4 @ Z2 ) )
= ( sup_sup_set_set_a @ Y4 @ ( sup_sup_set_set_a @ X5 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_865_inf__sup__aci_I7_J,axiom,
! [X5: set_a,Y4: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X5 @ ( sup_sup_set_a @ Y4 @ Z2 ) )
= ( sup_sup_set_a @ Y4 @ ( sup_sup_set_a @ X5 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_866_inf__sup__aci_I7_J,axiom,
! [X5: set_Product_prod_a_a,Y4: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X5 @ ( sup_su3048258781599657691od_a_a @ Y4 @ Z2 ) )
= ( sup_su3048258781599657691od_a_a @ Y4 @ ( sup_su3048258781599657691od_a_a @ X5 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_867_inf__sup__aci_I7_J,axiom,
! [X5: set_list_a,Y4: set_list_a,Z2: set_list_a] :
( ( sup_sup_set_list_a @ X5 @ ( sup_sup_set_list_a @ Y4 @ Z2 ) )
= ( sup_sup_set_list_a @ Y4 @ ( sup_sup_set_list_a @ X5 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_868_inf__sup__aci_I6_J,axiom,
! [X5: set_set_a,Y4: set_set_a,Z2: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ X5 @ Y4 ) @ Z2 )
= ( sup_sup_set_set_a @ X5 @ ( sup_sup_set_set_a @ Y4 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_869_inf__sup__aci_I6_J,axiom,
! [X5: set_a,Y4: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X5 @ Y4 ) @ Z2 )
= ( sup_sup_set_a @ X5 @ ( sup_sup_set_a @ Y4 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_870_inf__sup__aci_I6_J,axiom,
! [X5: set_Product_prod_a_a,Y4: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ X5 @ Y4 ) @ Z2 )
= ( sup_su3048258781599657691od_a_a @ X5 @ ( sup_su3048258781599657691od_a_a @ Y4 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_871_inf__sup__aci_I6_J,axiom,
! [X5: set_list_a,Y4: set_list_a,Z2: set_list_a] :
( ( sup_sup_set_list_a @ ( sup_sup_set_list_a @ X5 @ Y4 ) @ Z2 )
= ( sup_sup_set_list_a @ X5 @ ( sup_sup_set_list_a @ Y4 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_872_inf__sup__aci_I5_J,axiom,
( sup_sup_set_set_a
= ( ^ [X3: set_set_a,Y6: set_set_a] : ( sup_sup_set_set_a @ Y6 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_873_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X3: set_a,Y6: set_a] : ( sup_sup_set_a @ Y6 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_874_inf__sup__aci_I5_J,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [X3: set_Product_prod_a_a,Y6: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ Y6 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_875_inf__sup__aci_I5_J,axiom,
( sup_sup_set_list_a
= ( ^ [X3: set_list_a,Y6: set_list_a] : ( sup_sup_set_list_a @ Y6 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_876_sup_Oassoc,axiom,
! [A2: set_set_a,B7: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A2 @ B7 ) @ C2 )
= ( sup_sup_set_set_a @ A2 @ ( sup_sup_set_set_a @ B7 @ C2 ) ) ) ).
% sup.assoc
thf(fact_877_sup_Oassoc,axiom,
! [A2: set_a,B7: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B7 ) @ C2 )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B7 @ C2 ) ) ) ).
% sup.assoc
thf(fact_878_sup_Oassoc,axiom,
! [A2: set_Product_prod_a_a,B7: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ A2 @ B7 ) @ C2 )
= ( sup_su3048258781599657691od_a_a @ A2 @ ( sup_su3048258781599657691od_a_a @ B7 @ C2 ) ) ) ).
% sup.assoc
thf(fact_879_sup_Oassoc,axiom,
! [A2: set_list_a,B7: set_list_a,C2: set_list_a] :
( ( sup_sup_set_list_a @ ( sup_sup_set_list_a @ A2 @ B7 ) @ C2 )
= ( sup_sup_set_list_a @ A2 @ ( sup_sup_set_list_a @ B7 @ C2 ) ) ) ).
% sup.assoc
thf(fact_880_sup__assoc,axiom,
! [X5: set_set_a,Y4: set_set_a,Z2: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ X5 @ Y4 ) @ Z2 )
= ( sup_sup_set_set_a @ X5 @ ( sup_sup_set_set_a @ Y4 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_881_sup__assoc,axiom,
! [X5: set_a,Y4: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X5 @ Y4 ) @ Z2 )
= ( sup_sup_set_a @ X5 @ ( sup_sup_set_a @ Y4 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_882_sup__assoc,axiom,
! [X5: set_Product_prod_a_a,Y4: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ X5 @ Y4 ) @ Z2 )
= ( sup_su3048258781599657691od_a_a @ X5 @ ( sup_su3048258781599657691od_a_a @ Y4 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_883_sup__assoc,axiom,
! [X5: set_list_a,Y4: set_list_a,Z2: set_list_a] :
( ( sup_sup_set_list_a @ ( sup_sup_set_list_a @ X5 @ Y4 ) @ Z2 )
= ( sup_sup_set_list_a @ X5 @ ( sup_sup_set_list_a @ Y4 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_884_sup_Ocommute,axiom,
( sup_sup_set_set_a
= ( ^ [A7: set_set_a,B8: set_set_a] : ( sup_sup_set_set_a @ B8 @ A7 ) ) ) ).
% sup.commute
thf(fact_885_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A7: set_a,B8: set_a] : ( sup_sup_set_a @ B8 @ A7 ) ) ) ).
% sup.commute
thf(fact_886_sup_Ocommute,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [A7: set_Product_prod_a_a,B8: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ B8 @ A7 ) ) ) ).
% sup.commute
thf(fact_887_sup_Ocommute,axiom,
( sup_sup_set_list_a
= ( ^ [A7: set_list_a,B8: set_list_a] : ( sup_sup_set_list_a @ B8 @ A7 ) ) ) ).
% sup.commute
thf(fact_888_sup__commute,axiom,
( sup_sup_set_set_a
= ( ^ [X3: set_set_a,Y6: set_set_a] : ( sup_sup_set_set_a @ Y6 @ X3 ) ) ) ).
% sup_commute
thf(fact_889_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X3: set_a,Y6: set_a] : ( sup_sup_set_a @ Y6 @ X3 ) ) ) ).
% sup_commute
thf(fact_890_sup__commute,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [X3: set_Product_prod_a_a,Y6: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ Y6 @ X3 ) ) ) ).
% sup_commute
thf(fact_891_sup__commute,axiom,
( sup_sup_set_list_a
= ( ^ [X3: set_list_a,Y6: set_list_a] : ( sup_sup_set_list_a @ Y6 @ X3 ) ) ) ).
% sup_commute
thf(fact_892_sup_Oleft__commute,axiom,
! [B7: set_set_a,A2: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ B7 @ ( sup_sup_set_set_a @ A2 @ C2 ) )
= ( sup_sup_set_set_a @ A2 @ ( sup_sup_set_set_a @ B7 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_893_sup_Oleft__commute,axiom,
! [B7: set_a,A2: set_a,C2: set_a] :
( ( sup_sup_set_a @ B7 @ ( sup_sup_set_a @ A2 @ C2 ) )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B7 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_894_sup_Oleft__commute,axiom,
! [B7: set_Product_prod_a_a,A2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ B7 @ ( sup_su3048258781599657691od_a_a @ A2 @ C2 ) )
= ( sup_su3048258781599657691od_a_a @ A2 @ ( sup_su3048258781599657691od_a_a @ B7 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_895_sup_Oleft__commute,axiom,
! [B7: set_list_a,A2: set_list_a,C2: set_list_a] :
( ( sup_sup_set_list_a @ B7 @ ( sup_sup_set_list_a @ A2 @ C2 ) )
= ( sup_sup_set_list_a @ A2 @ ( sup_sup_set_list_a @ B7 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_896_sup__left__commute,axiom,
! [X5: set_set_a,Y4: set_set_a,Z2: set_set_a] :
( ( sup_sup_set_set_a @ X5 @ ( sup_sup_set_set_a @ Y4 @ Z2 ) )
= ( sup_sup_set_set_a @ Y4 @ ( sup_sup_set_set_a @ X5 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_897_sup__left__commute,axiom,
! [X5: set_a,Y4: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X5 @ ( sup_sup_set_a @ Y4 @ Z2 ) )
= ( sup_sup_set_a @ Y4 @ ( sup_sup_set_a @ X5 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_898_sup__left__commute,axiom,
! [X5: set_Product_prod_a_a,Y4: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X5 @ ( sup_su3048258781599657691od_a_a @ Y4 @ Z2 ) )
= ( sup_su3048258781599657691od_a_a @ Y4 @ ( sup_su3048258781599657691od_a_a @ X5 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_899_sup__left__commute,axiom,
! [X5: set_list_a,Y4: set_list_a,Z2: set_list_a] :
( ( sup_sup_set_list_a @ X5 @ ( sup_sup_set_list_a @ Y4 @ Z2 ) )
= ( sup_sup_set_list_a @ Y4 @ ( sup_sup_set_list_a @ X5 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_900_ulgraph_Onot__vert__adj,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_a,U: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ~ ( undire3510646817838285160_set_a @ Edges @ V @ U )
=> ~ ( member_set_set_a @ ( insert_set_a @ V @ ( insert_set_a @ U @ bot_bot_set_set_a ) ) @ Edges ) ) ) ).
% ulgraph.not_vert_adj
thf(fact_901_ulgraph_Onot__vert__adj,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V: product_prod_a_a,U: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ~ ( undire6135774327024169009od_a_a @ Edges @ V @ U )
=> ~ ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ V @ ( insert4534936382041156343od_a_a @ U @ bot_bo3357376287454694259od_a_a ) ) @ Edges ) ) ) ).
% ulgraph.not_vert_adj
thf(fact_902_ulgraph_Onot__vert__adj,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V: nat,U: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ~ ( undire1083030068171319366dj_nat @ Edges @ V @ U )
=> ~ ( member_set_nat @ ( insert_nat @ V @ ( insert_nat @ U @ bot_bot_set_nat ) ) @ Edges ) ) ) ).
% ulgraph.not_vert_adj
thf(fact_903_ulgraph_Onot__vert__adj,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a,U: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ~ ( undire397441198561214472_adj_a @ Edges @ V @ U )
=> ~ ( member_set_a @ ( insert_a @ V @ ( insert_a @ U @ bot_bot_set_a ) ) @ Edges ) ) ) ).
% ulgraph.not_vert_adj
thf(fact_904_ulgraph_Overt__adj__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V1: set_a,V2: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3510646817838285160_set_a @ Edges @ V1 @ V2 )
= ( member_set_set_a @ ( insert_set_a @ V1 @ ( insert_set_a @ V2 @ bot_bot_set_set_a ) ) @ Edges ) ) ) ).
% ulgraph.vert_adj_def
thf(fact_905_ulgraph_Overt__adj__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V1: product_prod_a_a,V2: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire6135774327024169009od_a_a @ Edges @ V1 @ V2 )
= ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ V1 @ ( insert4534936382041156343od_a_a @ V2 @ bot_bo3357376287454694259od_a_a ) ) @ Edges ) ) ) ).
% ulgraph.vert_adj_def
thf(fact_906_ulgraph_Overt__adj__def,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V1: nat,V2: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire1083030068171319366dj_nat @ Edges @ V1 @ V2 )
= ( member_set_nat @ ( insert_nat @ V1 @ ( insert_nat @ V2 @ bot_bot_set_nat ) ) @ Edges ) ) ) ).
% ulgraph.vert_adj_def
thf(fact_907_ulgraph_Overt__adj__def,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V2 )
= ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) @ Edges ) ) ) ).
% ulgraph.vert_adj_def
thf(fact_908_ulgraph_Ohas__loop__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire5774735625301615776_set_a @ Edges @ V )
= ( member_set_set_a @ ( insert_set_a @ V @ bot_bot_set_set_a ) @ Edges ) ) ) ).
% ulgraph.has_loop_def
thf(fact_909_ulgraph_Ohas__loop__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire7777398424729533289od_a_a @ Edges @ V )
= ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ V @ bot_bo3357376287454694259od_a_a ) @ Edges ) ) ) ).
% ulgraph.has_loop_def
thf(fact_910_ulgraph_Ohas__loop__def,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire5005864372999571214op_nat @ Edges @ V )
= ( member_set_nat @ ( insert_nat @ V @ bot_bot_set_nat ) @ Edges ) ) ) ).
% ulgraph.has_loop_def
thf(fact_911_ulgraph_Ohas__loop__def,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3617971648856834880loop_a @ Edges @ V )
= ( member_set_a @ ( insert_a @ V @ bot_bot_set_a ) @ Edges ) ) ) ).
% ulgraph.has_loop_def
thf(fact_912_ulgraph_Ois__isolated__vertex__degree0,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V )
=> ( ( undire8867928226783802224gree_a @ Edges @ V )
= zero_zero_nat ) ) ) ).
% ulgraph.is_isolated_vertex_degree0
thf(fact_913_comp__sgraph_Ois__loop__def,axiom,
( undire3618949687197220622_set_a
= ( ^ [E5: set_set_a] :
( ( finite_card_set_a @ E5 )
= one_one_nat ) ) ) ).
% comp_sgraph.is_loop_def
thf(fact_914_comp__sgraph_Ois__loop__def,axiom,
( undire3428022325429088215od_a_a
= ( ^ [E5: set_Product_prod_a_a] :
( ( finite4795055649997197647od_a_a @ E5 )
= one_one_nat ) ) ) ).
% comp_sgraph.is_loop_def
thf(fact_915_comp__sgraph_Ois__loop__def,axiom,
( undire643512044667278624op_nat
= ( ^ [E5: set_nat] :
( ( finite_card_nat @ E5 )
= one_one_nat ) ) ) ).
% comp_sgraph.is_loop_def
thf(fact_916_comp__sgraph_Ois__loop__def,axiom,
( undire2905028936066782638loop_a
= ( ^ [E5: set_a] :
( ( finite_card_a @ E5 )
= one_one_nat ) ) ) ).
% comp_sgraph.is_loop_def
thf(fact_917_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_918_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_919_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_920_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_921_ulgraph_Overt__adj__inc__edge__iff,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V1: set_a,V2: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3510646817838285160_set_a @ Edges @ V1 @ V2 )
= ( ( undire2320338297334612420_set_a @ V1 @ ( insert_set_a @ V1 @ ( insert_set_a @ V2 @ bot_bot_set_set_a ) ) )
& ( undire2320338297334612420_set_a @ V2 @ ( insert_set_a @ V1 @ ( insert_set_a @ V2 @ bot_bot_set_set_a ) ) )
& ( member_set_set_a @ ( insert_set_a @ V1 @ ( insert_set_a @ V2 @ bot_bot_set_set_a ) ) @ Edges ) ) ) ) ).
% ulgraph.vert_adj_inc_edge_iff
thf(fact_922_ulgraph_Overt__adj__inc__edge__iff,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V1: product_prod_a_a,V2: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire6135774327024169009od_a_a @ Edges @ V1 @ V2 )
= ( ( undire3369688177417741453od_a_a @ V1 @ ( insert4534936382041156343od_a_a @ V1 @ ( insert4534936382041156343od_a_a @ V2 @ bot_bo3357376287454694259od_a_a ) ) )
& ( undire3369688177417741453od_a_a @ V2 @ ( insert4534936382041156343od_a_a @ V1 @ ( insert4534936382041156343od_a_a @ V2 @ bot_bo3357376287454694259od_a_a ) ) )
& ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ V1 @ ( insert4534936382041156343od_a_a @ V2 @ bot_bo3357376287454694259od_a_a ) ) @ Edges ) ) ) ) ).
% ulgraph.vert_adj_inc_edge_iff
thf(fact_923_ulgraph_Overt__adj__inc__edge__iff,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V1: nat,V2: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire1083030068171319366dj_nat @ Edges @ V1 @ V2 )
= ( ( undire7858122600432113898nt_nat @ V1 @ ( insert_nat @ V1 @ ( insert_nat @ V2 @ bot_bot_set_nat ) ) )
& ( undire7858122600432113898nt_nat @ V2 @ ( insert_nat @ V1 @ ( insert_nat @ V2 @ bot_bot_set_nat ) ) )
& ( member_set_nat @ ( insert_nat @ V1 @ ( insert_nat @ V2 @ bot_bot_set_nat ) ) @ Edges ) ) ) ) ).
% ulgraph.vert_adj_inc_edge_iff
thf(fact_924_ulgraph_Overt__adj__inc__edge__iff,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( 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 ) ) ) ) ).
% ulgraph.vert_adj_inc_edge_iff
thf(fact_925_ulgraph_Oneighborhood__incident,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,U: set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( member_set_a @ U @ ( undire2074812191327625774_set_a @ Vertices @ Edges @ V ) )
= ( member_set_set_a @ ( insert_set_a @ U @ ( insert_set_a @ V @ bot_bot_set_set_a ) ) @ ( undire4631953023069350784_set_a @ Edges @ V ) ) ) ) ).
% ulgraph.neighborhood_incident
thf(fact_926_ulgraph_Oneighborhood__incident,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,U: product_prod_a_a,V: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( member1426531477525435216od_a_a @ U @ ( undire7963753511165915895od_a_a @ Vertices @ Edges @ V ) )
= ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ U @ ( insert4534936382041156343od_a_a @ V @ bot_bo3357376287454694259od_a_a ) ) @ ( undire8905369280470868553od_a_a @ Edges @ V ) ) ) ) ).
% ulgraph.neighborhood_incident
thf(fact_927_ulgraph_Oneighborhood__incident,axiom,
! [Vertices: set_nat,Edges: set_set_nat,U: nat,V: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( member_nat @ U @ ( undire8190396521545869824od_nat @ Vertices @ Edges @ V ) )
= ( member_set_nat @ ( insert_nat @ U @ ( insert_nat @ V @ bot_bot_set_nat ) ) @ ( undire4176300566717384750es_nat @ Edges @ V ) ) ) ) ).
% ulgraph.neighborhood_incident
thf(fact_928_ulgraph_Oneighborhood__incident,axiom,
! [Vertices: set_a,Edges: set_set_a,U: a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( 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 ) ) ) ) ).
% ulgraph.neighborhood_incident
thf(fact_929_ulgraph_Ois__loop__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,E: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3618949687197220622_set_a @ E )
= ( ( finite_card_set_a @ E )
= one_one_nat ) ) ) ).
% ulgraph.is_loop_def
thf(fact_930_ulgraph_Ois__loop__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,E: set_Product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire3428022325429088215od_a_a @ E )
= ( ( finite4795055649997197647od_a_a @ E )
= one_one_nat ) ) ) ).
% ulgraph.is_loop_def
thf(fact_931_ulgraph_Ois__loop__def,axiom,
! [Vertices: set_nat,Edges: set_set_nat,E: set_nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire643512044667278624op_nat @ E )
= ( ( finite_card_nat @ E )
= one_one_nat ) ) ) ).
% ulgraph.is_loop_def
thf(fact_932_ulgraph_Ois__loop__def,axiom,
! [Vertices: set_a,Edges: set_set_a,E: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire2905028936066782638loop_a @ E )
= ( ( finite_card_a @ E )
= one_one_nat ) ) ) ).
% ulgraph.is_loop_def
thf(fact_933_ulgraph_Ocard1__incident__imp__vert,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_a,E: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( ( undire2320338297334612420_set_a @ V @ E )
& ( ( finite_card_set_a @ E )
= one_one_nat ) )
=> ( E
= ( insert_set_a @ V @ bot_bot_set_set_a ) ) ) ) ).
% ulgraph.card1_incident_imp_vert
thf(fact_934_ulgraph_Ocard1__incident__imp__vert,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V: product_prod_a_a,E: set_Product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( ( undire3369688177417741453od_a_a @ V @ E )
& ( ( finite4795055649997197647od_a_a @ E )
= one_one_nat ) )
=> ( E
= ( insert4534936382041156343od_a_a @ V @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% ulgraph.card1_incident_imp_vert
thf(fact_935_ulgraph_Ocard1__incident__imp__vert,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V: nat,E: set_nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( ( undire7858122600432113898nt_nat @ V @ E )
& ( ( finite_card_nat @ E )
= one_one_nat ) )
=> ( E
= ( insert_nat @ V @ bot_bot_set_nat ) ) ) ) ).
% ulgraph.card1_incident_imp_vert
thf(fact_936_ulgraph_Ocard1__incident__imp__vert,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a,E: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( ( undire1521409233611534436dent_a @ V @ E )
& ( ( finite_card_a @ E )
= one_one_nat ) )
=> ( E
= ( insert_a @ V @ bot_bot_set_a ) ) ) ) ).
% ulgraph.card1_incident_imp_vert
thf(fact_937_ulgraph_Odegree0__neighborhood__empt__iff,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( finite7209287970140883943_set_a @ Edges )
=> ( ( ( undire8939077443744732368_set_a @ Edges @ V )
= zero_zero_nat )
= ( ( undire2074812191327625774_set_a @ Vertices @ Edges @ V )
= bot_bot_set_set_a ) ) ) ) ).
% ulgraph.degree0_neighborhood_empt_iff
thf(fact_938_ulgraph_Odegree0__neighborhood__empt__iff,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,V: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( finite8717734299975451184od_a_a @ Edges )
=> ( ( ( undire1436394852029823897od_a_a @ Edges @ V )
= zero_zero_nat )
= ( ( undire7963753511165915895od_a_a @ Vertices @ Edges @ V )
= bot_bo3357376287454694259od_a_a ) ) ) ) ).
% ulgraph.degree0_neighborhood_empt_iff
thf(fact_939_ulgraph_Odegree0__neighborhood__empt__iff,axiom,
! [Vertices: set_nat,Edges: set_set_nat,V: nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( finite1152437895449049373et_nat @ Edges )
=> ( ( ( undire6581030323043281630ee_nat @ Edges @ V )
= zero_zero_nat )
= ( ( undire8190396521545869824od_nat @ Vertices @ Edges @ V )
= bot_bot_set_nat ) ) ) ) ).
% ulgraph.degree0_neighborhood_empt_iff
thf(fact_940_ulgraph_Odegree0__neighborhood__empt__iff,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( finite_finite_set_a @ Edges )
=> ( ( ( undire8867928226783802224gree_a @ Edges @ V )
= zero_zero_nat )
= ( ( undire8504279938402040014hood_a @ Vertices @ Edges @ V )
= bot_bot_set_a ) ) ) ) ).
% ulgraph.degree0_neighborhood_empt_iff
thf(fact_941_ulgraph_Odegree0__inc__edges__empt__iff,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( finite_finite_set_a @ Edges )
=> ( ( ( undire8867928226783802224gree_a @ Edges @ V )
= zero_zero_nat )
= ( ( undire3231912044278729248dges_a @ Edges @ V )
= bot_bot_set_set_a ) ) ) ) ).
% ulgraph.degree0_inc_edges_empt_iff
thf(fact_942_ulgraph_Odegree__no__loops,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ~ ( undire3617971648856834880loop_a @ Edges @ V )
=> ( ( undire8867928226783802224gree_a @ Edges @ V )
= ( finite_card_set_a @ ( undire3231912044278729248dges_a @ Edges @ V ) ) ) ) ) ).
% ulgraph.degree_no_loops
thf(fact_943_sup_OcoboundedI2,axiom,
! [C2: set_Product_prod_a_a,B7: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C2 @ B7 )
=> ( ord_le746702958409616551od_a_a @ C2 @ ( sup_su3048258781599657691od_a_a @ A2 @ B7 ) ) ) ).
% sup.coboundedI2
thf(fact_944_sup_OcoboundedI2,axiom,
! [C2: set_list_a,B7: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ B7 )
=> ( ord_le8861187494160871172list_a @ C2 @ ( sup_sup_set_list_a @ A2 @ B7 ) ) ) ).
% sup.coboundedI2
thf(fact_945_sup_OcoboundedI2,axiom,
! [C2: set_set_a,B7: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ B7 )
=> ( ord_le3724670747650509150_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B7 ) ) ) ).
% sup.coboundedI2
thf(fact_946_sup_OcoboundedI2,axiom,
! [C2: set_a,B7: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ C2 @ B7 )
=> ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A2 @ B7 ) ) ) ).
% sup.coboundedI2
thf(fact_947_sup_OcoboundedI2,axiom,
! [C2: nat,B7: nat,A2: nat] :
( ( ord_less_eq_nat @ C2 @ B7 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A2 @ B7 ) ) ) ).
% sup.coboundedI2
thf(fact_948_sup_OcoboundedI1,axiom,
! [C2: set_Product_prod_a_a,A2: set_Product_prod_a_a,B7: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C2 @ A2 )
=> ( ord_le746702958409616551od_a_a @ C2 @ ( sup_su3048258781599657691od_a_a @ A2 @ B7 ) ) ) ).
% sup.coboundedI1
thf(fact_949_sup_OcoboundedI1,axiom,
! [C2: set_list_a,A2: set_list_a,B7: set_list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ A2 )
=> ( ord_le8861187494160871172list_a @ C2 @ ( sup_sup_set_list_a @ A2 @ B7 ) ) ) ).
% sup.coboundedI1
thf(fact_950_sup_OcoboundedI1,axiom,
! [C2: set_set_a,A2: set_set_a,B7: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ A2 )
=> ( ord_le3724670747650509150_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B7 ) ) ) ).
% sup.coboundedI1
thf(fact_951_sup_OcoboundedI1,axiom,
! [C2: set_a,A2: set_a,B7: set_a] :
( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A2 @ B7 ) ) ) ).
% sup.coboundedI1
thf(fact_952_sup_OcoboundedI1,axiom,
! [C2: nat,A2: nat,B7: nat] :
( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A2 @ B7 ) ) ) ).
% sup.coboundedI1
thf(fact_953_sup_Oabsorb__iff2,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A7: set_Product_prod_a_a,B8: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A7 @ B8 )
= B8 ) ) ) ).
% sup.absorb_iff2
thf(fact_954_sup_Oabsorb__iff2,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A7: set_list_a,B8: set_list_a] :
( ( sup_sup_set_list_a @ A7 @ B8 )
= B8 ) ) ) ).
% sup.absorb_iff2
thf(fact_955_sup_Oabsorb__iff2,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A7: set_set_a,B8: set_set_a] :
( ( sup_sup_set_set_a @ A7 @ B8 )
= B8 ) ) ) ).
% sup.absorb_iff2
thf(fact_956_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B8: set_a] :
( ( sup_sup_set_a @ A7 @ B8 )
= B8 ) ) ) ).
% sup.absorb_iff2
thf(fact_957_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A7: nat,B8: nat] :
( ( sup_sup_nat @ A7 @ B8 )
= B8 ) ) ) ).
% sup.absorb_iff2
thf(fact_958_sup_Oabsorb__iff1,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [B8: set_Product_prod_a_a,A7: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A7 @ B8 )
= A7 ) ) ) ).
% sup.absorb_iff1
thf(fact_959_sup_Oabsorb__iff1,axiom,
( ord_le8861187494160871172list_a
= ( ^ [B8: set_list_a,A7: set_list_a] :
( ( sup_sup_set_list_a @ A7 @ B8 )
= A7 ) ) ) ).
% sup.absorb_iff1
thf(fact_960_sup_Oabsorb__iff1,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B8: set_set_a,A7: set_set_a] :
( ( sup_sup_set_set_a @ A7 @ B8 )
= A7 ) ) ) ).
% sup.absorb_iff1
thf(fact_961_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B8: set_a,A7: set_a] :
( ( sup_sup_set_a @ A7 @ B8 )
= A7 ) ) ) ).
% sup.absorb_iff1
thf(fact_962_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B8: nat,A7: nat] :
( ( sup_sup_nat @ A7 @ B8 )
= A7 ) ) ) ).
% sup.absorb_iff1
thf(fact_963_sup_Ocobounded2,axiom,
! [B7: set_Product_prod_a_a,A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ B7 @ ( sup_su3048258781599657691od_a_a @ A2 @ B7 ) ) ).
% sup.cobounded2
thf(fact_964_sup_Ocobounded2,axiom,
! [B7: set_list_a,A2: set_list_a] : ( ord_le8861187494160871172list_a @ B7 @ ( sup_sup_set_list_a @ A2 @ B7 ) ) ).
% sup.cobounded2
thf(fact_965_sup_Ocobounded2,axiom,
! [B7: set_set_a,A2: set_set_a] : ( ord_le3724670747650509150_set_a @ B7 @ ( sup_sup_set_set_a @ A2 @ B7 ) ) ).
% sup.cobounded2
thf(fact_966_sup_Ocobounded2,axiom,
! [B7: set_a,A2: set_a] : ( ord_less_eq_set_a @ B7 @ ( sup_sup_set_a @ A2 @ B7 ) ) ).
% sup.cobounded2
thf(fact_967_sup_Ocobounded2,axiom,
! [B7: nat,A2: nat] : ( ord_less_eq_nat @ B7 @ ( sup_sup_nat @ A2 @ B7 ) ) ).
% sup.cobounded2
thf(fact_968_sup_Ocobounded1,axiom,
! [A2: set_Product_prod_a_a,B7: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A2 @ ( sup_su3048258781599657691od_a_a @ A2 @ B7 ) ) ).
% sup.cobounded1
thf(fact_969_sup_Ocobounded1,axiom,
! [A2: set_list_a,B7: set_list_a] : ( ord_le8861187494160871172list_a @ A2 @ ( sup_sup_set_list_a @ A2 @ B7 ) ) ).
% sup.cobounded1
thf(fact_970_sup_Ocobounded1,axiom,
! [A2: set_set_a,B7: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ ( sup_sup_set_set_a @ A2 @ B7 ) ) ).
% sup.cobounded1
thf(fact_971_sup_Ocobounded1,axiom,
! [A2: set_a,B7: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B7 ) ) ).
% sup.cobounded1
thf(fact_972_sup_Ocobounded1,axiom,
! [A2: nat,B7: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B7 ) ) ).
% sup.cobounded1
thf(fact_973_sup_Oorder__iff,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [B8: set_Product_prod_a_a,A7: set_Product_prod_a_a] :
( A7
= ( sup_su3048258781599657691od_a_a @ A7 @ B8 ) ) ) ) ).
% sup.order_iff
thf(fact_974_sup_Oorder__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [B8: set_list_a,A7: set_list_a] :
( A7
= ( sup_sup_set_list_a @ A7 @ B8 ) ) ) ) ).
% sup.order_iff
thf(fact_975_sup_Oorder__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B8: set_set_a,A7: set_set_a] :
( A7
= ( sup_sup_set_set_a @ A7 @ B8 ) ) ) ) ).
% sup.order_iff
thf(fact_976_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B8: set_a,A7: set_a] :
( A7
= ( sup_sup_set_a @ A7 @ B8 ) ) ) ) ).
% sup.order_iff
thf(fact_977_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B8: nat,A7: nat] :
( A7
= ( sup_sup_nat @ A7 @ B8 ) ) ) ) ).
% sup.order_iff
thf(fact_978_sup_OboundedI,axiom,
! [B7: set_Product_prod_a_a,A2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B7 @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ C2 @ A2 )
=> ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ B7 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_979_sup_OboundedI,axiom,
! [B7: set_list_a,A2: set_list_a,C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ B7 @ A2 )
=> ( ( ord_le8861187494160871172list_a @ C2 @ A2 )
=> ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ B7 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_980_sup_OboundedI,axiom,
! [B7: set_set_a,A2: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B7 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ C2 @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B7 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_981_sup_OboundedI,axiom,
! [B7: set_a,A2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B7 @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B7 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_982_sup_OboundedI,axiom,
! [B7: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B7 @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B7 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_983_sup_OboundedE,axiom,
! [B7: set_Product_prod_a_a,C2: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ B7 @ C2 ) @ A2 )
=> ~ ( ( ord_le746702958409616551od_a_a @ B7 @ A2 )
=> ~ ( ord_le746702958409616551od_a_a @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_984_sup_OboundedE,axiom,
! [B7: set_list_a,C2: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ B7 @ C2 ) @ A2 )
=> ~ ( ( ord_le8861187494160871172list_a @ B7 @ A2 )
=> ~ ( ord_le8861187494160871172list_a @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_985_sup_OboundedE,axiom,
! [B7: set_set_a,C2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B7 @ C2 ) @ A2 )
=> ~ ( ( ord_le3724670747650509150_set_a @ B7 @ A2 )
=> ~ ( ord_le3724670747650509150_set_a @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_986_sup_OboundedE,axiom,
! [B7: set_a,C2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B7 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq_set_a @ B7 @ A2 )
=> ~ ( ord_less_eq_set_a @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_987_sup_OboundedE,axiom,
! [B7: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B7 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B7 @ A2 )
=> ~ ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_988_sup__absorb2,axiom,
! [X5: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X5 @ Y4 )
=> ( ( sup_su3048258781599657691od_a_a @ X5 @ Y4 )
= Y4 ) ) ).
% sup_absorb2
thf(fact_989_sup__absorb2,axiom,
! [X5: set_list_a,Y4: set_list_a] :
( ( ord_le8861187494160871172list_a @ X5 @ Y4 )
=> ( ( sup_sup_set_list_a @ X5 @ Y4 )
= Y4 ) ) ).
% sup_absorb2
thf(fact_990_sup__absorb2,axiom,
! [X5: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ Y4 )
=> ( ( sup_sup_set_set_a @ X5 @ Y4 )
= Y4 ) ) ).
% sup_absorb2
thf(fact_991_sup__absorb2,axiom,
! [X5: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y4 )
=> ( ( sup_sup_set_a @ X5 @ Y4 )
= Y4 ) ) ).
% sup_absorb2
thf(fact_992_sup__absorb2,axiom,
! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ( sup_sup_nat @ X5 @ Y4 )
= Y4 ) ) ).
% sup_absorb2
thf(fact_993_sup__absorb1,axiom,
! [Y4: set_Product_prod_a_a,X5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y4 @ X5 )
=> ( ( sup_su3048258781599657691od_a_a @ X5 @ Y4 )
= X5 ) ) ).
% sup_absorb1
thf(fact_994_sup__absorb1,axiom,
! [Y4: set_list_a,X5: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y4 @ X5 )
=> ( ( sup_sup_set_list_a @ X5 @ Y4 )
= X5 ) ) ).
% sup_absorb1
thf(fact_995_sup__absorb1,axiom,
! [Y4: set_set_a,X5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y4 @ X5 )
=> ( ( sup_sup_set_set_a @ X5 @ Y4 )
= X5 ) ) ).
% sup_absorb1
thf(fact_996_sup__absorb1,axiom,
! [Y4: set_a,X5: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X5 )
=> ( ( sup_sup_set_a @ X5 @ Y4 )
= X5 ) ) ).
% sup_absorb1
thf(fact_997_sup__absorb1,axiom,
! [Y4: nat,X5: nat] :
( ( ord_less_eq_nat @ Y4 @ X5 )
=> ( ( sup_sup_nat @ X5 @ Y4 )
= X5 ) ) ).
% sup_absorb1
thf(fact_998_sup_Oabsorb2,axiom,
! [A2: set_Product_prod_a_a,B7: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B7 )
=> ( ( sup_su3048258781599657691od_a_a @ A2 @ B7 )
= B7 ) ) ).
% sup.absorb2
thf(fact_999_sup_Oabsorb2,axiom,
! [A2: set_list_a,B7: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B7 )
=> ( ( sup_sup_set_list_a @ A2 @ B7 )
= B7 ) ) ).
% sup.absorb2
thf(fact_1000_sup_Oabsorb2,axiom,
! [A2: set_set_a,B7: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B7 )
=> ( ( sup_sup_set_set_a @ A2 @ B7 )
= B7 ) ) ).
% sup.absorb2
thf(fact_1001_sup_Oabsorb2,axiom,
! [A2: set_a,B7: set_a] :
( ( ord_less_eq_set_a @ A2 @ B7 )
=> ( ( sup_sup_set_a @ A2 @ B7 )
= B7 ) ) ).
% sup.absorb2
thf(fact_1002_sup_Oabsorb2,axiom,
! [A2: nat,B7: nat] :
( ( ord_less_eq_nat @ A2 @ B7 )
=> ( ( sup_sup_nat @ A2 @ B7 )
= B7 ) ) ).
% sup.absorb2
thf(fact_1003_sup_Oabsorb1,axiom,
! [B7: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B7 @ A2 )
=> ( ( sup_su3048258781599657691od_a_a @ A2 @ B7 )
= A2 ) ) ).
% sup.absorb1
thf(fact_1004_sup_Oabsorb1,axiom,
! [B7: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ B7 @ A2 )
=> ( ( sup_sup_set_list_a @ A2 @ B7 )
= A2 ) ) ).
% sup.absorb1
thf(fact_1005_sup_Oabsorb1,axiom,
! [B7: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B7 @ A2 )
=> ( ( sup_sup_set_set_a @ A2 @ B7 )
= A2 ) ) ).
% sup.absorb1
thf(fact_1006_sup_Oabsorb1,axiom,
! [B7: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B7 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B7 )
= A2 ) ) ).
% sup.absorb1
thf(fact_1007_sup_Oabsorb1,axiom,
! [B7: nat,A2: nat] :
( ( ord_less_eq_nat @ B7 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B7 )
= A2 ) ) ).
% sup.absorb1
thf(fact_1008_sup__unique,axiom,
! [F3: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a,X5: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ! [X4: set_Product_prod_a_a,Y2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ X4 @ ( F3 @ X4 @ Y2 ) )
=> ( ! [X4: set_Product_prod_a_a,Y2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ Y2 @ ( F3 @ X4 @ Y2 ) )
=> ( ! [X4: set_Product_prod_a_a,Y2: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y2 @ X4 )
=> ( ( ord_le746702958409616551od_a_a @ Z3 @ X4 )
=> ( ord_le746702958409616551od_a_a @ ( F3 @ Y2 @ Z3 ) @ X4 ) ) )
=> ( ( sup_su3048258781599657691od_a_a @ X5 @ Y4 )
= ( F3 @ X5 @ Y4 ) ) ) ) ) ).
% sup_unique
thf(fact_1009_sup__unique,axiom,
! [F3: set_list_a > set_list_a > set_list_a,X5: set_list_a,Y4: set_list_a] :
( ! [X4: set_list_a,Y2: set_list_a] : ( ord_le8861187494160871172list_a @ X4 @ ( F3 @ X4 @ Y2 ) )
=> ( ! [X4: set_list_a,Y2: set_list_a] : ( ord_le8861187494160871172list_a @ Y2 @ ( F3 @ X4 @ Y2 ) )
=> ( ! [X4: set_list_a,Y2: set_list_a,Z3: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y2 @ X4 )
=> ( ( ord_le8861187494160871172list_a @ Z3 @ X4 )
=> ( ord_le8861187494160871172list_a @ ( F3 @ Y2 @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_set_list_a @ X5 @ Y4 )
= ( F3 @ X5 @ Y4 ) ) ) ) ) ).
% sup_unique
thf(fact_1010_sup__unique,axiom,
! [F3: set_set_a > set_set_a > set_set_a,X5: set_set_a,Y4: set_set_a] :
( ! [X4: set_set_a,Y2: set_set_a] : ( ord_le3724670747650509150_set_a @ X4 @ ( F3 @ X4 @ Y2 ) )
=> ( ! [X4: set_set_a,Y2: set_set_a] : ( ord_le3724670747650509150_set_a @ Y2 @ ( F3 @ X4 @ Y2 ) )
=> ( ! [X4: set_set_a,Y2: set_set_a,Z3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y2 @ X4 )
=> ( ( ord_le3724670747650509150_set_a @ Z3 @ X4 )
=> ( ord_le3724670747650509150_set_a @ ( F3 @ Y2 @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_set_set_a @ X5 @ Y4 )
= ( F3 @ X5 @ Y4 ) ) ) ) ) ).
% sup_unique
thf(fact_1011_sup__unique,axiom,
! [F3: set_a > set_a > set_a,X5: set_a,Y4: set_a] :
( ! [X4: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X4 @ ( F3 @ X4 @ Y2 ) )
=> ( ! [X4: set_a,Y2: set_a] : ( ord_less_eq_set_a @ Y2 @ ( F3 @ X4 @ Y2 ) )
=> ( ! [X4: set_a,Y2: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X4 )
=> ( ( ord_less_eq_set_a @ Z3 @ X4 )
=> ( ord_less_eq_set_a @ ( F3 @ Y2 @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_set_a @ X5 @ Y4 )
= ( F3 @ X5 @ Y4 ) ) ) ) ) ).
% sup_unique
thf(fact_1012_sup__unique,axiom,
! [F3: nat > nat > nat,X5: nat,Y4: nat] :
( ! [X4: nat,Y2: nat] : ( ord_less_eq_nat @ X4 @ ( F3 @ X4 @ Y2 ) )
=> ( ! [X4: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F3 @ X4 @ Y2 ) )
=> ( ! [X4: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y2 @ X4 )
=> ( ( ord_less_eq_nat @ Z3 @ X4 )
=> ( ord_less_eq_nat @ ( F3 @ Y2 @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_nat @ X5 @ Y4 )
= ( F3 @ X5 @ Y4 ) ) ) ) ) ).
% sup_unique
thf(fact_1013_sup_OorderI,axiom,
! [A2: set_Product_prod_a_a,B7: set_Product_prod_a_a] :
( ( A2
= ( sup_su3048258781599657691od_a_a @ A2 @ B7 ) )
=> ( ord_le746702958409616551od_a_a @ B7 @ A2 ) ) ).
% sup.orderI
thf(fact_1014_sup_OorderI,axiom,
! [A2: set_list_a,B7: set_list_a] :
( ( A2
= ( sup_sup_set_list_a @ A2 @ B7 ) )
=> ( ord_le8861187494160871172list_a @ B7 @ A2 ) ) ).
% sup.orderI
thf(fact_1015_sup_OorderI,axiom,
! [A2: set_set_a,B7: set_set_a] :
( ( A2
= ( sup_sup_set_set_a @ A2 @ B7 ) )
=> ( ord_le3724670747650509150_set_a @ B7 @ A2 ) ) ).
% sup.orderI
thf(fact_1016_sup_OorderI,axiom,
! [A2: set_a,B7: set_a] :
( ( A2
= ( sup_sup_set_a @ A2 @ B7 ) )
=> ( ord_less_eq_set_a @ B7 @ A2 ) ) ).
% sup.orderI
thf(fact_1017_sup_OorderI,axiom,
! [A2: nat,B7: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B7 ) )
=> ( ord_less_eq_nat @ B7 @ A2 ) ) ).
% sup.orderI
thf(fact_1018_sup_OorderE,axiom,
! [B7: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B7 @ A2 )
=> ( A2
= ( sup_su3048258781599657691od_a_a @ A2 @ B7 ) ) ) ).
% sup.orderE
thf(fact_1019_sup_OorderE,axiom,
! [B7: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ B7 @ A2 )
=> ( A2
= ( sup_sup_set_list_a @ A2 @ B7 ) ) ) ).
% sup.orderE
thf(fact_1020_sup_OorderE,axiom,
! [B7: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B7 @ A2 )
=> ( A2
= ( sup_sup_set_set_a @ A2 @ B7 ) ) ) ).
% sup.orderE
thf(fact_1021_sup_OorderE,axiom,
! [B7: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B7 @ A2 )
=> ( A2
= ( sup_sup_set_a @ A2 @ B7 ) ) ) ).
% sup.orderE
thf(fact_1022_sup_OorderE,axiom,
! [B7: nat,A2: nat] :
( ( ord_less_eq_nat @ B7 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B7 ) ) ) ).
% sup.orderE
thf(fact_1023_le__iff__sup,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [X3: set_Product_prod_a_a,Y6: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X3 @ Y6 )
= Y6 ) ) ) ).
% le_iff_sup
thf(fact_1024_le__iff__sup,axiom,
( ord_le8861187494160871172list_a
= ( ^ [X3: set_list_a,Y6: set_list_a] :
( ( sup_sup_set_list_a @ X3 @ Y6 )
= Y6 ) ) ) ).
% le_iff_sup
thf(fact_1025_le__iff__sup,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X3: set_set_a,Y6: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ Y6 )
= Y6 ) ) ) ).
% le_iff_sup
thf(fact_1026_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y6: set_a] :
( ( sup_sup_set_a @ X3 @ Y6 )
= Y6 ) ) ) ).
% le_iff_sup
thf(fact_1027_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y6: nat] :
( ( sup_sup_nat @ X3 @ Y6 )
= Y6 ) ) ) ).
% le_iff_sup
thf(fact_1028_sup__least,axiom,
! [Y4: set_Product_prod_a_a,X5: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y4 @ X5 )
=> ( ( ord_le746702958409616551od_a_a @ Z2 @ X5 )
=> ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ Y4 @ Z2 ) @ X5 ) ) ) ).
% sup_least
thf(fact_1029_sup__least,axiom,
! [Y4: set_list_a,X5: set_list_a,Z2: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y4 @ X5 )
=> ( ( ord_le8861187494160871172list_a @ Z2 @ X5 )
=> ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ Y4 @ Z2 ) @ X5 ) ) ) ).
% sup_least
thf(fact_1030_sup__least,axiom,
! [Y4: set_set_a,X5: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y4 @ X5 )
=> ( ( ord_le3724670747650509150_set_a @ Z2 @ X5 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ Y4 @ Z2 ) @ X5 ) ) ) ).
% sup_least
thf(fact_1031_sup__least,axiom,
! [Y4: set_a,X5: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X5 )
=> ( ( ord_less_eq_set_a @ Z2 @ X5 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y4 @ Z2 ) @ X5 ) ) ) ).
% sup_least
thf(fact_1032_sup__least,axiom,
! [Y4: nat,X5: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y4 @ X5 )
=> ( ( ord_less_eq_nat @ Z2 @ X5 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y4 @ Z2 ) @ X5 ) ) ) ).
% sup_least
thf(fact_1033_sup__mono,axiom,
! [A2: set_Product_prod_a_a,C2: set_Product_prod_a_a,B7: set_Product_prod_a_a,D2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ C2 )
=> ( ( ord_le746702958409616551od_a_a @ B7 @ D2 )
=> ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ A2 @ B7 ) @ ( sup_su3048258781599657691od_a_a @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_1034_sup__mono,axiom,
! [A2: set_list_a,C2: set_list_a,B7: set_list_a,D2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ C2 )
=> ( ( ord_le8861187494160871172list_a @ B7 @ D2 )
=> ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A2 @ B7 ) @ ( sup_sup_set_list_a @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_1035_sup__mono,axiom,
! [A2: set_set_a,C2: set_set_a,B7: set_set_a,D2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B7 @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B7 ) @ ( sup_sup_set_set_a @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_1036_sup__mono,axiom,
! [A2: set_a,C2: set_a,B7: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B7 @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B7 ) @ ( sup_sup_set_a @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_1037_sup__mono,axiom,
! [A2: nat,C2: nat,B7: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B7 @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B7 ) @ ( sup_sup_nat @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_1038_sup_Omono,axiom,
! [C2: set_Product_prod_a_a,A2: set_Product_prod_a_a,D2: set_Product_prod_a_a,B7: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C2 @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ D2 @ B7 )
=> ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ C2 @ D2 ) @ ( sup_su3048258781599657691od_a_a @ A2 @ B7 ) ) ) ) ).
% sup.mono
thf(fact_1039_sup_Omono,axiom,
! [C2: set_list_a,A2: set_list_a,D2: set_list_a,B7: set_list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ A2 )
=> ( ( ord_le8861187494160871172list_a @ D2 @ B7 )
=> ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ C2 @ D2 ) @ ( sup_sup_set_list_a @ A2 @ B7 ) ) ) ) ).
% sup.mono
thf(fact_1040_sup_Omono,axiom,
! [C2: set_set_a,A2: set_set_a,D2: set_set_a,B7: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ D2 @ B7 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ C2 @ D2 ) @ ( sup_sup_set_set_a @ A2 @ B7 ) ) ) ) ).
% sup.mono
thf(fact_1041_sup_Omono,axiom,
! [C2: set_a,A2: set_a,D2: set_a,B7: set_a] :
( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( ( ord_less_eq_set_a @ D2 @ B7 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C2 @ D2 ) @ ( sup_sup_set_a @ A2 @ B7 ) ) ) ) ).
% sup.mono
thf(fact_1042_sup_Omono,axiom,
! [C2: nat,A2: nat,D2: nat,B7: nat] :
( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ( ord_less_eq_nat @ D2 @ B7 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D2 ) @ ( sup_sup_nat @ A2 @ B7 ) ) ) ) ).
% sup.mono
thf(fact_1043_le__supI2,axiom,
! [X5: set_Product_prod_a_a,B7: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X5 @ B7 )
=> ( ord_le746702958409616551od_a_a @ X5 @ ( sup_su3048258781599657691od_a_a @ A2 @ B7 ) ) ) ).
% le_supI2
thf(fact_1044_le__supI2,axiom,
! [X5: set_list_a,B7: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X5 @ B7 )
=> ( ord_le8861187494160871172list_a @ X5 @ ( sup_sup_set_list_a @ A2 @ B7 ) ) ) ).
% le_supI2
thf(fact_1045_le__supI2,axiom,
! [X5: set_set_a,B7: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ B7 )
=> ( ord_le3724670747650509150_set_a @ X5 @ ( sup_sup_set_set_a @ A2 @ B7 ) ) ) ).
% le_supI2
thf(fact_1046_le__supI2,axiom,
! [X5: set_a,B7: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ X5 @ B7 )
=> ( ord_less_eq_set_a @ X5 @ ( sup_sup_set_a @ A2 @ B7 ) ) ) ).
% le_supI2
thf(fact_1047_le__supI2,axiom,
! [X5: nat,B7: nat,A2: nat] :
( ( ord_less_eq_nat @ X5 @ B7 )
=> ( ord_less_eq_nat @ X5 @ ( sup_sup_nat @ A2 @ B7 ) ) ) ).
% le_supI2
thf(fact_1048_le__supI1,axiom,
! [X5: set_Product_prod_a_a,A2: set_Product_prod_a_a,B7: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X5 @ A2 )
=> ( ord_le746702958409616551od_a_a @ X5 @ ( sup_su3048258781599657691od_a_a @ A2 @ B7 ) ) ) ).
% le_supI1
thf(fact_1049_le__supI1,axiom,
! [X5: set_list_a,A2: set_list_a,B7: set_list_a] :
( ( ord_le8861187494160871172list_a @ X5 @ A2 )
=> ( ord_le8861187494160871172list_a @ X5 @ ( sup_sup_set_list_a @ A2 @ B7 ) ) ) ).
% le_supI1
thf(fact_1050_le__supI1,axiom,
! [X5: set_set_a,A2: set_set_a,B7: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ A2 )
=> ( ord_le3724670747650509150_set_a @ X5 @ ( sup_sup_set_set_a @ A2 @ B7 ) ) ) ).
% le_supI1
thf(fact_1051_le__supI1,axiom,
! [X5: set_a,A2: set_a,B7: set_a] :
( ( ord_less_eq_set_a @ X5 @ A2 )
=> ( ord_less_eq_set_a @ X5 @ ( sup_sup_set_a @ A2 @ B7 ) ) ) ).
% le_supI1
thf(fact_1052_le__supI1,axiom,
! [X5: nat,A2: nat,B7: nat] :
( ( ord_less_eq_nat @ X5 @ A2 )
=> ( ord_less_eq_nat @ X5 @ ( sup_sup_nat @ A2 @ B7 ) ) ) ).
% le_supI1
thf(fact_1053_sup__ge2,axiom,
! [Y4: set_Product_prod_a_a,X5: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ Y4 @ ( sup_su3048258781599657691od_a_a @ X5 @ Y4 ) ) ).
% sup_ge2
thf(fact_1054_sup__ge2,axiom,
! [Y4: set_list_a,X5: set_list_a] : ( ord_le8861187494160871172list_a @ Y4 @ ( sup_sup_set_list_a @ X5 @ Y4 ) ) ).
% sup_ge2
thf(fact_1055_sup__ge2,axiom,
! [Y4: set_set_a,X5: set_set_a] : ( ord_le3724670747650509150_set_a @ Y4 @ ( sup_sup_set_set_a @ X5 @ Y4 ) ) ).
% sup_ge2
thf(fact_1056_sup__ge2,axiom,
! [Y4: set_a,X5: set_a] : ( ord_less_eq_set_a @ Y4 @ ( sup_sup_set_a @ X5 @ Y4 ) ) ).
% sup_ge2
thf(fact_1057_sup__ge2,axiom,
! [Y4: nat,X5: nat] : ( ord_less_eq_nat @ Y4 @ ( sup_sup_nat @ X5 @ Y4 ) ) ).
% sup_ge2
thf(fact_1058_sup__ge1,axiom,
! [X5: set_Product_prod_a_a,Y4: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ X5 @ ( sup_su3048258781599657691od_a_a @ X5 @ Y4 ) ) ).
% sup_ge1
thf(fact_1059_sup__ge1,axiom,
! [X5: set_list_a,Y4: set_list_a] : ( ord_le8861187494160871172list_a @ X5 @ ( sup_sup_set_list_a @ X5 @ Y4 ) ) ).
% sup_ge1
thf(fact_1060_sup__ge1,axiom,
! [X5: set_set_a,Y4: set_set_a] : ( ord_le3724670747650509150_set_a @ X5 @ ( sup_sup_set_set_a @ X5 @ Y4 ) ) ).
% sup_ge1
thf(fact_1061_sup__ge1,axiom,
! [X5: set_a,Y4: set_a] : ( ord_less_eq_set_a @ X5 @ ( sup_sup_set_a @ X5 @ Y4 ) ) ).
% sup_ge1
thf(fact_1062_sup__ge1,axiom,
! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ X5 @ ( sup_sup_nat @ X5 @ Y4 ) ) ).
% sup_ge1
thf(fact_1063_le__supI,axiom,
! [A2: set_Product_prod_a_a,X5: set_Product_prod_a_a,B7: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ X5 )
=> ( ( ord_le746702958409616551od_a_a @ B7 @ X5 )
=> ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ A2 @ B7 ) @ X5 ) ) ) ).
% le_supI
thf(fact_1064_le__supI,axiom,
! [A2: set_list_a,X5: set_list_a,B7: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ X5 )
=> ( ( ord_le8861187494160871172list_a @ B7 @ X5 )
=> ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A2 @ B7 ) @ X5 ) ) ) ).
% le_supI
thf(fact_1065_le__supI,axiom,
! [A2: set_set_a,X5: set_set_a,B7: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ X5 )
=> ( ( ord_le3724670747650509150_set_a @ B7 @ X5 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B7 ) @ X5 ) ) ) ).
% le_supI
thf(fact_1066_le__supI,axiom,
! [A2: set_a,X5: set_a,B7: set_a] :
( ( ord_less_eq_set_a @ A2 @ X5 )
=> ( ( ord_less_eq_set_a @ B7 @ X5 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B7 ) @ X5 ) ) ) ).
% le_supI
thf(fact_1067_le__supI,axiom,
! [A2: nat,X5: nat,B7: nat] :
( ( ord_less_eq_nat @ A2 @ X5 )
=> ( ( ord_less_eq_nat @ B7 @ X5 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B7 ) @ X5 ) ) ) ).
% le_supI
thf(fact_1068_le__supE,axiom,
! [A2: set_Product_prod_a_a,B7: set_Product_prod_a_a,X5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ A2 @ B7 ) @ X5 )
=> ~ ( ( ord_le746702958409616551od_a_a @ A2 @ X5 )
=> ~ ( ord_le746702958409616551od_a_a @ B7 @ X5 ) ) ) ).
% le_supE
thf(fact_1069_le__supE,axiom,
! [A2: set_list_a,B7: set_list_a,X5: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A2 @ B7 ) @ X5 )
=> ~ ( ( ord_le8861187494160871172list_a @ A2 @ X5 )
=> ~ ( ord_le8861187494160871172list_a @ B7 @ X5 ) ) ) ).
% le_supE
thf(fact_1070_le__supE,axiom,
! [A2: set_set_a,B7: set_set_a,X5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B7 ) @ X5 )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ X5 )
=> ~ ( ord_le3724670747650509150_set_a @ B7 @ X5 ) ) ) ).
% le_supE
thf(fact_1071_le__supE,axiom,
! [A2: set_a,B7: set_a,X5: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B7 ) @ X5 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ X5 )
=> ~ ( ord_less_eq_set_a @ B7 @ X5 ) ) ) ).
% le_supE
thf(fact_1072_le__supE,axiom,
! [A2: nat,B7: nat,X5: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B7 ) @ X5 )
=> ~ ( ( ord_less_eq_nat @ A2 @ X5 )
=> ~ ( ord_less_eq_nat @ B7 @ X5 ) ) ) ).
% le_supE
thf(fact_1073_inf__sup__ord_I3_J,axiom,
! [X5: set_Product_prod_a_a,Y4: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ X5 @ ( sup_su3048258781599657691od_a_a @ X5 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_1074_inf__sup__ord_I3_J,axiom,
! [X5: set_list_a,Y4: set_list_a] : ( ord_le8861187494160871172list_a @ X5 @ ( sup_sup_set_list_a @ X5 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_1075_inf__sup__ord_I3_J,axiom,
! [X5: set_set_a,Y4: set_set_a] : ( ord_le3724670747650509150_set_a @ X5 @ ( sup_sup_set_set_a @ X5 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_1076_inf__sup__ord_I3_J,axiom,
! [X5: set_a,Y4: set_a] : ( ord_less_eq_set_a @ X5 @ ( sup_sup_set_a @ X5 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_1077_inf__sup__ord_I3_J,axiom,
! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ X5 @ ( sup_sup_nat @ X5 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_1078_inf__sup__ord_I4_J,axiom,
! [Y4: set_Product_prod_a_a,X5: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ Y4 @ ( sup_su3048258781599657691od_a_a @ X5 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_1079_inf__sup__ord_I4_J,axiom,
! [Y4: set_list_a,X5: set_list_a] : ( ord_le8861187494160871172list_a @ Y4 @ ( sup_sup_set_list_a @ X5 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_1080_inf__sup__ord_I4_J,axiom,
! [Y4: set_set_a,X5: set_set_a] : ( ord_le3724670747650509150_set_a @ Y4 @ ( sup_sup_set_set_a @ X5 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_1081_inf__sup__ord_I4_J,axiom,
! [Y4: set_a,X5: set_a] : ( ord_less_eq_set_a @ Y4 @ ( sup_sup_set_a @ X5 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_1082_inf__sup__ord_I4_J,axiom,
! [Y4: nat,X5: nat] : ( ord_less_eq_nat @ Y4 @ ( sup_sup_nat @ X5 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_1083_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_1084_card__insert__disjoint,axiom,
! [A: set_list_a,X5: list_a] :
( ( finite_finite_list_a @ A )
=> ( ~ ( member_list_a @ X5 @ A )
=> ( ( finite_card_list_a @ ( insert_list_a @ X5 @ A ) )
= ( suc @ ( finite_card_list_a @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1085_card__insert__disjoint,axiom,
! [A: set_set_a,X5: set_a] :
( ( finite_finite_set_a @ A )
=> ( ~ ( member_set_a @ X5 @ A )
=> ( ( finite_card_set_a @ ( insert_set_a @ X5 @ A ) )
= ( suc @ ( finite_card_set_a @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1086_card__insert__disjoint,axiom,
! [A: set_a,X5: a] :
( ( finite_finite_a @ A )
=> ( ~ ( member_a @ X5 @ A )
=> ( ( finite_card_a @ ( insert_a @ X5 @ A ) )
= ( suc @ ( finite_card_a @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1087_card__insert__disjoint,axiom,
! [A: set_nat,X5: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X5 @ A )
=> ( ( finite_card_nat @ ( insert_nat @ X5 @ A ) )
= ( suc @ ( finite_card_nat @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1088_card__insert__disjoint,axiom,
! [A: set_Product_prod_a_a,X5: product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A )
=> ( ~ ( member1426531477525435216od_a_a @ X5 @ A )
=> ( ( finite4795055649997197647od_a_a @ ( insert4534936382041156343od_a_a @ X5 @ A ) )
= ( suc @ ( finite4795055649997197647od_a_a @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1089_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_1090_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1091_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1092_card__le__Suc__iff,axiom,
! [N: nat,A: set_list_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_list_a @ A ) )
= ( ? [A7: list_a,B5: set_list_a] :
( ( A
= ( insert_list_a @ A7 @ B5 ) )
& ~ ( member_list_a @ A7 @ B5 )
& ( ord_less_eq_nat @ N @ ( finite_card_list_a @ B5 ) )
& ( finite_finite_list_a @ B5 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_1093_card__le__Suc__iff,axiom,
! [N: nat,A: set_set_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_set_a @ A ) )
= ( ? [A7: set_a,B5: set_set_a] :
( ( A
= ( insert_set_a @ A7 @ B5 ) )
& ~ ( member_set_a @ A7 @ B5 )
& ( ord_less_eq_nat @ N @ ( finite_card_set_a @ B5 ) )
& ( finite_finite_set_a @ B5 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_1094_card__le__Suc__iff,axiom,
! [N: nat,A: set_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_a @ A ) )
= ( ? [A7: a,B5: set_a] :
( ( A
= ( insert_a @ A7 @ B5 ) )
& ~ ( member_a @ A7 @ B5 )
& ( ord_less_eq_nat @ N @ ( finite_card_a @ B5 ) )
& ( finite_finite_a @ B5 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_1095_card__le__Suc__iff,axiom,
! [N: nat,A: set_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_nat @ A ) )
= ( ? [A7: nat,B5: set_nat] :
( ( A
= ( insert_nat @ A7 @ B5 ) )
& ~ ( member_nat @ A7 @ B5 )
& ( ord_less_eq_nat @ N @ ( finite_card_nat @ B5 ) )
& ( finite_finite_nat @ B5 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_1096_card__le__Suc__iff,axiom,
! [N: nat,A: set_Product_prod_a_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite4795055649997197647od_a_a @ A ) )
= ( ? [A7: product_prod_a_a,B5: set_Product_prod_a_a] :
( ( A
= ( insert4534936382041156343od_a_a @ A7 @ B5 ) )
& ~ ( member1426531477525435216od_a_a @ A7 @ B5 )
& ( ord_less_eq_nat @ N @ ( finite4795055649997197647od_a_a @ B5 ) )
& ( finite6544458595007987280od_a_a @ B5 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_1097_card__le__Suc0__iff__eq,axiom,
! [A: set_list_a] :
( ( finite_finite_list_a @ A )
=> ( ( ord_less_eq_nat @ ( finite_card_list_a @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A )
=> ! [Y6: list_a] :
( ( member_list_a @ Y6 @ A )
=> ( X3 = Y6 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1098_card__le__Suc0__iff__eq,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ! [Y6: set_a] :
( ( member_set_a @ Y6 @ A )
=> ( X3 = Y6 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1099_card__le__Suc0__iff__eq,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ! [Y6: a] :
( ( member_a @ Y6 @ A )
=> ( X3 = Y6 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1100_card__le__Suc0__iff__eq,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ! [Y6: nat] :
( ( member_nat @ Y6 @ A )
=> ( X3 = Y6 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1101_card__le__Suc0__iff__eq,axiom,
! [A: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A )
=> ( ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A )
=> ! [Y6: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y6 @ A )
=> ( X3 = Y6 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1102_Suc__le__mono,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M4 ) )
= ( ord_less_eq_nat @ N @ M4 ) ) ).
% Suc_le_mono
thf(fact_1103_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_1104_Zero__not__Suc,axiom,
! [M4: nat] :
( zero_zero_nat
!= ( suc @ M4 ) ) ).
% Zero_not_Suc
thf(fact_1105_Zero__neq__Suc,axiom,
! [M4: nat] :
( zero_zero_nat
!= ( suc @ M4 ) ) ).
% Zero_neq_Suc
thf(fact_1106_Suc__neq__Zero,axiom,
! [M4: nat] :
( ( suc @ M4 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1107_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1108_diff__induct,axiom,
! [P: nat > nat > $o,M4: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X4: nat,Y2: nat] :
( ( P @ X4 @ Y2 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y2 ) ) )
=> ( P @ M4 @ N ) ) ) ) ).
% diff_induct
thf(fact_1109_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1110_old_Onat_Oexhaust,axiom,
! [Y4: nat] :
( ( Y4 != zero_zero_nat )
=> ~ ! [Nat: nat] :
( Y4
!= ( suc @ Nat ) ) ) ).
% old.nat.exhaust
thf(fact_1111_nat_OdiscI,axiom,
! [Nat2: nat,X22: nat] :
( ( Nat2
= ( suc @ X22 ) )
=> ( Nat2 != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1112_old_Onat_Odistinct_I1_J,axiom,
! [Nat3: nat] :
( zero_zero_nat
!= ( suc @ Nat3 ) ) ).
% old.nat.distinct(1)
thf(fact_1113_old_Onat_Odistinct_I2_J,axiom,
! [Nat3: nat] :
( ( suc @ Nat3 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1114_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1115_transitive__stepwise__le,axiom,
! [M4: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ! [X4: nat] : ( R2 @ X4 @ X4 )
=> ( ! [X4: nat,Y2: nat,Z3: nat] :
( ( R2 @ X4 @ Y2 )
=> ( ( R2 @ Y2 @ Z3 )
=> ( R2 @ X4 @ Z3 ) ) )
=> ( ! [N4: nat] : ( R2 @ N4 @ ( suc @ N4 ) )
=> ( R2 @ M4 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1116_nat__induct__at__least,axiom,
! [M4: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ( P @ M4 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1117_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N4 )
=> ( P @ M5 ) )
=> ( P @ N4 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1118_not__less__eq__eq,axiom,
! [M4: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M4 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M4 ) ) ).
% not_less_eq_eq
thf(fact_1119_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1120_le__Suc__eq,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M4 @ N )
| ( M4
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1121_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M3: nat] :
( M6
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_1122_le__SucI,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ord_less_eq_nat @ M4 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1123_le__SucE,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M4 @ N )
=> ( M4
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1124_Suc__leD,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N )
=> ( ord_less_eq_nat @ M4 @ N ) ) ).
% Suc_leD
thf(fact_1125_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1126_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B7: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B7 ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y7: nat] :
( ( P @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1127_nat__le__linear,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
| ( ord_less_eq_nat @ N @ M4 ) ) ).
% nat_le_linear
thf(fact_1128_le__antisym,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ( ord_less_eq_nat @ N @ M4 )
=> ( M4 = N ) ) ) ).
% le_antisym
thf(fact_1129_eq__imp__le,axiom,
! [M4: nat,N: nat] :
( ( M4 = N )
=> ( ord_less_eq_nat @ M4 @ N ) ) ).
% eq_imp_le
thf(fact_1130_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_1131_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1132_lift__Suc__antimono__le,axiom,
! [F3: nat > set_Product_prod_a_a,N: nat,N5: nat] :
( ! [N4: nat] : ( ord_le746702958409616551od_a_a @ ( F3 @ ( suc @ N4 ) ) @ ( F3 @ N4 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_le746702958409616551od_a_a @ ( F3 @ N5 ) @ ( F3 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1133_lift__Suc__antimono__le,axiom,
! [F3: nat > set_list_a,N: nat,N5: nat] :
( ! [N4: nat] : ( ord_le8861187494160871172list_a @ ( F3 @ ( suc @ N4 ) ) @ ( F3 @ N4 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_le8861187494160871172list_a @ ( F3 @ N5 ) @ ( F3 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1134_lift__Suc__antimono__le,axiom,
! [F3: nat > set_set_a,N: nat,N5: nat] :
( ! [N4: nat] : ( ord_le3724670747650509150_set_a @ ( F3 @ ( suc @ N4 ) ) @ ( F3 @ N4 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_le3724670747650509150_set_a @ ( F3 @ N5 ) @ ( F3 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1135_lift__Suc__antimono__le,axiom,
! [F3: nat > set_a,N: nat,N5: nat] :
( ! [N4: nat] : ( ord_less_eq_set_a @ ( F3 @ ( suc @ N4 ) ) @ ( F3 @ N4 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_set_a @ ( F3 @ N5 ) @ ( F3 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1136_lift__Suc__antimono__le,axiom,
! [F3: nat > nat,N: nat,N5: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F3 @ ( suc @ N4 ) ) @ ( F3 @ N4 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_nat @ ( F3 @ N5 ) @ ( F3 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1137_lift__Suc__mono__le,axiom,
! [F3: nat > set_Product_prod_a_a,N: nat,N5: nat] :
( ! [N4: nat] : ( ord_le746702958409616551od_a_a @ ( F3 @ N4 ) @ ( F3 @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_le746702958409616551od_a_a @ ( F3 @ N ) @ ( F3 @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1138_lift__Suc__mono__le,axiom,
! [F3: nat > set_list_a,N: nat,N5: nat] :
( ! [N4: nat] : ( ord_le8861187494160871172list_a @ ( F3 @ N4 ) @ ( F3 @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_le8861187494160871172list_a @ ( F3 @ N ) @ ( F3 @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1139_lift__Suc__mono__le,axiom,
! [F3: nat > set_set_a,N: nat,N5: nat] :
( ! [N4: nat] : ( ord_le3724670747650509150_set_a @ ( F3 @ N4 ) @ ( F3 @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_le3724670747650509150_set_a @ ( F3 @ N ) @ ( F3 @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1140_lift__Suc__mono__le,axiom,
! [F3: nat > set_a,N: nat,N5: nat] :
( ! [N4: nat] : ( ord_less_eq_set_a @ ( F3 @ N4 ) @ ( F3 @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_set_a @ ( F3 @ N ) @ ( F3 @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1141_lift__Suc__mono__le,axiom,
! [F3: nat > nat,N: nat,N5: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F3 @ N4 ) @ ( F3 @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_nat @ ( F3 @ N ) @ ( F3 @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1142_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1143_ulgraph_Ois__edge__or__loop,axiom,
! [Vertices: set_a,Edges: set_set_a,E: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( member_set_a @ E @ Edges )
=> ( ( undire2905028936066782638loop_a @ E )
| ( undire4917966558017083288edge_a @ E ) ) ) ) ).
% ulgraph.is_edge_or_loop
thf(fact_1144_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1145_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1146_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1147_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1148_comp__sgraph_Ois__edge__between__def,axiom,
( undire2578756059399487229_set_a
= ( ^ [X6: set_set_a,Y5: set_set_a,E5: set_set_a] :
? [X3: set_a,Y6: set_a] :
( ( E5
= ( insert_set_a @ X3 @ ( insert_set_a @ Y6 @ bot_bot_set_set_a ) ) )
& ( member_set_a @ X3 @ X6 )
& ( member_set_a @ Y6 @ Y5 ) ) ) ) ).
% comp_sgraph.is_edge_between_def
thf(fact_1149_comp__sgraph_Ois__edge__between__def,axiom,
( undire7011261089604658374od_a_a
= ( ^ [X6: set_Product_prod_a_a,Y5: set_Product_prod_a_a,E5: set_Product_prod_a_a] :
? [X3: product_prod_a_a,Y6: product_prod_a_a] :
( ( E5
= ( insert4534936382041156343od_a_a @ X3 @ ( insert4534936382041156343od_a_a @ Y6 @ bot_bo3357376287454694259od_a_a ) ) )
& ( member1426531477525435216od_a_a @ X3 @ X6 )
& ( member1426531477525435216od_a_a @ Y6 @ Y5 ) ) ) ) ).
% comp_sgraph.is_edge_between_def
thf(fact_1150_comp__sgraph_Ois__edge__between__def,axiom,
( undire6814325412647357297en_nat
= ( ^ [X6: set_nat,Y5: set_nat,E5: set_nat] :
? [X3: nat,Y6: nat] :
( ( E5
= ( insert_nat @ X3 @ ( insert_nat @ Y6 @ bot_bot_set_nat ) ) )
& ( member_nat @ X3 @ X6 )
& ( member_nat @ Y6 @ Y5 ) ) ) ) ).
% comp_sgraph.is_edge_between_def
thf(fact_1151_comp__sgraph_Ois__edge__between__def,axiom,
( undire8544646567961481629ween_a
= ( ^ [X6: set_a,Y5: set_a,E5: set_a] :
? [X3: a,Y6: a] :
( ( E5
= ( insert_a @ X3 @ ( insert_a @ Y6 @ bot_bot_set_a ) ) )
& ( member_a @ X3 @ X6 )
& ( member_a @ Y6 @ Y5 ) ) ) ) ).
% comp_sgraph.is_edge_between_def
thf(fact_1152_ulgraph_Ois__edge__between__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X2: set_set_a,Y: set_set_a,E: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire2578756059399487229_set_a @ X2 @ Y @ E )
= ( ? [X3: set_a,Y6: set_a] :
( ( E
= ( insert_set_a @ X3 @ ( insert_set_a @ Y6 @ bot_bot_set_set_a ) ) )
& ( member_set_a @ X3 @ X2 )
& ( member_set_a @ Y6 @ Y ) ) ) ) ) ).
% ulgraph.is_edge_between_def
thf(fact_1153_ulgraph_Ois__edge__between__def,axiom,
! [Vertices: set_Product_prod_a_a,Edges: set_se5735800977113168103od_a_a,X2: set_Product_prod_a_a,Y: set_Product_prod_a_a,E: set_Product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices @ Edges )
=> ( ( undire7011261089604658374od_a_a @ X2 @ Y @ E )
= ( ? [X3: product_prod_a_a,Y6: product_prod_a_a] :
( ( E
= ( insert4534936382041156343od_a_a @ X3 @ ( insert4534936382041156343od_a_a @ Y6 @ bot_bo3357376287454694259od_a_a ) ) )
& ( member1426531477525435216od_a_a @ X3 @ X2 )
& ( member1426531477525435216od_a_a @ Y6 @ Y ) ) ) ) ) ).
% ulgraph.is_edge_between_def
thf(fact_1154_ulgraph_Ois__edge__between__def,axiom,
! [Vertices: set_nat,Edges: set_set_nat,X2: set_nat,Y: set_nat,E: set_nat] :
( ( undire3269267262472140706ph_nat @ Vertices @ Edges )
=> ( ( undire6814325412647357297en_nat @ X2 @ Y @ E )
= ( ? [X3: nat,Y6: nat] :
( ( E
= ( insert_nat @ X3 @ ( insert_nat @ Y6 @ bot_bot_set_nat ) ) )
& ( member_nat @ X3 @ X2 )
& ( member_nat @ Y6 @ Y ) ) ) ) ) ).
% ulgraph.is_edge_between_def
thf(fact_1155_ulgraph_Ois__edge__between__def,axiom,
! [Vertices: set_a,Edges: set_set_a,X2: set_a,Y: set_a,E: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8544646567961481629ween_a @ X2 @ Y @ E )
= ( ? [X3: a,Y6: a] :
( ( E
= ( insert_a @ X3 @ ( insert_a @ Y6 @ bot_bot_set_a ) ) )
& ( member_a @ X3 @ X2 )
& ( member_a @ Y6 @ Y ) ) ) ) ) ).
% ulgraph.is_edge_between_def
thf(fact_1156_card__Suc__eq__finite,axiom,
! [A: set_list_a,K: nat] :
( ( ( finite_card_list_a @ A )
= ( suc @ K ) )
= ( ? [B8: list_a,B5: set_list_a] :
( ( A
= ( insert_list_a @ B8 @ B5 ) )
& ~ ( member_list_a @ B8 @ B5 )
& ( ( finite_card_list_a @ B5 )
= K )
& ( finite_finite_list_a @ B5 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_1157_card__Suc__eq__finite,axiom,
! [A: set_set_a,K: nat] :
( ( ( finite_card_set_a @ A )
= ( suc @ K ) )
= ( ? [B8: set_a,B5: set_set_a] :
( ( A
= ( insert_set_a @ B8 @ B5 ) )
& ~ ( member_set_a @ B8 @ B5 )
& ( ( finite_card_set_a @ B5 )
= K )
& ( finite_finite_set_a @ B5 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_1158_card__Suc__eq__finite,axiom,
! [A: set_a,K: nat] :
( ( ( finite_card_a @ A )
= ( suc @ K ) )
= ( ? [B8: a,B5: set_a] :
( ( A
= ( insert_a @ B8 @ B5 ) )
& ~ ( member_a @ B8 @ B5 )
& ( ( finite_card_a @ B5 )
= K )
& ( finite_finite_a @ B5 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_1159_card__Suc__eq__finite,axiom,
! [A: set_nat,K: nat] :
( ( ( finite_card_nat @ A )
= ( suc @ K ) )
= ( ? [B8: nat,B5: set_nat] :
( ( A
= ( insert_nat @ B8 @ B5 ) )
& ~ ( member_nat @ B8 @ B5 )
& ( ( finite_card_nat @ B5 )
= K )
& ( finite_finite_nat @ B5 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_1160_card__Suc__eq__finite,axiom,
! [A: set_Product_prod_a_a,K: nat] :
( ( ( finite4795055649997197647od_a_a @ A )
= ( suc @ K ) )
= ( ? [B8: product_prod_a_a,B5: set_Product_prod_a_a] :
( ( A
= ( insert4534936382041156343od_a_a @ B8 @ B5 ) )
& ~ ( member1426531477525435216od_a_a @ B8 @ B5 )
& ( ( finite4795055649997197647od_a_a @ B5 )
= K )
& ( finite6544458595007987280od_a_a @ B5 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_1161_card__insert__if,axiom,
! [A: set_list_a,X5: list_a] :
( ( finite_finite_list_a @ A )
=> ( ( ( member_list_a @ X5 @ A )
=> ( ( finite_card_list_a @ ( insert_list_a @ X5 @ A ) )
= ( finite_card_list_a @ A ) ) )
& ( ~ ( member_list_a @ X5 @ A )
=> ( ( finite_card_list_a @ ( insert_list_a @ X5 @ A ) )
= ( suc @ ( finite_card_list_a @ A ) ) ) ) ) ) ).
% card_insert_if
thf(fact_1162_card__insert__if,axiom,
! [A: set_set_a,X5: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( ( member_set_a @ X5 @ A )
=> ( ( finite_card_set_a @ ( insert_set_a @ X5 @ A ) )
= ( finite_card_set_a @ A ) ) )
& ( ~ ( member_set_a @ X5 @ A )
=> ( ( finite_card_set_a @ ( insert_set_a @ X5 @ A ) )
= ( suc @ ( finite_card_set_a @ A ) ) ) ) ) ) ).
% card_insert_if
thf(fact_1163_card__insert__if,axiom,
! [A: set_a,X5: a] :
( ( finite_finite_a @ A )
=> ( ( ( member_a @ X5 @ A )
=> ( ( finite_card_a @ ( insert_a @ X5 @ A ) )
= ( finite_card_a @ A ) ) )
& ( ~ ( member_a @ X5 @ A )
=> ( ( finite_card_a @ ( insert_a @ X5 @ A ) )
= ( suc @ ( finite_card_a @ A ) ) ) ) ) ) ).
% card_insert_if
thf(fact_1164_card__insert__if,axiom,
! [A: set_nat,X5: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( member_nat @ X5 @ A )
=> ( ( finite_card_nat @ ( insert_nat @ X5 @ A ) )
= ( finite_card_nat @ A ) ) )
& ( ~ ( member_nat @ X5 @ A )
=> ( ( finite_card_nat @ ( insert_nat @ X5 @ A ) )
= ( suc @ ( finite_card_nat @ A ) ) ) ) ) ) ).
% card_insert_if
thf(fact_1165_card__insert__if,axiom,
! [A: set_Product_prod_a_a,X5: product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A )
=> ( ( ( member1426531477525435216od_a_a @ X5 @ A )
=> ( ( finite4795055649997197647od_a_a @ ( insert4534936382041156343od_a_a @ X5 @ A ) )
= ( finite4795055649997197647od_a_a @ A ) ) )
& ( ~ ( member1426531477525435216od_a_a @ X5 @ A )
=> ( ( finite4795055649997197647od_a_a @ ( insert4534936382041156343od_a_a @ X5 @ A ) )
= ( suc @ ( finite4795055649997197647od_a_a @ A ) ) ) ) ) ) ).
% card_insert_if
thf(fact_1166_card__Suc__eq,axiom,
! [A: set_set_a,K: nat] :
( ( ( finite_card_set_a @ A )
= ( suc @ K ) )
= ( ? [B8: set_a,B5: set_set_a] :
( ( A
= ( insert_set_a @ B8 @ B5 ) )
& ~ ( member_set_a @ B8 @ B5 )
& ( ( finite_card_set_a @ B5 )
= K )
& ( ( K = zero_zero_nat )
=> ( B5 = bot_bot_set_set_a ) ) ) ) ) ).
% card_Suc_eq
thf(fact_1167_card__Suc__eq,axiom,
! [A: set_a,K: nat] :
( ( ( finite_card_a @ A )
= ( suc @ K ) )
= ( ? [B8: a,B5: set_a] :
( ( A
= ( insert_a @ B8 @ B5 ) )
& ~ ( member_a @ B8 @ B5 )
& ( ( finite_card_a @ B5 )
= K )
& ( ( K = zero_zero_nat )
=> ( B5 = bot_bot_set_a ) ) ) ) ) ).
% card_Suc_eq
thf(fact_1168_card__Suc__eq,axiom,
! [A: set_Product_prod_a_a,K: nat] :
( ( ( finite4795055649997197647od_a_a @ A )
= ( suc @ K ) )
= ( ? [B8: product_prod_a_a,B5: set_Product_prod_a_a] :
( ( A
= ( insert4534936382041156343od_a_a @ B8 @ B5 ) )
& ~ ( member1426531477525435216od_a_a @ B8 @ B5 )
& ( ( finite4795055649997197647od_a_a @ B5 )
= K )
& ( ( K = zero_zero_nat )
=> ( B5 = bot_bo3357376287454694259od_a_a ) ) ) ) ) ).
% card_Suc_eq
thf(fact_1169_card__Suc__eq,axiom,
! [A: set_nat,K: nat] :
( ( ( finite_card_nat @ A )
= ( suc @ K ) )
= ( ? [B8: nat,B5: set_nat] :
( ( A
= ( insert_nat @ B8 @ B5 ) )
& ~ ( member_nat @ B8 @ B5 )
& ( ( finite_card_nat @ B5 )
= K )
& ( ( K = zero_zero_nat )
=> ( B5 = bot_bot_set_nat ) ) ) ) ) ).
% card_Suc_eq
thf(fact_1170_card__eq__SucD,axiom,
! [A: set_set_a,K: nat] :
( ( ( finite_card_set_a @ A )
= ( suc @ K ) )
=> ? [B4: set_a,B2: set_set_a] :
( ( A
= ( insert_set_a @ B4 @ B2 ) )
& ~ ( member_set_a @ B4 @ B2 )
& ( ( finite_card_set_a @ B2 )
= K )
& ( ( K = zero_zero_nat )
=> ( B2 = bot_bot_set_set_a ) ) ) ) ).
% card_eq_SucD
thf(fact_1171_card__eq__SucD,axiom,
! [A: set_a,K: nat] :
( ( ( finite_card_a @ A )
= ( suc @ K ) )
=> ? [B4: a,B2: set_a] :
( ( A
= ( insert_a @ B4 @ B2 ) )
& ~ ( member_a @ B4 @ B2 )
& ( ( finite_card_a @ B2 )
= K )
& ( ( K = zero_zero_nat )
=> ( B2 = bot_bot_set_a ) ) ) ) ).
% card_eq_SucD
thf(fact_1172_card__eq__SucD,axiom,
! [A: set_Product_prod_a_a,K: nat] :
( ( ( finite4795055649997197647od_a_a @ A )
= ( suc @ K ) )
=> ? [B4: product_prod_a_a,B2: set_Product_prod_a_a] :
( ( A
= ( insert4534936382041156343od_a_a @ B4 @ B2 ) )
& ~ ( member1426531477525435216od_a_a @ B4 @ B2 )
& ( ( finite4795055649997197647od_a_a @ B2 )
= K )
& ( ( K = zero_zero_nat )
=> ( B2 = bot_bo3357376287454694259od_a_a ) ) ) ) ).
% card_eq_SucD
thf(fact_1173_card__eq__SucD,axiom,
! [A: set_nat,K: nat] :
( ( ( finite_card_nat @ A )
= ( suc @ K ) )
=> ? [B4: nat,B2: set_nat] :
( ( A
= ( insert_nat @ B4 @ B2 ) )
& ~ ( member_nat @ B4 @ B2 )
& ( ( finite_card_nat @ B2 )
= K )
& ( ( K = zero_zero_nat )
=> ( B2 = bot_bot_set_nat ) ) ) ) ).
% card_eq_SucD
thf(fact_1174_card__1__singleton__iff,axiom,
! [A: set_set_a] :
( ( ( finite_card_set_a @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X3: set_a] :
( A
= ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) ) ) ).
% card_1_singleton_iff
thf(fact_1175_card__1__singleton__iff,axiom,
! [A: set_a] :
( ( ( finite_card_a @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X3: a] :
( A
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% card_1_singleton_iff
thf(fact_1176_card__1__singleton__iff,axiom,
! [A: set_Product_prod_a_a] :
( ( ( finite4795055649997197647od_a_a @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X3: product_prod_a_a] :
( A
= ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% card_1_singleton_iff
thf(fact_1177_card__1__singleton__iff,axiom,
! [A: set_nat] :
( ( ( finite_card_nat @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X3: nat] :
( A
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% card_1_singleton_iff
thf(fact_1178_finite__ranking__induct,axiom,
! [S: set_set_a,P: set_set_a > $o,F3: set_a > nat] :
( ( finite_finite_set_a @ S )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X4: set_a,S2: set_set_a] :
( ( finite_finite_set_a @ S2 )
=> ( ! [Y7: set_a] :
( ( member_set_a @ Y7 @ S2 )
=> ( ord_less_eq_nat @ ( F3 @ Y7 ) @ ( F3 @ X4 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_set_a @ X4 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1179_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F3: a > nat] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X4: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y7: a] :
( ( member_a @ Y7 @ S2 )
=> ( ord_less_eq_nat @ ( F3 @ Y7 ) @ ( F3 @ X4 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X4 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1180_finite__ranking__induct,axiom,
! [S: set_Product_prod_a_a,P: set_Product_prod_a_a > $o,F3: product_prod_a_a > nat] :
( ( finite6544458595007987280od_a_a @ S )
=> ( ( P @ bot_bo3357376287454694259od_a_a )
=> ( ! [X4: 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 @ ( F3 @ Y7 ) @ ( F3 @ X4 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert4534936382041156343od_a_a @ X4 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1181_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F3: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X4: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y7: nat] :
( ( member_nat @ Y7 @ S2 )
=> ( ord_less_eq_nat @ ( F3 @ Y7 ) @ ( F3 @ X4 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat @ X4 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1182_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_1183_edge__density__eq0,axiom,
! [A: set_a,B: set_a,X2: set_a,Y: set_a] :
( ( ( undire8383842906760478443ween_a @ edges @ A @ B )
= bot_bo3357376287454694259od_a_a )
=> ( ( ord_less_eq_set_a @ X2 @ A )
=> ( ( ord_less_eq_set_a @ Y @ B )
=> ( ( undire297304480579013331sity_a @ edges @ X2 @ Y )
= zero_zero_real ) ) ) ) ).
% edge_density_eq0
thf(fact_1184__092_060open_062cycles_A_092_060subseteq_062_A_123xs_O_Aset_Axs_A_092_060subseteq_062_AV_A_092_060and_062_Alength_Axs_A_092_060le_062_ASuc_Aorder_125_092_060close_062,axiom,
( ord_le8861187494160871172list_a @ ( undire2685670433559799090cles_a @ vertices @ edges )
@ ( collect_list_a
@ ^ [Xs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ vertices )
& ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( suc @ ( finite_card_a @ vertices ) ) ) ) ) ) ).
% \<open>cycles \<subseteq> {xs. set xs \<subseteq> V \<and> length xs \<le> Suc order}\<close>
thf(fact_1185_card__all__edges__between__commute,axiom,
! [X2: set_a,Y: set_a] :
( ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y ) )
= ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ Y @ X2 ) ) ) ).
% card_all_edges_between_commute
thf(fact_1186_finite__all__edges__between_H,axiom,
! [X2: set_a,Y: set_a] : ( finite6544458595007987280od_a_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y ) ) ).
% finite_all_edges_between'
thf(fact_1187_all__edges__between__mono2,axiom,
! [Y: set_a,Z4: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ Z4 )
=> ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y ) @ ( undire8383842906760478443ween_a @ edges @ X2 @ Z4 ) ) ) ).
% all_edges_between_mono2
thf(fact_1188_all__edges__between__mono1,axiom,
! [Y: set_a,Z4: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ Z4 )
=> ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ edges @ Y @ X2 ) @ ( undire8383842906760478443ween_a @ edges @ Z4 @ X2 ) ) ) ).
% all_edges_between_mono1
thf(fact_1189__092_060open_062finite_A_123xs_O_Aset_Axs_A_092_060subseteq_062_AV_A_092_060and_062_Alength_Axs_A_092_060le_062_ASuc_Aorder_125_092_060close_062,axiom,
( finite_finite_list_a
@ ( collect_list_a
@ ^ [Xs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ vertices )
& ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( suc @ ( finite_card_a @ vertices ) ) ) ) ) ) ).
% \<open>finite {xs. set xs \<subseteq> V \<and> length xs \<le> Suc order}\<close>
thf(fact_1190_finite__all__edges__between,axiom,
! [X2: set_a,Y: set_a] :
( ( finite_finite_a @ X2 )
=> ( ( finite_finite_a @ Y )
=> ( finite6544458595007987280od_a_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y ) ) ) ) ).
% finite_all_edges_between
thf(fact_1191_all__edges__between__Un1,axiom,
! [X2: set_a,Y: set_a,Z4: set_a] :
( ( undire8383842906760478443ween_a @ edges @ ( sup_sup_set_a @ X2 @ Y ) @ Z4 )
= ( sup_su3048258781599657691od_a_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Z4 ) @ ( undire8383842906760478443ween_a @ edges @ Y @ Z4 ) ) ) ).
% all_edges_between_Un1
thf(fact_1192_all__edges__between__Un2,axiom,
! [X2: set_a,Y: set_a,Z4: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X2 @ ( sup_sup_set_a @ Y @ Z4 ) )
= ( sup_su3048258781599657691od_a_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y ) @ ( undire8383842906760478443ween_a @ edges @ X2 @ Z4 ) ) ) ).
% all_edges_between_Un2
thf(fact_1193_induced__edges__def,axiom,
! [V3: set_a] :
( ( undire7777452895879145676dges_a @ edges @ V3 )
= ( collect_set_a
@ ^ [E5: set_a] :
( ( member_set_a @ E5 @ edges )
& ( ord_less_eq_set_a @ E5 @ V3 ) ) ) ) ).
% induced_edges_def
thf(fact_1194_incident__edges__def,axiom,
! [V: a] :
( ( undire3231912044278729248dges_a @ edges @ V )
= ( collect_set_a
@ ^ [E5: set_a] :
( ( member_set_a @ E5 @ edges )
& ( undire1521409233611534436dent_a @ V @ E5 ) ) ) ) ).
% incident_edges_def
thf(fact_1195_incident__loops__def,axiom,
! [V: a] :
( ( undire4753905205749729249oops_a @ edges @ V )
= ( collect_set_a
@ ^ [E5: set_a] :
( ( member_set_a @ E5 @ edges )
& ( E5
= ( insert_a @ V @ bot_bot_set_a ) ) ) ) ) ).
% incident_loops_def
thf(fact_1196_all__edges__between__rem__wf,axiom,
! [X2: set_a,Y: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X2 @ Y )
= ( undire8383842906760478443ween_a @ edges @ ( inf_inf_set_a @ X2 @ vertices ) @ ( inf_inf_set_a @ Y @ vertices ) ) ) ).
% all_edges_between_rem_wf
thf(fact_1197_neighborhood__def,axiom,
! [X5: a] :
( ( undire8504279938402040014hood_a @ vertices @ edges @ X5 )
= ( collect_a
@ ^ [V6: a] :
( ( member_a @ V6 @ vertices )
& ( undire397441198561214472_adj_a @ edges @ X5 @ V6 ) ) ) ) ).
% neighborhood_def
thf(fact_1198_edges__split__loop,axiom,
( edges
= ( sup_sup_set_set_a
@ ( collect_set_a
@ ^ [E5: set_a] :
( ( member_set_a @ E5 @ edges )
& ( undire2905028936066782638loop_a @ E5 ) ) )
@ ( collect_set_a
@ ^ [E5: set_a] :
( ( member_set_a @ E5 @ edges )
& ( undire4917966558017083288edge_a @ E5 ) ) ) ) ) ).
% edges_split_loop
thf(fact_1199_incident__loops__alt,axiom,
! [V: a] :
( ( undire4753905205749729249oops_a @ edges @ V )
= ( collect_set_a
@ ^ [E5: set_a] :
( ( member_set_a @ E5 @ edges )
& ( undire1521409233611534436dent_a @ V @ E5 )
& ( ( finite_card_a @ E5 )
= one_one_nat ) ) ) ) ).
% incident_loops_alt
thf(fact_1200_card__is__has__loop__eq,axiom,
( ( finite_card_set_a
@ ( collect_set_a
@ ^ [E5: set_a] :
( ( member_set_a @ E5 @ edges )
& ( undire2905028936066782638loop_a @ E5 ) ) ) )
= ( finite_card_a
@ ( collect_a
@ ^ [V6: a] :
( ( member_a @ V6 @ vertices )
& ( undire3617971648856834880loop_a @ edges @ V6 ) ) ) ) ) ).
% card_is_has_loop_eq
thf(fact_1201_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_eq_nat @ I2 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_1202_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N6: nat] : ( ord_less_eq_nat @ N6 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1203_all__edges__between__empty_I2_J,axiom,
! [Z4: set_a] :
( ( undire8383842906760478443ween_a @ edges @ Z4 @ bot_bot_set_a )
= bot_bo3357376287454694259od_a_a ) ).
% all_edges_between_empty(2)
thf(fact_1204_all__edges__between__empty_I1_J,axiom,
! [Z4: set_a] :
( ( undire8383842906760478443ween_a @ edges @ bot_bot_set_a @ Z4 )
= bot_bo3357376287454694259od_a_a ) ).
% all_edges_between_empty(1)
thf(fact_1205_neighbors__ss__def,axiom,
! [X5: a,Y: set_a] :
( ( undire401937927514038589s_ss_a @ edges @ X5 @ Y )
= ( collect_a
@ ^ [Y6: a] :
( ( member_a @ Y6 @ Y )
& ( undire397441198561214472_adj_a @ edges @ X5 @ Y6 ) ) ) ) ).
% neighbors_ss_def
thf(fact_1206_finite__less__ub,axiom,
! [F3: nat > nat,U: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F3 @ N4 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N6: nat] : ( ord_less_eq_nat @ ( F3 @ N6 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1207_paths__def,axiom,
( ( undire1387732426225024653aths_a @ vertices @ edges )
= ( collect_list_a @ ( undire427332500224447920path_a @ vertices @ edges ) ) ) ).
% paths_def
thf(fact_1208_cycles__def,axiom,
( ( undire2685670433559799090cles_a @ vertices @ edges )
= ( collect_list_a @ ( undire2407311113669455967ycle_a @ vertices @ edges ) ) ) ).
% cycles_def
thf(fact_1209_incident__edges__neighbors__img,axiom,
! [V: a] :
( ( undire3231912044278729248dges_a @ edges @ V )
= ( image_a_set_a
@ ^ [U2: a] : ( insert_a @ V @ ( insert_a @ U2 @ bot_bot_set_a ) )
@ ( undire8504279938402040014hood_a @ vertices @ edges @ V ) ) ) ).
% incident_edges_neighbors_img
thf(fact_1210_gen__paths__def,axiom,
( ( undire6235733737954427521aths_a @ vertices @ edges )
= ( collect_list_a @ ( undire3562951555376170320path_a @ vertices @ edges ) ) ) ).
% gen_paths_def
thf(fact_1211_is__gen__path__cycle,axiom,
! [P2: list_a] :
( ( undire2407311113669455967ycle_a @ vertices @ edges @ P2 )
=> ( undire3562951555376170320path_a @ vertices @ edges @ P2 ) ) ).
% is_gen_path_cycle
thf(fact_1212_is__path__gen__path,axiom,
! [P2: list_a] :
( ( undire427332500224447920path_a @ vertices @ edges @ P2 )
=> ( undire3562951555376170320path_a @ vertices @ edges @ P2 ) ) ).
% is_path_gen_path
thf(fact_1213_edges__split__loop__inter__empty,axiom,
( bot_bot_set_set_a
= ( inf_inf_set_set_a
@ ( collect_set_a
@ ^ [E5: set_a] :
( ( member_set_a @ E5 @ edges )
& ( undire2905028936066782638loop_a @ E5 ) ) )
@ ( collect_set_a
@ ^ [E5: set_a] :
( ( member_set_a @ E5 @ edges )
& ( undire4917966558017083288edge_a @ E5 ) ) ) ) ) ).
% edges_split_loop_inter_empty
thf(fact_1214_max__all__edges__between,axiom,
! [X2: set_a,Y: set_a] :
( ( finite_finite_a @ X2 )
=> ( ( finite_finite_a @ Y )
=> ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y ) ) @ ( times_times_nat @ ( finite_card_a @ X2 ) @ ( finite_card_a @ Y ) ) ) ) ) ).
% max_all_edges_between
thf(fact_1215_card__all__edges__betw__neighbor,axiom,
! [X2: set_a,Y: set_a] :
( ( finite_finite_a @ X2 )
=> ( ( finite_finite_a @ Y )
=> ( ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y ) )
= ( groups6334556678337121940_a_nat
@ ^ [X3: a] : ( finite_card_a @ ( undire401937927514038589s_ss_a @ edges @ X3 @ Y ) )
@ X2 ) ) ) ) ).
% card_all_edges_betw_neighbor
thf(fact_1216_card__all__edges__between,axiom,
! [Y: set_a,X2: set_a] :
( ( finite_finite_a @ Y )
=> ( ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y ) )
= ( groups6334556678337121940_a_nat
@ ^ [Y6: a] : ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ ( insert_a @ Y6 @ bot_bot_set_a ) ) )
@ Y ) ) ) ).
% card_all_edges_between
thf(fact_1217_mult__is__0,axiom,
! [M4: nat,N: nat] :
( ( ( times_times_nat @ M4 @ N )
= zero_zero_nat )
= ( ( M4 = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1218_mult__0__right,axiom,
! [M4: nat] :
( ( times_times_nat @ M4 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1219_mult__cancel1,axiom,
! [K: nat,M4: nat,N: nat] :
( ( ( times_times_nat @ K @ M4 )
= ( times_times_nat @ K @ N ) )
= ( ( M4 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1220_mult__cancel2,axiom,
! [M4: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M4 @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M4 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1221_nat__mult__eq__1__iff,axiom,
! [M4: nat,N: nat] :
( ( ( times_times_nat @ M4 @ N )
= one_one_nat )
= ( ( M4 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1222_nat__1__eq__mult__iff,axiom,
! [M4: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M4 @ N ) )
= ( ( M4 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1223_one__eq__mult__iff,axiom,
! [M4: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M4 @ N ) )
= ( ( M4
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1224_mult__eq__1__iff,axiom,
! [M4: nat,N: nat] :
( ( ( times_times_nat @ M4 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M4
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1225_one__le__mult__iff,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M4 @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M4 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1226_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1227_le__cube,axiom,
! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ) ).
% le_cube
thf(fact_1228_le__square,axiom,
! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ).
% le_square
thf(fact_1229_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_1230_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_1231_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_1232_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1233_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1234_zero__notin__Suc__image,axiom,
! [A: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A ) ) ).
% zero_notin_Suc_image
thf(fact_1235_Suc__mult__le__cancel1,axiom,
! [K: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M4 ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M4 @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1236_mult__eq__self__implies__10,axiom,
! [M4: nat,N: nat] :
( ( M4
= ( times_times_nat @ M4 @ N ) )
=> ( ( N = one_one_nat )
| ( M4 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1237_all__edges__between__E__ss,axiom,
! [X2: set_a,Y: set_a] : ( ord_le3724670747650509150_set_a @ ( image_9052089385058188540_set_a @ undire6670514144573423676edge_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y ) ) @ edges ) ).
% all_edges_between_E_ss
thf(fact_1238_all__edges__between__set,axiom,
! [X2: set_a,Y: set_a] :
( ( image_9052089385058188540_set_a @ undire6670514144573423676edge_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y ) )
= ( collect_set_a
@ ^ [Uu: set_a] :
? [X3: a,Y6: a] :
( ( Uu
= ( insert_a @ X3 @ ( insert_a @ Y6 @ bot_bot_set_a ) ) )
& ( member_a @ X3 @ X2 )
& ( member_a @ Y6 @ Y )
& ( member_set_a @ ( insert_a @ X3 @ ( insert_a @ Y6 @ bot_bot_set_a ) ) @ edges ) ) ) ) ).
% all_edges_between_set
thf(fact_1239_incident__sedges__union,axiom,
( ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ ( undire1270416042309875431dges_a @ edges ) @ vertices ) )
= ( collect_set_a
@ ^ [E5: set_a] :
( ( member_set_a @ E5 @ edges )
& ( undire4917966558017083288edge_a @ E5 ) ) ) ) ).
% incident_sedges_union
thf(fact_1240_is__loop__set__alt,axiom,
( ( collect_set_a
@ ^ [Uu: set_a] :
? [V6: a] :
( ( Uu
= ( insert_a @ V6 @ bot_bot_set_a ) )
& ( undire3617971648856834880loop_a @ edges @ V6 ) ) )
= ( collect_set_a
@ ^ [E5: set_a] :
( ( member_set_a @ E5 @ edges )
& ( undire2905028936066782638loop_a @ E5 ) ) ) ) ).
% is_loop_set_alt
thf(fact_1241_incident__loops__union,axiom,
( ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ ( undire4753905205749729249oops_a @ edges ) @ vertices ) )
= ( collect_set_a
@ ^ [E5: set_a] :
( ( member_set_a @ E5 @ edges )
& ( undire2905028936066782638loop_a @ E5 ) ) ) ) ).
% incident_loops_union
thf(fact_1242_all__edges__between__Union2,axiom,
! [X2: set_a,Y8: set_set_a] :
( ( undire8383842906760478443ween_a @ edges @ X2 @ ( comple2307003609928055243_set_a @ Y8 ) )
= ( comple8421679170691845492od_a_a @ ( image_6165024369500519726od_a_a @ ( undire8383842906760478443ween_a @ edges @ X2 ) @ Y8 ) ) ) ).
% all_edges_between_Union2
thf(fact_1243_all__edges__between__Union1,axiom,
! [X7: set_set_a,Y: set_a] :
( ( undire8383842906760478443ween_a @ edges @ ( comple2307003609928055243_set_a @ X7 ) @ Y )
= ( comple8421679170691845492od_a_a
@ ( image_6165024369500519726od_a_a
@ ^ [X6: set_a] : ( undire8383842906760478443ween_a @ edges @ X6 @ Y )
@ X7 ) ) ) ).
% all_edges_between_Union1
thf(fact_1244_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1245_all__edges__between__subset__times,axiom,
! [X2: set_a,Y: set_a] :
( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y )
@ ( product_Sigma_a_a @ ( inf_inf_set_a @ X2 @ ( comple2307003609928055243_set_a @ edges ) )
@ ^ [Uu: a] : ( inf_inf_set_a @ Y @ ( comple2307003609928055243_set_a @ edges ) ) ) ) ).
% all_edges_between_subset_times
thf(fact_1246_local_Oinj__on__mk__edge,axiom,
! [X2: set_a,Y: set_a] :
( ( ( inf_inf_set_a @ X2 @ Y )
= bot_bot_set_a )
=> ( inj_on4851796814176604264_set_a @ undire6670514144573423676edge_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y ) ) ) ).
% local.inj_on_mk_edge
thf(fact_1247_all__edges__between__subset,axiom,
! [X2: set_a,Y: set_a] :
( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y )
@ ( product_Sigma_a_a @ X2
@ ^ [Uu: a] : Y ) ) ).
% all_edges_between_subset
thf(fact_1248_all__edges__betw__sigma__neighbor,axiom,
! [X2: set_a,Y: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X2 @ Y )
= ( product_Sigma_a_a @ X2
@ ^ [X3: a] : ( undire401937927514038589s_ss_a @ edges @ X3 @ Y ) ) ) ).
% all_edges_betw_sigma_neighbor
thf(fact_1249_all__edges__betw__I,axiom,
! [X5: a,X2: set_a,Y4: a,Y: set_a] :
( ( member_a @ X5 @ X2 )
=> ( ( member_a @ Y4 @ Y )
=> ( ( member_set_a @ ( insert_a @ X5 @ ( insert_a @ Y4 @ bot_bot_set_a ) ) @ edges )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y4 ) @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y ) ) ) ) ) ).
% all_edges_betw_I
thf(fact_1250_all__edges__betw__D3,axiom,
! [X5: a,Y4: a,X2: set_a,Y: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y4 ) @ ( undire8383842906760478443ween_a @ edges @ X2 @ Y ) )
=> ( member_set_a @ ( insert_a @ X5 @ ( insert_a @ Y4 @ bot_bot_set_a ) ) @ edges ) ) ).
% all_edges_betw_D3
thf(fact_1251_all__edges__betw__prod__def__neighbors,axiom,
! [X2: set_a,Y: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X2 @ Y )
= ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X3: a,Y6: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y6 )
@ ( product_Sigma_a_a @ X2
@ ^ [Uu: a] : Y ) )
& ( undire397441198561214472_adj_a @ edges @ X3 @ Y6 ) ) ) ) ) ).
% all_edges_betw_prod_def_neighbors
thf(fact_1252_is__walk__wf,axiom,
! [Xs2: list_a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs2 )
=> ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ vertices ) ) ).
% is_walk_wf
thf(fact_1253_is__path__walk,axiom,
! [Xs2: list_a] :
( ( undire427332500224447920path_a @ vertices @ edges @ Xs2 )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ Xs2 ) ) ).
% is_path_walk
thf(fact_1254_walks__def,axiom,
( ( undire3736599831911450577alks_a @ vertices @ edges )
= ( collect_list_a @ ( undire6133010728901294956walk_a @ vertices @ edges ) ) ) ).
% walks_def
thf(fact_1255_all__edges__between__def,axiom,
! [X2: set_a,Y: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X2 @ Y )
= ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X3: a,Y6: a] :
( ( member_a @ X3 @ X2 )
& ( member_a @ Y6 @ Y )
& ( member_set_a @ ( insert_a @ X3 @ ( insert_a @ Y6 @ bot_bot_set_a ) ) @ edges ) ) ) ) ) ).
% all_edges_between_def
thf(fact_1256_is__walk__hd__tl,axiom,
! [Y4: a,Ys: list_a,X5: a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ ( cons_a @ Y4 @ Ys ) )
=> ( ( member_set_a @ ( insert_a @ X5 @ ( insert_a @ Y4 @ bot_bot_set_a ) ) @ edges )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ ( cons_a @ X5 @ ( cons_a @ Y4 @ Ys ) ) ) ) ) ).
% is_walk_hd_tl
thf(fact_1257_is__walk__not__empty,axiom,
! [Xs2: list_a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs2 )
=> ( Xs2 != nil_a ) ) ).
% is_walk_not_empty
thf(fact_1258_walk__edges_Ocases,axiom,
! [X5: list_a] :
( ( X5 != nil_a )
=> ( ! [X4: a] :
( X5
!= ( cons_a @ X4 @ nil_a ) )
=> ~ ! [X4: a,Y2: a,Ys2: list_a] :
( X5
!= ( cons_a @ X4 @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ).
% walk_edges.cases
thf(fact_1259_all__edges__between__swap,axiom,
! [X2: set_a,Y: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X2 @ Y )
= ( image_4636654165204879301od_a_a
@ ( produc408267641121961211od_a_a
@ ^ [X3: a,Y6: a] : ( product_Pair_a_a @ Y6 @ X3 ) )
@ ( undire8383842906760478443ween_a @ edges @ Y @ X2 ) ) ) ).
% all_edges_between_swap
thf(fact_1260_is__walk__not__empty2,axiom,
~ ( undire6133010728901294956walk_a @ vertices @ edges @ nil_a ) ).
% is_walk_not_empty2
thf(fact_1261_is__walk__drop__hd,axiom,
! [Ys: list_a,Y4: a] :
( ( Ys != nil_a )
=> ( ( undire6133010728901294956walk_a @ vertices @ edges @ ( cons_a @ Y4 @ Ys ) )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ Ys ) ) ) ).
% is_walk_drop_hd
thf(fact_1262_is__walk__singleton,axiom,
! [U: a] :
( ( member_a @ U @ vertices )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ ( cons_a @ U @ nil_a ) ) ) ).
% is_walk_singleton
thf(fact_1263_is__gen__path__trivial,axiom,
! [X5: a] :
( ( member_a @ X5 @ vertices )
=> ( undire3562951555376170320path_a @ vertices @ edges @ ( cons_a @ X5 @ nil_a ) ) ) ).
% is_gen_path_trivial
thf(fact_1264_is__gen__path__options,axiom,
! [P2: list_a] :
( ( undire3562951555376170320path_a @ vertices @ edges @ P2 )
= ( ( undire2407311113669455967ycle_a @ vertices @ edges @ P2 )
| ( undire427332500224447920path_a @ vertices @ edges @ P2 )
| ? [X3: a] :
( ( member_a @ X3 @ vertices )
& ( P2
= ( cons_a @ X3 @ nil_a ) ) ) ) ) ).
% is_gen_path_options
thf(fact_1265_gen__paths__ss,axiom,
( ord_le8861187494160871172list_a @ ( undire6235733737954427521aths_a @ vertices @ edges )
@ ( sup_sup_set_list_a @ ( sup_sup_set_list_a @ ( undire2685670433559799090cles_a @ vertices @ edges ) @ ( undire1387732426225024653aths_a @ vertices @ edges ) )
@ ( collect_list_a
@ ^ [Uu: list_a] :
? [V6: a] :
( ( Uu
= ( cons_a @ V6 @ nil_a ) )
& ( member_a @ V6 @ vertices ) ) ) ) ) ).
% gen_paths_ss
thf(fact_1266_is__walk__def,axiom,
! [Xs2: list_a] :
( ( undire6133010728901294956walk_a @ vertices @ edges @ Xs2 )
= ( ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ vertices )
& ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Xs2 ) ) @ edges )
& ( Xs2 != nil_a ) ) ) ).
% is_walk_def
thf(fact_1267_is__walkI,axiom,
! [Xs2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ vertices )
=> ( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Xs2 ) ) @ edges )
=> ( ( Xs2 != nil_a )
=> ( undire6133010728901294956walk_a @ vertices @ edges @ Xs2 ) ) ) ) ).
% is_walkI
thf(fact_1268_walk__edges_Osimps_I1_J,axiom,
( ( undire7337870655677353998dges_a @ nil_a )
= nil_set_a ) ).
% walk_edges.simps(1)
thf(fact_1269_walk__edges_Osimps_I2_J,axiom,
! [X5: a] :
( ( undire7337870655677353998dges_a @ ( cons_a @ X5 @ nil_a ) )
= nil_set_a ) ).
% walk_edges.simps(2)
thf(fact_1270_walk__edges_Osimps_I3_J,axiom,
! [X5: a,Y4: a,Ys: list_a] :
( ( undire7337870655677353998dges_a @ ( cons_a @ X5 @ ( cons_a @ Y4 @ Ys ) ) )
= ( cons_set_a @ ( insert_a @ X5 @ ( insert_a @ Y4 @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ ( cons_a @ Y4 @ Ys ) ) ) ) ).
% walk_edges.simps(3)
thf(fact_1271_walk__edges_Oelims,axiom,
! [X5: list_a,Y4: list_set_a] :
( ( ( undire7337870655677353998dges_a @ X5 )
= Y4 )
=> ( ( ( X5 = nil_a )
=> ( Y4 != nil_set_a ) )
=> ( ( ? [X4: a] :
( X5
= ( cons_a @ X4 @ nil_a ) )
=> ( Y4 != nil_set_a ) )
=> ~ ! [X4: a,Y2: a,Ys2: list_a] :
( ( X5
= ( cons_a @ X4 @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( Y4
!= ( cons_set_a @ ( insert_a @ X4 @ ( insert_a @ Y2 @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ) ) ) ).
% walk_edges.elims
% Conjectures (1)
thf(conj_0,conjecture,
finite_finite_list_a @ ( undire2685670433559799090cles_a @ vertices @ edges ) ).
%------------------------------------------------------------------------------