TPTP Problem File: SLH0826^1.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/0014_Undirected_Graph_Basics/prob_00555_022195__13080562_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1455 ( 592 unt; 179 typ; 0 def)
% Number of atoms : 3615 (1330 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10553 ( 388 ~; 56 |; 287 &;8274 @)
% ( 0 <=>;1548 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 21 ( 20 usr)
% Number of type conns : 459 ( 459 >; 0 *; 0 +; 0 <<)
% Number of symbols : 160 ( 159 usr; 13 con; 0-3 aty)
% Number of variables : 3488 ( 198 ^;3198 !; 92 ?;3488 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:32:34.129
%------------------------------------------------------------------------------
% Could-be-implicit typings (20)
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thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_Itf__a_J,type,
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (159)
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thf(sy_c_Finite__Set_Ocard_001tf__a,type,
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thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001tf__a_001tf__a_001_Eo,type,
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thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
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thf(sy_c_Set_OCollect_001tf__a,type,
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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_Omk__edge_001tf__a,type,
undire6670514144573423676edge_a: product_prod_a_a > set_a ).
thf(sy_c_Undirected__Graph__Basics_Omk__triangle__set_001t__Nat__Onat,type,
undire4970100481470743719et_nat: produc7248412053542808358at_nat > set_nat ).
thf(sy_c_Undirected__Graph__Basics_Omk__triangle__set_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire2459242765783757584od_a_a: produc8857593507947890343od_a_a > set_Product_prod_a_a ).
thf(sy_c_Undirected__Graph__Basics_Omk__triangle__set_001t__Set__Oset_Itf__a_J,type,
undire4638465864238448455_set_a: produc3364680560414100336_set_a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Omk__triangle__set_001tf__a,type,
undire8536760333753235943_set_a: produc4044097585999906000od_a_a > set_a ).
thf(sy_c_Undirected__Graph__Basics_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_001t__Nat__Onat,type,
undire9168215380967949987en_nat: set_set_nat > set_nat > set_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oall__edges__between_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire4032395788819567636od_a_a: set_se5735800977113168103od_a_a > set_Product_prod_a_a > set_Product_prod_a_a > set_Pr8600417178894128327od_a_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oall__edges__between_001t__Set__Oset_Itf__a_J,type,
undire2462398226299384907_set_a: set_set_set_a > set_set_a > set_set_a > set_Pr5845495582615845127_set_a ).
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_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__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__Basics_Oulgraph__axioms_001tf__a,type,
undire2177556672586781897ioms_a: set_set_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member6330455413206600464od_a_a: produc3498347346309940967od_a_a > set_Pr8600417178894128327od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
member7983343339038529360_set_a: produc1703568184450464039_set_a > set_Pr5845495582615845127_set_a > $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__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_Z,type,
z: set_a ).
thf(sy_v_edges,type,
edges: set_set_a ).
thf(sy_v_vertices,type,
vertices: set_a ).
% Relevant facts (1275)
thf(fact_0_empty__not__edge,axiom,
~ ( member_set_a @ bot_bot_set_a @ edges ) ).
% empty_not_edge
thf(fact_1_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_2_ulgraph_Oall__edges__between_Ocong,axiom,
undire8383842906760478443ween_a = undire8383842906760478443ween_a ).
% ulgraph.all_edges_between.cong
thf(fact_3_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_4_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_5_empty__iff,axiom,
! [C: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ C @ bot_bo3357376287454694259od_a_a ) ).
% empty_iff
thf(fact_6_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_7_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_8_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X: a] :
~ ( member_a @ X @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_9_all__not__in__conv,axiom,
! [A: set_Product_prod_a_a] :
( ( ! [X: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X @ A ) )
= ( A = bot_bo3357376287454694259od_a_a ) ) ).
% all_not_in_conv
thf(fact_10_all__not__in__conv,axiom,
! [A: set_set_a] :
( ( ! [X: set_a] :
~ ( member_set_a @ X @ A ) )
= ( A = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_11_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X: nat] :
~ ( member_nat @ X @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_12_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X: a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_13_Collect__empty__eq,axiom,
! [P: product_prod_a_a > $o] :
( ( ( collec3336397797384452498od_a_a @ P )
= bot_bo3357376287454694259od_a_a )
= ( ! [X: product_prod_a_a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_14_Collect__empty__eq,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( ! [X: set_a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_15_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_16_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X: a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_17_empty__Collect__eq,axiom,
! [P: product_prod_a_a > $o] :
( ( bot_bo3357376287454694259od_a_a
= ( collec3336397797384452498od_a_a @ P ) )
= ( ! [X: product_prod_a_a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_18_empty__Collect__eq,axiom,
! [P: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P ) )
= ( ! [X: set_a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_19_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_20_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_21_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_22_emptyE,axiom,
! [A2: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ).
% emptyE
thf(fact_23_emptyE,axiom,
! [A2: set_a] :
~ ( member_set_a @ A2 @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_24_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_25_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_26_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_27_equals0D,axiom,
! [A: set_set_a,A2: set_a] :
( ( A = bot_bot_set_set_a )
=> ~ ( member_set_a @ A2 @ A ) ) ).
% equals0D
thf(fact_28_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_29_equals0I,axiom,
! [A: set_a] :
( ! [Y: a] :
~ ( member_a @ Y @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_30_equals0I,axiom,
! [A: set_Product_prod_a_a] :
( ! [Y: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ Y @ A )
=> ( A = bot_bo3357376287454694259od_a_a ) ) ).
% equals0I
thf(fact_31_equals0I,axiom,
! [A: set_set_a] :
( ! [Y: set_a] :
~ ( member_set_a @ Y @ A )
=> ( A = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_32_equals0I,axiom,
! [A: set_nat] :
( ! [Y: nat] :
~ ( member_nat @ Y @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_33_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X: a] : ( member_a @ X @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_34_ex__in__conv,axiom,
! [A: set_Product_prod_a_a] :
( ( ? [X: product_prod_a_a] : ( member1426531477525435216od_a_a @ X @ A ) )
= ( A != bot_bo3357376287454694259od_a_a ) ) ).
% ex_in_conv
thf(fact_35_ex__in__conv,axiom,
! [A: set_set_a] :
( ( ? [X: set_a] : ( member_set_a @ X @ A ) )
= ( A != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_36_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_37_IntI,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A )
=> ( ( member1426531477525435216od_a_a @ C @ B )
=> ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_38_IntI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_39_IntI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ( member_a @ C @ B )
=> ( member_a @ C @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_40_IntI,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A )
=> ( ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_41_Int__iff,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A @ B ) )
= ( ( member1426531477525435216od_a_a @ C @ A )
& ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_42_Int__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ( member_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_43_Int__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ( member_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_44_Int__iff,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
= ( ( member_set_a @ C @ A )
& ( member_set_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_45_IntE,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A @ B ) )
=> ~ ( ( member1426531477525435216od_a_a @ C @ A )
=> ~ ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% IntE
thf(fact_46_IntE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ~ ( member_nat @ C @ B ) ) ) ).
% IntE
thf(fact_47_IntE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ~ ( member_a @ C @ B ) ) ) ).
% IntE
thf(fact_48_IntE,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ~ ( ( member_set_a @ C @ A )
=> ~ ( member_set_a @ C @ B ) ) ) ).
% IntE
thf(fact_49_IntD1,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A @ B ) )
=> ( member1426531477525435216od_a_a @ C @ A ) ) ).
% IntD1
thf(fact_50_IntD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% IntD1
thf(fact_51_IntD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% IntD1
thf(fact_52_IntD1,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ( member_set_a @ C @ A ) ) ).
% IntD1
thf(fact_53_IntD2,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A @ B ) )
=> ( member1426531477525435216od_a_a @ C @ B ) ) ).
% IntD2
thf(fact_54_IntD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ B ) ) ).
% IntD2
thf(fact_55_IntD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ B ) ) ).
% IntD2
thf(fact_56_IntD2,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ( member_set_a @ C @ B ) ) ).
% IntD2
thf(fact_57_Int__assoc,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_58_Int__assoc,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_59_Int__absorb,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_60_Int__absorb,axiom,
! [A: set_set_a] :
( ( inf_inf_set_set_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_61_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A3: set_a,B2: set_a] : ( inf_inf_set_a @ B2 @ A3 ) ) ) ).
% Int_commute
thf(fact_62_Int__commute,axiom,
( inf_inf_set_set_a
= ( ^ [A3: set_set_a,B2: set_set_a] : ( inf_inf_set_set_a @ B2 @ A3 ) ) ) ).
% Int_commute
thf(fact_63_Int__left__absorb,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% Int_left_absorb
thf(fact_64_Int__left__absorb,axiom,
! [A: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ A @ B ) )
= ( inf_inf_set_set_a @ A @ B ) ) ).
% Int_left_absorb
thf(fact_65_Int__left__commute,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_66_Int__left__commute,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ B @ C2 ) )
= ( inf_inf_set_set_a @ B @ ( inf_inf_set_set_a @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_67_ulgraph_Overt__adj_Ocong,axiom,
undire397441198561214472_adj_a = undire397441198561214472_adj_a ).
% ulgraph.vert_adj.cong
thf(fact_68_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_69_bot__set__def,axiom,
( bot_bo3357376287454694259od_a_a
= ( collec3336397797384452498od_a_a @ bot_bo4160289986317612842_a_a_o ) ) ).
% bot_set_def
thf(fact_70_bot__set__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a @ bot_bot_set_a_o ) ) ).
% bot_set_def
thf(fact_71_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_72_disjoint__iff__not__equal,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ! [Y2: a] :
( ( member_a @ Y2 @ B )
=> ( X != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_73_disjoint__iff__not__equal,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ A @ B )
= bot_bo3357376287454694259od_a_a )
= ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A )
=> ! [Y2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y2 @ B )
=> ( X != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_74_disjoint__iff__not__equal,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ! [Y2: set_a] :
( ( member_set_a @ Y2 @ B )
=> ( X != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_75_disjoint__iff__not__equal,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ B )
=> ( X != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_76_Int__empty__right,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_77_Int__empty__right,axiom,
! [A: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% Int_empty_right
thf(fact_78_Int__empty__right,axiom,
! [A: set_set_a] :
( ( inf_inf_set_set_a @ A @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% Int_empty_right
thf(fact_79_Int__empty__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_80_Int__empty__left,axiom,
! [B: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_81_Int__empty__left,axiom,
! [B: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ B )
= bot_bo3357376287454694259od_a_a ) ).
% Int_empty_left
thf(fact_82_Int__empty__left,axiom,
! [B: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ B )
= bot_bot_set_set_a ) ).
% Int_empty_left
thf(fact_83_Int__empty__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_84_disjoint__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ~ ( member_a @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_85_disjoint__iff,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ A @ B )
= bot_bo3357376287454694259od_a_a )
= ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A )
=> ~ ( member1426531477525435216od_a_a @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_86_disjoint__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ~ ( member_set_a @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_87_disjoint__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ~ ( member_nat @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_88_Int__emptyI,axiom,
! [A: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ~ ( member_a @ X2 @ B ) )
=> ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_89_Int__emptyI,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A )
=> ~ ( member1426531477525435216od_a_a @ X2 @ B ) )
=> ( ( inf_in8905007599844390133od_a_a @ A @ B )
= bot_bo3357376287454694259od_a_a ) ) ).
% Int_emptyI
thf(fact_90_Int__emptyI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ~ ( member_set_a @ X2 @ B ) )
=> ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_91_Int__emptyI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ~ ( member_nat @ X2 @ B ) )
=> ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_92_graph__system_Oedge__adj_Ocong,axiom,
undire4022703626023482010_adj_a = undire4022703626023482010_adj_a ).
% graph_system.edge_adj.cong
thf(fact_93_vert__adj__edge__iff2,axiom,
! [V1: a,V2: a] :
( ( V1 != V2 )
=> ( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
= ( ? [X: set_a] :
( ( member_set_a @ X @ edges )
& ( undire1521409233611534436dent_a @ V1 @ X )
& ( undire1521409233611534436dent_a @ V2 @ X ) ) ) ) ) ).
% vert_adj_edge_iff2
thf(fact_94_boolean__algebra_Oconj__zero__right,axiom,
! [X3: set_a] :
( ( inf_inf_set_a @ X3 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_95_boolean__algebra_Oconj__zero__right,axiom,
! [X3: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X3 @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_96_boolean__algebra_Oconj__zero__right,axiom,
! [X3: set_set_a] :
( ( inf_inf_set_set_a @ X3 @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_97_boolean__algebra_Oconj__zero__right,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ X3 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_98_boolean__algebra_Oconj__zero__left,axiom,
! [X3: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X3 )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_99_boolean__algebra_Oconj__zero__left,axiom,
! [X3: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ X3 )
= bot_bo3357376287454694259od_a_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_100_boolean__algebra_Oconj__zero__left,axiom,
! [X3: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X3 )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_101_boolean__algebra_Oconj__zero__left,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X3 )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_102_inf__bot__right,axiom,
! [X3: set_a] :
( ( inf_inf_set_a @ X3 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_103_inf__bot__right,axiom,
! [X3: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X3 @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% inf_bot_right
thf(fact_104_inf__bot__right,axiom,
! [X3: set_set_a] :
( ( inf_inf_set_set_a @ X3 @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% inf_bot_right
thf(fact_105_inf__bot__right,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ X3 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_106_inf__bot__left,axiom,
! [X3: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X3 )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_107_inf__bot__left,axiom,
! [X3: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ X3 )
= bot_bo3357376287454694259od_a_a ) ).
% inf_bot_left
thf(fact_108_inf__bot__left,axiom,
! [X3: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X3 )
= bot_bot_set_set_a ) ).
% inf_bot_left
thf(fact_109_inf__bot__left,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X3 )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_110_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_111_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_112_inf_Oidem,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_113_inf_Oidem,axiom,
! [A2: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_114_inf__idem,axiom,
! [X3: set_a] :
( ( inf_inf_set_a @ X3 @ X3 )
= X3 ) ).
% inf_idem
thf(fact_115_inf__idem,axiom,
! [X3: set_set_a] :
( ( inf_inf_set_set_a @ X3 @ X3 )
= X3 ) ).
% inf_idem
thf(fact_116_inf_Oleft__idem,axiom,
! [A2: set_a,B3: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B3 ) )
= ( inf_inf_set_a @ A2 @ B3 ) ) ).
% inf.left_idem
thf(fact_117_inf_Oleft__idem,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ ( inf_inf_set_set_a @ A2 @ B3 ) )
= ( inf_inf_set_set_a @ A2 @ B3 ) ) ).
% inf.left_idem
thf(fact_118_inf__left__idem,axiom,
! [X3: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X3 @ ( inf_inf_set_a @ X3 @ Y3 ) )
= ( inf_inf_set_a @ X3 @ Y3 ) ) ).
% inf_left_idem
thf(fact_119_inf__left__idem,axiom,
! [X3: set_set_a,Y3: set_set_a] :
( ( inf_inf_set_set_a @ X3 @ ( inf_inf_set_set_a @ X3 @ Y3 ) )
= ( inf_inf_set_set_a @ X3 @ Y3 ) ) ).
% inf_left_idem
thf(fact_120_inf_Oright__idem,axiom,
! [A2: set_a,B3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ B3 )
= ( inf_inf_set_a @ A2 @ B3 ) ) ).
% inf.right_idem
thf(fact_121_inf_Oright__idem,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A2 @ B3 ) @ B3 )
= ( inf_inf_set_set_a @ A2 @ B3 ) ) ).
% inf.right_idem
thf(fact_122_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_123_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_124_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_125_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_126_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X: set_a] : ( member_set_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_127_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_128_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_129_Collect__mem__eq,axiom,
! [A: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X: product_prod_a_a] : ( member1426531477525435216od_a_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_130_Collect__cong,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X2: set_a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_set_a @ P )
= ( collect_set_a @ Q ) ) ) ).
% Collect_cong
thf(fact_131_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_132_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_133_Collect__cong,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X2: product_prod_a_a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collec3336397797384452498od_a_a @ P )
= ( collec3336397797384452498od_a_a @ Q ) ) ) ).
% Collect_cong
thf(fact_134_incident__def,axiom,
undire1521409233611534436dent_a = member_a ).
% incident_def
thf(fact_135_insertCI,axiom,
! [A2: set_a,B: set_set_a,B3: set_a] :
( ( ~ ( member_set_a @ A2 @ B )
=> ( A2 = B3 ) )
=> ( member_set_a @ A2 @ ( insert_set_a @ B3 @ B ) ) ) ).
% insertCI
thf(fact_136_insertCI,axiom,
! [A2: a,B: set_a,B3: a] :
( ( ~ ( member_a @ A2 @ B )
=> ( A2 = B3 ) )
=> ( member_a @ A2 @ ( insert_a @ B3 @ B ) ) ) ).
% insertCI
thf(fact_137_insertCI,axiom,
! [A2: product_prod_a_a,B: set_Product_prod_a_a,B3: product_prod_a_a] :
( ( ~ ( member1426531477525435216od_a_a @ A2 @ B )
=> ( A2 = B3 ) )
=> ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B3 @ B ) ) ) ).
% insertCI
thf(fact_138_insertCI,axiom,
! [A2: nat,B: set_nat,B3: nat] :
( ( ~ ( member_nat @ A2 @ B )
=> ( A2 = B3 ) )
=> ( member_nat @ A2 @ ( insert_nat @ B3 @ B ) ) ) ).
% insertCI
thf(fact_139_insert__iff,axiom,
! [A2: set_a,B3: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ ( insert_set_a @ B3 @ A ) )
= ( ( A2 = B3 )
| ( member_set_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_140_insert__iff,axiom,
! [A2: a,B3: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B3 @ A ) )
= ( ( A2 = B3 )
| ( member_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_141_insert__iff,axiom,
! [A2: product_prod_a_a,B3: product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B3 @ A ) )
= ( ( A2 = B3 )
| ( member1426531477525435216od_a_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_142_insert__iff,axiom,
! [A2: nat,B3: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B3 @ A ) )
= ( ( A2 = B3 )
| ( member_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_143_insert__absorb2,axiom,
! [X3: a,A: set_a] :
( ( insert_a @ X3 @ ( insert_a @ X3 @ A ) )
= ( insert_a @ X3 @ A ) ) ).
% insert_absorb2
thf(fact_144_insert__absorb2,axiom,
! [X3: set_a,A: set_set_a] :
( ( insert_set_a @ X3 @ ( insert_set_a @ X3 @ A ) )
= ( insert_set_a @ X3 @ A ) ) ).
% insert_absorb2
thf(fact_145_is__edge__between__def,axiom,
( undire8544646567961481629ween_a
= ( ^ [X4: set_a,Y4: set_a,E: set_a] :
? [X: a,Y2: a] :
( ( E
= ( insert_a @ X @ ( insert_a @ Y2 @ bot_bot_set_a ) ) )
& ( member_a @ X @ X4 )
& ( member_a @ Y2 @ Y4 ) ) ) ) ).
% is_edge_between_def
thf(fact_146_inf__right__idem,axiom,
! [X3: set_a,Y3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X3 @ Y3 ) @ Y3 )
= ( inf_inf_set_a @ X3 @ Y3 ) ) ).
% inf_right_idem
thf(fact_147_inf__right__idem,axiom,
! [X3: set_set_a,Y3: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X3 @ Y3 ) @ Y3 )
= ( inf_inf_set_set_a @ X3 @ Y3 ) ) ).
% inf_right_idem
thf(fact_148_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_149_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_150_singletonI,axiom,
! [A2: product_prod_a_a] : ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) ).
% singletonI
thf(fact_151_singletonI,axiom,
! [A2: set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_152_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_153_Int__insert__right__if1,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ A )
=> ( ( inf_in8905007599844390133od_a_a @ A @ ( insert4534936382041156343od_a_a @ A2 @ B ) )
= ( insert4534936382041156343od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_154_Int__insert__right__if1,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_155_Int__insert__right__if1,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_156_Int__insert__right__if1,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_157_Int__insert__right__if0,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A2 @ A )
=> ( ( inf_in8905007599844390133od_a_a @ A @ ( insert4534936382041156343od_a_a @ A2 @ B ) )
= ( inf_in8905007599844390133od_a_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_158_Int__insert__right__if0,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_159_Int__insert__right__if0,axiom,
! [A2: a,A: set_a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_160_Int__insert__right__if0,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( inf_inf_set_set_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_161_insert__inter__insert,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_162_insert__inter__insert,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ A ) @ ( insert_set_a @ A2 @ B ) )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_163_Int__insert__left__if1,axiom,
! [A2: product_prod_a_a,C2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ B ) @ C2 )
= ( insert4534936382041156343od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_164_Int__insert__left__if1,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_165_Int__insert__left__if1,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_166_Int__insert__left__if1,axiom,
! [A2: set_a,C2: set_set_a,B: set_set_a] :
( ( member_set_a @ A2 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C2 )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_167_Int__insert__left__if0,axiom,
! [A2: product_prod_a_a,C2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A2 @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ B ) @ C2 )
= ( inf_in8905007599844390133od_a_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_168_Int__insert__left__if0,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_nat @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_169_Int__insert__left__if0,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_170_Int__insert__left__if0,axiom,
! [A2: set_a,C2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A2 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_171_insert__disjoint_I1_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_172_insert__disjoint_I1_J,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ A ) @ B )
= bot_bo3357376287454694259od_a_a )
= ( ~ ( member1426531477525435216od_a_a @ A2 @ B )
& ( ( inf_in8905007599844390133od_a_a @ A @ B )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% insert_disjoint(1)
thf(fact_173_insert__disjoint_I1_J,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ A ) @ B )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A2 @ B )
& ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_174_insert__disjoint_I1_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B )
& ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_175_insert__disjoint_I2_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B ) )
= ( ~ ( member_a @ A2 @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_176_insert__disjoint_I2_J,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ A ) @ B ) )
= ( ~ ( member1426531477525435216od_a_a @ A2 @ B )
& ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_177_insert__disjoint_I2_J,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ A ) @ B ) )
= ( ~ ( member_set_a @ A2 @ B )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_178_insert__disjoint_I2_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B ) )
= ( ~ ( member_nat @ A2 @ B )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_179_disjoint__insert_I1_J,axiom,
! [B: set_a,A2: a,A: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A2 @ A ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ B @ A )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_180_disjoint__insert_I1_J,axiom,
! [B: set_Product_prod_a_a,A2: product_prod_a_a,A: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ B @ ( insert4534936382041156343od_a_a @ A2 @ A ) )
= bot_bo3357376287454694259od_a_a )
= ( ~ ( member1426531477525435216od_a_a @ A2 @ B )
& ( ( inf_in8905007599844390133od_a_a @ B @ A )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% disjoint_insert(1)
thf(fact_181_disjoint__insert_I1_J,axiom,
! [B: set_set_a,A2: set_a,A: set_set_a] :
( ( ( inf_inf_set_set_a @ B @ ( insert_set_a @ A2 @ A ) )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A2 @ B )
& ( ( inf_inf_set_set_a @ B @ A )
= bot_bot_set_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_182_disjoint__insert_I1_J,axiom,
! [B: set_nat,A2: nat,A: set_nat] :
( ( ( inf_inf_set_nat @ B @ ( insert_nat @ A2 @ A ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B )
& ( ( inf_inf_set_nat @ B @ A )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_183_disjoint__insert_I2_J,axiom,
! [A: set_a,B3: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A @ ( insert_a @ B3 @ B ) ) )
= ( ~ ( member_a @ B3 @ A )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_184_disjoint__insert_I2_J,axiom,
! [A: set_Product_prod_a_a,B3: product_prod_a_a,B: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A @ ( insert4534936382041156343od_a_a @ B3 @ B ) ) )
= ( ~ ( member1426531477525435216od_a_a @ B3 @ A )
& ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_185_disjoint__insert_I2_J,axiom,
! [A: set_set_a,B3: set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A @ ( insert_set_a @ B3 @ B ) ) )
= ( ~ ( member_set_a @ B3 @ A )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_186_disjoint__insert_I2_J,axiom,
! [A: set_nat,B3: nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ ( insert_nat @ B3 @ B ) ) )
= ( ~ ( member_nat @ B3 @ A )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_187_insertE,axiom,
! [A2: set_a,B3: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ ( insert_set_a @ B3 @ A ) )
=> ( ( A2 != B3 )
=> ( member_set_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_188_insertE,axiom,
! [A2: a,B3: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B3 @ A ) )
=> ( ( A2 != B3 )
=> ( member_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_189_insertE,axiom,
! [A2: product_prod_a_a,B3: product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B3 @ A ) )
=> ( ( A2 != B3 )
=> ( member1426531477525435216od_a_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_190_insertE,axiom,
! [A2: nat,B3: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B3 @ A ) )
=> ( ( A2 != B3 )
=> ( member_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_191_insertI1,axiom,
! [A2: set_a,B: set_set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ B ) ) ).
% insertI1
thf(fact_192_insertI1,axiom,
! [A2: a,B: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B ) ) ).
% insertI1
thf(fact_193_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_194_insertI1,axiom,
! [A2: nat,B: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_195_insertI2,axiom,
! [A2: set_a,B: set_set_a,B3: set_a] :
( ( member_set_a @ A2 @ B )
=> ( member_set_a @ A2 @ ( insert_set_a @ B3 @ B ) ) ) ).
% insertI2
thf(fact_196_insertI2,axiom,
! [A2: a,B: set_a,B3: a] :
( ( member_a @ A2 @ B )
=> ( member_a @ A2 @ ( insert_a @ B3 @ B ) ) ) ).
% insertI2
thf(fact_197_insertI2,axiom,
! [A2: product_prod_a_a,B: set_Product_prod_a_a,B3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ B )
=> ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B3 @ B ) ) ) ).
% insertI2
thf(fact_198_insertI2,axiom,
! [A2: nat,B: set_nat,B3: nat] :
( ( member_nat @ A2 @ B )
=> ( member_nat @ A2 @ ( insert_nat @ B3 @ B ) ) ) ).
% insertI2
thf(fact_199_Set_Oset__insert,axiom,
! [X3: set_a,A: set_set_a] :
( ( member_set_a @ X3 @ A )
=> ~ ! [B4: set_set_a] :
( ( A
= ( insert_set_a @ X3 @ B4 ) )
=> ( member_set_a @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_200_Set_Oset__insert,axiom,
! [X3: a,A: set_a] :
( ( member_a @ X3 @ A )
=> ~ ! [B4: set_a] :
( ( A
= ( insert_a @ X3 @ B4 ) )
=> ( member_a @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_201_Set_Oset__insert,axiom,
! [X3: product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A )
=> ~ ! [B4: set_Product_prod_a_a] :
( ( A
= ( insert4534936382041156343od_a_a @ X3 @ B4 ) )
=> ( member1426531477525435216od_a_a @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_202_Set_Oset__insert,axiom,
! [X3: nat,A: set_nat] :
( ( member_nat @ X3 @ A )
=> ~ ! [B4: set_nat] :
( ( A
= ( insert_nat @ X3 @ B4 ) )
=> ( member_nat @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_203_insert__ident,axiom,
! [X3: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X3 @ A )
=> ( ~ ( member_set_a @ X3 @ B )
=> ( ( ( insert_set_a @ X3 @ A )
= ( insert_set_a @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_204_insert__ident,axiom,
! [X3: a,A: set_a,B: set_a] :
( ~ ( member_a @ X3 @ A )
=> ( ~ ( member_a @ X3 @ B )
=> ( ( ( insert_a @ X3 @ A )
= ( insert_a @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_205_insert__ident,axiom,
! [X3: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X3 @ A )
=> ( ~ ( member1426531477525435216od_a_a @ X3 @ B )
=> ( ( ( insert4534936382041156343od_a_a @ X3 @ A )
= ( insert4534936382041156343od_a_a @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_206_insert__ident,axiom,
! [X3: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X3 @ A )
=> ( ~ ( member_nat @ X3 @ B )
=> ( ( ( insert_nat @ X3 @ A )
= ( insert_nat @ X3 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_207_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_208_insert__absorb,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_209_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_210_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_211_insert__eq__iff,axiom,
! [A2: set_a,A: set_set_a,B3: set_a,B: set_set_a] :
( ~ ( member_set_a @ A2 @ A )
=> ( ~ ( member_set_a @ B3 @ B )
=> ( ( ( insert_set_a @ A2 @ A )
= ( insert_set_a @ B3 @ B ) )
= ( ( ( A2 = B3 )
=> ( A = B ) )
& ( ( A2 != B3 )
=> ? [C3: set_set_a] :
( ( A
= ( insert_set_a @ B3 @ C3 ) )
& ~ ( member_set_a @ B3 @ C3 )
& ( B
= ( insert_set_a @ A2 @ C3 ) )
& ~ ( member_set_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_212_insert__eq__iff,axiom,
! [A2: a,A: set_a,B3: a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ~ ( member_a @ B3 @ B )
=> ( ( ( insert_a @ A2 @ A )
= ( insert_a @ B3 @ B ) )
= ( ( ( A2 = B3 )
=> ( A = B ) )
& ( ( A2 != B3 )
=> ? [C3: set_a] :
( ( A
= ( insert_a @ B3 @ C3 ) )
& ~ ( member_a @ B3 @ C3 )
& ( B
= ( insert_a @ A2 @ C3 ) )
& ~ ( member_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_213_insert__eq__iff,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a,B3: product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A2 @ A )
=> ( ~ ( member1426531477525435216od_a_a @ B3 @ B )
=> ( ( ( insert4534936382041156343od_a_a @ A2 @ A )
= ( insert4534936382041156343od_a_a @ B3 @ B ) )
= ( ( ( A2 = B3 )
=> ( A = B ) )
& ( ( A2 != B3 )
=> ? [C3: set_Product_prod_a_a] :
( ( A
= ( insert4534936382041156343od_a_a @ B3 @ C3 ) )
& ~ ( member1426531477525435216od_a_a @ B3 @ C3 )
& ( B
= ( insert4534936382041156343od_a_a @ A2 @ C3 ) )
& ~ ( member1426531477525435216od_a_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_214_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B3: nat,B: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ B3 @ B )
=> ( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B3 @ B ) )
= ( ( ( A2 = B3 )
=> ( A = B ) )
& ( ( A2 != B3 )
=> ? [C3: set_nat] :
( ( A
= ( insert_nat @ B3 @ C3 ) )
& ~ ( member_nat @ B3 @ C3 )
& ( B
= ( insert_nat @ A2 @ C3 ) )
& ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_215_insert__commute,axiom,
! [X3: a,Y3: a,A: set_a] :
( ( insert_a @ X3 @ ( insert_a @ Y3 @ A ) )
= ( insert_a @ Y3 @ ( insert_a @ X3 @ A ) ) ) ).
% insert_commute
thf(fact_216_insert__commute,axiom,
! [X3: set_a,Y3: set_a,A: set_set_a] :
( ( insert_set_a @ X3 @ ( insert_set_a @ Y3 @ A ) )
= ( insert_set_a @ Y3 @ ( insert_set_a @ X3 @ A ) ) ) ).
% insert_commute
thf(fact_217_mk__disjoint__insert,axiom,
! [A2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ? [B4: set_set_a] :
( ( A
= ( insert_set_a @ A2 @ B4 ) )
& ~ ( member_set_a @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_218_mk__disjoint__insert,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ? [B4: set_a] :
( ( A
= ( insert_a @ A2 @ B4 ) )
& ~ ( member_a @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_219_mk__disjoint__insert,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ A )
=> ? [B4: set_Product_prod_a_a] :
( ( A
= ( insert4534936382041156343od_a_a @ A2 @ B4 ) )
& ~ ( member1426531477525435216od_a_a @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_220_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ? [B4: set_nat] :
( ( A
= ( insert_nat @ A2 @ B4 ) )
& ~ ( member_nat @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_221_singletonD,axiom,
! [B3: a,A2: a] :
( ( member_a @ B3 @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_222_singletonD,axiom,
! [B3: product_prod_a_a,A2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B3 @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_223_singletonD,axiom,
! [B3: set_a,A2: set_a] :
( ( member_set_a @ B3 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_224_singletonD,axiom,
! [B3: nat,A2: nat] :
( ( member_nat @ B3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_225_singleton__iff,axiom,
! [B3: a,A2: a] :
( ( member_a @ B3 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_226_singleton__iff,axiom,
! [B3: product_prod_a_a,A2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B3 @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_227_singleton__iff,axiom,
! [B3: set_a,A2: set_a] :
( ( member_set_a @ B3 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_228_singleton__iff,axiom,
! [B3: nat,A2: nat] :
( ( member_nat @ B3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_229_doubleton__eq__iff,axiom,
! [A2: a,B3: a,C: a,D: a] :
( ( ( insert_a @ A2 @ ( insert_a @ B3 @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A2 = C )
& ( B3 = D ) )
| ( ( A2 = D )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_230_doubleton__eq__iff,axiom,
! [A2: product_prod_a_a,B3: product_prod_a_a,C: product_prod_a_a,D: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) )
= ( insert4534936382041156343od_a_a @ C @ ( insert4534936382041156343od_a_a @ D @ bot_bo3357376287454694259od_a_a ) ) )
= ( ( ( A2 = C )
& ( B3 = D ) )
| ( ( A2 = D )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_231_doubleton__eq__iff,axiom,
! [A2: set_a,B3: set_a,C: set_a,D: set_a] :
( ( ( insert_set_a @ A2 @ ( insert_set_a @ B3 @ bot_bot_set_set_a ) )
= ( insert_set_a @ C @ ( insert_set_a @ D @ bot_bot_set_set_a ) ) )
= ( ( ( A2 = C )
& ( B3 = D ) )
| ( ( A2 = D )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_232_doubleton__eq__iff,axiom,
! [A2: nat,B3: nat,C: nat,D: nat] :
( ( ( insert_nat @ A2 @ ( insert_nat @ B3 @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A2 = C )
& ( B3 = D ) )
| ( ( A2 = D )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_233_insert__not__empty,axiom,
! [A2: a,A: set_a] :
( ( insert_a @ A2 @ A )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_234_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_235_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_236_insert__not__empty,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ A )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_237_singleton__inject,axiom,
! [A2: a,B3: a] :
( ( ( insert_a @ A2 @ bot_bot_set_a )
= ( insert_a @ B3 @ bot_bot_set_a ) )
=> ( A2 = B3 ) ) ).
% singleton_inject
thf(fact_238_singleton__inject,axiom,
! [A2: product_prod_a_a,B3: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a )
= ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) )
=> ( A2 = B3 ) ) ).
% singleton_inject
thf(fact_239_singleton__inject,axiom,
! [A2: set_a,B3: set_a] :
( ( ( insert_set_a @ A2 @ bot_bot_set_set_a )
= ( insert_set_a @ B3 @ bot_bot_set_set_a ) )
=> ( A2 = B3 ) ) ).
% singleton_inject
thf(fact_240_singleton__inject,axiom,
! [A2: nat,B3: nat] :
( ( ( insert_nat @ A2 @ bot_bot_set_nat )
= ( insert_nat @ B3 @ bot_bot_set_nat ) )
=> ( A2 = B3 ) ) ).
% singleton_inject
thf(fact_241_Int__insert__right,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ A2 @ A )
=> ( ( inf_in8905007599844390133od_a_a @ A @ ( insert4534936382041156343od_a_a @ A2 @ B ) )
= ( insert4534936382041156343od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ A @ B ) ) ) )
& ( ~ ( member1426531477525435216od_a_a @ A2 @ A )
=> ( ( inf_in8905007599844390133od_a_a @ A @ ( insert4534936382041156343od_a_a @ A2 @ B ) )
= ( inf_in8905007599844390133od_a_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_242_Int__insert__right,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) )
& ( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_243_Int__insert__right,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) )
& ( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_244_Int__insert__right,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ A @ B ) ) ) )
& ( ~ ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_245_Int__insert__left,axiom,
! [A2: product_prod_a_a,C2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ A2 @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ B ) @ C2 )
= ( insert4534936382041156343od_a_a @ A2 @ ( inf_in8905007599844390133od_a_a @ B @ C2 ) ) ) )
& ( ~ ( member1426531477525435216od_a_a @ A2 @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ B ) @ C2 )
= ( inf_in8905007599844390133od_a_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_246_Int__insert__left,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B @ C2 ) ) ) )
& ( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_247_Int__insert__left,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_248_Int__insert__left,axiom,
! [A2: set_a,C2: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A2 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C2 )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_set_a @ A2 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_249_inf__left__commute,axiom,
! [X3: set_a,Y3: set_a,Z: set_a] :
( ( inf_inf_set_a @ X3 @ ( inf_inf_set_a @ Y3 @ Z ) )
= ( inf_inf_set_a @ Y3 @ ( inf_inf_set_a @ X3 @ Z ) ) ) ).
% inf_left_commute
thf(fact_250_inf__left__commute,axiom,
! [X3: set_set_a,Y3: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ X3 @ ( inf_inf_set_set_a @ Y3 @ Z ) )
= ( inf_inf_set_set_a @ Y3 @ ( inf_inf_set_set_a @ X3 @ Z ) ) ) ).
% inf_left_commute
thf(fact_251_inf_Oleft__commute,axiom,
! [B3: set_a,A2: set_a,C: set_a] :
( ( inf_inf_set_a @ B3 @ ( inf_inf_set_a @ A2 @ C ) )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B3 @ C ) ) ) ).
% inf.left_commute
thf(fact_252_inf_Oleft__commute,axiom,
! [B3: set_set_a,A2: set_set_a,C: set_set_a] :
( ( inf_inf_set_set_a @ B3 @ ( inf_inf_set_set_a @ A2 @ C ) )
= ( inf_inf_set_set_a @ A2 @ ( inf_inf_set_set_a @ B3 @ C ) ) ) ).
% inf.left_commute
thf(fact_253_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_a,K: set_a,B3: set_a,A2: set_a] :
( ( B
= ( inf_inf_set_a @ K @ B3 ) )
=> ( ( inf_inf_set_a @ A2 @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_254_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_set_a,K: set_set_a,B3: set_set_a,A2: set_set_a] :
( ( B
= ( inf_inf_set_set_a @ K @ B3 ) )
=> ( ( inf_inf_set_set_a @ A2 @ B )
= ( inf_inf_set_set_a @ K @ ( inf_inf_set_set_a @ A2 @ B3 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_255_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_a,K: set_a,A2: set_a,B3: set_a] :
( ( A
= ( inf_inf_set_a @ K @ A2 ) )
=> ( ( inf_inf_set_a @ A @ B3 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_256_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_set_a,K: set_set_a,A2: set_set_a,B3: set_set_a] :
( ( A
= ( inf_inf_set_set_a @ K @ A2 ) )
=> ( ( inf_inf_set_set_a @ A @ B3 )
= ( inf_inf_set_set_a @ K @ ( inf_inf_set_set_a @ A2 @ B3 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_257_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X ) ) ) ).
% inf_commute
thf(fact_258_inf__commute,axiom,
( inf_inf_set_set_a
= ( ^ [X: set_set_a,Y2: set_set_a] : ( inf_inf_set_set_a @ Y2 @ X ) ) ) ).
% inf_commute
thf(fact_259_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B5: set_a] : ( inf_inf_set_a @ B5 @ A4 ) ) ) ).
% inf.commute
thf(fact_260_inf_Ocommute,axiom,
( inf_inf_set_set_a
= ( ^ [A4: set_set_a,B5: set_set_a] : ( inf_inf_set_set_a @ B5 @ A4 ) ) ) ).
% inf.commute
thf(fact_261_inf__assoc,axiom,
! [X3: set_a,Y3: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X3 @ Y3 ) @ Z )
= ( inf_inf_set_a @ X3 @ ( inf_inf_set_a @ Y3 @ Z ) ) ) ).
% inf_assoc
thf(fact_262_inf__assoc,axiom,
! [X3: set_set_a,Y3: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X3 @ Y3 ) @ Z )
= ( inf_inf_set_set_a @ X3 @ ( inf_inf_set_set_a @ Y3 @ Z ) ) ) ).
% inf_assoc
thf(fact_263_inf_Oassoc,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ C )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B3 @ C ) ) ) ).
% inf.assoc
thf(fact_264_inf_Oassoc,axiom,
! [A2: set_set_a,B3: set_set_a,C: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A2 @ B3 ) @ C )
= ( inf_inf_set_set_a @ A2 @ ( inf_inf_set_set_a @ B3 @ C ) ) ) ).
% inf.assoc
thf(fact_265_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_266_inf__sup__aci_I1_J,axiom,
( inf_inf_set_set_a
= ( ^ [X: set_set_a,Y2: set_set_a] : ( inf_inf_set_set_a @ Y2 @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_267_inf__sup__aci_I2_J,axiom,
! [X3: set_a,Y3: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X3 @ Y3 ) @ Z )
= ( inf_inf_set_a @ X3 @ ( inf_inf_set_a @ Y3 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_268_inf__sup__aci_I2_J,axiom,
! [X3: set_set_a,Y3: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X3 @ Y3 ) @ Z )
= ( inf_inf_set_set_a @ X3 @ ( inf_inf_set_set_a @ Y3 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_269_inf__sup__aci_I3_J,axiom,
! [X3: set_a,Y3: set_a,Z: set_a] :
( ( inf_inf_set_a @ X3 @ ( inf_inf_set_a @ Y3 @ Z ) )
= ( inf_inf_set_a @ Y3 @ ( inf_inf_set_a @ X3 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_270_inf__sup__aci_I3_J,axiom,
! [X3: set_set_a,Y3: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ X3 @ ( inf_inf_set_set_a @ Y3 @ Z ) )
= ( inf_inf_set_set_a @ Y3 @ ( inf_inf_set_set_a @ X3 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_271_inf__sup__aci_I4_J,axiom,
! [X3: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X3 @ ( inf_inf_set_a @ X3 @ Y3 ) )
= ( inf_inf_set_a @ X3 @ Y3 ) ) ).
% inf_sup_aci(4)
thf(fact_272_inf__sup__aci_I4_J,axiom,
! [X3: set_set_a,Y3: set_set_a] :
( ( inf_inf_set_set_a @ X3 @ ( inf_inf_set_set_a @ X3 @ Y3 ) )
= ( inf_inf_set_set_a @ X3 @ Y3 ) ) ).
% inf_sup_aci(4)
thf(fact_273_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_274_all__edges__betw__D3,axiom,
! [X3: a,Y3: a,X5: set_a,Y5: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y5 ) )
=> ( member_set_a @ ( insert_a @ X3 @ ( insert_a @ Y3 @ bot_bot_set_a ) ) @ edges ) ) ).
% all_edges_betw_D3
thf(fact_275_all__edges__betw__I,axiom,
! [X3: a,X5: set_a,Y3: a,Y5: set_a] :
( ( member_a @ X3 @ X5 )
=> ( ( member_a @ Y3 @ Y5 )
=> ( ( member_set_a @ ( insert_a @ X3 @ ( insert_a @ Y3 @ bot_bot_set_a ) ) @ edges )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y5 ) ) ) ) ) ).
% all_edges_betw_I
thf(fact_276_the__elem__eq,axiom,
! [X3: a] :
( ( the_elem_a @ ( insert_a @ X3 @ bot_bot_set_a ) )
= X3 ) ).
% the_elem_eq
thf(fact_277_the__elem__eq,axiom,
! [X3: product_prod_a_a] :
( ( the_el8589169208993665564od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) )
= X3 ) ).
% the_elem_eq
thf(fact_278_the__elem__eq,axiom,
! [X3: set_a] :
( ( the_elem_set_a @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
= X3 ) ).
% the_elem_eq
thf(fact_279_the__elem__eq,axiom,
! [X3: nat] :
( ( the_elem_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= X3 ) ).
% the_elem_eq
thf(fact_280_all__edges__between__rem__wf,axiom,
! [X5: set_a,Y5: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X5 @ Y5 )
= ( undire8383842906760478443ween_a @ edges @ ( inf_inf_set_a @ X5 @ vertices ) @ ( inf_inf_set_a @ Y5 @ vertices ) ) ) ).
% all_edges_between_rem_wf
thf(fact_281_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X: a] : ( member_a @ X @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_282_bot__empty__eq,axiom,
( bot_bo4160289986317612842_a_a_o
= ( ^ [X: product_prod_a_a] : ( member1426531477525435216od_a_a @ X @ bot_bo3357376287454694259od_a_a ) ) ) ).
% bot_empty_eq
thf(fact_283_bot__empty__eq,axiom,
( bot_bot_set_a_o
= ( ^ [X: set_a] : ( member_set_a @ X @ bot_bot_set_set_a ) ) ) ).
% bot_empty_eq
thf(fact_284_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_285_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_286_Collect__empty__eq__bot,axiom,
! [P: product_prod_a_a > $o] :
( ( ( collec3336397797384452498od_a_a @ P )
= bot_bo3357376287454694259od_a_a )
= ( P = bot_bo4160289986317612842_a_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_287_Collect__empty__eq__bot,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( P = bot_bot_set_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_288_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_289_is__singletonI,axiom,
! [X3: a] : ( is_singleton_a @ ( insert_a @ X3 @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_290_is__singletonI,axiom,
! [X3: product_prod_a_a] : ( is_sin3171834905898671131od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) ) ).
% is_singletonI
thf(fact_291_is__singletonI,axiom,
! [X3: set_a] : ( is_singleton_set_a @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) ).
% is_singletonI
thf(fact_292_is__singletonI,axiom,
! [X3: nat] : ( is_singleton_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_293_wellformed__alt__fst,axiom,
! [X3: a,Y3: a] :
( ( member_set_a @ ( insert_a @ X3 @ ( insert_a @ Y3 @ bot_bot_set_a ) ) @ edges )
=> ( member_a @ X3 @ vertices ) ) ).
% wellformed_alt_fst
thf(fact_294_wellformed__alt__snd,axiom,
! [X3: a,Y3: a] :
( ( member_set_a @ ( insert_a @ X3 @ ( insert_a @ Y3 @ bot_bot_set_a ) ) @ edges )
=> ( member_a @ Y3 @ vertices ) ) ).
% wellformed_alt_snd
thf(fact_295_edge__adjacent__alt__def,axiom,
! [E1: set_a,E2: set_a] :
( ( member_set_a @ E1 @ edges )
=> ( ( member_set_a @ E2 @ edges )
=> ( ? [X6: a] :
( ( member_a @ X6 @ vertices )
& ( member_a @ X6 @ E1 )
& ( member_a @ X6 @ E2 ) )
=> ( undire4022703626023482010_adj_a @ edges @ E1 @ E2 ) ) ) ) ).
% edge_adjacent_alt_def
thf(fact_296_incident__edge__in__wf,axiom,
! [E3: set_a,V: a] :
( ( member_set_a @ E3 @ edges )
=> ( ( undire1521409233611534436dent_a @ V @ E3 )
=> ( member_a @ V @ vertices ) ) ) ).
% incident_edge_in_wf
thf(fact_297_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_298_has__loop__in__verts,axiom,
! [V: a] :
( ( undire3617971648856834880loop_a @ edges @ V )
=> ( member_a @ V @ vertices ) ) ).
% has_loop_in_verts
thf(fact_299_ulgraph__axioms,axiom,
undire7251896706689453996raph_a @ vertices @ edges ).
% ulgraph_axioms
thf(fact_300_mk__edge_Ocases,axiom,
! [X3: product_prod_a_a] :
~ ! [U2: a,V3: a] :
( X3
!= ( product_Pair_a_a @ U2 @ V3 ) ) ).
% mk_edge.cases
thf(fact_301_mk__triangle__set_Ocases,axiom,
! [X3: produc4044097585999906000od_a_a] :
~ ! [X2: a,Y: a,Z2: a] :
( X3
!= ( produc431845341423274048od_a_a @ X2 @ ( product_Pair_a_a @ Y @ Z2 ) ) ) ).
% mk_triangle_set.cases
thf(fact_302_ulgraph_Ohas__loop_Ocong,axiom,
undire3617971648856834880loop_a = undire3617971648856834880loop_a ).
% ulgraph.has_loop.cong
thf(fact_303_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A3: set_a] :
( A3
= ( insert_a @ ( the_elem_a @ A3 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_304_is__singleton__the__elem,axiom,
( is_sin3171834905898671131od_a_a
= ( ^ [A3: set_Product_prod_a_a] :
( A3
= ( insert4534936382041156343od_a_a @ ( the_el8589169208993665564od_a_a @ A3 ) @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_305_is__singleton__the__elem,axiom,
( is_singleton_set_a
= ( ^ [A3: set_set_a] :
( A3
= ( insert_set_a @ ( the_elem_set_a @ A3 ) @ bot_bot_set_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_306_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
( A3
= ( insert_nat @ ( the_elem_nat @ A3 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_307_is__singletonI_H,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
=> ( ! [X2: a,Y: a] :
( ( member_a @ X2 @ A )
=> ( ( member_a @ Y @ A )
=> ( X2 = Y ) ) )
=> ( is_singleton_a @ A ) ) ) ).
% is_singletonI'
thf(fact_308_is__singletonI_H,axiom,
! [A: set_Product_prod_a_a] :
( ( A != bot_bo3357376287454694259od_a_a )
=> ( ! [X2: product_prod_a_a,Y: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A )
=> ( ( member1426531477525435216od_a_a @ Y @ A )
=> ( X2 = Y ) ) )
=> ( is_sin3171834905898671131od_a_a @ A ) ) ) ).
% is_singletonI'
thf(fact_309_is__singletonI_H,axiom,
! [A: set_set_a] :
( ( A != bot_bot_set_set_a )
=> ( ! [X2: set_a,Y: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ( member_set_a @ Y @ A )
=> ( X2 = Y ) ) )
=> ( is_singleton_set_a @ A ) ) ) ).
% is_singletonI'
thf(fact_310_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X2: nat,Y: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y @ A )
=> ( X2 = Y ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_311_is__singletonE,axiom,
! [A: set_a] :
( ( is_singleton_a @ A )
=> ~ ! [X2: a] :
( A
!= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_312_is__singletonE,axiom,
! [A: set_Product_prod_a_a] :
( ( is_sin3171834905898671131od_a_a @ A )
=> ~ ! [X2: product_prod_a_a] :
( A
!= ( insert4534936382041156343od_a_a @ X2 @ bot_bo3357376287454694259od_a_a ) ) ) ).
% is_singletonE
thf(fact_313_is__singletonE,axiom,
! [A: set_set_a] :
( ( is_singleton_set_a @ A )
=> ~ ! [X2: set_a] :
( A
!= ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ) ).
% is_singletonE
thf(fact_314_is__singletonE,axiom,
! [A: set_nat] :
( ( is_singleton_nat @ A )
=> ~ ! [X2: nat] :
( A
!= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_315_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A3: set_a] :
? [X: a] :
( A3
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_316_is__singleton__def,axiom,
( is_sin3171834905898671131od_a_a
= ( ^ [A3: set_Product_prod_a_a] :
? [X: product_prod_a_a] :
( A3
= ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% is_singleton_def
thf(fact_317_is__singleton__def,axiom,
( is_singleton_set_a
= ( ^ [A3: set_set_a] :
? [X: set_a] :
( A3
= ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_318_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
? [X: nat] :
( A3
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_319_is__isolated__vertex__no__loop,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
=> ~ ( undire3617971648856834880loop_a @ edges @ V ) ) ).
% is_isolated_vertex_no_loop
thf(fact_320_is__isolated__vertex__def,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
= ( ( member_a @ V @ vertices )
& ! [X: a] :
( ( member_a @ X @ vertices )
=> ~ ( undire397441198561214472_adj_a @ edges @ X @ V ) ) ) ) ).
% is_isolated_vertex_def
thf(fact_321_is__isolated__vertex__edge,axiom,
! [V: a,E3: set_a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
=> ( ( member_set_a @ E3 @ edges )
=> ~ ( undire1521409233611534436dent_a @ V @ E3 ) ) ) ).
% is_isolated_vertex_edge
thf(fact_322_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_323_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_324_incident__edges__empty,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire3231912044278729248dges_a @ edges @ V )
= bot_bot_set_set_a ) ) ).
% incident_edges_empty
thf(fact_325_graph__system__axioms,axiom,
undire2554140024507503526stem_a @ vertices @ edges ).
% graph_system_axioms
thf(fact_326_wellformed,axiom,
! [E3: set_a] :
( ( member_set_a @ E3 @ edges )
=> ( ord_less_eq_set_a @ E3 @ vertices ) ) ).
% wellformed
thf(fact_327_mk__triangle__set_Osimps,axiom,
! [X3: a,Y3: a,Z: a] :
( ( undire8536760333753235943_set_a @ ( produc431845341423274048od_a_a @ X3 @ ( product_Pair_a_a @ Y3 @ Z ) ) )
= ( insert_a @ X3 @ ( insert_a @ Y3 @ ( insert_a @ Z @ bot_bot_set_a ) ) ) ) ).
% mk_triangle_set.simps
thf(fact_328_mk__triangle__set_Osimps,axiom,
! [X3: product_prod_a_a,Y3: product_prod_a_a,Z: product_prod_a_a] :
( ( undire2459242765783757584od_a_a @ ( produc4925843558922497303od_a_a @ X3 @ ( produc7886510207707329367od_a_a @ Y3 @ Z ) ) )
= ( insert4534936382041156343od_a_a @ X3 @ ( insert4534936382041156343od_a_a @ Y3 @ ( insert4534936382041156343od_a_a @ Z @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% mk_triangle_set.simps
thf(fact_329_mk__triangle__set_Osimps,axiom,
! [X3: set_a,Y3: set_a,Z: set_a] :
( ( undire4638465864238448455_set_a @ ( produc7299740244201487072_set_a @ X3 @ ( produc9088192753505129239_set_a @ Y3 @ Z ) ) )
= ( insert_set_a @ X3 @ ( insert_set_a @ Y3 @ ( insert_set_a @ Z @ bot_bot_set_set_a ) ) ) ) ).
% mk_triangle_set.simps
thf(fact_330_mk__triangle__set_Osimps,axiom,
! [X3: nat,Y3: nat,Z: nat] :
( ( undire4970100481470743719et_nat @ ( produc487386426758144856at_nat @ X3 @ ( product_Pair_nat_nat @ Y3 @ Z ) ) )
= ( insert_nat @ X3 @ ( insert_nat @ Y3 @ ( insert_nat @ Z @ bot_bot_set_nat ) ) ) ) ).
% mk_triangle_set.simps
thf(fact_331_mk__triangle__set_Oelims,axiom,
! [X3: produc4044097585999906000od_a_a,Y3: set_a] :
( ( ( undire8536760333753235943_set_a @ X3 )
= Y3 )
=> ~ ! [X2: a,Y: a,Z2: a] :
( ( X3
= ( produc431845341423274048od_a_a @ X2 @ ( product_Pair_a_a @ Y @ Z2 ) ) )
=> ( Y3
!= ( insert_a @ X2 @ ( insert_a @ Y @ ( insert_a @ Z2 @ bot_bot_set_a ) ) ) ) ) ) ).
% mk_triangle_set.elims
thf(fact_332_mk__triangle__set_Oelims,axiom,
! [X3: produc8857593507947890343od_a_a,Y3: set_Product_prod_a_a] :
( ( ( undire2459242765783757584od_a_a @ X3 )
= Y3 )
=> ~ ! [X2: product_prod_a_a,Y: product_prod_a_a,Z2: product_prod_a_a] :
( ( X3
= ( produc4925843558922497303od_a_a @ X2 @ ( produc7886510207707329367od_a_a @ Y @ Z2 ) ) )
=> ( Y3
!= ( insert4534936382041156343od_a_a @ X2 @ ( insert4534936382041156343od_a_a @ Y @ ( insert4534936382041156343od_a_a @ Z2 @ bot_bo3357376287454694259od_a_a ) ) ) ) ) ) ).
% mk_triangle_set.elims
thf(fact_333_mk__triangle__set_Oelims,axiom,
! [X3: produc3364680560414100336_set_a,Y3: set_set_a] :
( ( ( undire4638465864238448455_set_a @ X3 )
= Y3 )
=> ~ ! [X2: set_a,Y: set_a,Z2: set_a] :
( ( X3
= ( produc7299740244201487072_set_a @ X2 @ ( produc9088192753505129239_set_a @ Y @ Z2 ) ) )
=> ( Y3
!= ( insert_set_a @ X2 @ ( insert_set_a @ Y @ ( insert_set_a @ Z2 @ bot_bot_set_set_a ) ) ) ) ) ) ).
% mk_triangle_set.elims
thf(fact_334_mk__triangle__set_Oelims,axiom,
! [X3: produc7248412053542808358at_nat,Y3: set_nat] :
( ( ( undire4970100481470743719et_nat @ X3 )
= Y3 )
=> ~ ! [X2: nat,Y: nat,Z2: nat] :
( ( X3
= ( produc487386426758144856at_nat @ X2 @ ( product_Pair_nat_nat @ Y @ Z2 ) ) )
=> ( Y3
!= ( insert_nat @ X2 @ ( insert_nat @ Y @ ( insert_nat @ Z2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% mk_triangle_set.elims
thf(fact_335_order__refl,axiom,
! [X3: set_a] : ( ord_less_eq_set_a @ X3 @ X3 ) ).
% order_refl
thf(fact_336_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_337_order__refl,axiom,
! [X3: set_set_a] : ( ord_le3724670747650509150_set_a @ X3 @ X3 ) ).
% order_refl
thf(fact_338_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_339_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_340_dual__order_Orefl,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_341_subsetI,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A )
=> ( member1426531477525435216od_a_a @ X2 @ B ) )
=> ( ord_le746702958409616551od_a_a @ A @ B ) ) ).
% subsetI
thf(fact_342_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_343_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_344_subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( member_set_a @ X2 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% subsetI
thf(fact_345_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_346_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_347_inf_Obounded__iff,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B3 @ C ) )
= ( ( ord_less_eq_set_a @ A2 @ B3 )
& ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_348_inf_Obounded__iff,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B3 @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B3 )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_349_inf_Obounded__iff,axiom,
! [A2: set_set_a,B3: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( inf_inf_set_set_a @ B3 @ C ) )
= ( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
& ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_350_le__inf__iff,axiom,
! [X3: set_a,Y3: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X3 @ ( inf_inf_set_a @ Y3 @ Z ) )
= ( ( ord_less_eq_set_a @ X3 @ Y3 )
& ( ord_less_eq_set_a @ X3 @ Z ) ) ) ).
% le_inf_iff
thf(fact_351_le__inf__iff,axiom,
! [X3: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y3 @ Z ) )
= ( ( ord_less_eq_nat @ X3 @ Y3 )
& ( ord_less_eq_nat @ X3 @ Z ) ) ) ).
% le_inf_iff
thf(fact_352_le__inf__iff,axiom,
! [X3: set_set_a,Y3: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ ( inf_inf_set_set_a @ Y3 @ Z ) )
= ( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
& ( ord_le3724670747650509150_set_a @ X3 @ Z ) ) ) ).
% le_inf_iff
thf(fact_353_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_354_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_355_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_356_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_357_empty__subsetI,axiom,
! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ bot_bo3357376287454694259od_a_a @ A ) ).
% empty_subsetI
thf(fact_358_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_359_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_360_empty__subsetI,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% empty_subsetI
thf(fact_361_insert__subset,axiom,
! [X3: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ A ) @ B )
= ( ( member1426531477525435216od_a_a @ X3 @ B )
& ( ord_le746702958409616551od_a_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_362_insert__subset,axiom,
! [X3: nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ A ) @ B )
= ( ( member_nat @ X3 @ B )
& ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_363_insert__subset,axiom,
! [X3: a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X3 @ A ) @ B )
= ( ( member_a @ X3 @ B )
& ( ord_less_eq_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_364_insert__subset,axiom,
! [X3: set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X3 @ A ) @ B )
= ( ( member_set_a @ X3 @ B )
& ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_365_Int__subset__iff,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
= ( ( ord_less_eq_set_a @ C2 @ A )
& ( ord_less_eq_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_366_Int__subset__iff,axiom,
! [C2: set_set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A @ B ) )
= ( ( ord_le3724670747650509150_set_a @ C2 @ A )
& ( ord_le3724670747650509150_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_367_singleton__insert__inj__eq_H,axiom,
! [A2: product_prod_a_a,A: set_Product_prod_a_a,B3: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A2 @ A )
= ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) )
= ( ( A2 = B3 )
& ( ord_le746702958409616551od_a_a @ A @ ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_368_singleton__insert__inj__eq_H,axiom,
! [A2: nat,A: set_nat,B3: nat] :
( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B3 @ bot_bot_set_nat ) )
= ( ( A2 = B3 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B3 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_369_singleton__insert__inj__eq_H,axiom,
! [A2: a,A: set_a,B3: a] :
( ( ( insert_a @ A2 @ A )
= ( insert_a @ B3 @ bot_bot_set_a ) )
= ( ( A2 = B3 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B3 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_370_singleton__insert__inj__eq_H,axiom,
! [A2: set_a,A: set_set_a,B3: set_a] :
( ( ( insert_set_a @ A2 @ A )
= ( insert_set_a @ B3 @ bot_bot_set_set_a ) )
= ( ( A2 = B3 )
& ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ B3 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_371_singleton__insert__inj__eq,axiom,
! [B3: product_prod_a_a,A2: product_prod_a_a,A: set_Product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a )
= ( insert4534936382041156343od_a_a @ A2 @ A ) )
= ( ( A2 = B3 )
& ( ord_le746702958409616551od_a_a @ A @ ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_372_singleton__insert__inj__eq,axiom,
! [B3: nat,A2: nat,A: set_nat] :
( ( ( insert_nat @ B3 @ bot_bot_set_nat )
= ( insert_nat @ A2 @ A ) )
= ( ( A2 = B3 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B3 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_373_singleton__insert__inj__eq,axiom,
! [B3: a,A2: a,A: set_a] :
( ( ( insert_a @ B3 @ bot_bot_set_a )
= ( insert_a @ A2 @ A ) )
= ( ( A2 = B3 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B3 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_374_singleton__insert__inj__eq,axiom,
! [B3: set_a,A2: set_a,A: set_set_a] :
( ( ( insert_set_a @ B3 @ bot_bot_set_set_a )
= ( insert_set_a @ A2 @ A ) )
= ( ( A2 = B3 )
& ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ B3 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_375_ulgraph_Oincident__loops_Ocong,axiom,
undire4753905205749729249oops_a = undire4753905205749729249oops_a ).
% ulgraph.incident_loops.cong
thf(fact_376_nle__le,axiom,
! [A2: nat,B3: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B3 ) )
= ( ( ord_less_eq_nat @ B3 @ A2 )
& ( B3 != A2 ) ) ) ).
% nle_le
thf(fact_377_le__cases3,axiom,
! [X3: nat,Y3: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_378_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_a,Z3: set_a] : ( Y6 = Z3 ) )
= ( ^ [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
& ( ord_less_eq_set_a @ Y2 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_379_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_380_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_set_a,Z3: set_set_a] : ( Y6 = Z3 ) )
= ( ^ [X: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y2 )
& ( ord_le3724670747650509150_set_a @ Y2 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_381_ord__eq__le__trans,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( A2 = B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_382_ord__eq__le__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( A2 = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_383_ord__eq__le__trans,axiom,
! [A2: set_set_a,B3: set_set_a,C: set_set_a] :
( ( A2 = B3 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ C )
=> ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_384_ord__le__eq__trans,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_385_ord__le__eq__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_386_ord__le__eq__trans,axiom,
! [A2: set_set_a,B3: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_387_order__antisym,axiom,
! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% order_antisym
thf(fact_388_order__antisym,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% order_antisym
thf(fact_389_order__antisym,axiom,
! [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ( ord_le3724670747650509150_set_a @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% order_antisym
thf(fact_390_order_Otrans,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_391_order_Otrans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_392_order_Otrans,axiom,
! [A2: set_set_a,B3: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ C )
=> ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_393_order__trans,axiom,
! [X3: set_a,Y3: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ Z )
=> ( ord_less_eq_set_a @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_394_order__trans,axiom,
! [X3: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z )
=> ( ord_less_eq_nat @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_395_order__trans,axiom,
! [X3: set_set_a,Y3: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ( ord_le3724670747650509150_set_a @ Y3 @ Z )
=> ( ord_le3724670747650509150_set_a @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_396_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B3: nat] :
( ! [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: nat,B6: nat] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A2 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_397_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_a,Z3: set_a] : ( Y6 = Z3 ) )
= ( ^ [A4: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_398_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A4 )
& ( ord_less_eq_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_399_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_set_a,Z3: set_set_a] : ( Y6 = Z3 ) )
= ( ^ [A4: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ A4 )
& ( ord_le3724670747650509150_set_a @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_400_dual__order_Oantisym,axiom,
! [B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_401_dual__order_Oantisym,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_402_dual__order_Oantisym,axiom,
! [B3: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_403_dual__order_Otrans,axiom,
! [B3: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B3 )
=> ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_404_dual__order_Otrans,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_405_dual__order_Otrans,axiom,
! [B3: set_set_a,A2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ C @ B3 )
=> ( ord_le3724670747650509150_set_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_406_in__mono,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,X3: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( member1426531477525435216od_a_a @ X3 @ A )
=> ( member1426531477525435216od_a_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_407_in__mono,axiom,
! [A: set_nat,B: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_408_in__mono,axiom,
! [A: set_a,B: set_a,X3: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_409_in__mono,axiom,
! [A: set_set_a,B: set_set_a,X3: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ X3 @ A )
=> ( member_set_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_410_subsetD,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ( member1426531477525435216od_a_a @ C @ A )
=> ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% subsetD
thf(fact_411_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_412_subsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_413_subsetD,axiom,
! [A: set_set_a,B: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% subsetD
thf(fact_414_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_415_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_416_subset__eq,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A3 )
=> ( member1426531477525435216od_a_a @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_417_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( member_nat @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_418_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B2: set_a] :
! [X: a] :
( ( member_a @ X @ A3 )
=> ( member_a @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_419_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B2: set_set_a] :
! [X: set_a] :
( ( member_set_a @ X @ A3 )
=> ( member_set_a @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_420_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_421_equalityD1,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_422_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_423_equalityD2,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le3724670747650509150_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_424_subset__iff,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A3: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
! [T: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T @ A3 )
=> ( member1426531477525435216od_a_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_425_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_426_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B2: set_a] :
! [T: a] :
( ( member_a @ T @ A3 )
=> ( member_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_427_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B2: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A3 )
=> ( member_set_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_428_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_429_subset__refl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% subset_refl
thf(fact_430_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_431_Collect__mono,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X2: product_prod_a_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P ) @ ( collec3336397797384452498od_a_a @ Q ) ) ) ).
% Collect_mono
thf(fact_432_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_433_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X2: set_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_434_subset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_435_subset__trans,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ord_le3724670747650509150_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_436_antisym,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_437_antisym,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_438_antisym,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_439_set__eq__subset,axiom,
( ( ^ [Y6: set_a,Z3: set_a] : ( Y6 = Z3 ) )
= ( ^ [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
& ( ord_less_eq_set_a @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_440_set__eq__subset,axiom,
( ( ^ [Y6: set_set_a,Z3: set_set_a] : ( Y6 = Z3 ) )
= ( ^ [A3: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B2 )
& ( ord_le3724670747650509150_set_a @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_441_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_442_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 ) )
= ( ! [X: product_prod_a_a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_443_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_444_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X: set_a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_445_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_a,Z3: set_a] : ( Y6 = Z3 ) )
= ( ^ [A4: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_446_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
& ( ord_less_eq_nat @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_447_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_set_a,Z3: set_set_a] : ( Y6 = Z3 ) )
= ( ^ [A4: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_448_order__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_449_order__subst1,axiom,
! [A2: set_a,F: nat > set_a,B3: nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_450_order__subst1,axiom,
! [A2: set_a,F: set_set_a > set_a,B3: set_set_a,C: set_set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ C )
=> ( ! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_451_order__subst1,axiom,
! [A2: nat,F: set_a > nat,B3: set_a,C: set_a] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_452_order__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_453_order__subst1,axiom,
! [A2: nat,F: set_set_a > nat,B3: set_set_a,C: set_set_a] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ C )
=> ( ! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_454_order__subst1,axiom,
! [A2: set_set_a,F: set_a > set_set_a,B3: set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_455_order__subst1,axiom,
! [A2: set_set_a,F: nat > set_set_a,B3: nat,C: nat] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_456_order__subst1,axiom,
! [A2: set_set_a,F: set_set_a > set_set_a,B3: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ C )
=> ( ! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_457_order__subst2,axiom,
! [A2: set_a,B3: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_458_order__subst2,axiom,
! [A2: set_a,B3: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_459_order__subst2,axiom,
! [A2: set_a,B3: set_a,F: set_a > set_set_a,C: set_set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B3 ) @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_460_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_461_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_462_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > set_set_a,C: set_set_a] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_463_order__subst2,axiom,
! [A2: set_set_a,B3: set_set_a,F: set_set_a > set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C )
=> ( ! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_464_order__subst2,axiom,
! [A2: set_set_a,B3: set_set_a,F: set_set_a > nat,C: nat] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_465_order__subst2,axiom,
! [A2: set_set_a,B3: set_set_a,F: set_set_a > set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B3 ) @ C )
=> ( ! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_466_order__eq__refl,axiom,
! [X3: set_a,Y3: set_a] :
( ( X3 = Y3 )
=> ( ord_less_eq_set_a @ X3 @ Y3 ) ) ).
% order_eq_refl
thf(fact_467_order__eq__refl,axiom,
! [X3: nat,Y3: nat] :
( ( X3 = Y3 )
=> ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% order_eq_refl
thf(fact_468_order__eq__refl,axiom,
! [X3: set_set_a,Y3: set_set_a] :
( ( X3 = Y3 )
=> ( ord_le3724670747650509150_set_a @ X3 @ Y3 ) ) ).
% order_eq_refl
thf(fact_469_linorder__linear,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% linorder_linear
thf(fact_470_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_a > set_a,B3: set_a,C: set_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_471_ord__eq__le__subst,axiom,
! [A2: nat,F: set_a > nat,B3: set_a,C: set_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_472_ord__eq__le__subst,axiom,
! [A2: set_set_a,F: set_a > set_set_a,B3: set_a,C: set_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_473_ord__eq__le__subst,axiom,
! [A2: set_a,F: nat > set_a,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_474_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_475_ord__eq__le__subst,axiom,
! [A2: set_set_a,F: nat > set_set_a,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_476_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_set_a > set_a,B3: set_set_a,C: set_set_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ C )
=> ( ! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_477_ord__eq__le__subst,axiom,
! [A2: nat,F: set_set_a > nat,B3: set_set_a,C: set_set_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ C )
=> ( ! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_478_ord__eq__le__subst,axiom,
! [A2: set_set_a,F: set_set_a > set_set_a,B3: set_set_a,C: set_set_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ C )
=> ( ! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_479_ord__le__eq__subst,axiom,
! [A2: set_a,B3: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_480_ord__le__eq__subst,axiom,
! [A2: set_a,B3: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_481_ord__le__eq__subst,axiom,
! [A2: set_a,B3: set_a,F: set_a > set_set_a,C: set_set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_482_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_483_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_484_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > set_set_a,C: set_set_a] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_485_ord__le__eq__subst,axiom,
! [A2: set_set_a,B3: set_set_a,F: set_set_a > set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_486_ord__le__eq__subst,axiom,
! [A2: set_set_a,B3: set_set_a,F: set_set_a > nat,C: nat] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_487_ord__le__eq__subst,axiom,
! [A2: set_set_a,B3: set_set_a,F: set_set_a > set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_488_linorder__le__cases,axiom,
! [X3: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% linorder_le_cases
thf(fact_489_order__antisym__conv,axiom,
! [Y3: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X3 )
=> ( ( ord_less_eq_set_a @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_490_order__antisym__conv,axiom,
! [Y3: nat,X3: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_491_order__antisym__conv,axiom,
! [Y3: set_set_a,X3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y3 @ X3 )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_492_graph__system__def,axiom,
( undire7159349782766787846_set_a
= ( ^ [Vertices: set_set_a,Edges: set_set_set_a] :
! [E: set_set_a] :
( ( member_set_set_a @ E @ Edges )
=> ( ord_le3724670747650509150_set_a @ E @ Vertices ) ) ) ) ).
% graph_system_def
thf(fact_493_graph__system__def,axiom,
( undire2554140024507503526stem_a
= ( ^ [Vertices: set_a,Edges: set_set_a] :
! [E: set_a] :
( ( member_set_a @ E @ Edges )
=> ( ord_less_eq_set_a @ E @ Vertices ) ) ) ) ).
% graph_system_def
thf(fact_494_graph__system_Owellformed,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,E3: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices2 @ Edges2 )
=> ( ( member_set_set_a @ E3 @ Edges2 )
=> ( ord_le3724670747650509150_set_a @ E3 @ Vertices2 ) ) ) ).
% graph_system.wellformed
thf(fact_495_graph__system_Owellformed,axiom,
! [Vertices2: set_a,Edges2: set_set_a,E3: set_a] :
( ( undire2554140024507503526stem_a @ Vertices2 @ Edges2 )
=> ( ( member_set_a @ E3 @ Edges2 )
=> ( ord_less_eq_set_a @ E3 @ Vertices2 ) ) ) ).
% graph_system.wellformed
thf(fact_496_graph__system_Ointro,axiom,
! [Edges2: set_set_set_a,Vertices2: set_set_a] :
( ! [E4: set_set_a] :
( ( member_set_set_a @ E4 @ Edges2 )
=> ( ord_le3724670747650509150_set_a @ E4 @ Vertices2 ) )
=> ( undire7159349782766787846_set_a @ Vertices2 @ Edges2 ) ) ).
% graph_system.intro
thf(fact_497_graph__system_Ointro,axiom,
! [Edges2: set_set_a,Vertices2: set_a] :
( ! [E4: set_a] :
( ( member_set_a @ E4 @ Edges2 )
=> ( ord_less_eq_set_a @ E4 @ Vertices2 ) )
=> ( undire2554140024507503526stem_a @ Vertices2 @ Edges2 ) ) ).
% graph_system.intro
thf(fact_498_ulgraph_Oaxioms_I1_J,axiom,
! [Vertices2: set_a,Edges2: set_set_a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( undire2554140024507503526stem_a @ Vertices2 @ Edges2 ) ) ).
% ulgraph.axioms(1)
thf(fact_499_graph__system_Oincident__edges_Ocong,axiom,
undire3231912044278729248dges_a = undire3231912044278729248dges_a ).
% graph_system.incident_edges.cong
thf(fact_500_graph__system_Oincident__edges__empty,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,V: set_a] :
( ( undire7159349782766787846_set_a @ Vertices2 @ Edges2 )
=> ( ~ ( member_set_a @ V @ Vertices2 )
=> ( ( undire4631953023069350784_set_a @ Edges2 @ V )
= bot_bo3380559777022489994_set_a ) ) ) ).
% graph_system.incident_edges_empty
thf(fact_501_graph__system_Oincident__edges__empty,axiom,
! [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a,V: product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices2 @ Edges2 )
=> ( ~ ( member1426531477525435216od_a_a @ V @ Vertices2 )
=> ( ( undire8905369280470868553od_a_a @ Edges2 @ V )
= bot_bo777872063958040403od_a_a ) ) ) ).
% graph_system.incident_edges_empty
thf(fact_502_graph__system_Oincident__edges__empty,axiom,
! [Vertices2: set_nat,Edges2: set_set_nat,V: nat] :
( ( undire7481384412329822504em_nat @ Vertices2 @ Edges2 )
=> ( ~ ( member_nat @ V @ Vertices2 )
=> ( ( undire4176300566717384750es_nat @ Edges2 @ V )
= bot_bot_set_set_nat ) ) ) ).
% graph_system.incident_edges_empty
thf(fact_503_graph__system_Oincident__edges__empty,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V: a] :
( ( undire2554140024507503526stem_a @ Vertices2 @ Edges2 )
=> ( ~ ( member_a @ V @ Vertices2 )
=> ( ( undire3231912044278729248dges_a @ Edges2 @ V )
= bot_bot_set_set_a ) ) ) ).
% graph_system.incident_edges_empty
thf(fact_504_ulgraph_Ois__isolated__vertex_Ocong,axiom,
undire8931668460104145173rtex_a = undire8931668460104145173rtex_a ).
% ulgraph.is_isolated_vertex.cong
thf(fact_505_ulgraph_Oincident__loops__simp_I2_J,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ~ ( undire3617971648856834880loop_a @ Edges2 @ V )
=> ( ( undire4753905205749729249oops_a @ Edges2 @ V )
= bot_bot_set_set_a ) ) ) ).
% ulgraph.incident_loops_simp(2)
thf(fact_506_ulgraph_Ois__isolated__vertex__edge,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V: a,E3: set_a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ( undire8931668460104145173rtex_a @ Vertices2 @ Edges2 @ V )
=> ( ( member_set_a @ E3 @ Edges2 )
=> ~ ( undire1521409233611534436dent_a @ V @ E3 ) ) ) ) ).
% ulgraph.is_isolated_vertex_edge
thf(fact_507_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices2 @ Edges2 )
=> ( ( undire6879241558604981877_set_a @ Vertices2 @ Edges2 @ V )
= ( ( member_set_a @ V @ Vertices2 )
& ! [X: set_a] :
( ( member_set_a @ X @ Vertices2 )
=> ~ ( undire3510646817838285160_set_a @ Edges2 @ X @ V ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_508_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a,V: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices2 @ Edges2 )
=> ( ( undire3207556238582723646od_a_a @ Vertices2 @ Edges2 @ V )
= ( ( member1426531477525435216od_a_a @ V @ Vertices2 )
& ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ Vertices2 )
=> ~ ( undire6135774327024169009od_a_a @ Edges2 @ X @ V ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_509_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices2: set_nat,Edges2: set_set_nat,V: nat] :
( ( undire3269267262472140706ph_nat @ Vertices2 @ Edges2 )
=> ( ( undire5609513041723151865ex_nat @ Vertices2 @ Edges2 @ V )
= ( ( member_nat @ V @ Vertices2 )
& ! [X: nat] :
( ( member_nat @ X @ Vertices2 )
=> ~ ( undire1083030068171319366dj_nat @ Edges2 @ X @ V ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_510_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ( undire8931668460104145173rtex_a @ Vertices2 @ Edges2 @ V )
= ( ( member_a @ V @ Vertices2 )
& ! [X: a] :
( ( member_a @ X @ Vertices2 )
=> ~ ( undire397441198561214472_adj_a @ Edges2 @ X @ V ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_511_ulgraph_Ois__isolated__vertex__no__loop,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ( undire8931668460104145173rtex_a @ Vertices2 @ Edges2 @ V )
=> ~ ( undire3617971648856834880loop_a @ Edges2 @ V ) ) ) ).
% ulgraph.is_isolated_vertex_no_loop
thf(fact_512_ulgraph_Oempty__not__edge,axiom,
! [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices2 @ Edges2 )
=> ~ ( member1816616512716248880od_a_a @ bot_bo3357376287454694259od_a_a @ Edges2 ) ) ).
% ulgraph.empty_not_edge
thf(fact_513_ulgraph_Oempty__not__edge,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices2 @ Edges2 )
=> ~ ( member_set_set_a @ bot_bot_set_set_a @ Edges2 ) ) ).
% ulgraph.empty_not_edge
thf(fact_514_ulgraph_Oempty__not__edge,axiom,
! [Vertices2: set_nat,Edges2: set_set_nat] :
( ( undire3269267262472140706ph_nat @ Vertices2 @ Edges2 )
=> ~ ( member_set_nat @ bot_bot_set_nat @ Edges2 ) ) ).
% ulgraph.empty_not_edge
thf(fact_515_ulgraph_Oempty__not__edge,axiom,
! [Vertices2: set_a,Edges2: set_set_a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ~ ( member_set_a @ bot_bot_set_a @ Edges2 ) ) ).
% ulgraph.empty_not_edge
thf(fact_516_bot_Oextremum,axiom,
! [A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ bot_bo3357376287454694259od_a_a @ A2 ) ).
% bot.extremum
thf(fact_517_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_518_bot_Oextremum,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% bot.extremum
thf(fact_519_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_520_bot_Oextremum,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A2 ) ).
% bot.extremum
thf(fact_521_bot_Oextremum__unique,axiom,
! [A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ bot_bo3357376287454694259od_a_a )
= ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% bot.extremum_unique
thf(fact_522_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_523_bot_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_524_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_525_bot_Oextremum__unique,axiom,
! [A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a )
= ( A2 = bot_bot_set_set_a ) ) ).
% bot.extremum_unique
thf(fact_526_bot_Oextremum__uniqueI,axiom,
! [A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ bot_bo3357376287454694259od_a_a )
=> ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% bot.extremum_uniqueI
thf(fact_527_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_528_bot_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
=> ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_529_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_530_bot_Oextremum__uniqueI,axiom,
! [A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a )
=> ( A2 = bot_bot_set_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_531_inf_OcoboundedI2,axiom,
! [B3: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_532_inf_OcoboundedI2,axiom,
! [B3: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_533_inf_OcoboundedI2,axiom,
! [B3: set_set_a,C: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ C )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B3 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_534_inf_OcoboundedI1,axiom,
! [A2: set_a,C: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_535_inf_OcoboundedI1,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_536_inf_OcoboundedI1,axiom,
! [A2: set_set_a,C: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B3 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_537_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B5: set_a,A4: set_a] :
( ( inf_inf_set_a @ A4 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_538_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_539_inf_Oabsorb__iff2,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B5: set_set_a,A4: set_set_a] :
( ( inf_inf_set_set_a @ A4 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_540_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B5: set_a] :
( ( inf_inf_set_a @ A4 @ B5 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_541_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] :
( ( inf_inf_nat @ A4 @ B5 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_542_inf_Oabsorb__iff1,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B5: set_set_a] :
( ( inf_inf_set_set_a @ A4 @ B5 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_543_inf_Ocobounded2,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_544_inf_Ocobounded2,axiom,
! [A2: nat,B3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_545_inf_Ocobounded2,axiom,
! [A2: set_set_a,B3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_546_inf_Ocobounded1,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ A2 ) ).
% inf.cobounded1
thf(fact_547_inf_Ocobounded1,axiom,
! [A2: nat,B3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ A2 ) ).
% inf.cobounded1
thf(fact_548_inf_Ocobounded1,axiom,
! [A2: set_set_a,B3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B3 ) @ A2 ) ).
% inf.cobounded1
thf(fact_549_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B5: set_a] :
( A4
= ( inf_inf_set_a @ A4 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_550_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] :
( A4
= ( inf_inf_nat @ A4 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_551_inf_Oorder__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B5: set_set_a] :
( A4
= ( inf_inf_set_set_a @ A4 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_552_inf__greatest,axiom,
! [X3: set_a,Y3: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ( ord_less_eq_set_a @ X3 @ Z )
=> ( ord_less_eq_set_a @ X3 @ ( inf_inf_set_a @ Y3 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_553_inf__greatest,axiom,
! [X3: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ X3 @ Z )
=> ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y3 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_554_inf__greatest,axiom,
! [X3: set_set_a,Y3: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Z )
=> ( ord_le3724670747650509150_set_a @ X3 @ ( inf_inf_set_set_a @ Y3 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_555_inf_OboundedI,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ A2 @ C )
=> ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B3 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_556_inf_OboundedI,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B3 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_557_inf_OboundedI,axiom,
! [A2: set_set_a,B3: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( inf_inf_set_set_a @ B3 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_558_inf_OboundedE,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B3 @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ~ ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_559_inf_OboundedE,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B3 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B3 )
=> ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_560_inf_OboundedE,axiom,
! [A2: set_set_a,B3: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( inf_inf_set_set_a @ B3 @ C ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ~ ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_561_inf__absorb2,axiom,
! [Y3: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X3 )
=> ( ( inf_inf_set_a @ X3 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_562_inf__absorb2,axiom,
! [Y3: nat,X3: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( inf_inf_nat @ X3 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_563_inf__absorb2,axiom,
! [Y3: set_set_a,X3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y3 @ X3 )
=> ( ( inf_inf_set_set_a @ X3 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_564_inf__absorb1,axiom,
! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ( inf_inf_set_a @ X3 @ Y3 )
= X3 ) ) ).
% inf_absorb1
thf(fact_565_inf__absorb1,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( inf_inf_nat @ X3 @ Y3 )
= X3 ) ) ).
% inf_absorb1
thf(fact_566_inf__absorb1,axiom,
! [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ( inf_inf_set_set_a @ X3 @ Y3 )
= X3 ) ) ).
% inf_absorb1
thf(fact_567_inf_Oabsorb2,axiom,
! [B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_568_inf_Oabsorb2,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_569_inf_Oabsorb2,axiom,
! [B3: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_570_inf_Oabsorb1,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( inf_inf_set_a @ A2 @ B3 )
= A2 ) ) ).
% inf.absorb1
thf(fact_571_inf_Oabsorb1,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( inf_inf_nat @ A2 @ B3 )
= A2 ) ) ).
% inf.absorb1
thf(fact_572_inf_Oabsorb1,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( inf_inf_set_set_a @ A2 @ B3 )
= A2 ) ) ).
% inf.absorb1
thf(fact_573_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X @ Y2 )
= X ) ) ) ).
% le_iff_inf
thf(fact_574_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y2: nat] :
( ( inf_inf_nat @ X @ Y2 )
= X ) ) ) ).
% le_iff_inf
thf(fact_575_le__iff__inf,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X: set_set_a,Y2: set_set_a] :
( ( inf_inf_set_set_a @ X @ Y2 )
= X ) ) ) ).
% le_iff_inf
thf(fact_576_inf__unique,axiom,
! [F: set_a > set_a > set_a,X3: set_a,Y3: set_a] :
( ! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y ) @ X2 )
=> ( ! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y ) @ Y )
=> ( ! [X2: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ X2 @ Z2 )
=> ( ord_less_eq_set_a @ X2 @ ( F @ Y @ Z2 ) ) ) )
=> ( ( inf_inf_set_a @ X3 @ Y3 )
= ( F @ X3 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_577_inf__unique,axiom,
! [F: nat > nat > nat,X3: nat,Y3: nat] :
( ! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y ) @ X2 )
=> ( ! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y ) @ Y )
=> ( ! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ ( F @ Y @ Z2 ) ) ) )
=> ( ( inf_inf_nat @ X3 @ Y3 )
= ( F @ X3 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_578_inf__unique,axiom,
! [F: set_set_a > set_set_a > set_set_a,X3: set_set_a,Y3: set_set_a] :
( ! [X2: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F @ X2 @ Y ) @ X2 )
=> ( ! [X2: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F @ X2 @ Y ) @ Y )
=> ( ! [X2: set_set_a,Y: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Z2 )
=> ( ord_le3724670747650509150_set_a @ X2 @ ( F @ Y @ Z2 ) ) ) )
=> ( ( inf_inf_set_set_a @ X3 @ Y3 )
= ( F @ X3 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_579_inf_OorderI,axiom,
! [A2: set_a,B3: set_a] :
( ( A2
= ( inf_inf_set_a @ A2 @ B3 ) )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% inf.orderI
thf(fact_580_inf_OorderI,axiom,
! [A2: nat,B3: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B3 ) )
=> ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% inf.orderI
thf(fact_581_inf_OorderI,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( A2
= ( inf_inf_set_set_a @ A2 @ B3 ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B3 ) ) ).
% inf.orderI
thf(fact_582_inf_OorderE,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( A2
= ( inf_inf_set_a @ A2 @ B3 ) ) ) ).
% inf.orderE
thf(fact_583_inf_OorderE,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( A2
= ( inf_inf_nat @ A2 @ B3 ) ) ) ).
% inf.orderE
thf(fact_584_inf_OorderE,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( A2
= ( inf_inf_set_set_a @ A2 @ B3 ) ) ) ).
% inf.orderE
thf(fact_585_le__infI2,axiom,
! [B3: set_a,X3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ X3 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ X3 ) ) ).
% le_infI2
thf(fact_586_le__infI2,axiom,
! [B3: nat,X3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ X3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ X3 ) ) ).
% le_infI2
thf(fact_587_le__infI2,axiom,
! [B3: set_set_a,X3: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ X3 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B3 ) @ X3 ) ) ).
% le_infI2
thf(fact_588_le__infI1,axiom,
! [A2: set_a,X3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ X3 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ X3 ) ) ).
% le_infI1
thf(fact_589_le__infI1,axiom,
! [A2: nat,X3: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ X3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ X3 ) ) ).
% le_infI1
thf(fact_590_le__infI1,axiom,
! [A2: set_set_a,X3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ X3 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B3 ) @ X3 ) ) ).
% le_infI1
thf(fact_591_inf__mono,axiom,
! [A2: set_a,C: set_a,B3: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B3 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_592_inf__mono,axiom,
! [A2: nat,C: nat,B3: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B3 @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_593_inf__mono,axiom,
! [A2: set_set_a,C: set_set_a,B3: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ D )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B3 ) @ ( inf_inf_set_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_594_le__infI,axiom,
! [X3: set_a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ X3 @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ B3 )
=> ( ord_less_eq_set_a @ X3 @ ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% le_infI
thf(fact_595_le__infI,axiom,
! [X3: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ X3 @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ B3 )
=> ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A2 @ B3 ) ) ) ) ).
% le_infI
thf(fact_596_le__infI,axiom,
! [X3: set_set_a,A2: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ B3 )
=> ( ord_le3724670747650509150_set_a @ X3 @ ( inf_inf_set_set_a @ A2 @ B3 ) ) ) ) ).
% le_infI
thf(fact_597_le__infE,axiom,
! [X3: set_a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ X3 @ ( inf_inf_set_a @ A2 @ B3 ) )
=> ~ ( ( ord_less_eq_set_a @ X3 @ A2 )
=> ~ ( ord_less_eq_set_a @ X3 @ B3 ) ) ) ).
% le_infE
thf(fact_598_le__infE,axiom,
! [X3: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A2 @ B3 ) )
=> ~ ( ( ord_less_eq_nat @ X3 @ A2 )
=> ~ ( ord_less_eq_nat @ X3 @ B3 ) ) ) ).
% le_infE
thf(fact_599_le__infE,axiom,
! [X3: set_set_a,A2: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ ( inf_inf_set_set_a @ A2 @ B3 ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ X3 @ A2 )
=> ~ ( ord_le3724670747650509150_set_a @ X3 @ B3 ) ) ) ).
% le_infE
thf(fact_600_inf__le2,axiom,
! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X3 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_601_inf__le2,axiom,
! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_602_inf__le2,axiom,
! [X3: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X3 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_603_inf__le1,axiom,
! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X3 @ Y3 ) @ X3 ) ).
% inf_le1
thf(fact_604_inf__le1,axiom,
! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y3 ) @ X3 ) ).
% inf_le1
thf(fact_605_inf__le1,axiom,
! [X3: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X3 @ Y3 ) @ X3 ) ).
% inf_le1
thf(fact_606_inf__sup__ord_I1_J,axiom,
! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X3 @ Y3 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_607_inf__sup__ord_I1_J,axiom,
! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y3 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_608_inf__sup__ord_I1_J,axiom,
! [X3: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X3 @ Y3 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_609_inf__sup__ord_I2_J,axiom,
! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X3 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_610_inf__sup__ord_I2_J,axiom,
! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_611_inf__sup__ord_I2_J,axiom,
! [X3: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X3 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_612_insert__mono,axiom,
! [C2: set_a,D2: set_a,A2: a] :
( ( ord_less_eq_set_a @ C2 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A2 @ C2 ) @ ( insert_a @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_613_insert__mono,axiom,
! [C2: set_set_a,D2: set_set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ A2 @ C2 ) @ ( insert_set_a @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_614_subset__insert,axiom,
! [X3: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X3 @ A )
=> ( ( ord_le746702958409616551od_a_a @ A @ ( insert4534936382041156343od_a_a @ X3 @ B ) )
= ( ord_le746702958409616551od_a_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_615_subset__insert,axiom,
! [X3: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X3 @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X3 @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_616_subset__insert,axiom,
! [X3: a,A: set_a,B: set_a] :
( ~ ( member_a @ X3 @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a @ X3 @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_617_subset__insert,axiom,
! [X3: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X3 @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X3 @ B ) )
= ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_618_subset__insertI,axiom,
! [B: set_a,A2: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A2 @ B ) ) ).
% subset_insertI
thf(fact_619_subset__insertI,axiom,
! [B: set_set_a,A2: set_a] : ( ord_le3724670747650509150_set_a @ B @ ( insert_set_a @ A2 @ B ) ) ).
% subset_insertI
thf(fact_620_subset__insertI2,axiom,
! [A: set_a,B: set_a,B3: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ B3 @ B ) ) ) ).
% subset_insertI2
thf(fact_621_subset__insertI2,axiom,
! [A: set_set_a,B: set_set_a,B3: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ B3 @ B ) ) ) ).
% subset_insertI2
thf(fact_622_Int__Collect__mono,axiom,
! [A: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_623_Int__Collect__mono,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ A @ B )
=> ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ ( collec3336397797384452498od_a_a @ P ) ) @ ( inf_in8905007599844390133od_a_a @ B @ ( collec3336397797384452498od_a_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_624_Int__Collect__mono,axiom,
! [A: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_625_Int__Collect__mono,axiom,
! [A: set_set_a,B: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_626_Int__greatest,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_627_Int__greatest,axiom,
! [C2: set_set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ A )
=> ( ( ord_le3724670747650509150_set_a @ C2 @ B )
=> ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_628_Int__absorb2,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_629_Int__absorb2,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( inf_inf_set_set_a @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_630_Int__absorb1,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_631_Int__absorb1,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( inf_inf_set_set_a @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_632_Int__lower2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_633_Int__lower2,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_634_Int__lower1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_635_Int__lower1,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_636_Int__mono,axiom,
! [A: set_a,C2: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_637_Int__mono,axiom,
! [A: set_set_a,C2: set_set_a,B: set_set_a,D2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ ( inf_inf_set_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_638_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,V1: set_a,V2: set_a] :
( ( undire6886684016831807756_set_a @ Vertices2 @ Edges2 )
=> ( ( undire3510646817838285160_set_a @ Edges2 @ V1 @ V2 )
=> ( ( member_set_a @ V1 @ Vertices2 )
& ( member_set_a @ V2 @ Vertices2 ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_639_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a,V1: product_prod_a_a,V2: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices2 @ Edges2 )
=> ( ( undire6135774327024169009od_a_a @ Edges2 @ V1 @ V2 )
=> ( ( member1426531477525435216od_a_a @ V1 @ Vertices2 )
& ( member1426531477525435216od_a_a @ V2 @ Vertices2 ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_640_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices2: set_nat,Edges2: set_set_nat,V1: nat,V2: nat] :
( ( undire3269267262472140706ph_nat @ Vertices2 @ Edges2 )
=> ( ( undire1083030068171319366dj_nat @ Edges2 @ V1 @ V2 )
=> ( ( member_nat @ V1 @ Vertices2 )
& ( member_nat @ V2 @ Vertices2 ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_641_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V1: a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ( undire397441198561214472_adj_a @ Edges2 @ V1 @ V2 )
=> ( ( member_a @ V1 @ Vertices2 )
& ( member_a @ V2 @ Vertices2 ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_642_ulgraph_Overt__adj__sym,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V1: a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ( undire397441198561214472_adj_a @ Edges2 @ V1 @ V2 )
= ( undire397441198561214472_adj_a @ Edges2 @ V2 @ V1 ) ) ) ).
% ulgraph.vert_adj_sym
thf(fact_643_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices2 @ Edges2 )
=> ( ( undire5774735625301615776_set_a @ Edges2 @ V )
=> ( member_set_a @ V @ Vertices2 ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_644_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a,V: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices2 @ Edges2 )
=> ( ( undire7777398424729533289od_a_a @ Edges2 @ V )
=> ( member1426531477525435216od_a_a @ V @ Vertices2 ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_645_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices2: set_nat,Edges2: set_set_nat,V: nat] :
( ( undire3269267262472140706ph_nat @ Vertices2 @ Edges2 )
=> ( ( undire5005864372999571214op_nat @ Edges2 @ V )
=> ( member_nat @ V @ Vertices2 ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_646_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ( undire3617971648856834880loop_a @ Edges2 @ V )
=> ( member_a @ V @ Vertices2 ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_647_graph__system_Oincident__edge__in__wf,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,E3: set_set_a,V: set_a] :
( ( undire7159349782766787846_set_a @ Vertices2 @ Edges2 )
=> ( ( member_set_set_a @ E3 @ Edges2 )
=> ( ( undire2320338297334612420_set_a @ V @ E3 )
=> ( member_set_a @ V @ Vertices2 ) ) ) ) ).
% graph_system.incident_edge_in_wf
thf(fact_648_graph__system_Oincident__edge__in__wf,axiom,
! [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a,E3: set_Product_prod_a_a,V: product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices2 @ Edges2 )
=> ( ( member1816616512716248880od_a_a @ E3 @ Edges2 )
=> ( ( undire3369688177417741453od_a_a @ V @ E3 )
=> ( member1426531477525435216od_a_a @ V @ Vertices2 ) ) ) ) ).
% graph_system.incident_edge_in_wf
thf(fact_649_graph__system_Oincident__edge__in__wf,axiom,
! [Vertices2: set_nat,Edges2: set_set_nat,E3: set_nat,V: nat] :
( ( undire7481384412329822504em_nat @ Vertices2 @ Edges2 )
=> ( ( member_set_nat @ E3 @ Edges2 )
=> ( ( undire7858122600432113898nt_nat @ V @ E3 )
=> ( member_nat @ V @ Vertices2 ) ) ) ) ).
% graph_system.incident_edge_in_wf
thf(fact_650_graph__system_Oincident__edge__in__wf,axiom,
! [Vertices2: set_a,Edges2: set_set_a,E3: set_a,V: a] :
( ( undire2554140024507503526stem_a @ Vertices2 @ Edges2 )
=> ( ( member_set_a @ E3 @ Edges2 )
=> ( ( undire1521409233611534436dent_a @ V @ E3 )
=> ( member_a @ V @ Vertices2 ) ) ) ) ).
% graph_system.incident_edge_in_wf
thf(fact_651_graph__system_Oincident__def,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,V: set_a,E3: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices2 @ Edges2 )
=> ( ( undire2320338297334612420_set_a @ V @ E3 )
= ( member_set_a @ V @ E3 ) ) ) ).
% graph_system.incident_def
thf(fact_652_graph__system_Oincident__def,axiom,
! [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a,V: product_prod_a_a,E3: set_Product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices2 @ Edges2 )
=> ( ( undire3369688177417741453od_a_a @ V @ E3 )
= ( member1426531477525435216od_a_a @ V @ E3 ) ) ) ).
% graph_system.incident_def
thf(fact_653_graph__system_Oincident__def,axiom,
! [Vertices2: set_nat,Edges2: set_set_nat,V: nat,E3: set_nat] :
( ( undire7481384412329822504em_nat @ Vertices2 @ Edges2 )
=> ( ( undire7858122600432113898nt_nat @ V @ E3 )
= ( member_nat @ V @ E3 ) ) ) ).
% graph_system.incident_def
thf(fact_654_graph__system_Oincident__def,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V: a,E3: set_a] :
( ( undire2554140024507503526stem_a @ Vertices2 @ Edges2 )
=> ( ( undire1521409233611534436dent_a @ V @ E3 )
= ( member_a @ V @ E3 ) ) ) ).
% graph_system.incident_def
thf(fact_655_ulgraph_Oincident__loops__simp_I1_J,axiom,
! [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a,V: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices2 @ Edges2 )
=> ( ( undire7777398424729533289od_a_a @ Edges2 @ V )
=> ( ( undire3049230956220217098od_a_a @ Edges2 @ V )
= ( insert914553114930139863od_a_a @ ( insert4534936382041156343od_a_a @ V @ bot_bo3357376287454694259od_a_a ) @ bot_bo777872063958040403od_a_a ) ) ) ) ).
% ulgraph.incident_loops_simp(1)
thf(fact_656_ulgraph_Oincident__loops__simp_I1_J,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices2 @ Edges2 )
=> ( ( undire5774735625301615776_set_a @ Edges2 @ V )
=> ( ( undire7215034953758041409_set_a @ Edges2 @ 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_657_ulgraph_Oincident__loops__simp_I1_J,axiom,
! [Vertices2: set_nat,Edges2: set_set_nat,V: nat] :
( ( undire3269267262472140706ph_nat @ Vertices2 @ Edges2 )
=> ( ( undire5005864372999571214op_nat @ Edges2 @ V )
=> ( ( undire1050940535076293677ps_nat @ Edges2 @ V )
= ( insert_set_nat @ ( insert_nat @ V @ bot_bot_set_nat ) @ bot_bot_set_set_nat ) ) ) ) ).
% ulgraph.incident_loops_simp(1)
thf(fact_658_ulgraph_Oincident__loops__simp_I1_J,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ( undire3617971648856834880loop_a @ Edges2 @ V )
=> ( ( undire4753905205749729249oops_a @ Edges2 @ V )
= ( insert_set_a @ ( insert_a @ V @ bot_bot_set_a ) @ bot_bot_set_set_a ) ) ) ) ).
% ulgraph.incident_loops_simp(1)
thf(fact_659_subset__singleton__iff,axiom,
! [X5: set_Product_prod_a_a,A2: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X5 @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) )
= ( ( X5 = bot_bo3357376287454694259od_a_a )
| ( X5
= ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_660_subset__singleton__iff,axiom,
! [X5: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ X5 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
| ( X5
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_661_subset__singleton__iff,axiom,
! [X5: set_a,A2: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_662_subset__singleton__iff,axiom,
! [X5: set_set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
= ( ( X5 = bot_bot_set_set_a )
| ( X5
= ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_663_subset__singletonD,axiom,
! [A: set_Product_prod_a_a,X3: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) )
=> ( ( A = bot_bo3357376287454694259od_a_a )
| ( A
= ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% subset_singletonD
thf(fact_664_subset__singletonD,axiom,
! [A: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ( A = bot_bot_set_nat )
| ( A
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_665_subset__singletonD,axiom,
! [A: set_a,X3: a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X3 @ bot_bot_set_a ) )
=> ( ( A = bot_bot_set_a )
| ( A
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_666_subset__singletonD,axiom,
! [A: set_set_a,X3: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
=> ( ( A = bot_bot_set_set_a )
| ( A
= ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_667_graph__system_Oedge__adjacent__alt__def,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,E1: set_set_a,E2: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices2 @ Edges2 )
=> ( ( member_set_set_a @ E1 @ Edges2 )
=> ( ( member_set_set_a @ E2 @ Edges2 )
=> ( ? [X6: set_a] :
( ( member_set_a @ X6 @ Vertices2 )
& ( member_set_a @ X6 @ E1 )
& ( member_set_a @ X6 @ E2 ) )
=> ( undire3485422320110889978_set_a @ Edges2 @ E1 @ E2 ) ) ) ) ) ).
% graph_system.edge_adjacent_alt_def
thf(fact_668_graph__system_Oedge__adjacent__alt__def,axiom,
! [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a,E1: set_Product_prod_a_a,E2: set_Product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices2 @ Edges2 )
=> ( ( member1816616512716248880od_a_a @ E1 @ Edges2 )
=> ( ( member1816616512716248880od_a_a @ E2 @ Edges2 )
=> ( ? [X6: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X6 @ Vertices2 )
& ( member1426531477525435216od_a_a @ X6 @ E1 )
& ( member1426531477525435216od_a_a @ X6 @ E2 ) )
=> ( undire9186443406341554371od_a_a @ Edges2 @ E1 @ E2 ) ) ) ) ) ).
% graph_system.edge_adjacent_alt_def
thf(fact_669_graph__system_Oedge__adjacent__alt__def,axiom,
! [Vertices2: set_nat,Edges2: set_set_nat,E1: set_nat,E2: set_nat] :
( ( undire7481384412329822504em_nat @ Vertices2 @ Edges2 )
=> ( ( member_set_nat @ E1 @ Edges2 )
=> ( ( member_set_nat @ E2 @ Edges2 )
=> ( ? [X6: nat] :
( ( member_nat @ X6 @ Vertices2 )
& ( member_nat @ X6 @ E1 )
& ( member_nat @ X6 @ E2 ) )
=> ( undire1664191744716346676dj_nat @ Edges2 @ E1 @ E2 ) ) ) ) ) ).
% graph_system.edge_adjacent_alt_def
thf(fact_670_graph__system_Oedge__adjacent__alt__def,axiom,
! [Vertices2: set_a,Edges2: set_set_a,E1: set_a,E2: set_a] :
( ( undire2554140024507503526stem_a @ Vertices2 @ Edges2 )
=> ( ( member_set_a @ E1 @ Edges2 )
=> ( ( member_set_a @ E2 @ Edges2 )
=> ( ? [X6: a] :
( ( member_a @ X6 @ Vertices2 )
& ( member_a @ X6 @ E1 )
& ( member_a @ X6 @ E2 ) )
=> ( undire4022703626023482010_adj_a @ Edges2 @ E1 @ E2 ) ) ) ) ) ).
% graph_system.edge_adjacent_alt_def
thf(fact_671_graph__system_Oedge__adj__inE,axiom,
! [Vertices2: set_a,Edges2: set_set_a,E1: set_a,E2: set_a] :
( ( undire2554140024507503526stem_a @ Vertices2 @ Edges2 )
=> ( ( undire4022703626023482010_adj_a @ Edges2 @ E1 @ E2 )
=> ( ( member_set_a @ E1 @ Edges2 )
& ( member_set_a @ E2 @ Edges2 ) ) ) ) ).
% graph_system.edge_adj_inE
thf(fact_672_ulgraph_Oall__edges__between__rem__wf,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,X5: set_set_a,Y5: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices2 @ Edges2 )
=> ( ( undire2462398226299384907_set_a @ Edges2 @ X5 @ Y5 )
= ( undire2462398226299384907_set_a @ Edges2 @ ( inf_inf_set_set_a @ X5 @ Vertices2 ) @ ( inf_inf_set_set_a @ Y5 @ Vertices2 ) ) ) ) ).
% ulgraph.all_edges_between_rem_wf
thf(fact_673_ulgraph_Oall__edges__between__rem__wf,axiom,
! [Vertices2: set_a,Edges2: set_set_a,X5: set_a,Y5: set_a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ( undire8383842906760478443ween_a @ Edges2 @ X5 @ Y5 )
= ( undire8383842906760478443ween_a @ Edges2 @ ( inf_inf_set_a @ X5 @ Vertices2 ) @ ( inf_inf_set_a @ Y5 @ Vertices2 ) ) ) ) ).
% ulgraph.all_edges_between_rem_wf
thf(fact_674_ulgraph_Overt__adj__edge__iff2,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V1: a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ( V1 != V2 )
=> ( ( undire397441198561214472_adj_a @ Edges2 @ V1 @ V2 )
= ( ? [X: set_a] :
( ( member_set_a @ X @ Edges2 )
& ( undire1521409233611534436dent_a @ V1 @ X )
& ( undire1521409233611534436dent_a @ V2 @ X ) ) ) ) ) ) ).
% ulgraph.vert_adj_edge_iff2
thf(fact_675_graph__system_Owellformed__alt__fst,axiom,
! [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a,X3: product_prod_a_a,Y3: product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices2 @ Edges2 )
=> ( ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ ( insert4534936382041156343od_a_a @ Y3 @ bot_bo3357376287454694259od_a_a ) ) @ Edges2 )
=> ( member1426531477525435216od_a_a @ X3 @ Vertices2 ) ) ) ).
% graph_system.wellformed_alt_fst
thf(fact_676_graph__system_Owellformed__alt__fst,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,X3: set_a,Y3: set_a] :
( ( undire7159349782766787846_set_a @ Vertices2 @ Edges2 )
=> ( ( member_set_set_a @ ( insert_set_a @ X3 @ ( insert_set_a @ Y3 @ bot_bot_set_set_a ) ) @ Edges2 )
=> ( member_set_a @ X3 @ Vertices2 ) ) ) ).
% graph_system.wellformed_alt_fst
thf(fact_677_graph__system_Owellformed__alt__fst,axiom,
! [Vertices2: set_nat,Edges2: set_set_nat,X3: nat,Y3: nat] :
( ( undire7481384412329822504em_nat @ Vertices2 @ Edges2 )
=> ( ( member_set_nat @ ( insert_nat @ X3 @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) @ Edges2 )
=> ( member_nat @ X3 @ Vertices2 ) ) ) ).
% graph_system.wellformed_alt_fst
thf(fact_678_graph__system_Owellformed__alt__fst,axiom,
! [Vertices2: set_a,Edges2: set_set_a,X3: a,Y3: a] :
( ( undire2554140024507503526stem_a @ Vertices2 @ Edges2 )
=> ( ( member_set_a @ ( insert_a @ X3 @ ( insert_a @ Y3 @ bot_bot_set_a ) ) @ Edges2 )
=> ( member_a @ X3 @ Vertices2 ) ) ) ).
% graph_system.wellformed_alt_fst
thf(fact_679_graph__system_Owellformed__alt__snd,axiom,
! [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a,X3: product_prod_a_a,Y3: product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices2 @ Edges2 )
=> ( ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ ( insert4534936382041156343od_a_a @ Y3 @ bot_bo3357376287454694259od_a_a ) ) @ Edges2 )
=> ( member1426531477525435216od_a_a @ Y3 @ Vertices2 ) ) ) ).
% graph_system.wellformed_alt_snd
thf(fact_680_graph__system_Owellformed__alt__snd,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,X3: set_a,Y3: set_a] :
( ( undire7159349782766787846_set_a @ Vertices2 @ Edges2 )
=> ( ( member_set_set_a @ ( insert_set_a @ X3 @ ( insert_set_a @ Y3 @ bot_bot_set_set_a ) ) @ Edges2 )
=> ( member_set_a @ Y3 @ Vertices2 ) ) ) ).
% graph_system.wellformed_alt_snd
thf(fact_681_graph__system_Owellformed__alt__snd,axiom,
! [Vertices2: set_nat,Edges2: set_set_nat,X3: nat,Y3: nat] :
( ( undire7481384412329822504em_nat @ Vertices2 @ Edges2 )
=> ( ( member_set_nat @ ( insert_nat @ X3 @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) @ Edges2 )
=> ( member_nat @ Y3 @ Vertices2 ) ) ) ).
% graph_system.wellformed_alt_snd
thf(fact_682_graph__system_Owellformed__alt__snd,axiom,
! [Vertices2: set_a,Edges2: set_set_a,X3: a,Y3: a] :
( ( undire2554140024507503526stem_a @ Vertices2 @ Edges2 )
=> ( ( member_set_a @ ( insert_a @ X3 @ ( insert_a @ Y3 @ bot_bot_set_a ) ) @ Edges2 )
=> ( member_a @ Y3 @ Vertices2 ) ) ) ).
% graph_system.wellformed_alt_snd
thf(fact_683_ulgraph_Overt__adj__def,axiom,
! [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a,V1: product_prod_a_a,V2: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices2 @ Edges2 )
=> ( ( undire6135774327024169009od_a_a @ Edges2 @ V1 @ V2 )
= ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ V1 @ ( insert4534936382041156343od_a_a @ V2 @ bot_bo3357376287454694259od_a_a ) ) @ Edges2 ) ) ) ).
% ulgraph.vert_adj_def
thf(fact_684_ulgraph_Overt__adj__def,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,V1: set_a,V2: set_a] :
( ( undire6886684016831807756_set_a @ Vertices2 @ Edges2 )
=> ( ( undire3510646817838285160_set_a @ Edges2 @ V1 @ V2 )
= ( member_set_set_a @ ( insert_set_a @ V1 @ ( insert_set_a @ V2 @ bot_bot_set_set_a ) ) @ Edges2 ) ) ) ).
% ulgraph.vert_adj_def
thf(fact_685_ulgraph_Overt__adj__def,axiom,
! [Vertices2: set_nat,Edges2: set_set_nat,V1: nat,V2: nat] :
( ( undire3269267262472140706ph_nat @ Vertices2 @ Edges2 )
=> ( ( undire1083030068171319366dj_nat @ Edges2 @ V1 @ V2 )
= ( member_set_nat @ ( insert_nat @ V1 @ ( insert_nat @ V2 @ bot_bot_set_nat ) ) @ Edges2 ) ) ) ).
% ulgraph.vert_adj_def
thf(fact_686_ulgraph_Overt__adj__def,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V1: a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ( undire397441198561214472_adj_a @ Edges2 @ V1 @ V2 )
= ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) @ Edges2 ) ) ) ).
% ulgraph.vert_adj_def
thf(fact_687_ulgraph_Onot__vert__adj,axiom,
! [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a,V: product_prod_a_a,U: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices2 @ Edges2 )
=> ( ~ ( undire6135774327024169009od_a_a @ Edges2 @ V @ U )
=> ~ ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ V @ ( insert4534936382041156343od_a_a @ U @ bot_bo3357376287454694259od_a_a ) ) @ Edges2 ) ) ) ).
% ulgraph.not_vert_adj
thf(fact_688_ulgraph_Onot__vert__adj,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,V: set_a,U: set_a] :
( ( undire6886684016831807756_set_a @ Vertices2 @ Edges2 )
=> ( ~ ( undire3510646817838285160_set_a @ Edges2 @ V @ U )
=> ~ ( member_set_set_a @ ( insert_set_a @ V @ ( insert_set_a @ U @ bot_bot_set_set_a ) ) @ Edges2 ) ) ) ).
% ulgraph.not_vert_adj
thf(fact_689_ulgraph_Onot__vert__adj,axiom,
! [Vertices2: set_nat,Edges2: set_set_nat,V: nat,U: nat] :
( ( undire3269267262472140706ph_nat @ Vertices2 @ Edges2 )
=> ( ~ ( undire1083030068171319366dj_nat @ Edges2 @ V @ U )
=> ~ ( member_set_nat @ ( insert_nat @ V @ ( insert_nat @ U @ bot_bot_set_nat ) ) @ Edges2 ) ) ) ).
% ulgraph.not_vert_adj
thf(fact_690_ulgraph_Onot__vert__adj,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V: a,U: a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ~ ( undire397441198561214472_adj_a @ Edges2 @ V @ U )
=> ~ ( member_set_a @ ( insert_a @ V @ ( insert_a @ U @ bot_bot_set_a ) ) @ Edges2 ) ) ) ).
% ulgraph.not_vert_adj
thf(fact_691_ulgraph_Ohas__loop__def,axiom,
! [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a,V: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices2 @ Edges2 )
=> ( ( undire7777398424729533289od_a_a @ Edges2 @ V )
= ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ V @ bot_bo3357376287454694259od_a_a ) @ Edges2 ) ) ) ).
% ulgraph.has_loop_def
thf(fact_692_ulgraph_Ohas__loop__def,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices2 @ Edges2 )
=> ( ( undire5774735625301615776_set_a @ Edges2 @ V )
= ( member_set_set_a @ ( insert_set_a @ V @ bot_bot_set_set_a ) @ Edges2 ) ) ) ).
% ulgraph.has_loop_def
thf(fact_693_ulgraph_Ohas__loop__def,axiom,
! [Vertices2: set_nat,Edges2: set_set_nat,V: nat] :
( ( undire3269267262472140706ph_nat @ Vertices2 @ Edges2 )
=> ( ( undire5005864372999571214op_nat @ Edges2 @ V )
= ( member_set_nat @ ( insert_nat @ V @ bot_bot_set_nat ) @ Edges2 ) ) ) ).
% ulgraph.has_loop_def
thf(fact_694_ulgraph_Ohas__loop__def,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ( undire3617971648856834880loop_a @ Edges2 @ V )
= ( member_set_a @ ( insert_a @ V @ bot_bot_set_a ) @ Edges2 ) ) ) ).
% ulgraph.has_loop_def
thf(fact_695_ulgraph_Ois__edge__between__def,axiom,
! [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a,X5: set_Product_prod_a_a,Y5: set_Product_prod_a_a,E3: set_Product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices2 @ Edges2 )
=> ( ( undire7011261089604658374od_a_a @ X5 @ Y5 @ E3 )
= ( ? [X: product_prod_a_a,Y2: product_prod_a_a] :
( ( E3
= ( insert4534936382041156343od_a_a @ X @ ( insert4534936382041156343od_a_a @ Y2 @ bot_bo3357376287454694259od_a_a ) ) )
& ( member1426531477525435216od_a_a @ X @ X5 )
& ( member1426531477525435216od_a_a @ Y2 @ Y5 ) ) ) ) ) ).
% ulgraph.is_edge_between_def
thf(fact_696_ulgraph_Ois__edge__between__def,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,X5: set_set_a,Y5: set_set_a,E3: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices2 @ Edges2 )
=> ( ( undire2578756059399487229_set_a @ X5 @ Y5 @ E3 )
= ( ? [X: set_a,Y2: set_a] :
( ( E3
= ( insert_set_a @ X @ ( insert_set_a @ Y2 @ bot_bot_set_set_a ) ) )
& ( member_set_a @ X @ X5 )
& ( member_set_a @ Y2 @ Y5 ) ) ) ) ) ).
% ulgraph.is_edge_between_def
thf(fact_697_ulgraph_Ois__edge__between__def,axiom,
! [Vertices2: set_nat,Edges2: set_set_nat,X5: set_nat,Y5: set_nat,E3: set_nat] :
( ( undire3269267262472140706ph_nat @ Vertices2 @ Edges2 )
=> ( ( undire6814325412647357297en_nat @ X5 @ Y5 @ E3 )
= ( ? [X: nat,Y2: nat] :
( ( E3
= ( insert_nat @ X @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
& ( member_nat @ X @ X5 )
& ( member_nat @ Y2 @ Y5 ) ) ) ) ) ).
% ulgraph.is_edge_between_def
thf(fact_698_ulgraph_Ois__edge__between__def,axiom,
! [Vertices2: set_a,Edges2: set_set_a,X5: set_a,Y5: set_a,E3: set_a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ( undire8544646567961481629ween_a @ X5 @ Y5 @ E3 )
= ( ? [X: a,Y2: a] :
( ( E3
= ( insert_a @ X @ ( insert_a @ Y2 @ bot_bot_set_a ) ) )
& ( member_a @ X @ X5 )
& ( member_a @ Y2 @ Y5 ) ) ) ) ) ).
% ulgraph.is_edge_between_def
thf(fact_699_ulgraph_Oall__edges__betw__I,axiom,
! [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a,X3: product_prod_a_a,X5: set_Product_prod_a_a,Y3: product_prod_a_a,Y5: set_Product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices2 @ Edges2 )
=> ( ( member1426531477525435216od_a_a @ X3 @ X5 )
=> ( ( member1426531477525435216od_a_a @ Y3 @ Y5 )
=> ( ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ ( insert4534936382041156343od_a_a @ Y3 @ bot_bo3357376287454694259od_a_a ) ) @ Edges2 )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X3 @ Y3 ) @ ( undire4032395788819567636od_a_a @ Edges2 @ X5 @ Y5 ) ) ) ) ) ) ).
% ulgraph.all_edges_betw_I
thf(fact_700_ulgraph_Oall__edges__betw__I,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,X3: set_a,X5: set_set_a,Y3: set_a,Y5: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices2 @ Edges2 )
=> ( ( member_set_a @ X3 @ X5 )
=> ( ( member_set_a @ Y3 @ Y5 )
=> ( ( member_set_set_a @ ( insert_set_a @ X3 @ ( insert_set_a @ Y3 @ bot_bot_set_set_a ) ) @ Edges2 )
=> ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X3 @ Y3 ) @ ( undire2462398226299384907_set_a @ Edges2 @ X5 @ Y5 ) ) ) ) ) ) ).
% ulgraph.all_edges_betw_I
thf(fact_701_ulgraph_Oall__edges__betw__I,axiom,
! [Vertices2: set_nat,Edges2: set_set_nat,X3: nat,X5: set_nat,Y3: nat,Y5: set_nat] :
( ( undire3269267262472140706ph_nat @ Vertices2 @ Edges2 )
=> ( ( member_nat @ X3 @ X5 )
=> ( ( member_nat @ Y3 @ Y5 )
=> ( ( member_set_nat @ ( insert_nat @ X3 @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) @ Edges2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( undire9168215380967949987en_nat @ Edges2 @ X5 @ Y5 ) ) ) ) ) ) ).
% ulgraph.all_edges_betw_I
thf(fact_702_ulgraph_Oall__edges__betw__I,axiom,
! [Vertices2: set_a,Edges2: set_set_a,X3: a,X5: set_a,Y3: a,Y5: set_a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ( member_a @ X3 @ X5 )
=> ( ( member_a @ Y3 @ Y5 )
=> ( ( member_set_a @ ( insert_a @ X3 @ ( insert_a @ Y3 @ bot_bot_set_a ) ) @ Edges2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( undire8383842906760478443ween_a @ Edges2 @ X5 @ Y5 ) ) ) ) ) ) ).
% ulgraph.all_edges_betw_I
thf(fact_703_ulgraph_Oall__edges__betw__D3,axiom,
! [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a,X3: product_prod_a_a,Y3: product_prod_a_a,X5: set_Product_prod_a_a,Y5: set_Product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices2 @ Edges2 )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X3 @ Y3 ) @ ( undire4032395788819567636od_a_a @ Edges2 @ X5 @ Y5 ) )
=> ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ ( insert4534936382041156343od_a_a @ Y3 @ bot_bo3357376287454694259od_a_a ) ) @ Edges2 ) ) ) ).
% ulgraph.all_edges_betw_D3
thf(fact_704_ulgraph_Oall__edges__betw__D3,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,X3: set_a,Y3: set_a,X5: set_set_a,Y5: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices2 @ Edges2 )
=> ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X3 @ Y3 ) @ ( undire2462398226299384907_set_a @ Edges2 @ X5 @ Y5 ) )
=> ( member_set_set_a @ ( insert_set_a @ X3 @ ( insert_set_a @ Y3 @ bot_bot_set_set_a ) ) @ Edges2 ) ) ) ).
% ulgraph.all_edges_betw_D3
thf(fact_705_ulgraph_Oall__edges__betw__D3,axiom,
! [Vertices2: set_nat,Edges2: set_set_nat,X3: nat,Y3: nat,X5: set_nat,Y5: set_nat] :
( ( undire3269267262472140706ph_nat @ Vertices2 @ Edges2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( undire9168215380967949987en_nat @ Edges2 @ X5 @ Y5 ) )
=> ( member_set_nat @ ( insert_nat @ X3 @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) @ Edges2 ) ) ) ).
% ulgraph.all_edges_betw_D3
thf(fact_706_ulgraph_Oall__edges__betw__D3,axiom,
! [Vertices2: set_a,Edges2: set_set_a,X3: a,Y3: a,X5: set_a,Y5: set_a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ ( undire8383842906760478443ween_a @ Edges2 @ X5 @ Y5 ) )
=> ( member_set_a @ ( insert_a @ X3 @ ( insert_a @ Y3 @ bot_bot_set_a ) ) @ Edges2 ) ) ) ).
% ulgraph.all_edges_betw_D3
thf(fact_707_ulgraph_Overt__adj__inc__edge__iff,axiom,
! [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a,V1: product_prod_a_a,V2: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices2 @ Edges2 )
=> ( ( undire6135774327024169009od_a_a @ Edges2 @ 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 ) ) @ Edges2 ) ) ) ) ).
% ulgraph.vert_adj_inc_edge_iff
thf(fact_708_ulgraph_Overt__adj__inc__edge__iff,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,V1: set_a,V2: set_a] :
( ( undire6886684016831807756_set_a @ Vertices2 @ Edges2 )
=> ( ( undire3510646817838285160_set_a @ Edges2 @ 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 ) ) @ Edges2 ) ) ) ) ).
% ulgraph.vert_adj_inc_edge_iff
thf(fact_709_ulgraph_Overt__adj__inc__edge__iff,axiom,
! [Vertices2: set_nat,Edges2: set_set_nat,V1: nat,V2: nat] :
( ( undire3269267262472140706ph_nat @ Vertices2 @ Edges2 )
=> ( ( undire1083030068171319366dj_nat @ Edges2 @ 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 ) ) @ Edges2 ) ) ) ) ).
% ulgraph.vert_adj_inc_edge_iff
thf(fact_710_ulgraph_Overt__adj__inc__edge__iff,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V1: a,V2: a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ( undire397441198561214472_adj_a @ Edges2 @ 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 ) ) @ Edges2 ) ) ) ) ).
% ulgraph.vert_adj_inc_edge_iff
thf(fact_711_graph__system_Oedge__adj__def,axiom,
! [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a,E1: set_Product_prod_a_a,E2: set_Product_prod_a_a] :
( ( undire1860116983885411791od_a_a @ Vertices2 @ Edges2 )
=> ( ( undire9186443406341554371od_a_a @ Edges2 @ E1 @ E2 )
= ( ( ( inf_in8905007599844390133od_a_a @ E1 @ E2 )
!= bot_bo3357376287454694259od_a_a )
& ( member1816616512716248880od_a_a @ E1 @ Edges2 )
& ( member1816616512716248880od_a_a @ E2 @ Edges2 ) ) ) ) ).
% graph_system.edge_adj_def
thf(fact_712_graph__system_Oedge__adj__def,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,E1: set_set_a,E2: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices2 @ Edges2 )
=> ( ( undire3485422320110889978_set_a @ Edges2 @ E1 @ E2 )
= ( ( ( inf_inf_set_set_a @ E1 @ E2 )
!= bot_bot_set_set_a )
& ( member_set_set_a @ E1 @ Edges2 )
& ( member_set_set_a @ E2 @ Edges2 ) ) ) ) ).
% graph_system.edge_adj_def
thf(fact_713_graph__system_Oedge__adj__def,axiom,
! [Vertices2: set_nat,Edges2: set_set_nat,E1: set_nat,E2: set_nat] :
( ( undire7481384412329822504em_nat @ Vertices2 @ Edges2 )
=> ( ( undire1664191744716346676dj_nat @ Edges2 @ E1 @ E2 )
= ( ( ( inf_inf_set_nat @ E1 @ E2 )
!= bot_bot_set_nat )
& ( member_set_nat @ E1 @ Edges2 )
& ( member_set_nat @ E2 @ Edges2 ) ) ) ) ).
% graph_system.edge_adj_def
thf(fact_714_graph__system_Oedge__adj__def,axiom,
! [Vertices2: set_a,Edges2: set_set_a,E1: set_a,E2: set_a] :
( ( undire2554140024507503526stem_a @ Vertices2 @ Edges2 )
=> ( ( undire4022703626023482010_adj_a @ Edges2 @ E1 @ E2 )
= ( ( ( inf_inf_set_a @ E1 @ E2 )
!= bot_bot_set_a )
& ( member_set_a @ E1 @ Edges2 )
& ( member_set_a @ E2 @ Edges2 ) ) ) ) ).
% graph_system.edge_adj_def
thf(fact_715_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_716_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_717_incident__sedges__empty,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire1270416042309875431dges_a @ edges @ V )
= bot_bot_set_set_a ) ) ).
% incident_sedges_empty
thf(fact_718_incident__edges__sedges,axiom,
! [V: a] :
( ~ ( undire3617971648856834880loop_a @ edges @ V )
=> ( ( undire3231912044278729248dges_a @ edges @ V )
= ( undire1270416042309875431dges_a @ edges @ V ) ) ) ).
% incident_edges_sedges
thf(fact_719_finite__incident__loops,axiom,
! [V: a] : ( finite_finite_set_a @ ( undire4753905205749729249oops_a @ edges @ V ) ) ).
% finite_incident_loops
thf(fact_720_finite__incident__edges,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( finite_finite_set_a @ ( undire3231912044278729248dges_a @ edges @ V ) ) ) ).
% finite_incident_edges
thf(fact_721_ulgraph_Oneighborhood__incident,axiom,
! [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a,U: product_prod_a_a,V: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices2 @ Edges2 )
=> ( ( member1426531477525435216od_a_a @ U @ ( undire7963753511165915895od_a_a @ Vertices2 @ Edges2 @ V ) )
= ( member1816616512716248880od_a_a @ ( insert4534936382041156343od_a_a @ U @ ( insert4534936382041156343od_a_a @ V @ bot_bo3357376287454694259od_a_a ) ) @ ( undire8905369280470868553od_a_a @ Edges2 @ V ) ) ) ) ).
% ulgraph.neighborhood_incident
thf(fact_722_ulgraph_Oneighborhood__incident,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,U: set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices2 @ Edges2 )
=> ( ( member_set_a @ U @ ( undire2074812191327625774_set_a @ Vertices2 @ Edges2 @ V ) )
= ( member_set_set_a @ ( insert_set_a @ U @ ( insert_set_a @ V @ bot_bot_set_set_a ) ) @ ( undire4631953023069350784_set_a @ Edges2 @ V ) ) ) ) ).
% ulgraph.neighborhood_incident
thf(fact_723_ulgraph_Oneighborhood__incident,axiom,
! [Vertices2: set_nat,Edges2: set_set_nat,U: nat,V: nat] :
( ( undire3269267262472140706ph_nat @ Vertices2 @ Edges2 )
=> ( ( member_nat @ U @ ( undire8190396521545869824od_nat @ Vertices2 @ Edges2 @ V ) )
= ( member_set_nat @ ( insert_nat @ U @ ( insert_nat @ V @ bot_bot_set_nat ) ) @ ( undire4176300566717384750es_nat @ Edges2 @ V ) ) ) ) ).
% ulgraph.neighborhood_incident
thf(fact_724_ulgraph_Oneighborhood__incident,axiom,
! [Vertices2: set_a,Edges2: set_set_a,U: a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ( member_a @ U @ ( undire8504279938402040014hood_a @ Vertices2 @ Edges2 @ V ) )
= ( member_set_a @ ( insert_a @ U @ ( insert_a @ V @ bot_bot_set_a ) ) @ ( undire3231912044278729248dges_a @ Edges2 @ V ) ) ) ) ).
% ulgraph.neighborhood_incident
thf(fact_725_old_Oprod_Oinject,axiom,
! [A2: a,B3: a,A6: a,B7: a] :
( ( ( product_Pair_a_a @ A2 @ B3 )
= ( product_Pair_a_a @ A6 @ B7 ) )
= ( ( A2 = A6 )
& ( B3 = B7 ) ) ) ).
% old.prod.inject
thf(fact_726_prod_Oinject,axiom,
! [X1: a,X22: a,Y1: a,Y22: a] :
( ( ( product_Pair_a_a @ X1 @ X22 )
= ( product_Pair_a_a @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_727_finite__inc__sedges,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( finite_finite_set_a @ ( undire1270416042309875431dges_a @ edges @ V ) ) ) ).
% finite_inc_sedges
thf(fact_728_ulgraph_Oneighborhood_Ocong,axiom,
undire8504279938402040014hood_a = undire8504279938402040014hood_a ).
% ulgraph.neighborhood.cong
thf(fact_729_ulgraph_Oincident__sedges_Ocong,axiom,
undire1270416042309875431dges_a = undire1270416042309875431dges_a ).
% ulgraph.incident_sedges.cong
thf(fact_730_ulgraph_Ofinite__inc__sedges,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ( finite_finite_set_a @ Edges2 )
=> ( finite_finite_set_a @ ( undire1270416042309875431dges_a @ Edges2 @ V ) ) ) ) ).
% ulgraph.finite_inc_sedges
thf(fact_731_subrelI,axiom,
! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ! [X2: a,Y: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ R )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ S ) )
=> ( ord_le746702958409616551od_a_a @ R @ S ) ) ).
% subrelI
thf(fact_732_ulgraph_Ofinite__incident__loops,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( finite_finite_set_a @ ( undire4753905205749729249oops_a @ Edges2 @ V ) ) ) ).
% ulgraph.finite_incident_loops
thf(fact_733_graph__system_Ofinite__incident__edges,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V: a] :
( ( undire2554140024507503526stem_a @ Vertices2 @ Edges2 )
=> ( ( finite_finite_set_a @ Edges2 )
=> ( finite_finite_set_a @ ( undire3231912044278729248dges_a @ Edges2 @ V ) ) ) ) ).
% graph_system.finite_incident_edges
thf(fact_734_old_Oprod_Oexhaust,axiom,
! [Y3: product_prod_a_a] :
~ ! [A5: a,B6: a] :
( Y3
!= ( product_Pair_a_a @ A5 @ B6 ) ) ).
% old.prod.exhaust
thf(fact_735_surj__pair,axiom,
! [P2: product_prod_a_a] :
? [X2: a,Y: a] :
( P2
= ( product_Pair_a_a @ X2 @ Y ) ) ).
% surj_pair
thf(fact_736_prod__cases,axiom,
! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
( ! [A5: a,B6: a] : ( P @ ( product_Pair_a_a @ A5 @ B6 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_737_Pair__inject,axiom,
! [A2: a,B3: a,A6: a,B7: a] :
( ( ( product_Pair_a_a @ A2 @ B3 )
= ( product_Pair_a_a @ A6 @ B7 ) )
=> ~ ( ( A2 = A6 )
=> ( B3 != B7 ) ) ) ).
% Pair_inject
thf(fact_738_ulgraph_Oincident__sedges__empty,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices2 @ Edges2 )
=> ( ~ ( member_set_a @ V @ Vertices2 )
=> ( ( undire5844230293943614535_set_a @ Edges2 @ V )
= bot_bo3380559777022489994_set_a ) ) ) ).
% ulgraph.incident_sedges_empty
thf(fact_739_ulgraph_Oincident__sedges__empty,axiom,
! [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a,V: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices2 @ Edges2 )
=> ( ~ ( member1426531477525435216od_a_a @ V @ Vertices2 )
=> ( ( undire1583524423955984400od_a_a @ Edges2 @ V )
= bot_bo777872063958040403od_a_a ) ) ) ).
% ulgraph.incident_sedges_empty
thf(fact_740_ulgraph_Oincident__sedges__empty,axiom,
! [Vertices2: set_nat,Edges2: set_set_nat,V: nat] :
( ( undire3269267262472140706ph_nat @ Vertices2 @ Edges2 )
=> ( ~ ( member_nat @ V @ Vertices2 )
=> ( ( undire996053960663353255es_nat @ Edges2 @ V )
= bot_bot_set_set_nat ) ) ) ).
% ulgraph.incident_sedges_empty
thf(fact_741_ulgraph_Oincident__sedges__empty,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ~ ( member_a @ V @ Vertices2 )
=> ( ( undire1270416042309875431dges_a @ Edges2 @ V )
= bot_bot_set_set_a ) ) ) ).
% ulgraph.incident_sedges_empty
thf(fact_742_ulgraph_Oiso__vertex__empty__neighborhood,axiom,
! [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a,V: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices2 @ Edges2 )
=> ( ( undire3207556238582723646od_a_a @ Vertices2 @ Edges2 @ V )
=> ( ( undire7963753511165915895od_a_a @ Vertices2 @ Edges2 @ V )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% ulgraph.iso_vertex_empty_neighborhood
thf(fact_743_ulgraph_Oiso__vertex__empty__neighborhood,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices2 @ Edges2 )
=> ( ( undire6879241558604981877_set_a @ Vertices2 @ Edges2 @ V )
=> ( ( undire2074812191327625774_set_a @ Vertices2 @ Edges2 @ V )
= bot_bot_set_set_a ) ) ) ).
% ulgraph.iso_vertex_empty_neighborhood
thf(fact_744_ulgraph_Oiso__vertex__empty__neighborhood,axiom,
! [Vertices2: set_nat,Edges2: set_set_nat,V: nat] :
( ( undire3269267262472140706ph_nat @ Vertices2 @ Edges2 )
=> ( ( undire5609513041723151865ex_nat @ Vertices2 @ Edges2 @ V )
=> ( ( undire8190396521545869824od_nat @ Vertices2 @ Edges2 @ V )
= bot_bot_set_nat ) ) ) ).
% ulgraph.iso_vertex_empty_neighborhood
thf(fact_745_ulgraph_Oiso__vertex__empty__neighborhood,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ( undire8931668460104145173rtex_a @ Vertices2 @ Edges2 @ V )
=> ( ( undire8504279938402040014hood_a @ Vertices2 @ Edges2 @ V )
= bot_bot_set_a ) ) ) ).
% ulgraph.iso_vertex_empty_neighborhood
thf(fact_746_ulgraph_Oincident__edges__sedges,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ~ ( undire3617971648856834880loop_a @ Edges2 @ V )
=> ( ( undire3231912044278729248dges_a @ Edges2 @ V )
= ( undire1270416042309875431dges_a @ Edges2 @ V ) ) ) ) ).
% ulgraph.incident_edges_sedges
thf(fact_747_finite__Int,axiom,
! [F2: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_748_finite__Int,axiom,
! [F2: set_a,G: set_a] :
( ( ( finite_finite_a @ F2 )
| ( finite_finite_a @ G ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_749_finite__Int,axiom,
! [F2: set_set_a,G: set_set_a] :
( ( ( finite_finite_set_a @ F2 )
| ( finite_finite_set_a @ G ) )
=> ( finite_finite_set_a @ ( inf_inf_set_set_a @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_750_finite__insert,axiom,
! [A2: a,A: set_a] :
( ( finite_finite_a @ ( insert_a @ A2 @ A ) )
= ( finite_finite_a @ A ) ) ).
% finite_insert
thf(fact_751_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_752_finite__insert,axiom,
! [A2: nat,A: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A2 @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_insert
thf(fact_753_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_754_finite__subset__induct_H,axiom,
! [F2: set_Product_prod_a_a,A: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ F2 )
=> ( ( ord_le746702958409616551od_a_a @ F2 @ A )
=> ( ( P @ bot_bo3357376287454694259od_a_a )
=> ( ! [A5: product_prod_a_a,F3: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F3 )
=> ( ( member1426531477525435216od_a_a @ A5 @ A )
=> ( ( ord_le746702958409616551od_a_a @ F3 @ A )
=> ( ~ ( member1426531477525435216od_a_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert4534936382041156343od_a_a @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_755_finite__subset__induct_H,axiom,
! [F2: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A5: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A5 @ A )
=> ( ( ord_less_eq_set_nat @ F3 @ A )
=> ( ~ ( member_nat @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_756_finite__subset__induct_H,axiom,
! [F2: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A5 @ A )
=> ( ( ord_less_eq_set_a @ F3 @ A )
=> ( ~ ( member_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_757_finite__subset__induct_H,axiom,
! [F2: set_set_a,A: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ F2 @ A )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A5: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A5 @ A )
=> ( ( ord_le3724670747650509150_set_a @ F3 @ A )
=> ( ~ ( member_set_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_758_finite__subset__induct,axiom,
! [F2: set_Product_prod_a_a,A: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ F2 )
=> ( ( ord_le746702958409616551od_a_a @ F2 @ A )
=> ( ( P @ bot_bo3357376287454694259od_a_a )
=> ( ! [A5: product_prod_a_a,F3: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F3 )
=> ( ( member1426531477525435216od_a_a @ A5 @ A )
=> ( ~ ( member1426531477525435216od_a_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert4534936382041156343od_a_a @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_759_finite__subset__induct,axiom,
! [F2: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A5: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A5 @ A )
=> ( ~ ( member_nat @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_760_finite__subset__induct,axiom,
! [F2: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A5 @ A )
=> ( ~ ( member_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_761_finite__subset__induct,axiom,
! [F2: set_set_a,A: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ F2 @ A )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A5: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A5 @ A )
=> ( ~ ( member_set_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_762_finite__ranking__induct,axiom,
! [S2: set_a,P: set_a > $o,F: a > nat] :
( ( finite_finite_a @ S2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,S3: set_a] :
( ( finite_finite_a @ S3 )
=> ( ! [Y7: a] :
( ( member_a @ Y7 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y7 ) @ ( F @ X2 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_a @ X2 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_763_finite__ranking__induct,axiom,
! [S2: set_Product_prod_a_a,P: set_Product_prod_a_a > $o,F: product_prod_a_a > nat] :
( ( finite6544458595007987280od_a_a @ S2 )
=> ( ( P @ bot_bo3357376287454694259od_a_a )
=> ( ! [X2: product_prod_a_a,S3: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ S3 )
=> ( ! [Y7: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y7 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y7 ) @ ( F @ X2 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert4534936382041156343od_a_a @ X2 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_764_finite__ranking__induct,axiom,
! [S2: set_set_a,P: set_set_a > $o,F: set_a > nat] :
( ( finite_finite_set_a @ S2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X2: set_a,S3: set_set_a] :
( ( finite_finite_set_a @ S3 )
=> ( ! [Y7: set_a] :
( ( member_set_a @ Y7 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y7 ) @ ( F @ X2 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_set_a @ X2 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_765_finite__ranking__induct,axiom,
! [S2: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [Y7: nat] :
( ( member_nat @ Y7 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y7 ) @ ( F @ X2 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_nat @ X2 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_766_sup_Oidem,axiom,
! [A2: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_767_sup__idem,axiom,
! [X3: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_768_sup_Oleft__idem,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ ( sup_sup_set_set_a @ A2 @ B3 ) )
= ( sup_sup_set_set_a @ A2 @ B3 ) ) ).
% sup.left_idem
thf(fact_769_sup__left__idem,axiom,
! [X3: set_set_a,Y3: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ ( sup_sup_set_set_a @ X3 @ Y3 ) )
= ( sup_sup_set_set_a @ X3 @ Y3 ) ) ).
% sup_left_idem
thf(fact_770_sup_Oright__idem,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A2 @ B3 ) @ B3 )
= ( sup_sup_set_set_a @ A2 @ B3 ) ) ).
% sup.right_idem
thf(fact_771_UnCI,axiom,
! [C: a,B: set_a,A: set_a] :
( ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ A ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_772_UnCI,axiom,
! [C: product_prod_a_a,B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ~ ( member1426531477525435216od_a_a @ C @ B )
=> ( member1426531477525435216od_a_a @ C @ A ) )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B ) ) ) ).
% UnCI
thf(fact_773_UnCI,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ A ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_774_UnCI,axiom,
! [C: set_a,B: set_set_a,A: set_set_a] :
( ( ~ ( member_set_a @ C @ B )
=> ( member_set_a @ C @ A ) )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_775_Un__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
| ( member_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_776_Un__iff,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B ) )
= ( ( member1426531477525435216od_a_a @ C @ A )
| ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_777_Un__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
| ( member_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_778_Un__iff,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) )
= ( ( member_set_a @ C @ A )
| ( member_set_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_779_le__sup__iff,axiom,
! [X3: set_a,Y3: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X3 @ Y3 ) @ Z )
= ( ( ord_less_eq_set_a @ X3 @ Z )
& ( ord_less_eq_set_a @ Y3 @ Z ) ) ) ).
% le_sup_iff
thf(fact_780_le__sup__iff,axiom,
! [X3: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ Y3 ) @ Z )
= ( ( ord_less_eq_nat @ X3 @ Z )
& ( ord_less_eq_nat @ Y3 @ Z ) ) ) ).
% le_sup_iff
thf(fact_781_le__sup__iff,axiom,
! [X3: set_set_a,Y3: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ X3 @ Y3 ) @ Z )
= ( ( ord_le3724670747650509150_set_a @ X3 @ Z )
& ( ord_le3724670747650509150_set_a @ Y3 @ Z ) ) ) ).
% le_sup_iff
thf(fact_782_sup_Obounded__iff,axiom,
! [B3: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B3 @ C ) @ A2 )
= ( ( ord_less_eq_set_a @ B3 @ A2 )
& ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_783_sup_Obounded__iff,axiom,
! [B3: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B3 @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_784_sup_Obounded__iff,axiom,
! [B3: set_set_a,C: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B3 @ C ) @ A2 )
= ( ( ord_le3724670747650509150_set_a @ B3 @ A2 )
& ( ord_le3724670747650509150_set_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_785_sup__bot__left,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_786_sup__bot__left,axiom,
! [X3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ bot_bo3357376287454694259od_a_a @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_787_sup__bot__left,axiom,
! [X3: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_788_sup__bot__left,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_789_sup__bot__right,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ X3 @ bot_bot_set_a )
= X3 ) ).
% sup_bot_right
thf(fact_790_sup__bot__right,axiom,
! [X3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X3 @ bot_bo3357376287454694259od_a_a )
= X3 ) ).
% sup_bot_right
thf(fact_791_sup__bot__right,axiom,
! [X3: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ bot_bot_set_set_a )
= X3 ) ).
% sup_bot_right
thf(fact_792_sup__bot__right,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ bot_bot_set_nat )
= X3 ) ).
% sup_bot_right
thf(fact_793_bot__eq__sup__iff,axiom,
! [X3: set_a,Y3: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X3 @ Y3 ) )
= ( ( X3 = bot_bot_set_a )
& ( Y3 = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_794_bot__eq__sup__iff,axiom,
! [X3: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( sup_su3048258781599657691od_a_a @ X3 @ Y3 ) )
= ( ( X3 = bot_bo3357376287454694259od_a_a )
& ( Y3 = bot_bo3357376287454694259od_a_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_795_bot__eq__sup__iff,axiom,
! [X3: set_set_a,Y3: set_set_a] :
( ( bot_bot_set_set_a
= ( sup_sup_set_set_a @ X3 @ Y3 ) )
= ( ( X3 = bot_bot_set_set_a )
& ( Y3 = bot_bot_set_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_796_bot__eq__sup__iff,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X3 @ Y3 ) )
= ( ( X3 = bot_bot_set_nat )
& ( Y3 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_797_sup__eq__bot__iff,axiom,
! [X3: set_a,Y3: set_a] :
( ( ( sup_sup_set_a @ X3 @ Y3 )
= bot_bot_set_a )
= ( ( X3 = bot_bot_set_a )
& ( Y3 = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_798_sup__eq__bot__iff,axiom,
! [X3: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( ( sup_su3048258781599657691od_a_a @ X3 @ Y3 )
= bot_bo3357376287454694259od_a_a )
= ( ( X3 = bot_bo3357376287454694259od_a_a )
& ( Y3 = bot_bo3357376287454694259od_a_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_799_sup__eq__bot__iff,axiom,
! [X3: set_set_a,Y3: set_set_a] :
( ( ( sup_sup_set_set_a @ X3 @ Y3 )
= bot_bot_set_set_a )
= ( ( X3 = bot_bot_set_set_a )
& ( Y3 = bot_bot_set_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_800_sup__eq__bot__iff,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( ( sup_sup_set_nat @ X3 @ Y3 )
= bot_bot_set_nat )
= ( ( X3 = bot_bot_set_nat )
& ( Y3 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_801_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_a,B3: set_a] :
( ( ( sup_sup_set_a @ A2 @ B3 )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B3 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_802_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( ( sup_su3048258781599657691od_a_a @ A2 @ B3 )
= bot_bo3357376287454694259od_a_a )
= ( ( A2 = bot_bo3357376287454694259od_a_a )
& ( B3 = bot_bo3357376287454694259od_a_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_803_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( ( sup_sup_set_set_a @ A2 @ B3 )
= bot_bot_set_set_a )
= ( ( A2 = bot_bot_set_set_a )
& ( B3 = bot_bot_set_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_804_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B3 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B3 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_805_sup__bot_Oleft__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_806_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_807_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_808_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_809_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_a,B3: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A2 @ B3 ) )
= ( ( A2 = bot_bot_set_a )
& ( B3 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_810_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( sup_su3048258781599657691od_a_a @ A2 @ B3 ) )
= ( ( A2 = bot_bo3357376287454694259od_a_a )
& ( B3 = bot_bo3357376287454694259od_a_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_811_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( bot_bot_set_set_a
= ( sup_sup_set_set_a @ A2 @ B3 ) )
= ( ( A2 = bot_bot_set_set_a )
& ( B3 = bot_bot_set_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_812_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat,B3: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A2 @ B3 ) )
= ( ( A2 = bot_bot_set_nat )
& ( B3 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_813_sup__bot_Oright__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_814_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_815_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_816_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_817_sup__inf__absorb,axiom,
! [X3: set_a,Y3: set_a] :
( ( sup_sup_set_a @ X3 @ ( inf_inf_set_a @ X3 @ Y3 ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_818_sup__inf__absorb,axiom,
! [X3: set_set_a,Y3: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ ( inf_inf_set_set_a @ X3 @ Y3 ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_819_inf__sup__absorb,axiom,
! [X3: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y3 ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_820_inf__sup__absorb,axiom,
! [X3: set_set_a,Y3: set_set_a] :
( ( inf_inf_set_set_a @ X3 @ ( sup_sup_set_set_a @ X3 @ Y3 ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_821_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_822_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_823_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_824_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_825_finite__Un,axiom,
! [F2: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) )
= ( ( finite_finite_nat @ F2 )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_826_finite__Un,axiom,
! [F2: set_set_a,G: set_set_a] :
( ( finite_finite_set_a @ ( sup_sup_set_set_a @ F2 @ G ) )
= ( ( finite_finite_set_a @ F2 )
& ( finite_finite_set_a @ G ) ) ) ).
% finite_Un
thf(fact_827_Un__subset__iff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_a @ A @ C2 )
& ( ord_less_eq_set_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_828_Un__subset__iff,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A @ B ) @ C2 )
= ( ( ord_le3724670747650509150_set_a @ A @ C2 )
& ( ord_le3724670747650509150_set_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_829_Un__insert__left,axiom,
! [A2: a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_830_Un__insert__left,axiom,
! [A2: set_a,B: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ ( insert_set_a @ A2 @ B ) @ C2 )
= ( insert_set_a @ A2 @ ( sup_sup_set_set_a @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_831_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_832_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_833_Un__Int__eq_I1_J,axiom,
! [S2: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S2 @ T2 ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_834_Un__Int__eq_I1_J,axiom,
! [S2: set_set_a,T2: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ S2 @ T2 ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_835_Un__Int__eq_I2_J,axiom,
! [S2: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S2 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_836_Un__Int__eq_I2_J,axiom,
! [S2: set_set_a,T2: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ S2 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_837_Un__Int__eq_I3_J,axiom,
! [S2: set_a,T2: set_a] :
( ( inf_inf_set_a @ S2 @ ( sup_sup_set_a @ S2 @ T2 ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_838_Un__Int__eq_I3_J,axiom,
! [S2: set_set_a,T2: set_set_a] :
( ( inf_inf_set_set_a @ S2 @ ( sup_sup_set_set_a @ S2 @ T2 ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_839_Un__Int__eq_I4_J,axiom,
! [T2: set_a,S2: set_a] :
( ( inf_inf_set_a @ T2 @ ( sup_sup_set_a @ S2 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_840_Un__Int__eq_I4_J,axiom,
! [T2: set_set_a,S2: set_set_a] :
( ( inf_inf_set_set_a @ T2 @ ( sup_sup_set_set_a @ S2 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_841_Int__Un__eq_I1_J,axiom,
! [S2: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S2 @ T2 ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_842_Int__Un__eq_I1_J,axiom,
! [S2: set_set_a,T2: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ S2 @ T2 ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_843_Int__Un__eq_I2_J,axiom,
! [S2: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S2 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_844_Int__Un__eq_I2_J,axiom,
! [S2: set_set_a,T2: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ S2 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_845_Int__Un__eq_I3_J,axiom,
! [S2: set_a,T2: set_a] :
( ( sup_sup_set_a @ S2 @ ( inf_inf_set_a @ S2 @ T2 ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_846_Int__Un__eq_I3_J,axiom,
! [S2: set_set_a,T2: set_set_a] :
( ( sup_sup_set_set_a @ S2 @ ( inf_inf_set_set_a @ S2 @ T2 ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_847_Int__Un__eq_I4_J,axiom,
! [T2: set_a,S2: set_a] :
( ( sup_sup_set_a @ T2 @ ( inf_inf_set_a @ S2 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_848_Int__Un__eq_I4_J,axiom,
! [T2: set_set_a,S2: set_set_a] :
( ( sup_sup_set_set_a @ T2 @ ( inf_inf_set_set_a @ S2 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_849_finite__UnI,axiom,
! [F2: set_nat,G: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( finite_finite_nat @ G )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_850_finite__UnI,axiom,
! [F2: set_set_a,G: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ( finite_finite_set_a @ G )
=> ( finite_finite_set_a @ ( sup_sup_set_set_a @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_851_Un__infinite,axiom,
! [S2: set_nat,T2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S2 @ T2 ) ) ) ).
% Un_infinite
thf(fact_852_Un__infinite,axiom,
! [S2: set_set_a,T2: set_set_a] :
( ~ ( finite_finite_set_a @ S2 )
=> ~ ( finite_finite_set_a @ ( sup_sup_set_set_a @ S2 @ T2 ) ) ) ).
% Un_infinite
thf(fact_853_infinite__Un,axiom,
! [S2: set_nat,T2: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S2 @ T2 ) ) )
= ( ~ ( finite_finite_nat @ S2 )
| ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_854_infinite__Un,axiom,
! [S2: set_set_a,T2: set_set_a] :
( ( ~ ( finite_finite_set_a @ ( sup_sup_set_set_a @ S2 @ T2 ) ) )
= ( ~ ( finite_finite_set_a @ S2 )
| ~ ( finite_finite_set_a @ T2 ) ) ) ).
% infinite_Un
thf(fact_855_UnE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
=> ( ~ ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% UnE
thf(fact_856_UnE,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B ) )
=> ( ~ ( member1426531477525435216od_a_a @ C @ A )
=> ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% UnE
thf(fact_857_UnE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
=> ( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% UnE
thf(fact_858_UnE,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) )
=> ( ~ ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% UnE
thf(fact_859_UnI1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_860_UnI1,axiom,
! [C: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B ) ) ) ).
% UnI1
thf(fact_861_UnI1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_862_UnI1,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_863_UnI2,axiom,
! [C: a,B: set_a,A: set_a] :
( ( member_a @ C @ B )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_864_UnI2,axiom,
! [C: product_prod_a_a,B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ B )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B ) ) ) ).
% UnI2
thf(fact_865_UnI2,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_866_UnI2,axiom,
! [C: set_a,B: set_set_a,A: set_set_a] :
( ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_867_bex__Un,axiom,
! [A: set_set_a,B: set_set_a,P: set_a > $o] :
( ( ? [X: set_a] :
( ( member_set_a @ X @ ( sup_sup_set_set_a @ A @ B ) )
& ( P @ X ) ) )
= ( ? [X: set_a] :
( ( member_set_a @ X @ A )
& ( P @ X ) )
| ? [X: set_a] :
( ( member_set_a @ X @ B )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_868_ball__Un,axiom,
! [A: set_set_a,B: set_set_a,P: set_a > $o] :
( ( ! [X: set_a] :
( ( member_set_a @ X @ ( sup_sup_set_set_a @ A @ B ) )
=> ( P @ X ) ) )
= ( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( P @ X ) )
& ! [X: set_a] :
( ( member_set_a @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_869_Un__assoc,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A @ B ) @ C2 )
= ( sup_sup_set_set_a @ A @ ( sup_sup_set_set_a @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_870_Un__absorb,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_871_Un__commute,axiom,
( sup_sup_set_set_a
= ( ^ [A3: set_set_a,B2: set_set_a] : ( sup_sup_set_set_a @ B2 @ A3 ) ) ) ).
% Un_commute
thf(fact_872_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_873_Un__left__commute,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ A @ ( sup_sup_set_set_a @ B @ C2 ) )
= ( sup_sup_set_set_a @ B @ ( sup_sup_set_set_a @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_874_inf__sup__aci_I8_J,axiom,
! [X3: set_set_a,Y3: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ ( sup_sup_set_set_a @ X3 @ Y3 ) )
= ( sup_sup_set_set_a @ X3 @ Y3 ) ) ).
% inf_sup_aci(8)
thf(fact_875_inf__sup__aci_I7_J,axiom,
! [X3: set_set_a,Y3: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ ( sup_sup_set_set_a @ Y3 @ Z ) )
= ( sup_sup_set_set_a @ Y3 @ ( sup_sup_set_set_a @ X3 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_876_inf__sup__aci_I6_J,axiom,
! [X3: set_set_a,Y3: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ X3 @ Y3 ) @ Z )
= ( sup_sup_set_set_a @ X3 @ ( sup_sup_set_set_a @ Y3 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_877_inf__sup__aci_I5_J,axiom,
( sup_sup_set_set_a
= ( ^ [X: set_set_a,Y2: set_set_a] : ( sup_sup_set_set_a @ Y2 @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_878_sup_Oassoc,axiom,
! [A2: set_set_a,B3: set_set_a,C: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A2 @ B3 ) @ C )
= ( sup_sup_set_set_a @ A2 @ ( sup_sup_set_set_a @ B3 @ C ) ) ) ).
% sup.assoc
thf(fact_879_sup__assoc,axiom,
! [X3: set_set_a,Y3: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ X3 @ Y3 ) @ Z )
= ( sup_sup_set_set_a @ X3 @ ( sup_sup_set_set_a @ Y3 @ Z ) ) ) ).
% sup_assoc
thf(fact_880_sup_Ocommute,axiom,
( sup_sup_set_set_a
= ( ^ [A4: set_set_a,B5: set_set_a] : ( sup_sup_set_set_a @ B5 @ A4 ) ) ) ).
% sup.commute
thf(fact_881_sup__commute,axiom,
( sup_sup_set_set_a
= ( ^ [X: set_set_a,Y2: set_set_a] : ( sup_sup_set_set_a @ Y2 @ X ) ) ) ).
% sup_commute
thf(fact_882_boolean__algebra__cancel_Osup1,axiom,
! [A: set_set_a,K: set_set_a,A2: set_set_a,B3: set_set_a] :
( ( A
= ( sup_sup_set_set_a @ K @ A2 ) )
=> ( ( sup_sup_set_set_a @ A @ B3 )
= ( sup_sup_set_set_a @ K @ ( sup_sup_set_set_a @ A2 @ B3 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_883_boolean__algebra__cancel_Osup2,axiom,
! [B: set_set_a,K: set_set_a,B3: set_set_a,A2: set_set_a] :
( ( B
= ( sup_sup_set_set_a @ K @ B3 ) )
=> ( ( sup_sup_set_set_a @ A2 @ B )
= ( sup_sup_set_set_a @ K @ ( sup_sup_set_set_a @ A2 @ B3 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_884_sup_Oleft__commute,axiom,
! [B3: set_set_a,A2: set_set_a,C: set_set_a] :
( ( sup_sup_set_set_a @ B3 @ ( sup_sup_set_set_a @ A2 @ C ) )
= ( sup_sup_set_set_a @ A2 @ ( sup_sup_set_set_a @ B3 @ C ) ) ) ).
% sup.left_commute
thf(fact_885_sup__left__commute,axiom,
! [X3: set_set_a,Y3: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ ( sup_sup_set_set_a @ Y3 @ Z ) )
= ( sup_sup_set_set_a @ Y3 @ ( sup_sup_set_set_a @ X3 @ Z ) ) ) ).
% sup_left_commute
thf(fact_886_inf__sup__ord_I4_J,axiom,
! [Y3: set_a,X3: set_a] : ( ord_less_eq_set_a @ Y3 @ ( sup_sup_set_a @ X3 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_887_inf__sup__ord_I4_J,axiom,
! [Y3: nat,X3: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X3 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_888_inf__sup__ord_I4_J,axiom,
! [Y3: set_set_a,X3: set_set_a] : ( ord_le3724670747650509150_set_a @ Y3 @ ( sup_sup_set_set_a @ X3 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_889_inf__sup__ord_I3_J,axiom,
! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_890_inf__sup__ord_I3_J,axiom,
! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_891_inf__sup__ord_I3_J,axiom,
! [X3: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ X3 @ ( sup_sup_set_set_a @ X3 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_892_le__supE,axiom,
! [A2: set_a,B3: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B3 ) @ X3 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ X3 )
=> ~ ( ord_less_eq_set_a @ B3 @ X3 ) ) ) ).
% le_supE
thf(fact_893_le__supE,axiom,
! [A2: nat,B3: nat,X3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ X3 )
=> ~ ( ( ord_less_eq_nat @ A2 @ X3 )
=> ~ ( ord_less_eq_nat @ B3 @ X3 ) ) ) ).
% le_supE
thf(fact_894_le__supE,axiom,
! [A2: set_set_a,B3: set_set_a,X3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B3 ) @ X3 )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ X3 )
=> ~ ( ord_le3724670747650509150_set_a @ B3 @ X3 ) ) ) ).
% le_supE
thf(fact_895_le__supI,axiom,
! [A2: set_a,X3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ X3 )
=> ( ( ord_less_eq_set_a @ B3 @ X3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B3 ) @ X3 ) ) ) ).
% le_supI
thf(fact_896_le__supI,axiom,
! [A2: nat,X3: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ X3 )
=> ( ( ord_less_eq_nat @ B3 @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ X3 ) ) ) ).
% le_supI
thf(fact_897_le__supI,axiom,
! [A2: set_set_a,X3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ X3 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ X3 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B3 ) @ X3 ) ) ) ).
% le_supI
thf(fact_898_sup__ge1,axiom,
! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y3 ) ) ).
% sup_ge1
thf(fact_899_sup__ge1,axiom,
! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y3 ) ) ).
% sup_ge1
thf(fact_900_sup__ge1,axiom,
! [X3: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ X3 @ ( sup_sup_set_set_a @ X3 @ Y3 ) ) ).
% sup_ge1
thf(fact_901_sup__ge2,axiom,
! [Y3: set_a,X3: set_a] : ( ord_less_eq_set_a @ Y3 @ ( sup_sup_set_a @ X3 @ Y3 ) ) ).
% sup_ge2
thf(fact_902_sup__ge2,axiom,
! [Y3: nat,X3: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X3 @ Y3 ) ) ).
% sup_ge2
thf(fact_903_sup__ge2,axiom,
! [Y3: set_set_a,X3: set_set_a] : ( ord_le3724670747650509150_set_a @ Y3 @ ( sup_sup_set_set_a @ X3 @ Y3 ) ) ).
% sup_ge2
thf(fact_904_le__supI1,axiom,
! [X3: set_a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ X3 @ ( sup_sup_set_a @ A2 @ B3 ) ) ) ).
% le_supI1
thf(fact_905_le__supI1,axiom,
! [X3: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% le_supI1
thf(fact_906_le__supI1,axiom,
! [X3: set_set_a,A2: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ A2 )
=> ( ord_le3724670747650509150_set_a @ X3 @ ( sup_sup_set_set_a @ A2 @ B3 ) ) ) ).
% le_supI1
thf(fact_907_le__supI2,axiom,
! [X3: set_a,B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ X3 @ B3 )
=> ( ord_less_eq_set_a @ X3 @ ( sup_sup_set_a @ A2 @ B3 ) ) ) ).
% le_supI2
thf(fact_908_le__supI2,axiom,
! [X3: nat,B3: nat,A2: nat] :
( ( ord_less_eq_nat @ X3 @ B3 )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% le_supI2
thf(fact_909_le__supI2,axiom,
! [X3: set_set_a,B3: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ B3 )
=> ( ord_le3724670747650509150_set_a @ X3 @ ( sup_sup_set_set_a @ A2 @ B3 ) ) ) ).
% le_supI2
thf(fact_910_sup_Omono,axiom,
! [C: set_a,A2: set_a,D: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ D @ B3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D ) @ ( sup_sup_set_a @ A2 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_911_sup_Omono,axiom,
! [C: nat,A2: nat,D: nat,B3: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D @ B3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A2 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_912_sup_Omono,axiom,
! [C: set_set_a,A2: set_set_a,D: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ D @ B3 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ C @ D ) @ ( sup_sup_set_set_a @ A2 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_913_sup__mono,axiom,
! [A2: set_a,C: set_a,B3: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B3 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B3 ) @ ( sup_sup_set_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_914_sup__mono,axiom,
! [A2: nat,C: nat,B3: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B3 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_915_sup__mono,axiom,
! [A2: set_set_a,C: set_set_a,B3: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ D )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B3 ) @ ( sup_sup_set_set_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_916_sup__least,axiom,
! [Y3: set_a,X3: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X3 )
=> ( ( ord_less_eq_set_a @ Z @ X3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y3 @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_917_sup__least,axiom,
! [Y3: nat,X3: nat,Z: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_nat @ Z @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y3 @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_918_sup__least,axiom,
! [Y3: set_set_a,X3: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y3 @ X3 )
=> ( ( ord_le3724670747650509150_set_a @ Z @ X3 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ Y3 @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_919_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X: set_a,Y2: set_a] :
( ( sup_sup_set_a @ X @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_920_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y2: nat] :
( ( sup_sup_nat @ X @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_921_le__iff__sup,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X: set_set_a,Y2: set_set_a] :
( ( sup_sup_set_set_a @ X @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_922_sup_OorderE,axiom,
! [B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( A2
= ( sup_sup_set_a @ A2 @ B3 ) ) ) ).
% sup.orderE
thf(fact_923_sup_OorderE,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% sup.orderE
thf(fact_924_sup_OorderE,axiom,
! [B3: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A2 )
=> ( A2
= ( sup_sup_set_set_a @ A2 @ B3 ) ) ) ).
% sup.orderE
thf(fact_925_sup_OorderI,axiom,
! [A2: set_a,B3: set_a] :
( ( A2
= ( sup_sup_set_a @ A2 @ B3 ) )
=> ( ord_less_eq_set_a @ B3 @ A2 ) ) ).
% sup.orderI
thf(fact_926_sup_OorderI,axiom,
! [A2: nat,B3: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B3 ) )
=> ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% sup.orderI
thf(fact_927_sup_OorderI,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( A2
= ( sup_sup_set_set_a @ A2 @ B3 ) )
=> ( ord_le3724670747650509150_set_a @ B3 @ A2 ) ) ).
% sup.orderI
thf(fact_928_sup__unique,axiom,
! [F: set_a > set_a > set_a,X3: set_a,Y3: set_a] :
( ! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ X2 @ ( F @ X2 @ Y ) )
=> ( ! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ Y @ ( F @ X2 @ Y ) )
=> ( ! [X2: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ( ( ord_less_eq_set_a @ Z2 @ X2 )
=> ( ord_less_eq_set_a @ ( F @ Y @ Z2 ) @ X2 ) ) )
=> ( ( sup_sup_set_a @ X3 @ Y3 )
= ( F @ X3 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_929_sup__unique,axiom,
! [F: nat > nat > nat,X3: nat,Y3: nat] :
( ! [X2: nat,Y: nat] : ( ord_less_eq_nat @ X2 @ ( F @ X2 @ Y ) )
=> ( ! [X2: nat,Y: nat] : ( ord_less_eq_nat @ Y @ ( F @ X2 @ Y ) )
=> ( ! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ( ord_less_eq_nat @ ( F @ Y @ Z2 ) @ X2 ) ) )
=> ( ( sup_sup_nat @ X3 @ Y3 )
= ( F @ X3 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_930_sup__unique,axiom,
! [F: set_set_a > set_set_a > set_set_a,X3: set_set_a,Y3: set_set_a] :
( ! [X2: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ X2 @ ( F @ X2 @ Y ) )
=> ( ! [X2: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ Y @ ( F @ X2 @ Y ) )
=> ( ! [X2: set_set_a,Y: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X2 )
=> ( ( ord_le3724670747650509150_set_a @ Z2 @ X2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ Y @ Z2 ) @ X2 ) ) )
=> ( ( sup_sup_set_set_a @ X3 @ Y3 )
= ( F @ X3 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_931_sup_Oabsorb1,axiom,
! [B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B3 )
= A2 ) ) ).
% sup.absorb1
thf(fact_932_sup_Oabsorb1,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B3 )
= A2 ) ) ).
% sup.absorb1
thf(fact_933_sup_Oabsorb1,axiom,
! [B3: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A2 )
=> ( ( sup_sup_set_set_a @ A2 @ B3 )
= A2 ) ) ).
% sup.absorb1
thf(fact_934_sup_Oabsorb2,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( sup_sup_set_a @ A2 @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_935_sup_Oabsorb2,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( sup_sup_nat @ A2 @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_936_sup_Oabsorb2,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( sup_sup_set_set_a @ A2 @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_937_sup__absorb1,axiom,
! [Y3: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X3 )
=> ( ( sup_sup_set_a @ X3 @ Y3 )
= X3 ) ) ).
% sup_absorb1
thf(fact_938_sup__absorb1,axiom,
! [Y3: nat,X3: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( sup_sup_nat @ X3 @ Y3 )
= X3 ) ) ).
% sup_absorb1
thf(fact_939_sup__absorb1,axiom,
! [Y3: set_set_a,X3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y3 @ X3 )
=> ( ( sup_sup_set_set_a @ X3 @ Y3 )
= X3 ) ) ).
% sup_absorb1
thf(fact_940_sup__absorb2,axiom,
! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ( sup_sup_set_a @ X3 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_941_sup__absorb2,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( sup_sup_nat @ X3 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_942_sup__absorb2,axiom,
! [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ( sup_sup_set_set_a @ X3 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_943_sup_OboundedE,axiom,
! [B3: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B3 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_a @ B3 @ A2 )
=> ~ ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_944_sup_OboundedE,axiom,
! [B3: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B3 @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_945_sup_OboundedE,axiom,
! [B3: set_set_a,C: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B3 @ C ) @ A2 )
=> ~ ( ( ord_le3724670747650509150_set_a @ B3 @ A2 )
=> ~ ( ord_le3724670747650509150_set_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_946_sup_OboundedI,axiom,
! [B3: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B3 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_947_sup_OboundedI,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_948_sup_OboundedI,axiom,
! [B3: set_set_a,A2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ C @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B3 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_949_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B5: set_a,A4: set_a] :
( A4
= ( sup_sup_set_a @ A4 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_950_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_951_sup_Oorder__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B5: set_set_a,A4: set_set_a] :
( A4
= ( sup_sup_set_set_a @ A4 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_952_sup_Ocobounded1,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B3 ) ) ).
% sup.cobounded1
thf(fact_953_sup_Ocobounded1,axiom,
! [A2: nat,B3: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B3 ) ) ).
% sup.cobounded1
thf(fact_954_sup_Ocobounded1,axiom,
! [A2: set_set_a,B3: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ ( sup_sup_set_set_a @ A2 @ B3 ) ) ).
% sup.cobounded1
thf(fact_955_sup_Ocobounded2,axiom,
! [B3: set_a,A2: set_a] : ( ord_less_eq_set_a @ B3 @ ( sup_sup_set_a @ A2 @ B3 ) ) ).
% sup.cobounded2
thf(fact_956_sup_Ocobounded2,axiom,
! [B3: nat,A2: nat] : ( ord_less_eq_nat @ B3 @ ( sup_sup_nat @ A2 @ B3 ) ) ).
% sup.cobounded2
thf(fact_957_sup_Ocobounded2,axiom,
! [B3: set_set_a,A2: set_set_a] : ( ord_le3724670747650509150_set_a @ B3 @ ( sup_sup_set_set_a @ A2 @ B3 ) ) ).
% sup.cobounded2
thf(fact_958_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B5: set_a,A4: set_a] :
( ( sup_sup_set_a @ A4 @ B5 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_959_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B5 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_960_sup_Oabsorb__iff1,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B5: set_set_a,A4: set_set_a] :
( ( sup_sup_set_set_a @ A4 @ B5 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_961_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B5: set_a] :
( ( sup_sup_set_a @ A4 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_962_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] :
( ( sup_sup_nat @ A4 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_963_sup_Oabsorb__iff2,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B5: set_set_a] :
( ( sup_sup_set_set_a @ A4 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_964_sup_OcoboundedI1,axiom,
! [C: set_a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_965_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B3: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_966_sup_OcoboundedI1,axiom,
! [C: set_set_a,A2: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ A2 )
=> ( ord_le3724670747650509150_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_967_sup_OcoboundedI2,axiom,
! [C: set_a,B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ C @ B3 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_968_sup_OcoboundedI2,axiom,
! [C: nat,B3: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_969_sup_OcoboundedI2,axiom,
! [C: set_set_a,B3: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ B3 )
=> ( ord_le3724670747650509150_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_970_boolean__algebra_Odisj__zero__right,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ X3 @ bot_bot_set_a )
= X3 ) ).
% boolean_algebra.disj_zero_right
thf(fact_971_boolean__algebra_Odisj__zero__right,axiom,
! [X3: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X3 @ bot_bo3357376287454694259od_a_a )
= X3 ) ).
% boolean_algebra.disj_zero_right
thf(fact_972_boolean__algebra_Odisj__zero__right,axiom,
! [X3: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ bot_bot_set_set_a )
= X3 ) ).
% boolean_algebra.disj_zero_right
thf(fact_973_boolean__algebra_Odisj__zero__right,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ bot_bot_set_nat )
= X3 ) ).
% boolean_algebra.disj_zero_right
thf(fact_974_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y3: set_a,Z: set_a,X3: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y3 @ Z ) @ X3 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y3 @ X3 ) @ ( sup_sup_set_a @ Z @ X3 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_975_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y3: set_set_a,Z: set_set_a,X3: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ Y3 @ Z ) @ X3 )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ Y3 @ X3 ) @ ( sup_sup_set_set_a @ Z @ X3 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_976_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y3: set_a,Z: set_a,X3: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y3 @ Z ) @ X3 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y3 @ X3 ) @ ( inf_inf_set_a @ Z @ X3 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_977_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y3: set_set_a,Z: set_set_a,X3: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ Y3 @ Z ) @ X3 )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ Y3 @ X3 ) @ ( inf_inf_set_set_a @ Z @ X3 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_978_boolean__algebra_Odisj__conj__distrib,axiom,
! [X3: set_a,Y3: set_a,Z: set_a] :
( ( sup_sup_set_a @ X3 @ ( inf_inf_set_a @ Y3 @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X3 @ Y3 ) @ ( sup_sup_set_a @ X3 @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_979_boolean__algebra_Odisj__conj__distrib,axiom,
! [X3: set_set_a,Y3: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ ( inf_inf_set_set_a @ Y3 @ Z ) )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ X3 @ Y3 ) @ ( sup_sup_set_set_a @ X3 @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_980_boolean__algebra_Oconj__disj__distrib,axiom,
! [X3: set_a,Y3: set_a,Z: set_a] :
( ( inf_inf_set_a @ X3 @ ( sup_sup_set_a @ Y3 @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X3 @ Y3 ) @ ( inf_inf_set_a @ X3 @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_981_boolean__algebra_Oconj__disj__distrib,axiom,
! [X3: set_set_a,Y3: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ X3 @ ( sup_sup_set_set_a @ Y3 @ Z ) )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ X3 @ Y3 ) @ ( inf_inf_set_set_a @ X3 @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_982_sup__inf__distrib2,axiom,
! [Y3: set_a,Z: set_a,X3: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y3 @ Z ) @ X3 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y3 @ X3 ) @ ( sup_sup_set_a @ Z @ X3 ) ) ) ).
% sup_inf_distrib2
thf(fact_983_sup__inf__distrib2,axiom,
! [Y3: set_set_a,Z: set_set_a,X3: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ Y3 @ Z ) @ X3 )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ Y3 @ X3 ) @ ( sup_sup_set_set_a @ Z @ X3 ) ) ) ).
% sup_inf_distrib2
thf(fact_984_sup__inf__distrib1,axiom,
! [X3: set_a,Y3: set_a,Z: set_a] :
( ( sup_sup_set_a @ X3 @ ( inf_inf_set_a @ Y3 @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X3 @ Y3 ) @ ( sup_sup_set_a @ X3 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_985_sup__inf__distrib1,axiom,
! [X3: set_set_a,Y3: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ ( inf_inf_set_set_a @ Y3 @ Z ) )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ X3 @ Y3 ) @ ( sup_sup_set_set_a @ X3 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_986_inf__sup__distrib2,axiom,
! [Y3: set_a,Z: set_a,X3: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y3 @ Z ) @ X3 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y3 @ X3 ) @ ( inf_inf_set_a @ Z @ X3 ) ) ) ).
% inf_sup_distrib2
thf(fact_987_inf__sup__distrib2,axiom,
! [Y3: set_set_a,Z: set_set_a,X3: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ Y3 @ Z ) @ X3 )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ Y3 @ X3 ) @ ( inf_inf_set_set_a @ Z @ X3 ) ) ) ).
% inf_sup_distrib2
thf(fact_988_inf__sup__distrib1,axiom,
! [X3: set_a,Y3: set_a,Z: set_a] :
( ( inf_inf_set_a @ X3 @ ( sup_sup_set_a @ Y3 @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X3 @ Y3 ) @ ( inf_inf_set_a @ X3 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_989_inf__sup__distrib1,axiom,
! [X3: set_set_a,Y3: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ X3 @ ( sup_sup_set_set_a @ Y3 @ Z ) )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ X3 @ Y3 ) @ ( inf_inf_set_set_a @ X3 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_990_distrib__imp2,axiom,
! [X3: set_a,Y3: set_a,Z: set_a] :
( ! [X2: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y ) @ ( sup_sup_set_a @ X2 @ Z2 ) ) )
=> ( ( inf_inf_set_a @ X3 @ ( sup_sup_set_a @ Y3 @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X3 @ Y3 ) @ ( inf_inf_set_a @ X3 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_991_distrib__imp2,axiom,
! [X3: set_set_a,Y3: set_set_a,Z: set_set_a] :
( ! [X2: set_set_a,Y: set_set_a,Z2: set_set_a] :
( ( sup_sup_set_set_a @ X2 @ ( inf_inf_set_set_a @ Y @ Z2 ) )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ X2 @ Y ) @ ( sup_sup_set_set_a @ X2 @ Z2 ) ) )
=> ( ( inf_inf_set_set_a @ X3 @ ( sup_sup_set_set_a @ Y3 @ Z ) )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ X3 @ Y3 ) @ ( inf_inf_set_set_a @ X3 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_992_distrib__imp1,axiom,
! [X3: set_a,Y3: set_a,Z: set_a] :
( ! [X2: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ ( inf_inf_set_a @ X2 @ Z2 ) ) )
=> ( ( sup_sup_set_a @ X3 @ ( inf_inf_set_a @ Y3 @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X3 @ Y3 ) @ ( sup_sup_set_a @ X3 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_993_distrib__imp1,axiom,
! [X3: set_set_a,Y3: set_set_a,Z: set_set_a] :
( ! [X2: set_set_a,Y: set_set_a,Z2: set_set_a] :
( ( inf_inf_set_set_a @ X2 @ ( sup_sup_set_set_a @ Y @ Z2 ) )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ X2 @ Y ) @ ( inf_inf_set_set_a @ X2 @ Z2 ) ) )
=> ( ( sup_sup_set_set_a @ X3 @ ( inf_inf_set_set_a @ Y3 @ Z ) )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ X3 @ Y3 ) @ ( sup_sup_set_set_a @ X3 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_994_Un__empty__left,axiom,
! [B: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_995_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_996_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_997_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_998_Un__empty__right,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% Un_empty_right
thf(fact_999_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_1000_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_1001_Un__empty__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Un_empty_right
thf(fact_1002_Un__mono,axiom,
! [A: set_a,C2: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_1003_Un__mono,axiom,
! [A: set_set_a,C2: set_set_a,B: set_set_a,D2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A @ B ) @ ( sup_sup_set_set_a @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_1004_Un__least,axiom,
! [A: set_a,C2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_1005_Un__least,axiom,
! [A: set_set_a,C2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_1006_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_1007_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_1008_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_1009_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_1010_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_1011_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_1012_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_1013_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_1014_subset__UnE,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
=> ~ ! [A7: set_a] :
( ( ord_less_eq_set_a @ A7 @ A )
=> ! [B8: set_a] :
( ( ord_less_eq_set_a @ B8 @ B )
=> ( C2
!= ( sup_sup_set_a @ A7 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_1015_subset__UnE,axiom,
! [C2: set_set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ ( sup_sup_set_set_a @ A @ B ) )
=> ~ ! [A7: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A7 @ A )
=> ! [B8: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B8 @ B )
=> ( C2
!= ( sup_sup_set_set_a @ A7 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_1016_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B2: set_a] :
( ( sup_sup_set_a @ A3 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_1017_subset__Un__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B2: set_set_a] :
( ( sup_sup_set_set_a @ A3 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_1018_Un__Int__distrib2,axiom,
! [B: set_a,C2: set_a,A: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B @ C2 ) @ A )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B @ A ) @ ( sup_sup_set_a @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_1019_Un__Int__distrib2,axiom,
! [B: set_set_a,C2: set_set_a,A: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ B @ C2 ) @ A )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ B @ A ) @ ( sup_sup_set_set_a @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_1020_Int__Un__distrib2,axiom,
! [B: set_a,C2: set_a,A: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B @ C2 ) @ A )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B @ A ) @ ( inf_inf_set_a @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_1021_Int__Un__distrib2,axiom,
! [B: set_set_a,C2: set_set_a,A: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ B @ C2 ) @ A )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ B @ A ) @ ( inf_inf_set_set_a @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_1022_Un__Int__distrib,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1023_Un__Int__distrib,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ A @ ( inf_inf_set_set_a @ B @ C2 ) )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ A @ B ) @ ( sup_sup_set_set_a @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1024_Int__Un__distrib,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1025_Int__Un__distrib,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( inf_inf_set_set_a @ A @ ( sup_sup_set_set_a @ B @ C2 ) )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ A @ B ) @ ( inf_inf_set_set_a @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1026_Un__Int__crazy,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ B @ C2 ) ) @ ( inf_inf_set_a @ C2 @ A ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ B @ C2 ) ) @ ( sup_sup_set_a @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_1027_Un__Int__crazy,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ A @ B ) @ ( inf_inf_set_set_a @ B @ C2 ) ) @ ( inf_inf_set_set_a @ C2 @ A ) )
= ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ A @ B ) @ ( sup_sup_set_set_a @ B @ C2 ) ) @ ( sup_sup_set_set_a @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_1028_distrib__sup__le,axiom,
! [X3: set_a,Y3: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X3 @ ( inf_inf_set_a @ Y3 @ Z ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X3 @ Y3 ) @ ( sup_sup_set_a @ X3 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_1029_distrib__sup__le,axiom,
! [X3: nat,Y3: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ ( inf_inf_nat @ Y3 @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X3 @ Y3 ) @ ( sup_sup_nat @ X3 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_1030_distrib__sup__le,axiom,
! [X3: set_set_a,Y3: set_set_a,Z: set_set_a] : ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ X3 @ ( inf_inf_set_set_a @ Y3 @ Z ) ) @ ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ X3 @ Y3 ) @ ( sup_sup_set_set_a @ X3 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_1031_distrib__inf__le,axiom,
! [X3: set_a,Y3: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X3 @ Y3 ) @ ( inf_inf_set_a @ X3 @ Z ) ) @ ( inf_inf_set_a @ X3 @ ( sup_sup_set_a @ Y3 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_1032_distrib__inf__le,axiom,
! [X3: nat,Y3: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X3 @ Y3 ) @ ( inf_inf_nat @ X3 @ Z ) ) @ ( inf_inf_nat @ X3 @ ( sup_sup_nat @ Y3 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_1033_distrib__inf__le,axiom,
! [X3: set_set_a,Y3: set_set_a,Z: set_set_a] : ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ X3 @ Y3 ) @ ( inf_inf_set_set_a @ X3 @ Z ) ) @ ( inf_inf_set_set_a @ X3 @ ( sup_sup_set_set_a @ Y3 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_1034_singleton__Un__iff,axiom,
! [X3: a,A: set_a,B: set_a] :
( ( ( insert_a @ X3 @ bot_bot_set_a )
= ( sup_sup_set_a @ A @ B ) )
= ( ( ( A = bot_bot_set_a )
& ( B
= ( insert_a @ X3 @ bot_bot_set_a ) ) )
| ( ( A
= ( insert_a @ X3 @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A
= ( insert_a @ X3 @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1035_singleton__Un__iff,axiom,
! [X3: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a )
= ( sup_su3048258781599657691od_a_a @ A @ B ) )
= ( ( ( A = bot_bo3357376287454694259od_a_a )
& ( B
= ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) ) )
| ( ( A
= ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) )
& ( B = bot_bo3357376287454694259od_a_a ) )
| ( ( A
= ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) )
& ( B
= ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1036_singleton__Un__iff,axiom,
! [X3: set_a,A: set_set_a,B: set_set_a] :
( ( ( insert_set_a @ X3 @ bot_bot_set_set_a )
= ( sup_sup_set_set_a @ A @ B ) )
= ( ( ( A = bot_bot_set_set_a )
& ( B
= ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) )
| ( ( A
= ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
& ( B = bot_bot_set_set_a ) )
| ( ( A
= ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
& ( B
= ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1037_singleton__Un__iff,axiom,
! [X3: nat,A: set_nat,B: set_nat] :
( ( ( insert_nat @ X3 @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A @ B ) )
= ( ( ( A = bot_bot_set_nat )
& ( B
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat @ X3 @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat @ X3 @ bot_bot_set_nat ) )
& ( B
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1038_Un__singleton__iff,axiom,
! [A: set_a,B: set_a,X3: a] :
( ( ( sup_sup_set_a @ A @ B )
= ( insert_a @ X3 @ bot_bot_set_a ) )
= ( ( ( A = bot_bot_set_a )
& ( B
= ( insert_a @ X3 @ bot_bot_set_a ) ) )
| ( ( A
= ( insert_a @ X3 @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A
= ( insert_a @ X3 @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1039_Un__singleton__iff,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,X3: product_prod_a_a] :
( ( ( sup_su3048258781599657691od_a_a @ A @ B )
= ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) )
= ( ( ( A = bot_bo3357376287454694259od_a_a )
& ( B
= ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) ) )
| ( ( A
= ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) )
& ( B = bot_bo3357376287454694259od_a_a ) )
| ( ( A
= ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) )
& ( B
= ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1040_Un__singleton__iff,axiom,
! [A: set_set_a,B: set_set_a,X3: set_a] :
( ( ( sup_sup_set_set_a @ A @ B )
= ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
= ( ( ( A = bot_bot_set_set_a )
& ( B
= ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) )
| ( ( A
= ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
& ( B = bot_bot_set_set_a ) )
| ( ( A
= ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
& ( B
= ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1041_Un__singleton__iff,axiom,
! [A: set_nat,B: set_nat,X3: nat] :
( ( ( sup_sup_set_nat @ A @ B )
= ( insert_nat @ X3 @ bot_bot_set_nat ) )
= ( ( ( A = bot_bot_set_nat )
& ( B
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat @ X3 @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat @ X3 @ bot_bot_set_nat ) )
& ( B
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1042_insert__is__Un,axiom,
( insert_a
= ( ^ [A4: a] : ( sup_sup_set_a @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_1043_insert__is__Un,axiom,
( insert4534936382041156343od_a_a
= ( ^ [A4: product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ ( insert4534936382041156343od_a_a @ A4 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% insert_is_Un
thf(fact_1044_insert__is__Un,axiom,
( insert_set_a
= ( ^ [A4: set_a] : ( sup_sup_set_set_a @ ( insert_set_a @ A4 @ bot_bot_set_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_1045_insert__is__Un,axiom,
( insert_nat
= ( ^ [A4: nat] : ( sup_sup_set_nat @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_1046_Un__Int__assoc__eq,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) )
= ( ord_less_eq_set_a @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1047_Un__Int__assoc__eq,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_set_a @ A @ ( sup_sup_set_set_a @ B @ C2 ) ) )
= ( ord_le3724670747650509150_set_a @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1048_ulgraph_Oincident__edges__union,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ( undire3231912044278729248dges_a @ Edges2 @ V )
= ( sup_sup_set_set_a @ ( undire1270416042309875431dges_a @ Edges2 @ V ) @ ( undire4753905205749729249oops_a @ Edges2 @ V ) ) ) ) ).
% ulgraph.incident_edges_union
thf(fact_1049_finite__has__maximal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( ord_less_eq_set_a @ A2 @ X2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1050_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ A2 @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1051_finite__has__maximal2,axiom,
! [A: set_set_set_a,A2: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( member_set_set_a @ A2 @ A )
=> ? [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A )
& ( ord_le3724670747650509150_set_a @ A2 @ X2 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1052_finite__has__minimal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( ord_less_eq_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1053_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1054_finite__has__minimal2,axiom,
! [A: set_set_set_a,A2: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( member_set_set_a @ A2 @ A )
=> ? [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A )
& ( ord_le3724670747650509150_set_a @ X2 @ A2 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A )
=> ( ( ord_le3724670747650509150_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1055_infinite__imp__nonempty,axiom,
! [S2: set_a] :
( ~ ( finite_finite_a @ S2 )
=> ( S2 != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_1056_infinite__imp__nonempty,axiom,
! [S2: set_Product_prod_a_a] :
( ~ ( finite6544458595007987280od_a_a @ S2 )
=> ( S2 != bot_bo3357376287454694259od_a_a ) ) ).
% infinite_imp_nonempty
thf(fact_1057_infinite__imp__nonempty,axiom,
! [S2: set_set_a] :
( ~ ( finite_finite_set_a @ S2 )
=> ( S2 != bot_bot_set_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_1058_infinite__imp__nonempty,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( S2 != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_1059_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_1060_finite_OemptyI,axiom,
finite6544458595007987280od_a_a @ bot_bo3357376287454694259od_a_a ).
% finite.emptyI
thf(fact_1061_finite_OemptyI,axiom,
finite_finite_set_a @ bot_bot_set_set_a ).
% finite.emptyI
thf(fact_1062_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_1063_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_1064_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_1065_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_1066_infinite__super,axiom,
! [S2: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_1067_infinite__super,axiom,
! [S2: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S2 @ T2 )
=> ( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_1068_infinite__super,axiom,
! [S2: set_set_a,T2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ S2 @ T2 )
=> ( ~ ( finite_finite_set_a @ S2 )
=> ~ ( finite_finite_set_a @ T2 ) ) ) ).
% infinite_super
thf(fact_1069_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_1070_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_1071_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_1072_finite_OinsertI,axiom,
! [A: set_a,A2: a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( insert_a @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_1073_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_1074_finite_OinsertI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( insert_nat @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_1075_finite__has__minimal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1076_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1077_finite__has__minimal,axiom,
! [A: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ? [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A )
=> ( ( ord_le3724670747650509150_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1078_finite__has__maximal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1079_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1080_finite__has__maximal,axiom,
! [A: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ? [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1081_finite_Ocases,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( A2 != bot_bot_set_a )
=> ~ ! [A8: set_a] :
( ? [A5: a] :
( A2
= ( insert_a @ A5 @ A8 ) )
=> ~ ( finite_finite_a @ A8 ) ) ) ) ).
% finite.cases
thf(fact_1082_finite_Ocases,axiom,
! [A2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A2 )
=> ( ( A2 != bot_bo3357376287454694259od_a_a )
=> ~ ! [A8: set_Product_prod_a_a] :
( ? [A5: product_prod_a_a] :
( A2
= ( insert4534936382041156343od_a_a @ A5 @ A8 ) )
=> ~ ( finite6544458595007987280od_a_a @ A8 ) ) ) ) ).
% finite.cases
thf(fact_1083_finite_Ocases,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ~ ! [A8: set_set_a] :
( ? [A5: set_a] :
( A2
= ( insert_set_a @ A5 @ A8 ) )
=> ~ ( finite_finite_set_a @ A8 ) ) ) ) ).
% finite.cases
thf(fact_1084_finite_Ocases,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ~ ! [A8: set_nat] :
( ? [A5: nat] :
( A2
= ( insert_nat @ A5 @ A8 ) )
=> ~ ( finite_finite_nat @ A8 ) ) ) ) ).
% finite.cases
thf(fact_1085_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A4: set_a] :
( ( A4 = bot_bot_set_a )
| ? [A3: set_a,B5: a] :
( ( A4
= ( insert_a @ B5 @ A3 ) )
& ( finite_finite_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_1086_finite_Osimps,axiom,
( finite6544458595007987280od_a_a
= ( ^ [A4: set_Product_prod_a_a] :
( ( A4 = bot_bo3357376287454694259od_a_a )
| ? [A3: set_Product_prod_a_a,B5: product_prod_a_a] :
( ( A4
= ( insert4534936382041156343od_a_a @ B5 @ A3 ) )
& ( finite6544458595007987280od_a_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_1087_finite_Osimps,axiom,
( finite_finite_set_a
= ( ^ [A4: set_set_a] :
( ( A4 = bot_bot_set_set_a )
| ? [A3: set_set_a,B5: set_a] :
( ( A4
= ( insert_set_a @ B5 @ A3 ) )
& ( finite_finite_set_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_1088_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A4: set_nat] :
( ( A4 = bot_bot_set_nat )
| ? [A3: set_nat,B5: nat] :
( ( A4
= ( insert_nat @ B5 @ A3 ) )
& ( finite_finite_nat @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_1089_finite__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1090_finite__induct,axiom,
! [F2: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ F2 )
=> ( ( P @ bot_bo3357376287454694259od_a_a )
=> ( ! [X2: product_prod_a_a,F3: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F3 )
=> ( ~ ( member1426531477525435216od_a_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert4534936382041156343od_a_a @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1091_finite__induct,axiom,
! [F2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X2: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1092_finite__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1093_finite__ne__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ! [X2: a] : ( P @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1094_finite__ne__induct,axiom,
! [F2: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ F2 )
=> ( ( F2 != bot_bo3357376287454694259od_a_a )
=> ( ! [X2: product_prod_a_a] : ( P @ ( insert4534936382041156343od_a_a @ X2 @ bot_bo3357376287454694259od_a_a ) )
=> ( ! [X2: product_prod_a_a,F3: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F3 )
=> ( ( F3 != bot_bo3357376287454694259od_a_a )
=> ( ~ ( member1426531477525435216od_a_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert4534936382041156343od_a_a @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1095_finite__ne__induct,axiom,
! [F2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( F2 != bot_bot_set_set_a )
=> ( ! [X2: set_a] : ( P @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
=> ( ! [X2: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( F3 != bot_bot_set_set_a )
=> ( ~ ( member_set_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1096_finite__ne__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X2: nat] : ( P @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1097_infinite__finite__induct,axiom,
! [P: set_a > $o,A: set_a] :
( ! [A8: set_a] :
( ~ ( finite_finite_a @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_1098_infinite__finite__induct,axiom,
! [P: set_Product_prod_a_a > $o,A: set_Product_prod_a_a] :
( ! [A8: set_Product_prod_a_a] :
( ~ ( finite6544458595007987280od_a_a @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bo3357376287454694259od_a_a )
=> ( ! [X2: product_prod_a_a,F3: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F3 )
=> ( ~ ( member1426531477525435216od_a_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert4534936382041156343od_a_a @ X2 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_1099_infinite__finite__induct,axiom,
! [P: set_set_a > $o,A: set_set_a] :
( ! [A8: set_set_a] :
( ~ ( finite_finite_set_a @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X2: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X2 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_1100_infinite__finite__induct,axiom,
! [P: set_nat > $o,A: set_nat] :
( ! [A8: set_nat] :
( ~ ( finite_finite_nat @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_1101_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_1102_less__by__empty,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A = bot_bo3357376287454694259od_a_a )
=> ( ord_le746702958409616551od_a_a @ A @ B ) ) ).
% less_by_empty
thf(fact_1103_arg__min__least,axiom,
! [S2: set_a,Y3: a,F: a > nat] :
( ( finite_finite_a @ S2 )
=> ( ( S2 != bot_bot_set_a )
=> ( ( member_a @ Y3 @ S2 )
=> ( ord_less_eq_nat @ ( F @ ( lattic6340287419671400565_a_nat @ F @ S2 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% arg_min_least
thf(fact_1104_arg__min__least,axiom,
! [S2: set_Product_prod_a_a,Y3: product_prod_a_a,F: product_prod_a_a > nat] :
( ( finite6544458595007987280od_a_a @ S2 )
=> ( ( S2 != bot_bo3357376287454694259od_a_a )
=> ( ( member1426531477525435216od_a_a @ Y3 @ S2 )
=> ( ord_less_eq_nat @ ( F @ ( lattic806887198133436574_a_nat @ F @ S2 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% arg_min_least
thf(fact_1105_arg__min__least,axiom,
! [S2: set_set_a,Y3: set_a,F: set_a > nat] :
( ( finite_finite_set_a @ S2 )
=> ( ( S2 != bot_bot_set_set_a )
=> ( ( member_set_a @ Y3 @ S2 )
=> ( ord_less_eq_nat @ ( F @ ( lattic4678118661306933717_a_nat @ F @ S2 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% arg_min_least
thf(fact_1106_arg__min__least,axiom,
! [S2: set_nat,Y3: nat,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ( ( member_nat @ Y3 @ S2 )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S2 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% arg_min_least
thf(fact_1107_insert__subsetI,axiom,
! [X3: product_prod_a_a,A: set_Product_prod_a_a,X5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A )
=> ( ( ord_le746702958409616551od_a_a @ X5 @ A )
=> ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ X3 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1108_insert__subsetI,axiom,
! [X3: nat,A: set_nat,X5: set_nat] :
( ( member_nat @ X3 @ A )
=> ( ( ord_less_eq_set_nat @ X5 @ A )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1109_insert__subsetI,axiom,
! [X3: a,A: set_a,X5: set_a] :
( ( member_a @ X3 @ A )
=> ( ( ord_less_eq_set_a @ X5 @ A )
=> ( ord_less_eq_set_a @ ( insert_a @ X3 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1110_insert__subsetI,axiom,
! [X3: set_a,A: set_set_a,X5: set_set_a] :
( ( member_set_a @ X3 @ A )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ A )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X3 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1111_subset__emptyI,axiom,
! [A: set_Product_prod_a_a] :
( ! [X2: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X2 @ A )
=> ( ord_le746702958409616551od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ) ).
% subset_emptyI
thf(fact_1112_subset__emptyI,axiom,
! [A: set_nat] :
( ! [X2: nat] :
~ ( member_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_1113_subset__emptyI,axiom,
! [A: set_a] :
( ! [X2: a] :
~ ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ A @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_1114_subset__emptyI,axiom,
! [A: set_set_a] :
( ! [X2: set_a] :
~ ( member_set_a @ X2 @ A )
=> ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a ) ) ).
% subset_emptyI
thf(fact_1115_finite__transitivity__chain,axiom,
! [A: set_a,R2: a > a > $o] :
( ( finite_finite_a @ A )
=> ( ! [X2: a] :
~ ( R2 @ X2 @ X2 )
=> ( ! [X2: a,Y: a,Z2: a] :
( ( R2 @ X2 @ Y )
=> ( ( R2 @ Y @ Z2 )
=> ( R2 @ X2 @ Z2 ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ? [Y7: a] :
( ( member_a @ Y7 @ A )
& ( R2 @ X2 @ Y7 ) ) )
=> ( A = bot_bot_set_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1116_finite__transitivity__chain,axiom,
! [A: set_Product_prod_a_a,R2: product_prod_a_a > product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ A )
=> ( ! [X2: product_prod_a_a] :
~ ( R2 @ X2 @ X2 )
=> ( ! [X2: product_prod_a_a,Y: product_prod_a_a,Z2: product_prod_a_a] :
( ( R2 @ X2 @ Y )
=> ( ( R2 @ Y @ Z2 )
=> ( R2 @ X2 @ Z2 ) ) )
=> ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A )
=> ? [Y7: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y7 @ A )
& ( R2 @ X2 @ Y7 ) ) )
=> ( A = bot_bo3357376287454694259od_a_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1117_finite__transitivity__chain,axiom,
! [A: set_set_a,R2: set_a > set_a > $o] :
( ( finite_finite_set_a @ A )
=> ( ! [X2: set_a] :
~ ( R2 @ X2 @ X2 )
=> ( ! [X2: set_a,Y: set_a,Z2: set_a] :
( ( R2 @ X2 @ Y )
=> ( ( R2 @ Y @ Z2 )
=> ( R2 @ X2 @ Z2 ) ) )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ? [Y7: set_a] :
( ( member_set_a @ Y7 @ A )
& ( R2 @ X2 @ Y7 ) ) )
=> ( A = bot_bot_set_set_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1118_finite__transitivity__chain,axiom,
! [A: set_nat,R2: nat > nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [X2: nat] :
~ ( R2 @ X2 @ X2 )
=> ( ! [X2: nat,Y: nat,Z2: nat] :
( ( R2 @ X2 @ Y )
=> ( ( R2 @ Y @ Z2 )
=> ( R2 @ X2 @ Z2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Y7: nat] :
( ( member_nat @ Y7 @ A )
& ( R2 @ X2 @ Y7 ) ) )
=> ( A = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1119_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_1120_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_1121_degree__none,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire8867928226783802224gree_a @ edges @ V )
= zero_zero_nat ) ) ).
% degree_none
thf(fact_1122_ulgraph_Odegree_Ocong,axiom,
undire8867928226783802224gree_a = undire8867928226783802224gree_a ).
% ulgraph.degree.cong
thf(fact_1123_ulgraph_Odegree__none,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices2 @ Edges2 )
=> ( ~ ( member_set_a @ V @ Vertices2 )
=> ( ( undire8939077443744732368_set_a @ Edges2 @ V )
= zero_zero_nat ) ) ) ).
% ulgraph.degree_none
thf(fact_1124_ulgraph_Odegree__none,axiom,
! [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a,V: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices2 @ Edges2 )
=> ( ~ ( member1426531477525435216od_a_a @ V @ Vertices2 )
=> ( ( undire1436394852029823897od_a_a @ Edges2 @ V )
= zero_zero_nat ) ) ) ).
% ulgraph.degree_none
thf(fact_1125_ulgraph_Odegree__none,axiom,
! [Vertices2: set_nat,Edges2: set_set_nat,V: nat] :
( ( undire3269267262472140706ph_nat @ Vertices2 @ Edges2 )
=> ( ~ ( member_nat @ V @ Vertices2 )
=> ( ( undire6581030323043281630ee_nat @ Edges2 @ V )
= zero_zero_nat ) ) ) ).
% ulgraph.degree_none
thf(fact_1126_ulgraph_Odegree__none,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ~ ( member_a @ V @ Vertices2 )
=> ( ( undire8867928226783802224gree_a @ Edges2 @ V )
= zero_zero_nat ) ) ) ).
% ulgraph.degree_none
thf(fact_1127_ulgraph_Ois__isolated__vertex__degree0,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ( undire8931668460104145173rtex_a @ Vertices2 @ Edges2 @ V )
=> ( ( undire8867928226783802224gree_a @ Edges2 @ V )
= zero_zero_nat ) ) ) ).
% ulgraph.is_isolated_vertex_degree0
thf(fact_1128_ulgraph_Odegree0__neighborhood__empt__iff,axiom,
! [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a,V: product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices2 @ Edges2 )
=> ( ( finite8717734299975451184od_a_a @ Edges2 )
=> ( ( ( undire1436394852029823897od_a_a @ Edges2 @ V )
= zero_zero_nat )
= ( ( undire7963753511165915895od_a_a @ Vertices2 @ Edges2 @ V )
= bot_bo3357376287454694259od_a_a ) ) ) ) ).
% ulgraph.degree0_neighborhood_empt_iff
thf(fact_1129_ulgraph_Odegree0__neighborhood__empt__iff,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices2 @ Edges2 )
=> ( ( finite7209287970140883943_set_a @ Edges2 )
=> ( ( ( undire8939077443744732368_set_a @ Edges2 @ V )
= zero_zero_nat )
= ( ( undire2074812191327625774_set_a @ Vertices2 @ Edges2 @ V )
= bot_bot_set_set_a ) ) ) ) ).
% ulgraph.degree0_neighborhood_empt_iff
thf(fact_1130_ulgraph_Odegree0__neighborhood__empt__iff,axiom,
! [Vertices2: set_nat,Edges2: set_set_nat,V: nat] :
( ( undire3269267262472140706ph_nat @ Vertices2 @ Edges2 )
=> ( ( finite1152437895449049373et_nat @ Edges2 )
=> ( ( ( undire6581030323043281630ee_nat @ Edges2 @ V )
= zero_zero_nat )
= ( ( undire8190396521545869824od_nat @ Vertices2 @ Edges2 @ V )
= bot_bot_set_nat ) ) ) ) ).
% ulgraph.degree0_neighborhood_empt_iff
thf(fact_1131_ulgraph_Odegree0__neighborhood__empt__iff,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ( finite_finite_set_a @ Edges2 )
=> ( ( ( undire8867928226783802224gree_a @ Edges2 @ V )
= zero_zero_nat )
= ( ( undire8504279938402040014hood_a @ Vertices2 @ Edges2 @ V )
= bot_bot_set_a ) ) ) ) ).
% ulgraph.degree0_neighborhood_empt_iff
thf(fact_1132_ulgraph_Odegree0__inc__edges__empt__iff,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ( finite_finite_set_a @ Edges2 )
=> ( ( ( undire8867928226783802224gree_a @ Edges2 @ V )
= zero_zero_nat )
= ( ( undire3231912044278729248dges_a @ Edges2 @ V )
= bot_bot_set_set_a ) ) ) ) ).
% ulgraph.degree0_inc_edges_empt_iff
thf(fact_1133_ssubst__Pair__rhs,axiom,
! [R: a,S: a,R2: set_Product_prod_a_a,S4: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ R @ S ) @ R2 )
=> ( ( S4 = S )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ R @ S4 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_1134_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1135_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1136_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1137_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_1138_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1139_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_1140_card_Oempty,axiom,
( ( finite4795055649997197647od_a_a @ bot_bo3357376287454694259od_a_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_1141_card_Oempty,axiom,
( ( finite_card_set_a @ bot_bot_set_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_1142_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_1143_card_Oinfinite,axiom,
! [A: set_a] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_card_a @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_1144_card_Oinfinite,axiom,
! [A: set_set_a] :
( ~ ( finite_finite_set_a @ A )
=> ( ( finite_card_set_a @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_1145_card_Oinfinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_card_nat @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_1146_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_1147_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_1148_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_1149_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_1150_infinite__nat__iff__unbounded__le,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M: nat] :
? [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( member_nat @ N2 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_1151_card__insert__le,axiom,
! [A: set_set_a,X3: set_a] : ( ord_less_eq_nat @ ( finite_card_set_a @ A ) @ ( finite_card_set_a @ ( insert_set_a @ X3 @ A ) ) ) ).
% card_insert_le
thf(fact_1152_card__insert__le,axiom,
! [A: set_a,X3: a] : ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ ( insert_a @ X3 @ A ) ) ) ).
% card_insert_le
thf(fact_1153_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1154_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_1155_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_1156_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_1157_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_1158_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B3 ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y7: nat] :
( ( P @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1159_ulgraph_Ocard__incident__sedges__neighborhood,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices2 @ Edges2 )
=> ( ( finite6524359278146944486_set_a @ ( undire4631953023069350784_set_a @ Edges2 @ V ) )
= ( finite_card_set_a @ ( undire2074812191327625774_set_a @ Vertices2 @ Edges2 @ V ) ) ) ) ).
% ulgraph.card_incident_sedges_neighborhood
thf(fact_1160_ulgraph_Ocard__incident__sedges__neighborhood,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ( finite_card_set_a @ ( undire3231912044278729248dges_a @ Edges2 @ V ) )
= ( finite_card_a @ ( undire8504279938402040014hood_a @ Vertices2 @ Edges2 @ V ) ) ) ) ).
% ulgraph.card_incident_sedges_neighborhood
thf(fact_1161_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_1162_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_1163_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_1164_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_1165_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_1166_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_1167_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_1168_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_1169_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_1170_exists__subset__between,axiom,
! [A: set_nat,N: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
& ( ord_less_eq_set_nat @ B4 @ C2 )
& ( ( finite_card_nat @ B4 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_1171_exists__subset__between,axiom,
! [A: set_a,N: nat,C2: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_a @ C2 ) )
=> ( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( finite_finite_a @ C2 )
=> ? [B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
& ( ord_less_eq_set_a @ B4 @ C2 )
& ( ( finite_card_a @ B4 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_1172_exists__subset__between,axiom,
! [A: set_set_a,N: nat,C2: set_set_a] :
( ( ord_less_eq_nat @ ( finite_card_set_a @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_set_a @ C2 ) )
=> ( ( ord_le3724670747650509150_set_a @ A @ C2 )
=> ( ( finite_finite_set_a @ C2 )
=> ? [B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ C2 )
& ( ( finite_card_set_a @ B4 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_1173_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S2 ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S2 )
=> ( ( ( finite_card_nat @ T3 )
= N )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_1174_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ S2 ) )
=> ~ ! [T3: set_a] :
( ( ord_less_eq_set_a @ T3 @ S2 )
=> ( ( ( finite_card_a @ T3 )
= N )
=> ~ ( finite_finite_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_1175_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_set_a @ S2 ) )
=> ~ ! [T3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ T3 @ S2 )
=> ( ( ( finite_card_set_a @ T3 )
= N )
=> ~ ( finite_finite_set_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_1176_infinite__arbitrarily__large,axiom,
! [A: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ( finite_card_nat @ B4 )
= N )
& ( ord_less_eq_set_nat @ B4 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_1177_infinite__arbitrarily__large,axiom,
! [A: set_a,N: nat] :
( ~ ( finite_finite_a @ A )
=> ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ( finite_card_a @ B4 )
= N )
& ( ord_less_eq_set_a @ B4 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_1178_infinite__arbitrarily__large,axiom,
! [A: set_set_a,N: nat] :
( ~ ( finite_finite_set_a @ A )
=> ? [B4: set_set_a] :
( ( finite_finite_set_a @ B4 )
& ( ( finite_card_set_a @ B4 )
= N )
& ( ord_le3724670747650509150_set_a @ B4 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_1179_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C2: nat] :
( ! [G2: set_nat] :
( ( ord_less_eq_set_nat @ G2 @ F2 )
=> ( ( finite_finite_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_1180_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_a,C2: nat] :
( ! [G2: set_a] :
( ( ord_less_eq_set_a @ G2 @ F2 )
=> ( ( finite_finite_a @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ G2 ) @ C2 ) ) )
=> ( ( finite_finite_a @ F2 )
& ( ord_less_eq_nat @ ( finite_card_a @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_1181_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_set_a,C2: nat] :
( ! [G2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ G2 @ F2 )
=> ( ( finite_finite_set_a @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ G2 ) @ C2 ) ) )
=> ( ( finite_finite_set_a @ F2 )
& ( ord_less_eq_nat @ ( finite_card_set_a @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_1182_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_1183_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_1184_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_1185_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_1186_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1187_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1188_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_1189_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_1190_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1191_ulgraph_Odegree__no__loops,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ~ ( undire3617971648856834880loop_a @ Edges2 @ V )
=> ( ( undire8867928226783802224gree_a @ Edges2 @ V )
= ( finite_card_set_a @ ( undire3231912044278729248dges_a @ Edges2 @ V ) ) ) ) ) ).
% ulgraph.degree_no_loops
thf(fact_1192_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_1193_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_1194_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_1195_card__Ex__subset,axiom,
! [K: nat,M3: set_a] :
( ( ord_less_eq_nat @ K @ ( finite_card_a @ M3 ) )
=> ? [N3: set_a] :
( ( ord_less_eq_set_a @ N3 @ M3 )
& ( ( finite_card_a @ N3 )
= K ) ) ) ).
% card_Ex_subset
thf(fact_1196_card__Ex__subset,axiom,
! [K: nat,M3: set_set_a] :
( ( ord_less_eq_nat @ K @ ( finite_card_set_a @ M3 ) )
=> ? [N3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ N3 @ M3 )
& ( ( finite_card_set_a @ N3 )
= K ) ) ) ).
% card_Ex_subset
thf(fact_1197_card__le__if__inj__on__rel,axiom,
! [B: set_a,A: set_nat,R: nat > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ? [B9: a] :
( ( member_a @ B9 @ B )
& ( R @ A5 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B6: a] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_a @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1198_card__le__if__inj__on__rel,axiom,
! [B: set_a,A: set_a,R: a > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A5: a] :
( ( member_a @ A5 @ A )
=> ? [B9: a] :
( ( member_a @ B9 @ B )
& ( R @ A5 @ B9 ) ) )
=> ( ! [A1: a,A22: a,B6: a] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_a @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1199_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B )
& ( R @ A5 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B6: nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_nat @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1200_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_a,R: a > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A5: a] :
( ( member_a @ A5 @ A )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B )
& ( R @ A5 @ B9 ) ) )
=> ( ! [A1: a,A22: a,B6: nat] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_nat @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1201_card__le__if__inj__on__rel,axiom,
! [B: set_a,A: set_set_a,R: set_a > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A5: set_a] :
( ( member_set_a @ A5 @ A )
=> ? [B9: a] :
( ( member_a @ B9 @ B )
& ( R @ A5 @ B9 ) ) )
=> ( ! [A1: set_a,A22: set_a,B6: a] :
( ( member_set_a @ A1 @ A )
=> ( ( member_set_a @ A22 @ A )
=> ( ( member_a @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1202_card__le__if__inj__on__rel,axiom,
! [B: set_set_a,A: set_nat,R: nat > set_a > $o] :
( ( finite_finite_set_a @ B )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ? [B9: set_a] :
( ( member_set_a @ B9 @ B )
& ( R @ A5 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B6: set_a] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_set_a @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_set_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1203_card__le__if__inj__on__rel,axiom,
! [B: set_set_a,A: set_a,R: a > set_a > $o] :
( ( finite_finite_set_a @ B )
=> ( ! [A5: a] :
( ( member_a @ A5 @ A )
=> ? [B9: set_a] :
( ( member_set_a @ B9 @ B )
& ( R @ A5 @ B9 ) ) )
=> ( ! [A1: a,A22: a,B6: set_a] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_set_a @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_set_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1204_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_set_a,R: set_a > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A5: set_a] :
( ( member_set_a @ A5 @ A )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B )
& ( R @ A5 @ B9 ) ) )
=> ( ! [A1: set_a,A22: set_a,B6: nat] :
( ( member_set_a @ A1 @ A )
=> ( ( member_set_a @ A22 @ A )
=> ( ( member_nat @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1205_card__le__if__inj__on__rel,axiom,
! [B: set_Product_prod_a_a,A: set_nat,R: nat > product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ B )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ? [B9: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B9 @ B )
& ( R @ A5 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B6: product_prod_a_a] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member1426531477525435216od_a_a @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite4795055649997197647od_a_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1206_card__le__if__inj__on__rel,axiom,
! [B: set_a,A: set_Product_prod_a_a,R: product_prod_a_a > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A5 @ A )
=> ? [B9: a] :
( ( member_a @ B9 @ B )
& ( R @ A5 @ B9 ) ) )
=> ( ! [A1: product_prod_a_a,A22: product_prod_a_a,B6: a] :
( ( member1426531477525435216od_a_a @ A1 @ A )
=> ( ( member1426531477525435216od_a_a @ A22 @ A )
=> ( ( member_a @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1207_card1__incident__imp__vert,axiom,
! [V: a,E3: set_a] :
( ( ( undire1521409233611534436dent_a @ V @ E3 )
& ( ( finite_card_a @ E3 )
= one_one_nat ) )
=> ( E3
= ( insert_a @ V @ bot_bot_set_a ) ) ) ).
% card1_incident_imp_vert
thf(fact_1208_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1209_is__singleton__altdef,axiom,
( is_singleton_set_a
= ( ^ [A3: set_set_a] :
( ( finite_card_set_a @ A3 )
= one_one_nat ) ) ) ).
% is_singleton_altdef
thf(fact_1210_is__singleton__altdef,axiom,
( is_singleton_a
= ( ^ [A3: set_a] :
( ( finite_card_a @ A3 )
= one_one_nat ) ) ) ).
% is_singleton_altdef
thf(fact_1211_card__1__singletonE,axiom,
! [A: set_a] :
( ( ( finite_card_a @ A )
= one_one_nat )
=> ~ ! [X2: a] :
( A
!= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ).
% card_1_singletonE
thf(fact_1212_card__1__singletonE,axiom,
! [A: set_Product_prod_a_a] :
( ( ( finite4795055649997197647od_a_a @ A )
= one_one_nat )
=> ~ ! [X2: product_prod_a_a] :
( A
!= ( insert4534936382041156343od_a_a @ X2 @ bot_bo3357376287454694259od_a_a ) ) ) ).
% card_1_singletonE
thf(fact_1213_card__1__singletonE,axiom,
! [A: set_set_a] :
( ( ( finite_card_set_a @ A )
= one_one_nat )
=> ~ ! [X2: set_a] :
( A
!= ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ) ).
% card_1_singletonE
thf(fact_1214_card__1__singletonE,axiom,
! [A: set_nat] :
( ( ( finite_card_nat @ A )
= one_one_nat )
=> ~ ! [X2: nat] :
( A
!= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% card_1_singletonE
thf(fact_1215_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M: nat] :
! [X: nat] :
( ( member_nat @ X @ N4 )
=> ( ord_less_eq_nat @ X @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1216_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M3: nat] :
( ( P @ X3 )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M3 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1217_ulgraph_Oincident__loops__card,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ ( undire4753905205749729249oops_a @ Edges2 @ V ) ) @ one_one_nat ) ) ).
% ulgraph.incident_loops_card
thf(fact_1218_ulgraph_Ocard1__incident__imp__vert,axiom,
! [Vertices2: set_Product_prod_a_a,Edges2: set_se5735800977113168103od_a_a,V: product_prod_a_a,E3: set_Product_prod_a_a] :
( ( undire4585262585102564309od_a_a @ Vertices2 @ Edges2 )
=> ( ( ( undire3369688177417741453od_a_a @ V @ E3 )
& ( ( finite4795055649997197647od_a_a @ E3 )
= one_one_nat ) )
=> ( E3
= ( insert4534936382041156343od_a_a @ V @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% ulgraph.card1_incident_imp_vert
thf(fact_1219_ulgraph_Ocard1__incident__imp__vert,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,V: set_a,E3: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices2 @ Edges2 )
=> ( ( ( undire2320338297334612420_set_a @ V @ E3 )
& ( ( finite_card_set_a @ E3 )
= one_one_nat ) )
=> ( E3
= ( insert_set_a @ V @ bot_bot_set_set_a ) ) ) ) ).
% ulgraph.card1_incident_imp_vert
thf(fact_1220_ulgraph_Ocard1__incident__imp__vert,axiom,
! [Vertices2: set_nat,Edges2: set_set_nat,V: nat,E3: set_nat] :
( ( undire3269267262472140706ph_nat @ Vertices2 @ Edges2 )
=> ( ( ( undire7858122600432113898nt_nat @ V @ E3 )
& ( ( finite_card_nat @ E3 )
= one_one_nat ) )
=> ( E3
= ( insert_nat @ V @ bot_bot_set_nat ) ) ) ) ).
% ulgraph.card1_incident_imp_vert
thf(fact_1221_ulgraph_Ocard1__incident__imp__vert,axiom,
! [Vertices2: set_a,Edges2: set_set_a,V: a,E3: set_a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ( ( undire1521409233611534436dent_a @ V @ E3 )
& ( ( finite_card_a @ E3 )
= one_one_nat ) )
=> ( E3
= ( insert_a @ V @ bot_bot_set_a ) ) ) ) ).
% ulgraph.card1_incident_imp_vert
thf(fact_1222_is__loop__def,axiom,
( undire2905028936066782638loop_a
= ( ^ [E: set_a] :
( ( finite_card_a @ E )
= one_one_nat ) ) ) ).
% is_loop_def
thf(fact_1223_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1224_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_1225_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_1226_is__edge__or__loop,axiom,
! [E3: set_a] :
( ( member_set_a @ E3 @ edges )
=> ( ( undire2905028936066782638loop_a @ E3 )
| ( undire4917966558017083288edge_a @ E3 ) ) ) ).
% is_edge_or_loop
thf(fact_1227_ulgraph_Ois__loop__def,axiom,
! [Vertices2: set_set_a,Edges2: set_set_set_a,E3: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices2 @ Edges2 )
=> ( ( undire3618949687197220622_set_a @ E3 )
= ( ( finite_card_set_a @ E3 )
= one_one_nat ) ) ) ).
% ulgraph.is_loop_def
thf(fact_1228_ulgraph_Ois__loop__def,axiom,
! [Vertices2: set_a,Edges2: set_set_a,E3: set_a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ( undire2905028936066782638loop_a @ E3 )
= ( ( finite_card_a @ E3 )
= one_one_nat ) ) ) ).
% ulgraph.is_loop_def
thf(fact_1229_ulgraph_Ois__edge__or__loop,axiom,
! [Vertices2: set_a,Edges2: set_set_a,E3: set_a] :
( ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 )
=> ( ( member_set_a @ E3 @ Edges2 )
=> ( ( undire2905028936066782638loop_a @ E3 )
| ( undire4917966558017083288edge_a @ E3 ) ) ) ) ).
% ulgraph.is_edge_or_loop
thf(fact_1230_ulgraph__def,axiom,
( undire7251896706689453996raph_a
= ( ^ [Vertices: set_a,Edges: set_set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
& ( undire2177556672586781897ioms_a @ Edges ) ) ) ) ).
% ulgraph_def
thf(fact_1231_ulgraph_Ointro,axiom,
! [Vertices2: set_a,Edges2: set_set_a] :
( ( undire2554140024507503526stem_a @ Vertices2 @ Edges2 )
=> ( ( undire2177556672586781897ioms_a @ Edges2 )
=> ( undire7251896706689453996raph_a @ Vertices2 @ Edges2 ) ) ) ).
% ulgraph.intro
thf(fact_1232_mk__edge_Oelims,axiom,
! [X3: product_prod_a_a,Y3: set_a] :
( ( ( undire6670514144573423676edge_a @ X3 )
= Y3 )
=> ~ ! [U2: a,V3: a] :
( ( X3
= ( product_Pair_a_a @ U2 @ V3 ) )
=> ( Y3
!= ( insert_a @ U2 @ ( insert_a @ V3 @ bot_bot_set_a ) ) ) ) ) ).
% mk_edge.elims
thf(fact_1233_all__edges__between__E__ss,axiom,
! [X5: set_a,Y5: set_a] : ( ord_le3724670747650509150_set_a @ ( image_9052089385058188540_set_a @ undire6670514144573423676edge_a @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y5 ) ) @ edges ) ).
% all_edges_between_E_ss
thf(fact_1234_incident__edges__neighbors__img,axiom,
! [V: a] :
( ( undire3231912044278729248dges_a @ edges @ V )
= ( image_a_set_a
@ ^ [U3: a] : ( insert_a @ V @ ( insert_a @ U3 @ bot_bot_set_a ) )
@ ( undire8504279938402040014hood_a @ vertices @ edges @ V ) ) ) ).
% incident_edges_neighbors_img
thf(fact_1235_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1236_incident__edges__def,axiom,
! [V: a] :
( ( undire3231912044278729248dges_a @ edges @ V )
= ( collect_set_a
@ ^ [E: set_a] :
( ( member_set_a @ E @ edges )
& ( undire1521409233611534436dent_a @ V @ E ) ) ) ) ).
% incident_edges_def
thf(fact_1237_incident__loops__def,axiom,
! [V: a] :
( ( undire4753905205749729249oops_a @ edges @ V )
= ( collect_set_a
@ ^ [E: set_a] :
( ( member_set_a @ E @ edges )
& ( E
= ( insert_a @ V @ bot_bot_set_a ) ) ) ) ) ).
% incident_loops_def
thf(fact_1238_edges__split__loop__inter__empty,axiom,
( bot_bot_set_set_a
= ( inf_inf_set_set_a
@ ( collect_set_a
@ ^ [E: set_a] :
( ( member_set_a @ E @ edges )
& ( undire2905028936066782638loop_a @ E ) ) )
@ ( collect_set_a
@ ^ [E: set_a] :
( ( member_set_a @ E @ edges )
& ( undire4917966558017083288edge_a @ E ) ) ) ) ) ).
% edges_split_loop_inter_empty
thf(fact_1239_neighborhood__def,axiom,
! [X3: a] :
( ( undire8504279938402040014hood_a @ vertices @ edges @ X3 )
= ( collect_a
@ ^ [V4: a] :
( ( member_a @ V4 @ vertices )
& ( undire397441198561214472_adj_a @ edges @ X3 @ V4 ) ) ) ) ).
% neighborhood_def
thf(fact_1240_incident__loops__alt,axiom,
! [V: a] :
( ( undire4753905205749729249oops_a @ edges @ V )
= ( collect_set_a
@ ^ [E: set_a] :
( ( member_set_a @ E @ edges )
& ( undire1521409233611534436dent_a @ V @ E )
& ( ( finite_card_a @ E )
= one_one_nat ) ) ) ) ).
% incident_loops_alt
thf(fact_1241_edges__split__loop,axiom,
( edges
= ( sup_sup_set_set_a
@ ( collect_set_a
@ ^ [E: set_a] :
( ( member_set_a @ E @ edges )
& ( undire2905028936066782638loop_a @ E ) ) )
@ ( collect_set_a
@ ^ [E: set_a] :
( ( member_set_a @ E @ edges )
& ( undire4917966558017083288edge_a @ E ) ) ) ) ) ).
% edges_split_loop
thf(fact_1242_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1243_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_1244_neighbors__ss__def,axiom,
! [X3: a,Y5: set_a] :
( ( undire401937927514038589s_ss_a @ edges @ X3 @ Y5 )
= ( collect_a
@ ^ [Y2: a] :
( ( member_a @ Y2 @ Y5 )
& ( undire397441198561214472_adj_a @ edges @ X3 @ Y2 ) ) ) ) ).
% neighbors_ss_def
thf(fact_1245_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_1246_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1247_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N5: nat] : ( ord_less_eq_nat @ N5 @ ( F @ N5 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1248_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1249_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1250_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1251_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1252_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_1253_diff__less__mono,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B3 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1254_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1255_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1256_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B3 ) )
= ( ord_less_eq_nat @ B3 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1257_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_1258_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1259_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1260_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1261_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1262_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1263_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_1264_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less_nat @ K3 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_1265_bounded__nat__set__is__finite,axiom,
! [N6: set_nat,N: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ N6 )
=> ( ord_less_nat @ X2 @ N ) )
=> ( finite_finite_nat @ N6 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1266_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M: nat] :
! [X: nat] :
( ( member_nat @ X @ N4 )
=> ( ord_less_nat @ X @ M ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1267_infinite__nat__iff__unbounded,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M: nat] :
? [N2: nat] :
( ( ord_less_nat @ M @ N2 )
& ( member_nat @ N2 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_1268_unbounded__k__infinite,axiom,
! [K: nat,S2: set_nat] :
( ! [M4: nat] :
( ( ord_less_nat @ K @ M4 )
=> ? [N7: nat] :
( ( ord_less_nat @ M4 @ N7 )
& ( member_nat @ N7 @ S2 ) ) )
=> ~ ( finite_finite_nat @ S2 ) ) ).
% unbounded_k_infinite
thf(fact_1269_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1270_incident__sedges__union,axiom,
( ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ ( undire1270416042309875431dges_a @ edges ) @ vertices ) )
= ( collect_set_a
@ ^ [E: set_a] :
( ( member_set_a @ E @ edges )
& ( undire4917966558017083288edge_a @ E ) ) ) ) ).
% incident_sedges_union
thf(fact_1271_incident__loops__union,axiom,
( ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ ( undire4753905205749729249oops_a @ edges ) @ vertices ) )
= ( collect_set_a
@ ^ [E: set_a] :
( ( member_set_a @ E @ edges )
& ( undire2905028936066782638loop_a @ E ) ) ) ) ).
% incident_loops_union
thf(fact_1272_is__loop__set__alt,axiom,
( ( collect_set_a
@ ^ [Uu: set_a] :
? [V4: a] :
( ( Uu
= ( insert_a @ V4 @ bot_bot_set_a ) )
& ( undire3617971648856834880loop_a @ edges @ V4 ) ) )
= ( collect_set_a
@ ^ [E: set_a] :
( ( member_set_a @ E @ edges )
& ( undire2905028936066782638loop_a @ E ) ) ) ) ).
% is_loop_set_alt
thf(fact_1273_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1274_all__edges__between__def,axiom,
! [X5: set_a,Y5: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X5 @ Y5 )
= ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X: a,Y2: a] :
( ( member_a @ X @ X5 )
& ( member_a @ Y2 @ Y5 )
& ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y2 @ bot_bot_set_a ) ) @ edges ) ) ) ) ) ).
% all_edges_between_def
% Conjectures (1)
thf(conj_0,conjecture,
( ( undire8383842906760478443ween_a @ edges @ bot_bot_set_a @ z )
= bot_bo3357376287454694259od_a_a ) ).
%------------------------------------------------------------------------------