TPTP Problem File: SLH0033^1.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_00959_037551__13126020_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1407 ( 567 unt; 128 typ; 0 def)
% Number of atoms : 3636 (1154 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 11088 ( 383 ~; 52 |; 250 &;8787 @)
% ( 0 <=>;1616 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 487 ( 487 >; 0 *; 0 +; 0 <<)
% Number of symbols : 124 ( 123 usr; 11 con; 0-4 aty)
% Number of variables : 3428 ( 240 ^;3116 !; 72 ?;3428 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:32:58.317
%------------------------------------------------------------------------------
% Could-be-implicit typings (5)
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thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_tf__a,type,
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% Explicit typings (123)
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thf(sy_c_Zorn_Ochains_001tf__a,type,
chains_a: set_set_a > set_set_set_a ).
thf(sy_c_Zorn_Opred__on_Ochain_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
pred_chain_set_set_a: set_set_set_a > ( set_set_a > set_set_a > $o ) > set_set_set_a > $o ).
thf(sy_c_Zorn_Opred__on_Ochain_001t__Set__Oset_Itf__a_J,type,
pred_chain_set_a: set_set_a > ( set_a > set_a > $o ) > set_set_a > $o ).
thf(sy_c_Zorn_Opred__on_Ochain_001tf__a,type,
pred_chain_a: set_a > ( a > a > $o ) > set_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
member_set_set_set_a: set_set_set_a > set_set_set_set_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_EH1,type,
eH1: set_set_a ).
thf(sy_v_EH2,type,
eH2: set_set_a ).
thf(sy_v_S,type,
s: set_a ).
thf(sy_v_T,type,
t: set_a ).
thf(sy_v_VH1,type,
vH1: set_a ).
thf(sy_v_VH2,type,
vH2: set_a ).
thf(sy_v_edges,type,
edges: set_set_a ).
thf(sy_v_vertices,type,
vertices: set_a ).
% Relevant facts (1278)
thf(fact_0_induced__graph_Osubgraph__refl,axiom,
! [V: set_a] : ( undire7103218114511261257raph_a @ V @ ( undire7777452895879145676dges_a @ edges @ V ) @ V @ ( undire7777452895879145676dges_a @ edges @ V ) ) ).
% induced_graph.subgraph_refl
thf(fact_1_assms_I5_J,axiom,
undire7103218114511261257raph_a @ ( sup_sup_set_a @ vH1 @ vH2 ) @ ( sup_sup_set_set_a @ eH1 @ eH2 ) @ ( sup_sup_set_a @ s @ t ) @ ( undire7777452895879145676dges_a @ edges @ ( sup_sup_set_a @ s @ t ) ) ).
% assms(5)
thf(fact_2_assms_I2_J,axiom,
ord_less_eq_set_a @ vH2 @ t ).
% assms(2)
thf(fact_3_assms_I1_J,axiom,
ord_less_eq_set_a @ vH1 @ s ).
% assms(1)
thf(fact_4_assms_I4_J,axiom,
undire2554140024507503526stem_a @ vH2 @ eH2 ).
% assms(4)
thf(fact_5_assms_I3_J,axiom,
undire2554140024507503526stem_a @ vH1 @ eH1 ).
% assms(3)
thf(fact_6_subgraph_Osubgraph__antisym,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a,V: set_a,E: set_set_a,V2: set_a,E2: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ( undire7103218114511261257raph_a @ V @ E @ V2 @ E2 )
=> ( ( undire7103218114511261257raph_a @ V2 @ E2 @ V @ E )
=> ( ( V2 = V )
& ( E2 = E ) ) ) ) ) ).
% subgraph.subgraph_antisym
thf(fact_7_graph__system_Oinduced__edges_Ocong,axiom,
undire7777452895879145676dges_a = undire7777452895879145676dges_a ).
% graph_system.induced_edges.cong
thf(fact_8_induced__graph_Oedge__adj__inE,axiom,
! [V: set_a,E1: set_a,E22: set_a] :
( ( undire4022703626023482010_adj_a @ ( undire7777452895879145676dges_a @ edges @ V ) @ E1 @ E22 )
=> ( ( member_set_a @ E1 @ ( undire7777452895879145676dges_a @ edges @ V ) )
& ( member_set_a @ E22 @ ( undire7777452895879145676dges_a @ edges @ V ) ) ) ) ).
% induced_graph.edge_adj_inE
thf(fact_9_induced__graph_Oedge__adjacent__alt__def,axiom,
! [E1: set_a,V: set_a,E22: set_a] :
( ( member_set_a @ E1 @ ( undire7777452895879145676dges_a @ edges @ V ) )
=> ( ( member_set_a @ E22 @ ( undire7777452895879145676dges_a @ edges @ V ) )
=> ( ? [X: a] :
( ( member_a @ X @ V )
& ( member_a @ X @ E1 )
& ( member_a @ X @ E22 ) )
=> ( undire4022703626023482010_adj_a @ ( undire7777452895879145676dges_a @ edges @ V ) @ E1 @ E22 ) ) ) ) ).
% induced_graph.edge_adjacent_alt_def
thf(fact_10_edge__adj__inE,axiom,
! [E1: set_a,E22: set_a] :
( ( undire4022703626023482010_adj_a @ edges @ E1 @ E22 )
=> ( ( member_set_a @ E1 @ edges )
& ( member_set_a @ E22 @ edges ) ) ) ).
% edge_adj_inE
thf(fact_11_induced__graph_Oincident__edge__in__wf,axiom,
! [E3: set_a,V: set_a,V3: a] :
( ( member_set_a @ E3 @ ( undire7777452895879145676dges_a @ edges @ V ) )
=> ( ( undire1521409233611534436dent_a @ V3 @ E3 )
=> ( member_a @ V3 @ V ) ) ) ).
% induced_graph.incident_edge_in_wf
thf(fact_12_subgraph__refl,axiom,
undire7103218114511261257raph_a @ vertices @ edges @ vertices @ edges ).
% subgraph_refl
thf(fact_13_induced__graph_Owellformed,axiom,
! [E3: set_a,V: set_a] :
( ( member_set_a @ E3 @ ( undire7777452895879145676dges_a @ edges @ V ) )
=> ( ord_less_eq_set_a @ E3 @ V ) ) ).
% induced_graph.wellformed
thf(fact_14_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_15_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_16_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_17_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_18_sup_Oidem,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_19_sup_Oidem,axiom,
! [A2: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_20_sup__idem,axiom,
! [X2: set_a] :
( ( sup_sup_set_a @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_21_sup__idem,axiom,
! [X2: set_set_a] :
( ( sup_sup_set_set_a @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_22_sup_Oleft__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_23_sup_Oleft__idem,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ ( sup_sup_set_set_a @ A2 @ B2 ) )
= ( sup_sup_set_set_a @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_24_sup__left__idem,axiom,
! [X2: set_a,Y: set_a] :
( ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y ) )
= ( sup_sup_set_a @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_25_sup__left__idem,axiom,
! [X2: set_set_a,Y: set_set_a] :
( ( sup_sup_set_set_a @ X2 @ ( sup_sup_set_set_a @ X2 @ Y ) )
= ( sup_sup_set_set_a @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_26_incident__def,axiom,
undire1521409233611534436dent_a = member_a ).
% incident_def
thf(fact_27_graph__system__axioms,axiom,
undire2554140024507503526stem_a @ vertices @ edges ).
% graph_system_axioms
thf(fact_28_induced__is__graph__sys,axiom,
! [V: set_a] : ( undire2554140024507503526stem_a @ V @ ( undire7777452895879145676dges_a @ edges @ V ) ) ).
% induced_is_graph_sys
thf(fact_29_induced__graph_Oinduced__is__graph__sys,axiom,
! [V: set_a,V_a: set_a] : ( undire2554140024507503526stem_a @ V @ ( undire7777452895879145676dges_a @ ( undire7777452895879145676dges_a @ edges @ V_a ) @ V ) ) ).
% induced_graph.induced_is_graph_sys
thf(fact_30_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_31_subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( member_set_a @ X3 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% subsetI
thf(fact_32_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_33_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_34_sup_Oright__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_35_sup_Oright__idem,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_set_a @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_36_wellformed,axiom,
! [E3: set_a] :
( ( member_set_a @ E3 @ edges )
=> ( ord_less_eq_set_a @ E3 @ vertices ) ) ).
% wellformed
thf(fact_37_edge__adjacent__alt__def,axiom,
! [E1: set_a,E22: set_a] :
( ( member_set_a @ E1 @ edges )
=> ( ( member_set_a @ E22 @ edges )
=> ( ? [X: a] :
( ( member_a @ X @ vertices )
& ( member_a @ X @ E1 )
& ( member_a @ X @ E22 ) )
=> ( undire4022703626023482010_adj_a @ edges @ E1 @ E22 ) ) ) ) ).
% edge_adjacent_alt_def
thf(fact_38_incident__edge__in__wf,axiom,
! [E3: set_a,V3: a] :
( ( member_set_a @ E3 @ edges )
=> ( ( undire1521409233611534436dent_a @ V3 @ E3 )
=> ( member_a @ V3 @ vertices ) ) ) ).
% incident_edge_in_wf
thf(fact_39_induced__is__subgraph,axiom,
! [V: set_a] :
( ( ord_less_eq_set_a @ V @ vertices )
=> ( undire7103218114511261257raph_a @ V @ ( undire7777452895879145676dges_a @ edges @ V ) @ vertices @ edges ) ) ).
% induced_is_subgraph
thf(fact_40_sup_Obounded__iff,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_set_a @ B2 @ A2 )
& ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_41_sup_Obounded__iff,axiom,
! [B2: set_set_a,C: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B2 @ C ) @ A2 )
= ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
& ( ord_le3724670747650509150_set_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_42_le__sup__iff,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X2 @ Y ) @ Z )
= ( ( ord_less_eq_set_a @ X2 @ Z )
& ( ord_less_eq_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_43_le__sup__iff,axiom,
! [X2: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ X2 @ Y ) @ Z )
= ( ( ord_le3724670747650509150_set_a @ X2 @ Z )
& ( ord_le3724670747650509150_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_44_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_45_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_46_is__subgraphI,axiom,
! [V: set_set_a,V2: set_set_a,E: set_set_set_a,E2: set_set_set_a] :
( ( ord_le3724670747650509150_set_a @ V @ V2 )
=> ( ( ord_le5722252365846178494_set_a @ E @ E2 )
=> ( ( undire7159349782766787846_set_a @ V @ E )
=> ( ( undire7159349782766787846_set_a @ V2 @ E2 )
=> ( undire1186139521737116585_set_a @ V @ E @ V2 @ E2 ) ) ) ) ) ).
% is_subgraphI
thf(fact_47_is__subgraphI,axiom,
! [V: set_a,V2: set_a,E: set_set_a,E2: set_set_a] :
( ( ord_less_eq_set_a @ V @ V2 )
=> ( ( ord_le3724670747650509150_set_a @ E @ E2 )
=> ( ( undire2554140024507503526stem_a @ V @ E )
=> ( ( undire2554140024507503526stem_a @ V2 @ E2 )
=> ( undire7103218114511261257raph_a @ V @ E @ V2 @ E2 ) ) ) ) ) ).
% is_subgraphI
thf(fact_48_graph__system_Ointro,axiom,
! [Edges: set_set_set_a,Vertices: set_set_a] :
( ! [E4: set_set_a] :
( ( member_set_set_a @ E4 @ Edges )
=> ( ord_le3724670747650509150_set_a @ E4 @ Vertices ) )
=> ( undire7159349782766787846_set_a @ Vertices @ Edges ) ) ).
% graph_system.intro
thf(fact_49_graph__system_Ointro,axiom,
! [Edges: set_set_a,Vertices: set_a] :
( ! [E4: set_a] :
( ( member_set_a @ E4 @ Edges )
=> ( ord_less_eq_set_a @ E4 @ Vertices ) )
=> ( undire2554140024507503526stem_a @ Vertices @ Edges ) ) ).
% graph_system.intro
thf(fact_50_graph__system_Owellformed,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,E3: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( member_set_set_a @ E3 @ Edges )
=> ( ord_le3724670747650509150_set_a @ E3 @ Vertices ) ) ) ).
% graph_system.wellformed
thf(fact_51_graph__system_Owellformed,axiom,
! [Vertices: set_a,Edges: set_set_a,E3: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( member_set_a @ E3 @ Edges )
=> ( ord_less_eq_set_a @ E3 @ Vertices ) ) ) ).
% graph_system.wellformed
thf(fact_52_comp__sgraph_Oincident__def,axiom,
undire2320338297334612420_set_a = member_set_a ).
% comp_sgraph.incident_def
thf(fact_53_comp__sgraph_Oincident__def,axiom,
undire1521409233611534436dent_a = member_a ).
% comp_sgraph.incident_def
thf(fact_54_graph__system_Oedge__adj__inE,axiom,
! [Vertices: set_a,Edges: set_set_a,E1: set_a,E22: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( undire4022703626023482010_adj_a @ Edges @ E1 @ E22 )
=> ( ( member_set_a @ E1 @ Edges )
& ( member_set_a @ E22 @ Edges ) ) ) ) ).
% graph_system.edge_adj_inE
thf(fact_55_graph__system_Oincident__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V3: set_a,E3: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( undire2320338297334612420_set_a @ V3 @ E3 )
= ( member_set_a @ V3 @ E3 ) ) ) ).
% graph_system.incident_def
thf(fact_56_graph__system_Oincident__def,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a,E3: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( undire1521409233611534436dent_a @ V3 @ E3 )
= ( member_a @ V3 @ E3 ) ) ) ).
% graph_system.incident_def
thf(fact_57_graph__system_Oinduced__edges__ss,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( ord_le3724670747650509150_set_a @ V @ Vertices )
=> ( ord_le5722252365846178494_set_a @ ( undire7854589003810675244_set_a @ Edges @ V ) @ Edges ) ) ) ).
% graph_system.induced_edges_ss
thf(fact_58_graph__system_Oinduced__edges__ss,axiom,
! [Vertices: set_a,Edges: set_set_a,V: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( ord_less_eq_set_a @ V @ Vertices )
=> ( ord_le3724670747650509150_set_a @ ( undire7777452895879145676dges_a @ Edges @ V ) @ Edges ) ) ) ).
% graph_system.induced_edges_ss
thf(fact_59_graph__system_Oincident__edge__in__wf,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,E3: set_set_a,V3: set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( member_set_set_a @ E3 @ Edges )
=> ( ( undire2320338297334612420_set_a @ V3 @ E3 )
=> ( member_set_a @ V3 @ Vertices ) ) ) ) ).
% graph_system.incident_edge_in_wf
thf(fact_60_graph__system_Oincident__edge__in__wf,axiom,
! [Vertices: set_a,Edges: set_set_a,E3: set_a,V3: a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( member_set_a @ E3 @ Edges )
=> ( ( undire1521409233611534436dent_a @ V3 @ E3 )
=> ( member_a @ V3 @ Vertices ) ) ) ) ).
% graph_system.incident_edge_in_wf
thf(fact_61_graph__system_Oedge__adjacent__alt__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,E1: set_set_a,E22: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( member_set_set_a @ E1 @ Edges )
=> ( ( member_set_set_a @ E22 @ Edges )
=> ( ? [X: set_a] :
( ( member_set_a @ X @ Vertices )
& ( member_set_a @ X @ E1 )
& ( member_set_a @ X @ E22 ) )
=> ( undire3485422320110889978_set_a @ Edges @ E1 @ E22 ) ) ) ) ) ).
% graph_system.edge_adjacent_alt_def
thf(fact_62_graph__system_Oedge__adjacent__alt__def,axiom,
! [Vertices: set_a,Edges: set_set_a,E1: set_a,E22: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( member_set_a @ E1 @ Edges )
=> ( ( member_set_a @ E22 @ Edges )
=> ( ? [X: a] :
( ( member_a @ X @ Vertices )
& ( member_a @ X @ E1 )
& ( member_a @ X @ E22 ) )
=> ( undire4022703626023482010_adj_a @ Edges @ E1 @ E22 ) ) ) ) ) ).
% graph_system.edge_adjacent_alt_def
thf(fact_63_in__mono,axiom,
! [A: set_a,B: set_a,X2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_64_in__mono,axiom,
! [A: set_set_a,B: set_set_a,X2: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ X2 @ A )
=> ( member_set_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_65_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_66_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_67_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X4: set_a] : ( member_set_a @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_68_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_69_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_70_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_71_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_72_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_73_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [X4: a] :
( ( member_a @ X4 @ A3 )
=> ( member_a @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_74_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
! [X4: set_a] :
( ( member_set_a @ X4 @ A3 )
=> ( member_set_a @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_75_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_76_equalityD1,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_77_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_78_equalityD2,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le3724670747650509150_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_79_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [T: a] :
( ( member_a @ T @ A3 )
=> ( member_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_80_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A3 )
=> ( member_set_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_81_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_82_subset__refl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% subset_refl
thf(fact_83_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_84_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X3: set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_85_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_86_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_87_set__eq__subset,axiom,
( ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_88_set__eq__subset,axiom,
( ( ^ [Y2: set_set_a,Z2: set_set_a] : ( Y2 = Z2 ) )
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_89_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_90_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X4: set_a] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_91_graph__system__def,axiom,
( undire7159349782766787846_set_a
= ( ^ [Vertices2: set_set_a,Edges2: set_set_set_a] :
! [E5: set_set_a] :
( ( member_set_set_a @ E5 @ Edges2 )
=> ( ord_le3724670747650509150_set_a @ E5 @ Vertices2 ) ) ) ) ).
% graph_system_def
thf(fact_92_graph__system__def,axiom,
( undire2554140024507503526stem_a
= ( ^ [Vertices2: set_a,Edges2: set_set_a] :
! [E5: set_a] :
( ( member_set_a @ E5 @ Edges2 )
=> ( ord_less_eq_set_a @ E5 @ Vertices2 ) ) ) ) ).
% graph_system_def
thf(fact_93_graph__system_Oedge__adj_Ocong,axiom,
undire4022703626023482010_adj_a = undire4022703626023482010_adj_a ).
% graph_system.edge_adj.cong
thf(fact_94_sup_OcoboundedI2,axiom,
! [C: set_a,B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_95_sup_OcoboundedI2,axiom,
! [C: set_set_a,B2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ B2 )
=> ( ord_le3724670747650509150_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_96_sup_OcoboundedI1,axiom,
! [C: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_97_sup_OcoboundedI1,axiom,
! [C: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ A2 )
=> ( ord_le3724670747650509150_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_98_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( sup_sup_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_99_sup_Oabsorb__iff2,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( sup_sup_set_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_100_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( sup_sup_set_a @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_101_sup_Oabsorb__iff1,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B4: set_set_a,A4: set_set_a] :
( ( sup_sup_set_set_a @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_102_sup_Ocobounded2,axiom,
! [B2: set_a,A2: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_103_sup_Ocobounded2,axiom,
! [B2: set_set_a,A2: set_set_a] : ( ord_le3724670747650509150_set_a @ B2 @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_104_sup_Ocobounded1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_105_sup_Ocobounded1,axiom,
! [A2: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_106_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( A4
= ( sup_sup_set_a @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_107_sup_Oorder__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B4: set_set_a,A4: set_set_a] :
( A4
= ( sup_sup_set_set_a @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_108_sup_OboundedI,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_109_sup_OboundedI,axiom,
! [B2: set_set_a,A2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ C @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_110_sup_OboundedE,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ~ ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_111_sup_OboundedE,axiom,
! [B2: set_set_a,C: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ~ ( ord_le3724670747650509150_set_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_112_sup__absorb2,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( sup_sup_set_a @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_113_sup__absorb2,axiom,
! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ( sup_sup_set_set_a @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_114_sup__absorb1,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ( ( sup_sup_set_a @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_115_sup__absorb1,axiom,
! [Y: set_set_a,X2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X2 )
=> ( ( sup_sup_set_set_a @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_116_sup_Oabsorb2,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_117_sup_Oabsorb2,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_set_a @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_118_sup_Oabsorb1,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_119_sup_Oabsorb1,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_set_a @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_120_sup__unique,axiom,
! [F: set_a > set_a > set_a,X2: set_a,Y: set_a] :
( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_a,Y3: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X3 )
=> ( ( ord_less_eq_set_a @ Z3 @ X3 )
=> ( ord_less_eq_set_a @ ( F @ Y3 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_set_a @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_121_sup__unique,axiom,
! [F: set_set_a > set_set_a > set_set_a,X2: set_set_a,Y: set_set_a] :
( ! [X3: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_set_a,Y3: set_set_a,Z3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y3 @ X3 )
=> ( ( ord_le3724670747650509150_set_a @ Z3 @ X3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ Y3 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_set_set_a @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_122_sup_OorderI,axiom,
! [A2: set_a,B2: set_a] :
( ( A2
= ( sup_sup_set_a @ A2 @ B2 ) )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_123_sup_OorderI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2
= ( sup_sup_set_set_a @ A2 @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_124_sup_OorderE,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_125_sup_OorderE,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_126_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X4: set_a,Y4: set_a] :
( ( sup_sup_set_a @ X4 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_127_le__iff__sup,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X4: set_set_a,Y4: set_set_a] :
( ( sup_sup_set_set_a @ X4 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_128_sup__least,axiom,
! [Y: set_a,X2: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ( ( ord_less_eq_set_a @ Z @ X2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_129_sup__least,axiom,
! [Y: set_set_a,X2: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X2 )
=> ( ( ord_le3724670747650509150_set_a @ Z @ X2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ Y @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_130_sup__mono,axiom,
! [A2: set_a,C: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_131_sup__mono,axiom,
! [A2: set_set_a,C: set_set_a,B2: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ D )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ ( sup_sup_set_set_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_132_sup_Omono,axiom,
! [C: set_a,A2: set_a,D: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ D @ B2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D ) @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_133_sup_Omono,axiom,
! [C: set_set_a,A2: set_set_a,D: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ D @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ C @ D ) @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_134_le__supI2,axiom,
! [X2: set_a,B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ X2 @ B2 )
=> ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_135_le__supI2,axiom,
! [X2: set_set_a,B2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ X2 @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_136_le__supI1,axiom,
! [X2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_137_le__supI1,axiom,
! [X2: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ A2 )
=> ( ord_le3724670747650509150_set_a @ X2 @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_138_sup__ge2,axiom,
! [Y: set_a,X2: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_139_sup__ge2,axiom,
! [Y: set_set_a,X2: set_set_a] : ( ord_le3724670747650509150_set_a @ Y @ ( sup_sup_set_set_a @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_140_sup__ge1,axiom,
! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_141_sup__ge1,axiom,
! [X2: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ X2 @ ( sup_sup_set_set_a @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_142_le__supI,axiom,
! [A2: set_a,X2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ X2 )
=> ( ( ord_less_eq_set_a @ B2 @ X2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_143_le__supI,axiom,
! [A2: set_set_a,X2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ X2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ X2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_144_le__supE,axiom,
! [A2: set_a,B2: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ X2 )
=> ~ ( ord_less_eq_set_a @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_145_le__supE,axiom,
! [A2: set_set_a,B2: set_set_a,X2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ X2 )
=> ~ ( ord_le3724670747650509150_set_a @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_146_inf__sup__ord_I3_J,axiom,
! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_147_inf__sup__ord_I3_J,axiom,
! [X2: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ X2 @ ( sup_sup_set_set_a @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_148_inf__sup__ord_I4_J,axiom,
! [Y: set_a,X2: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_149_inf__sup__ord_I4_J,axiom,
! [Y: set_set_a,X2: set_set_a] : ( ord_le3724670747650509150_set_a @ Y @ ( sup_sup_set_set_a @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_150_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( sup_sup_set_a @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_151_subset__Un__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( sup_sup_set_set_a @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_152_subset__UnE,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B ) )
=> ~ ! [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ A )
=> ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ B )
=> ( C2
!= ( sup_sup_set_a @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_153_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 ) )
=> ~ ! [A5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ A )
=> ! [B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ B )
=> ( C2
!= ( sup_sup_set_set_a @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_154_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_155_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_156_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_157_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_158_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_159_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_160_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_161_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_162_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_163_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_164_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_165_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_166_graph__system_Oinduced__edges__union,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,VH1: set_set_a,S: set_set_a,VH2: set_set_a,T2: set_set_a,EH1: set_set_set_a,EH2: set_set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( ord_le3724670747650509150_set_a @ VH1 @ S )
=> ( ( ord_le3724670747650509150_set_a @ VH2 @ T2 )
=> ( ( undire7159349782766787846_set_a @ VH1 @ EH1 )
=> ( ( undire7159349782766787846_set_a @ VH2 @ EH2 )
=> ( ( ord_le5722252365846178494_set_a @ ( sup_su2076012971530813682_set_a @ EH1 @ EH2 ) @ ( undire7854589003810675244_set_a @ Edges @ ( sup_sup_set_set_a @ S @ T2 ) ) )
=> ( ord_le5722252365846178494_set_a @ EH1 @ ( undire7854589003810675244_set_a @ Edges @ S ) ) ) ) ) ) ) ) ).
% graph_system.induced_edges_union
thf(fact_167_graph__system_Oinduced__edges__union,axiom,
! [Vertices: set_a,Edges: set_set_a,VH1: set_a,S: set_a,VH2: set_a,T2: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T2 )
=> ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
=> ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
=> ( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( undire7777452895879145676dges_a @ Edges @ ( sup_sup_set_a @ S @ T2 ) ) )
=> ( ord_le3724670747650509150_set_a @ EH1 @ ( undire7777452895879145676dges_a @ Edges @ S ) ) ) ) ) ) ) ) ).
% graph_system.induced_edges_union
thf(fact_168_graph__system_Oinduced__is__subgraph,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( ord_le3724670747650509150_set_a @ V @ Vertices )
=> ( undire1186139521737116585_set_a @ V @ ( undire7854589003810675244_set_a @ Edges @ V ) @ Vertices @ Edges ) ) ) ).
% graph_system.induced_is_subgraph
thf(fact_169_graph__system_Oinduced__is__subgraph,axiom,
! [Vertices: set_a,Edges: set_set_a,V: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( ord_less_eq_set_a @ V @ Vertices )
=> ( undire7103218114511261257raph_a @ V @ ( undire7777452895879145676dges_a @ Edges @ V ) @ Vertices @ Edges ) ) ) ).
% graph_system.induced_is_subgraph
thf(fact_170_graph__system_Oinduced__is__graph__sys,axiom,
! [Vertices: set_a,Edges: set_set_a,V: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( undire2554140024507503526stem_a @ V @ ( undire7777452895879145676dges_a @ Edges @ V ) ) ) ).
% graph_system.induced_is_graph_sys
thf(fact_171_subgraph_Overts__ss,axiom,
! [V_H: set_set_a,E_H: set_set_set_a,V_G: set_set_a,E_G: set_set_set_a] :
( ( undire1186139521737116585_set_a @ V_H @ E_H @ V_G @ E_G )
=> ( ord_le3724670747650509150_set_a @ V_H @ V_G ) ) ).
% subgraph.verts_ss
thf(fact_172_subgraph_Overts__ss,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ord_less_eq_set_a @ V_H @ V_G ) ) ).
% subgraph.verts_ss
thf(fact_173_graph__system_Osubgraph__refl,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( undire7103218114511261257raph_a @ Vertices @ Edges @ Vertices @ Edges ) ) ).
% graph_system.subgraph_refl
thf(fact_174_subgraph_Osubgraph__trans,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a,V2: set_a,E2: set_set_a,V: set_a,E: set_set_a,V4: set_a,E6: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ( undire2554140024507503526stem_a @ V2 @ E2 )
=> ( ( undire2554140024507503526stem_a @ V @ E )
=> ( ( undire2554140024507503526stem_a @ V4 @ E6 )
=> ( ( undire7103218114511261257raph_a @ V4 @ E6 @ V @ E )
=> ( ( undire7103218114511261257raph_a @ V @ E @ V2 @ E2 )
=> ( undire7103218114511261257raph_a @ V4 @ E6 @ V2 @ E2 ) ) ) ) ) ) ) ).
% subgraph.subgraph_trans
thf(fact_175_subgraph_Oaxioms_I1_J,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( undire2554140024507503526stem_a @ V_H @ E_H ) ) ).
% subgraph.axioms(1)
thf(fact_176_subgraph_Oaxioms_I2_J,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( undire2554140024507503526stem_a @ V_G @ E_G ) ) ).
% subgraph.axioms(2)
thf(fact_177_sup__left__commute,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X2 @ Z ) ) ) ).
% sup_left_commute
thf(fact_178_sup__left__commute,axiom,
! [X2: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ X2 @ ( sup_sup_set_set_a @ Y @ Z ) )
= ( sup_sup_set_set_a @ Y @ ( sup_sup_set_set_a @ X2 @ Z ) ) ) ).
% sup_left_commute
thf(fact_179_sup_Oleft__commute,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A2 @ C ) )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_180_sup_Oleft__commute,axiom,
! [B2: set_set_a,A2: set_set_a,C: set_set_a] :
( ( sup_sup_set_set_a @ B2 @ ( sup_sup_set_set_a @ A2 @ C ) )
= ( sup_sup_set_set_a @ A2 @ ( sup_sup_set_set_a @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_181_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X4: set_a,Y4: set_a] : ( sup_sup_set_a @ Y4 @ X4 ) ) ) ).
% sup_commute
thf(fact_182_sup__commute,axiom,
( sup_sup_set_set_a
= ( ^ [X4: set_set_a,Y4: set_set_a] : ( sup_sup_set_set_a @ Y4 @ X4 ) ) ) ).
% sup_commute
thf(fact_183_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B4: set_a] : ( sup_sup_set_a @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_184_sup_Ocommute,axiom,
( sup_sup_set_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] : ( sup_sup_set_set_a @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_185_sup__assoc,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X2 @ Y ) @ Z )
= ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_186_sup__assoc,axiom,
! [X2: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ X2 @ Y ) @ Z )
= ( sup_sup_set_set_a @ X2 @ ( sup_sup_set_set_a @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_187_sup_Oassoc,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_188_sup_Oassoc,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ C )
= ( sup_sup_set_set_a @ A2 @ ( sup_sup_set_set_a @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_189_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X4: set_a,Y4: set_a] : ( sup_sup_set_a @ Y4 @ X4 ) ) ) ).
% inf_sup_aci(5)
thf(fact_190_inf__sup__aci_I5_J,axiom,
( sup_sup_set_set_a
= ( ^ [X4: set_set_a,Y4: set_set_a] : ( sup_sup_set_set_a @ Y4 @ X4 ) ) ) ).
% inf_sup_aci(5)
thf(fact_191_inf__sup__aci_I6_J,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X2 @ Y ) @ Z )
= ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_192_inf__sup__aci_I6_J,axiom,
! [X2: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ X2 @ Y ) @ Z )
= ( sup_sup_set_set_a @ X2 @ ( sup_sup_set_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_193_inf__sup__aci_I7_J,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_194_inf__sup__aci_I7_J,axiom,
! [X2: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ X2 @ ( sup_sup_set_set_a @ Y @ Z ) )
= ( sup_sup_set_set_a @ Y @ ( sup_sup_set_set_a @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_195_inf__sup__aci_I8_J,axiom,
! [X2: set_a,Y: set_a] :
( ( sup_sup_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y ) )
= ( sup_sup_set_a @ X2 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_196_inf__sup__aci_I8_J,axiom,
! [X2: set_set_a,Y: set_set_a] :
( ( sup_sup_set_set_a @ X2 @ ( sup_sup_set_set_a @ X2 @ Y ) )
= ( sup_sup_set_set_a @ X2 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_197_Un__left__commute,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) )
= ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_198_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_199_Un__left__absorb,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_a @ A @ B ) ) ).
% Un_left_absorb
thf(fact_200_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_201_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_202_Un__commute,axiom,
( sup_sup_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] : ( sup_sup_set_set_a @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_203_Un__absorb,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_204_Un__absorb,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_205_Un__assoc,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_206_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_207_ball__Un,axiom,
! [A: set_a,B: set_a,P: a > $o] :
( ( ! [X4: a] :
( ( member_a @ X4 @ ( sup_sup_set_a @ A @ B ) )
=> ( P @ X4 ) ) )
= ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( P @ X4 ) )
& ! [X4: a] :
( ( member_a @ X4 @ B )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_208_ball__Un,axiom,
! [A: set_set_a,B: set_set_a,P: set_a > $o] :
( ( ! [X4: set_a] :
( ( member_set_a @ X4 @ ( sup_sup_set_set_a @ A @ B ) )
=> ( P @ X4 ) ) )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ( P @ X4 ) )
& ! [X4: set_a] :
( ( member_set_a @ X4 @ B )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_209_bex__Un,axiom,
! [A: set_a,B: set_a,P: a > $o] :
( ( ? [X4: a] :
( ( member_a @ X4 @ ( sup_sup_set_a @ A @ B ) )
& ( P @ X4 ) ) )
= ( ? [X4: a] :
( ( member_a @ X4 @ A )
& ( P @ X4 ) )
| ? [X4: a] :
( ( member_a @ X4 @ B )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_210_bex__Un,axiom,
! [A: set_set_a,B: set_set_a,P: set_a > $o] :
( ( ? [X4: set_a] :
( ( member_set_a @ X4 @ ( sup_sup_set_set_a @ A @ B ) )
& ( P @ X4 ) ) )
= ( ? [X4: set_a] :
( ( member_set_a @ X4 @ A )
& ( P @ X4 ) )
| ? [X4: set_a] :
( ( member_set_a @ X4 @ B )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_211_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_212_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_213_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_214_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_215_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_216_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_217_induced__edges__union,axiom,
! [VH1: set_a,S: set_a,VH2: set_a,T2: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T2 )
=> ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
=> ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
=> ( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( undire7777452895879145676dges_a @ edges @ ( sup_sup_set_a @ S @ T2 ) ) )
=> ( ord_le3724670747650509150_set_a @ EH1 @ ( undire7777452895879145676dges_a @ edges @ S ) ) ) ) ) ) ) ).
% induced_edges_union
thf(fact_218_induced__edges__ss,axiom,
! [V: set_a] :
( ( ord_less_eq_set_a @ V @ vertices )
=> ( ord_le3724670747650509150_set_a @ ( undire7777452895879145676dges_a @ edges @ V ) @ edges ) ) ).
% induced_edges_ss
thf(fact_219_induced__graph_Oinduced__edges__ss,axiom,
! [V: set_a,V_a: set_a] :
( ( ord_less_eq_set_a @ V @ V_a )
=> ( ord_le3724670747650509150_set_a @ ( undire7777452895879145676dges_a @ ( undire7777452895879145676dges_a @ edges @ V_a ) @ V ) @ ( undire7777452895879145676dges_a @ edges @ V_a ) ) ) ).
% induced_graph.induced_edges_ss
thf(fact_220_order__refl,axiom,
! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_221_order__refl,axiom,
! [X2: set_set_a] : ( ord_le3724670747650509150_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_222_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_223_dual__order_Orefl,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_224_subgraph__def,axiom,
( undire7103218114511261257raph_a
= ( ^ [V_H2: set_a,E_H2: set_set_a,V_G2: set_a,E_G2: set_set_a] :
( ( undire2554140024507503526stem_a @ V_H2 @ E_H2 )
& ( undire2554140024507503526stem_a @ V_G2 @ E_G2 )
& ( undire4675926955456076134ioms_a @ V_H2 @ E_H2 @ V_G2 @ E_G2 ) ) ) ) ).
% subgraph_def
thf(fact_225_subgraph_Ointro,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a] :
( ( undire2554140024507503526stem_a @ V_H @ E_H )
=> ( ( undire2554140024507503526stem_a @ V_G @ E_G )
=> ( ( undire4675926955456076134ioms_a @ V_H @ E_H @ V_G @ E_G )
=> ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G ) ) ) ) ).
% subgraph.intro
thf(fact_226_comp__sgraph_Oinduced__edges__union,axiom,
! [VH1: set_set_a,S: set_set_a,VH2: set_set_a,T2: set_set_a,EH1: set_set_set_a,EH2: set_set_set_a,Sa: set_set_a] :
( ( ord_le3724670747650509150_set_a @ VH1 @ S )
=> ( ( ord_le3724670747650509150_set_a @ VH2 @ T2 )
=> ( ( undire7159349782766787846_set_a @ VH1 @ EH1 )
=> ( ( undire7159349782766787846_set_a @ VH2 @ EH2 )
=> ( ( ord_le5722252365846178494_set_a @ ( sup_su2076012971530813682_set_a @ EH1 @ EH2 ) @ ( undire7854589003810675244_set_a @ ( undire8247866692393712962_set_a @ Sa ) @ ( sup_sup_set_set_a @ S @ T2 ) ) )
=> ( ord_le5722252365846178494_set_a @ EH1 @ ( undire7854589003810675244_set_a @ ( undire8247866692393712962_set_a @ Sa ) @ S ) ) ) ) ) ) ) ).
% comp_sgraph.induced_edges_union
thf(fact_227_comp__sgraph_Oinduced__edges__union,axiom,
! [VH1: set_a,S: set_a,VH2: set_a,T2: set_a,EH1: set_set_a,EH2: set_set_a,Sa: set_a] :
( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T2 )
=> ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
=> ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
=> ( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( undire7777452895879145676dges_a @ ( undire2918257014606996450dges_a @ Sa ) @ ( sup_sup_set_a @ S @ T2 ) ) )
=> ( ord_le3724670747650509150_set_a @ EH1 @ ( undire7777452895879145676dges_a @ ( undire2918257014606996450dges_a @ Sa ) @ S ) ) ) ) ) ) ) ).
% comp_sgraph.induced_edges_union
thf(fact_228_subgraph_Oaxioms_I3_J,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( undire4675926955456076134ioms_a @ V_H @ E_H @ V_G @ E_G ) ) ).
% subgraph.axioms(3)
thf(fact_229_induced__graph_Oedge__adj__def,axiom,
! [V: set_a,E1: set_a,E22: set_a] :
( ( undire4022703626023482010_adj_a @ ( undire7777452895879145676dges_a @ edges @ V ) @ E1 @ E22 )
= ( ( ( inf_inf_set_a @ E1 @ E22 )
!= bot_bot_set_a )
& ( member_set_a @ E1 @ ( undire7777452895879145676dges_a @ edges @ V ) )
& ( member_set_a @ E22 @ ( undire7777452895879145676dges_a @ edges @ V ) ) ) ) ).
% induced_graph.edge_adj_def
thf(fact_230_boolean__algebra__cancel_Osup2,axiom,
! [B: set_a,K: set_a,B2: set_a,A2: set_a] :
( ( B
= ( sup_sup_set_a @ K @ B2 ) )
=> ( ( sup_sup_set_a @ A2 @ B )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_231_boolean__algebra__cancel_Osup2,axiom,
! [B: set_set_a,K: set_set_a,B2: set_set_a,A2: set_set_a] :
( ( B
= ( sup_sup_set_set_a @ K @ B2 ) )
=> ( ( sup_sup_set_set_a @ A2 @ B )
= ( sup_sup_set_set_a @ K @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_232_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_233_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_234_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X4: a] :
~ ( member_a @ X4 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_235_all__not__in__conv,axiom,
! [A: set_set_a] :
( ( ! [X4: set_a] :
~ ( member_set_a @ X4 @ A ) )
= ( A = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_236_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X4: a] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_237_Collect__empty__eq,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( ! [X4: set_a] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_238_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X4: a] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_239_empty__Collect__eq,axiom,
! [P: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P ) )
= ( ! [X4: set_a] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_240_inf_Oidem,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_241_inf__idem,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_242_inf_Oleft__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_243_inf__left__idem,axiom,
! [X2: set_a,Y: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ X2 @ Y ) )
= ( inf_inf_set_a @ X2 @ Y ) ) ).
% inf_left_idem
thf(fact_244_inf_Oright__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 )
= ( inf_inf_set_a @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_245_inf__right__idem,axiom,
! [X2: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ Y )
= ( inf_inf_set_a @ X2 @ Y ) ) ).
% inf_right_idem
thf(fact_246_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_247_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_248_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_249_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_250_edge__adj__def,axiom,
! [E1: set_a,E22: set_a] :
( ( undire4022703626023482010_adj_a @ edges @ E1 @ E22 )
= ( ( ( inf_inf_set_a @ E1 @ E22 )
!= bot_bot_set_a )
& ( member_set_a @ E1 @ edges )
& ( member_set_a @ E22 @ edges ) ) ) ).
% edge_adj_def
thf(fact_251_inf_Obounded__iff,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
= ( ( ord_less_eq_set_a @ A2 @ B2 )
& ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_252_inf_Obounded__iff,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( inf_inf_set_set_a @ B2 @ C ) )
= ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
& ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_253_le__inf__iff,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y @ Z ) )
= ( ( ord_less_eq_set_a @ X2 @ Y )
& ( ord_less_eq_set_a @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_254_le__inf__iff,axiom,
! [X2: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ ( inf_inf_set_set_a @ Y @ Z ) )
= ( ( ord_le3724670747650509150_set_a @ X2 @ Y )
& ( ord_le3724670747650509150_set_a @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_255_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_256_empty__subsetI,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% empty_subsetI
thf(fact_257_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_258_subset__empty,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_259_inf__bot__left,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X2 )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_260_inf__bot__left,axiom,
! [X2: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X2 )
= bot_bot_set_set_a ) ).
% inf_bot_left
thf(fact_261_inf__bot__right,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_262_inf__bot__right,axiom,
! [X2: set_set_a] :
( ( inf_inf_set_set_a @ X2 @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% inf_bot_right
thf(fact_263_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_264_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_set_a] :
( ( inf_inf_set_set_a @ X2 @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_265_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X2 )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_266_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X2 )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_267_sup__bot__left,axiom,
! [X2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_268_sup__bot__left,axiom,
! [X2: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_269_sup__bot__right,axiom,
! [X2: set_a] :
( ( sup_sup_set_a @ X2 @ bot_bot_set_a )
= X2 ) ).
% sup_bot_right
thf(fact_270_sup__bot__right,axiom,
! [X2: set_set_a] :
( ( sup_sup_set_set_a @ X2 @ bot_bot_set_set_a )
= X2 ) ).
% sup_bot_right
thf(fact_271_bot__eq__sup__iff,axiom,
! [X2: set_a,Y: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X2 @ Y ) )
= ( ( X2 = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_272_bot__eq__sup__iff,axiom,
! [X2: set_set_a,Y: set_set_a] :
( ( bot_bot_set_set_a
= ( sup_sup_set_set_a @ X2 @ Y ) )
= ( ( X2 = bot_bot_set_set_a )
& ( Y = bot_bot_set_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_273_sup__eq__bot__iff,axiom,
! [X2: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X2 @ Y )
= bot_bot_set_a )
= ( ( X2 = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_274_sup__eq__bot__iff,axiom,
! [X2: set_set_a,Y: set_set_a] :
( ( ( sup_sup_set_set_a @ X2 @ Y )
= bot_bot_set_set_a )
= ( ( X2 = bot_bot_set_set_a )
& ( Y = bot_bot_set_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_275_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_276_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ( sup_sup_set_set_a @ A2 @ B2 )
= bot_bot_set_set_a )
= ( ( A2 = bot_bot_set_set_a )
& ( B2 = bot_bot_set_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_277_sup__bot_Oleft__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_278_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_279_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_280_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( bot_bot_set_set_a
= ( sup_sup_set_set_a @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_set_a )
& ( B2 = bot_bot_set_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_281_sup__bot_Oright__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_282_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_283_inf__sup__absorb,axiom,
! [X2: set_a,Y: set_a] :
( ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_284_inf__sup__absorb,axiom,
! [X2: set_set_a,Y: set_set_a] :
( ( inf_inf_set_set_a @ X2 @ ( sup_sup_set_set_a @ X2 @ Y ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_285_sup__inf__absorb,axiom,
! [X2: set_a,Y: set_a] :
( ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ X2 @ Y ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_286_sup__inf__absorb,axiom,
! [X2: set_set_a,Y: set_set_a] :
( ( sup_sup_set_set_a @ X2 @ ( inf_inf_set_set_a @ X2 @ Y ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_287_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_288_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_289_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_290_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_291_Int__Un__eq_I4_J,axiom,
! [T2: set_a,S: set_a] :
( ( sup_sup_set_a @ T2 @ ( inf_inf_set_a @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_292_Int__Un__eq_I4_J,axiom,
! [T2: set_set_a,S: set_set_a] :
( ( sup_sup_set_set_a @ T2 @ ( inf_inf_set_set_a @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_293_Int__Un__eq_I3_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_294_Int__Un__eq_I3_J,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( sup_sup_set_set_a @ S @ ( inf_inf_set_set_a @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_295_Int__Un__eq_I2_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_296_Int__Un__eq_I2_J,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_297_Int__Un__eq_I1_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_298_Int__Un__eq_I1_J,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_299_Un__Int__eq_I4_J,axiom,
! [T2: set_a,S: set_a] :
( ( inf_inf_set_a @ T2 @ ( sup_sup_set_a @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_300_Un__Int__eq_I4_J,axiom,
! [T2: set_set_a,S: set_set_a] :
( ( inf_inf_set_set_a @ T2 @ ( sup_sup_set_set_a @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_301_Un__Int__eq_I3_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_302_Un__Int__eq_I3_J,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( inf_inf_set_set_a @ S @ ( sup_sup_set_set_a @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_303_Un__Int__eq_I2_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_304_Un__Int__eq_I2_J,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_305_Un__Int__eq_I1_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_306_Un__Int__eq_I1_J,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_307_inf__sup__aci_I4_J,axiom,
! [X2: set_a,Y: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ X2 @ Y ) )
= ( inf_inf_set_a @ X2 @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_308_inf__sup__aci_I3_J,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X2 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_309_inf__sup__aci_I2_J,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ Z )
= ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_310_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X4: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X4 ) ) ) ).
% inf_sup_aci(1)
thf(fact_311_inf_Oassoc,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_312_inf__assoc,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ Z )
= ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_313_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B4: set_a] : ( inf_inf_set_a @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_314_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X4: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X4 ) ) ) ).
% inf_commute
thf(fact_315_inf_Oleft__commute,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A2 @ C ) )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_316_inf__left__commute,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X2 @ Z ) ) ) ).
% inf_left_commute
thf(fact_317_comp__sgraph_Oedge__adj__def,axiom,
! [S: set_set_a,E1: set_set_a,E22: set_set_a] :
( ( undire3485422320110889978_set_a @ ( undire8247866692393712962_set_a @ S ) @ E1 @ E22 )
= ( ( ( inf_inf_set_set_a @ E1 @ E22 )
!= bot_bot_set_set_a )
& ( member_set_set_a @ E1 @ ( undire8247866692393712962_set_a @ S ) )
& ( member_set_set_a @ E22 @ ( undire8247866692393712962_set_a @ S ) ) ) ) ).
% comp_sgraph.edge_adj_def
thf(fact_318_comp__sgraph_Oedge__adj__def,axiom,
! [S: set_a,E1: set_a,E22: set_a] :
( ( undire4022703626023482010_adj_a @ ( undire2918257014606996450dges_a @ S ) @ E1 @ E22 )
= ( ( ( inf_inf_set_a @ E1 @ E22 )
!= bot_bot_set_a )
& ( member_set_a @ E1 @ ( undire2918257014606996450dges_a @ S ) )
& ( member_set_a @ E22 @ ( undire2918257014606996450dges_a @ S ) ) ) ) ).
% comp_sgraph.edge_adj_def
thf(fact_319_comp__sgraph_Owellformed__all__edges,axiom,
! [S: set_a] : ( ord_le3724670747650509150_set_a @ ( undire2918257014606996450dges_a @ S ) @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.wellformed_all_edges
thf(fact_320_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_321_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_322_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_323_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_324_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_325_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_326_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_327_emptyE,axiom,
! [A2: set_a] :
~ ( member_set_a @ A2 @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_328_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_329_equals0D,axiom,
! [A: set_set_a,A2: set_a] :
( ( A = bot_bot_set_set_a )
=> ~ ( member_set_a @ A2 @ A ) ) ).
% equals0D
thf(fact_330_equals0I,axiom,
! [A: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_331_equals0I,axiom,
! [A: set_set_a] :
( ! [Y3: set_a] :
~ ( member_set_a @ Y3 @ A )
=> ( A = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_332_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_333_Int__absorb,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_334_Int__emptyI,axiom,
! [A: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ~ ( member_a @ X3 @ B ) )
=> ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_335_Int__emptyI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ~ ( member_set_a @ X3 @ B ) )
=> ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_336_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X4: a] : ( member_a @ X4 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_337_ex__in__conv,axiom,
! [A: set_set_a] :
( ( ? [X4: set_a] : ( member_set_a @ X4 @ A ) )
= ( A != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_338_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A3: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_339_disjoint__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ~ ( member_a @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_340_disjoint__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ~ ( member_set_a @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_341_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_342_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_343_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_344_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_345_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_346_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_347_disjoint__iff__not__equal,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ! [Y4: a] :
( ( member_a @ Y4 @ B )
=> ( X4 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_348_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 )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ B )
=> ( X4 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_349_all__edges__disjoint,axiom,
! [S: set_a,T2: set_a] :
( ( ( inf_inf_set_a @ S @ T2 )
= bot_bot_set_a )
=> ( ( inf_inf_set_set_a @ ( undire2918257014606996450dges_a @ S ) @ ( undire2918257014606996450dges_a @ T2 ) )
= bot_bot_set_set_a ) ) ).
% all_edges_disjoint
thf(fact_350_all__edges__disjoint,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( ( inf_inf_set_set_a @ S @ T2 )
= bot_bot_set_set_a )
=> ( ( inf_in1205276777018777868_set_a @ ( undire8247866692393712962_set_a @ S ) @ ( undire8247866692393712962_set_a @ T2 ) )
= bot_bo3380559777022489994_set_a ) ) ).
% all_edges_disjoint
thf(fact_351_comp__sgraph_Oempty__not__edge,axiom,
! [S: set_a] :
~ ( member_set_a @ bot_bot_set_a @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.empty_not_edge
thf(fact_352_comp__sgraph_Oempty__not__edge,axiom,
! [S: set_set_a] :
~ ( member_set_set_a @ bot_bot_set_set_a @ ( undire8247866692393712962_set_a @ S ) ) ).
% comp_sgraph.empty_not_edge
thf(fact_353_comp__sgraph_Oe__in__all__edges,axiom,
! [E3: set_a,S: set_a] :
( ( member_set_a @ E3 @ ( undire2918257014606996450dges_a @ S ) )
=> ( member_set_a @ E3 @ ( undire2918257014606996450dges_a @ S ) ) ) ).
% comp_sgraph.e_in_all_edges
thf(fact_354_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_a,K: set_a,B2: set_a,A2: set_a] :
( ( B
= ( inf_inf_set_a @ K @ B2 ) )
=> ( ( inf_inf_set_a @ A2 @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_355_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_a,K: set_a,A2: set_a,B2: set_a] :
( ( A
= ( inf_inf_set_a @ K @ A2 ) )
=> ( ( inf_inf_set_a @ A @ B2 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_356_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_357_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_358_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_359_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_360_bot_Oextremum,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% bot.extremum
thf(fact_361_bot_Oextremum,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A2 ) ).
% bot.extremum
thf(fact_362_boolean__algebra_Oconj__disj__distrib,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ ( inf_inf_set_a @ X2 @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_363_boolean__algebra_Oconj__disj__distrib,axiom,
! [X2: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ X2 @ ( sup_sup_set_set_a @ Y @ Z ) )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ X2 @ Y ) @ ( inf_inf_set_set_a @ X2 @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_364_boolean__algebra_Odisj__conj__distrib,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y ) @ ( sup_sup_set_a @ X2 @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_365_boolean__algebra_Odisj__conj__distrib,axiom,
! [X2: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ X2 @ ( inf_inf_set_set_a @ Y @ Z ) )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ X2 @ Y ) @ ( sup_sup_set_set_a @ X2 @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_366_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_a,Z: set_a,X2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X2 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X2 ) @ ( inf_inf_set_a @ Z @ X2 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_367_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_set_a,Z: set_set_a,X2: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ Y @ Z ) @ X2 )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ Y @ X2 ) @ ( inf_inf_set_set_a @ Z @ X2 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_368_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_a,Z: set_a,X2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z ) @ X2 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X2 ) @ ( sup_sup_set_a @ Z @ X2 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_369_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_set_a,Z: set_set_a,X2: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ Y @ Z ) @ X2 )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ Y @ X2 ) @ ( sup_sup_set_set_a @ Z @ X2 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_370_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_a] :
( ( sup_sup_set_a @ X2 @ bot_bot_set_a )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_371_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_set_a] :
( ( sup_sup_set_set_a @ X2 @ bot_bot_set_set_a )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_372_inf_OcoboundedI2,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_373_inf_OcoboundedI2,axiom,
! [B2: set_set_a,C: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_374_inf_OcoboundedI1,axiom,
! [A2: set_a,C: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_375_inf_OcoboundedI1,axiom,
! [A2: set_set_a,C: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_376_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( inf_inf_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_377_inf_Oabsorb__iff2,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B4: set_set_a,A4: set_set_a] :
( ( inf_inf_set_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_378_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( inf_inf_set_a @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_379_inf_Oabsorb__iff1,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( inf_inf_set_set_a @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_380_inf_Ocobounded2,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_381_inf_Ocobounded2,axiom,
! [A2: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_382_inf_Ocobounded1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_383_inf_Ocobounded1,axiom,
! [A2: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_384_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( A4
= ( inf_inf_set_a @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_385_inf_Oorder__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
( A4
= ( inf_inf_set_set_a @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_386_inf__greatest,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ X2 @ Z )
=> ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_387_inf__greatest,axiom,
! [X2: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Z )
=> ( ord_le3724670747650509150_set_a @ X2 @ ( inf_inf_set_set_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_388_inf_OboundedI,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ C )
=> ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_389_inf_OboundedI,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( inf_inf_set_set_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_390_inf_OboundedE,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_391_inf_OboundedE,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( inf_inf_set_set_a @ B2 @ C ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ~ ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_392_inf__absorb2,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ( ( inf_inf_set_a @ X2 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_393_inf__absorb2,axiom,
! [Y: set_set_a,X2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X2 )
=> ( ( inf_inf_set_set_a @ X2 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_394_inf__absorb1,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( inf_inf_set_a @ X2 @ Y )
= X2 ) ) ).
% inf_absorb1
thf(fact_395_inf__absorb1,axiom,
! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ( inf_inf_set_set_a @ X2 @ Y )
= X2 ) ) ).
% inf_absorb1
thf(fact_396_inf_Oabsorb2,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_397_inf_Oabsorb2,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_398_inf_Oabsorb1,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_399_inf_Oabsorb1,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_set_a @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_400_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X4: set_a,Y4: set_a] :
( ( inf_inf_set_a @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_401_le__iff__inf,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X4: set_set_a,Y4: set_set_a] :
( ( inf_inf_set_set_a @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_402_inf__unique,axiom,
! [F: set_a > set_a > set_a,X2: set_a,Y: set_a] :
( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: set_a,Y3: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ( ord_less_eq_set_a @ X3 @ Z3 )
=> ( ord_less_eq_set_a @ X3 @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_403_inf__unique,axiom,
! [F: set_set_a > set_set_a > set_set_a,X2: set_set_a,Y: set_set_a] :
( ! [X3: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: set_set_a,Y3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: set_set_a,Y3: set_set_a,Z3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Z3 )
=> ( ord_le3724670747650509150_set_a @ X3 @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_set_set_a @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_404_inf_OorderI,axiom,
! [A2: set_a,B2: set_a] :
( ( A2
= ( inf_inf_set_a @ A2 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_405_inf_OorderI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2
= ( inf_inf_set_set_a @ A2 @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_406_inf_OorderE,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_407_inf_OorderE,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_408_le__infI2,axiom,
! [B2: set_a,X2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ X2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_409_le__infI2,axiom,
! [B2: set_set_a,X2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ X2 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_410_le__infI1,axiom,
! [A2: set_a,X2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ X2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_411_le__infI1,axiom,
! [A2: set_set_a,X2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ X2 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_412_inf__mono,axiom,
! [A2: set_a,C: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_413_inf__mono,axiom,
! [A2: set_set_a,C: set_set_a,B2: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ D )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B2 ) @ ( inf_inf_set_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_414_le__infI,axiom,
! [X2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X2 @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ B2 )
=> ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_415_le__infI,axiom,
! [X2: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ X2 @ ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_416_le__infE,axiom,
! [X2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_a @ X2 @ A2 )
=> ~ ( ord_less_eq_set_a @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_417_le__infE,axiom,
! [X2: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ ( inf_inf_set_set_a @ A2 @ B2 ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ X2 @ A2 )
=> ~ ( ord_le3724670747650509150_set_a @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_418_inf__le2,axiom,
! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_419_inf__le2,axiom,
! [X2: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_420_inf__le1,axiom,
! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_421_inf__le1,axiom,
! [X2: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_422_inf__sup__ord_I1_J,axiom,
! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_423_inf__sup__ord_I1_J,axiom,
! [X2: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_424_inf__sup__ord_I2_J,axiom,
! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_425_inf__sup__ord_I2_J,axiom,
! [X2: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_426_all__edges__mono,axiom,
! [Vs: set_a,Ws: set_a] :
( ( ord_less_eq_set_a @ Vs @ Ws )
=> ( ord_le3724670747650509150_set_a @ ( undire2918257014606996450dges_a @ Vs ) @ ( undire2918257014606996450dges_a @ Ws ) ) ) ).
% all_edges_mono
thf(fact_427_all__edges__mono,axiom,
! [Vs: set_set_a,Ws: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Vs @ Ws )
=> ( ord_le5722252365846178494_set_a @ ( undire8247866692393712962_set_a @ Vs ) @ ( undire8247866692393712962_set_a @ Ws ) ) ) ).
% all_edges_mono
thf(fact_428_comp__sgraph_Oe__in__all__edges__ss,axiom,
! [E3: set_a,S: set_a,V: set_a] :
( ( member_set_a @ E3 @ ( undire2918257014606996450dges_a @ S ) )
=> ( ( ord_less_eq_set_a @ E3 @ V )
=> ( ( ord_less_eq_set_a @ V @ S )
=> ( member_set_a @ E3 @ ( undire2918257014606996450dges_a @ V ) ) ) ) ) ).
% comp_sgraph.e_in_all_edges_ss
thf(fact_429_comp__sgraph_Oe__in__all__edges__ss,axiom,
! [E3: set_set_a,S: set_set_a,V: set_set_a] :
( ( member_set_set_a @ E3 @ ( undire8247866692393712962_set_a @ S ) )
=> ( ( ord_le3724670747650509150_set_a @ E3 @ V )
=> ( ( ord_le3724670747650509150_set_a @ V @ S )
=> ( member_set_set_a @ E3 @ ( undire8247866692393712962_set_a @ V ) ) ) ) ) ).
% comp_sgraph.e_in_all_edges_ss
thf(fact_430_comp__sgraph_Owellformed,axiom,
! [E3: set_a,S: set_a] :
( ( member_set_a @ E3 @ ( undire2918257014606996450dges_a @ S ) )
=> ( ord_less_eq_set_a @ E3 @ S ) ) ).
% comp_sgraph.wellformed
thf(fact_431_comp__sgraph_Owellformed,axiom,
! [E3: set_set_a,S: set_set_a] :
( ( member_set_set_a @ E3 @ ( undire8247866692393712962_set_a @ S ) )
=> ( ord_le3724670747650509150_set_a @ E3 @ S ) ) ).
% comp_sgraph.wellformed
thf(fact_432_distrib__imp1,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ! [X3: set_a,Y3: set_a,Z3: set_a] :
( ( inf_inf_set_a @ X3 @ ( sup_sup_set_a @ Y3 @ Z3 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X3 @ Y3 ) @ ( inf_inf_set_a @ X3 @ Z3 ) ) )
=> ( ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y ) @ ( sup_sup_set_a @ X2 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_433_distrib__imp1,axiom,
! [X2: set_set_a,Y: set_set_a,Z: set_set_a] :
( ! [X3: set_set_a,Y3: set_set_a,Z3: set_set_a] :
( ( inf_inf_set_set_a @ X3 @ ( sup_sup_set_set_a @ Y3 @ Z3 ) )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ X3 @ Y3 ) @ ( inf_inf_set_set_a @ X3 @ Z3 ) ) )
=> ( ( sup_sup_set_set_a @ X2 @ ( inf_inf_set_set_a @ Y @ Z ) )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ X2 @ Y ) @ ( sup_sup_set_set_a @ X2 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_434_distrib__imp2,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ! [X3: set_a,Y3: set_a,Z3: set_a] :
( ( sup_sup_set_a @ X3 @ ( inf_inf_set_a @ Y3 @ Z3 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X3 @ Y3 ) @ ( sup_sup_set_a @ X3 @ Z3 ) ) )
=> ( ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ ( inf_inf_set_a @ X2 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_435_distrib__imp2,axiom,
! [X2: set_set_a,Y: set_set_a,Z: set_set_a] :
( ! [X3: set_set_a,Y3: set_set_a,Z3: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ ( inf_inf_set_set_a @ Y3 @ Z3 ) )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ X3 @ Y3 ) @ ( sup_sup_set_set_a @ X3 @ Z3 ) ) )
=> ( ( inf_inf_set_set_a @ X2 @ ( sup_sup_set_set_a @ Y @ Z ) )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ X2 @ Y ) @ ( inf_inf_set_set_a @ X2 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_436_inf__sup__distrib1,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ ( inf_inf_set_a @ X2 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_437_inf__sup__distrib1,axiom,
! [X2: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ X2 @ ( sup_sup_set_set_a @ Y @ Z ) )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ X2 @ Y ) @ ( inf_inf_set_set_a @ X2 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_438_inf__sup__distrib2,axiom,
! [Y: set_a,Z: set_a,X2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X2 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X2 ) @ ( inf_inf_set_a @ Z @ X2 ) ) ) ).
% inf_sup_distrib2
thf(fact_439_inf__sup__distrib2,axiom,
! [Y: set_set_a,Z: set_set_a,X2: set_set_a] :
( ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ Y @ Z ) @ X2 )
= ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ Y @ X2 ) @ ( inf_inf_set_set_a @ Z @ X2 ) ) ) ).
% inf_sup_distrib2
thf(fact_440_sup__inf__distrib1,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y ) @ ( sup_sup_set_a @ X2 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_441_sup__inf__distrib1,axiom,
! [X2: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ X2 @ ( inf_inf_set_set_a @ Y @ Z ) )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ X2 @ Y ) @ ( sup_sup_set_set_a @ X2 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_442_sup__inf__distrib2,axiom,
! [Y: set_a,Z: set_a,X2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z ) @ X2 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X2 ) @ ( sup_sup_set_a @ Z @ X2 ) ) ) ).
% sup_inf_distrib2
thf(fact_443_sup__inf__distrib2,axiom,
! [Y: set_set_a,Z: set_set_a,X2: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ Y @ Z ) @ X2 )
= ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ Y @ X2 ) @ ( sup_sup_set_set_a @ Z @ X2 ) ) ) ).
% sup_inf_distrib2
thf(fact_444_Int__Collect__mono,axiom,
! [A: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( 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_445_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 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( 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_446_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_447_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_448_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_449_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_450_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_451_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_452_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_453_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_454_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_455_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_456_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_457_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_458_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_459_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_460_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_461_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_462_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_463_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_464_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_465_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_466_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_467_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_468_comp__sgraph_Ograph__system__axioms,axiom,
! [S: set_a] : ( undire2554140024507503526stem_a @ S @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.graph_system_axioms
thf(fact_469_comp__sgraph_Oinduced__edges__self,axiom,
! [S: set_a] :
( ( undire7777452895879145676dges_a @ ( undire2918257014606996450dges_a @ S ) @ S )
= ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.induced_edges_self
thf(fact_470_graph__system_Oedge__adj__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,E1: set_set_a,E22: set_set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( undire3485422320110889978_set_a @ Edges @ E1 @ E22 )
= ( ( ( inf_inf_set_set_a @ E1 @ E22 )
!= bot_bot_set_set_a )
& ( member_set_set_a @ E1 @ Edges )
& ( member_set_set_a @ E22 @ Edges ) ) ) ) ).
% graph_system.edge_adj_def
thf(fact_471_graph__system_Oedge__adj__def,axiom,
! [Vertices: set_a,Edges: set_set_a,E1: set_a,E22: set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( undire4022703626023482010_adj_a @ Edges @ E1 @ E22 )
= ( ( ( inf_inf_set_a @ E1 @ E22 )
!= bot_bot_set_a )
& ( member_set_a @ E1 @ Edges )
& ( member_set_a @ E22 @ Edges ) ) ) ) ).
% graph_system.edge_adj_def
thf(fact_472_Un__empty__left,axiom,
! [B: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_473_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_474_Un__empty__right,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% Un_empty_right
thf(fact_475_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_476_comp__sgraph_Osubgraph__complete,axiom,
! [S: set_a] : ( undire7103218114511261257raph_a @ S @ ( undire2918257014606996450dges_a @ S ) @ S @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.subgraph_complete
thf(fact_477_comp__sgraph_Oincident__edge__in__wf,axiom,
! [E3: set_set_a,S: set_set_a,V3: set_a] :
( ( member_set_set_a @ E3 @ ( undire8247866692393712962_set_a @ S ) )
=> ( ( undire2320338297334612420_set_a @ V3 @ E3 )
=> ( member_set_a @ V3 @ S ) ) ) ).
% comp_sgraph.incident_edge_in_wf
thf(fact_478_comp__sgraph_Oincident__edge__in__wf,axiom,
! [E3: set_a,S: set_a,V3: a] :
( ( member_set_a @ E3 @ ( undire2918257014606996450dges_a @ S ) )
=> ( ( undire1521409233611534436dent_a @ V3 @ E3 )
=> ( member_a @ V3 @ S ) ) ) ).
% comp_sgraph.incident_edge_in_wf
thf(fact_479_comp__sgraph_Oedge__adjacent__alt__def,axiom,
! [E1: set_set_a,S: set_set_a,E22: set_set_a] :
( ( member_set_set_a @ E1 @ ( undire8247866692393712962_set_a @ S ) )
=> ( ( member_set_set_a @ E22 @ ( undire8247866692393712962_set_a @ S ) )
=> ( ? [X: set_a] :
( ( member_set_a @ X @ S )
& ( member_set_a @ X @ E1 )
& ( member_set_a @ X @ E22 ) )
=> ( undire3485422320110889978_set_a @ ( undire8247866692393712962_set_a @ S ) @ E1 @ E22 ) ) ) ) ).
% comp_sgraph.edge_adjacent_alt_def
thf(fact_480_comp__sgraph_Oedge__adjacent__alt__def,axiom,
! [E1: set_a,S: set_a,E22: set_a] :
( ( member_set_a @ E1 @ ( undire2918257014606996450dges_a @ S ) )
=> ( ( member_set_a @ E22 @ ( undire2918257014606996450dges_a @ S ) )
=> ( ? [X: a] :
( ( member_a @ X @ S )
& ( member_a @ X @ E1 )
& ( member_a @ X @ E22 ) )
=> ( undire4022703626023482010_adj_a @ ( undire2918257014606996450dges_a @ S ) @ E1 @ E22 ) ) ) ) ).
% comp_sgraph.edge_adjacent_alt_def
thf(fact_481_comp__sgraph_Oedge__adj__inE,axiom,
! [S: set_a,E1: set_a,E22: set_a] :
( ( undire4022703626023482010_adj_a @ ( undire2918257014606996450dges_a @ S ) @ E1 @ E22 )
=> ( ( member_set_a @ E1 @ ( undire2918257014606996450dges_a @ S ) )
& ( member_set_a @ E22 @ ( undire2918257014606996450dges_a @ S ) ) ) ) ).
% comp_sgraph.edge_adj_inE
thf(fact_482_subgraph_Oedges__ss,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ord_le3724670747650509150_set_a @ E_H @ E_G ) ) ).
% subgraph.edges_ss
thf(fact_483_distrib__sup__le,axiom,
! [X2: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y @ Z ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y ) @ ( sup_sup_set_a @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_484_distrib__sup__le,axiom,
! [X2: set_set_a,Y: set_set_a,Z: set_set_a] : ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ X2 @ ( inf_inf_set_set_a @ Y @ Z ) ) @ ( inf_inf_set_set_a @ ( sup_sup_set_set_a @ X2 @ Y ) @ ( sup_sup_set_set_a @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_485_distrib__inf__le,axiom,
! [X2: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ ( inf_inf_set_a @ X2 @ Z ) ) @ ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_486_distrib__inf__le,axiom,
! [X2: set_set_a,Y: set_set_a,Z: set_set_a] : ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ X2 @ Y ) @ ( inf_inf_set_set_a @ X2 @ Z ) ) @ ( inf_inf_set_set_a @ X2 @ ( sup_sup_set_set_a @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_487_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_488_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_489_comp__sgraph_Oinduced__edges__ss,axiom,
! [V: set_set_a,S: set_set_a] :
( ( ord_le3724670747650509150_set_a @ V @ S )
=> ( ord_le5722252365846178494_set_a @ ( undire7854589003810675244_set_a @ ( undire8247866692393712962_set_a @ S ) @ V ) @ ( undire8247866692393712962_set_a @ S ) ) ) ).
% comp_sgraph.induced_edges_ss
thf(fact_490_comp__sgraph_Oinduced__edges__ss,axiom,
! [V: set_a,S: set_a] :
( ( ord_less_eq_set_a @ V @ S )
=> ( ord_le3724670747650509150_set_a @ ( undire7777452895879145676dges_a @ ( undire2918257014606996450dges_a @ S ) @ V ) @ ( undire2918257014606996450dges_a @ S ) ) ) ).
% comp_sgraph.induced_edges_ss
thf(fact_491_comp__sgraph_Oinduced__is__graph__sys,axiom,
! [V: set_a,S: set_a] : ( undire2554140024507503526stem_a @ V @ ( undire7777452895879145676dges_a @ ( undire2918257014606996450dges_a @ S ) @ V ) ) ).
% comp_sgraph.induced_is_graph_sys
thf(fact_492_subgraph__axioms__def,axiom,
( undire4675926955456076134ioms_a
= ( ^ [V_H2: set_a,E_H2: set_set_a,V_G2: set_a,E_G2: set_set_a] :
( ( ord_less_eq_set_a @ V_H2 @ V_G2 )
& ( ord_le3724670747650509150_set_a @ E_H2 @ E_G2 ) ) ) ) ).
% subgraph_axioms_def
thf(fact_493_subgraph__axioms__def,axiom,
( undire7690874192998179526_set_a
= ( ^ [V_H2: set_set_a,E_H2: set_set_set_a,V_G2: set_set_a,E_G2: set_set_set_a] :
( ( ord_le3724670747650509150_set_a @ V_H2 @ V_G2 )
& ( ord_le5722252365846178494_set_a @ E_H2 @ E_G2 ) ) ) ) ).
% subgraph_axioms_def
thf(fact_494_subgraph__axioms_Ointro,axiom,
! [V_H: set_a,V_G: set_a,E_H: set_set_a,E_G: set_set_a] :
( ( ord_less_eq_set_a @ V_H @ V_G )
=> ( ( ord_le3724670747650509150_set_a @ E_H @ E_G )
=> ( undire4675926955456076134ioms_a @ V_H @ E_H @ V_G @ E_G ) ) ) ).
% subgraph_axioms.intro
thf(fact_495_subgraph__axioms_Ointro,axiom,
! [V_H: set_set_a,V_G: set_set_a,E_H: set_set_set_a,E_G: set_set_set_a] :
( ( ord_le3724670747650509150_set_a @ V_H @ V_G )
=> ( ( ord_le5722252365846178494_set_a @ E_H @ E_G )
=> ( undire7690874192998179526_set_a @ V_H @ E_H @ V_G @ E_G ) ) ) ).
% subgraph_axioms.intro
thf(fact_496_order__antisym__conv,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ( ( ord_less_eq_set_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_497_order__antisym__conv,axiom,
! [Y: set_set_a,X2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X2 )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_498_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_499_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_set_a,C: set_set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_500_ord__le__eq__subst,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_501_ord__le__eq__subst,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_502_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_503_ord__eq__le__subst,axiom,
! [A2: set_set_a,F: set_a > set_set_a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_504_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_set_a > set_a,B2: set_set_a,C: set_set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_505_ord__eq__le__subst,axiom,
! [A2: set_set_a,F: set_set_a > set_set_a,B2: set_set_a,C: set_set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_506_order__eq__refl,axiom,
! [X2: set_a,Y: set_a] :
( ( X2 = Y )
=> ( ord_less_eq_set_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_507_order__eq__refl,axiom,
! [X2: set_set_a,Y: set_set_a] :
( ( X2 = Y )
=> ( ord_le3724670747650509150_set_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_508_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_509_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_set_a,C: set_set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B2 ) @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_510_order__subst2,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_511_order__subst2,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B2 ) @ C )
=> ( ! [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_512_order__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_513_order__subst1,axiom,
! [A2: set_a,F: set_set_a > set_a,B2: set_set_a,C: set_set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_514_order__subst1,axiom,
! [A2: set_set_a,F: set_a > set_set_a,B2: set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_515_order__subst1,axiom,
! [A2: set_set_a,F: set_set_a > set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_516_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_517_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_set_a,Z2: set_set_a] : ( Y2 = Z2 ) )
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_518_antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_519_antisym,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_520_dual__order_Otrans,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_521_dual__order_Otrans,axiom,
! [B2: set_set_a,A2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ C @ B2 )
=> ( ord_le3724670747650509150_set_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_522_dual__order_Oantisym,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_523_dual__order_Oantisym,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_524_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_525_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_set_a,Z2: set_set_a] : ( Y2 = Z2 ) )
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A4 )
& ( ord_le3724670747650509150_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_526_order__trans,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_eq_set_a @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_527_order__trans,axiom,
! [X2: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ Z )
=> ( ord_le3724670747650509150_set_a @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_528_order_Otrans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_529_order_Otrans,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_530_order__antisym,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_531_order__antisym,axiom,
! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_532_ord__le__eq__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_533_ord__le__eq__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_534_ord__eq__le__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_535_ord__eq__le__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( A2 = B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_536_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 ) )
= ( ^ [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_537_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_set_a,Z2: set_set_a] : ( Y2 = Z2 ) )
= ( ^ [X4: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y4 )
& ( ord_le3724670747650509150_set_a @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_538_comp__sgraph_Oinduced__is__subgraph,axiom,
! [V: set_set_a,S: set_set_a] :
( ( ord_le3724670747650509150_set_a @ V @ S )
=> ( undire1186139521737116585_set_a @ V @ ( undire7854589003810675244_set_a @ ( undire8247866692393712962_set_a @ S ) @ V ) @ S @ ( undire8247866692393712962_set_a @ S ) ) ) ).
% comp_sgraph.induced_is_subgraph
thf(fact_539_comp__sgraph_Oinduced__is__subgraph,axiom,
! [V: set_a,S: set_a] :
( ( ord_less_eq_set_a @ V @ S )
=> ( undire7103218114511261257raph_a @ V @ ( undire7777452895879145676dges_a @ ( undire2918257014606996450dges_a @ S ) @ V ) @ S @ ( undire2918257014606996450dges_a @ S ) ) ) ).
% comp_sgraph.induced_is_subgraph
thf(fact_540_boolean__algebra__cancel_Osup1,axiom,
! [A: set_a,K: set_a,A2: set_a,B2: set_a] :
( ( A
= ( sup_sup_set_a @ K @ A2 ) )
=> ( ( sup_sup_set_a @ A @ B2 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_541_boolean__algebra__cancel_Osup1,axiom,
! [A: set_set_a,K: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( A
= ( sup_sup_set_set_a @ K @ A2 ) )
=> ( ( sup_sup_set_set_a @ A @ B2 )
= ( sup_sup_set_set_a @ K @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_542_wellformed__alt__fst,axiom,
! [X2: a,Y: a] :
( ( member_set_a @ ( insert_a @ X2 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges )
=> ( member_a @ X2 @ vertices ) ) ).
% wellformed_alt_fst
thf(fact_543_wellformed__alt__snd,axiom,
! [X2: a,Y: a] :
( ( member_set_a @ ( insert_a @ X2 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges )
=> ( member_a @ Y @ vertices ) ) ).
% wellformed_alt_snd
thf(fact_544_induced__graph_Owellformed__alt__fst,axiom,
! [X2: a,Y: a,V: set_a] :
( ( member_set_a @ ( insert_a @ X2 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ ( undire7777452895879145676dges_a @ edges @ V ) )
=> ( member_a @ X2 @ V ) ) ).
% induced_graph.wellformed_alt_fst
thf(fact_545_induced__graph_Owellformed__alt__snd,axiom,
! [X2: a,Y: a,V: set_a] :
( ( member_set_a @ ( insert_a @ X2 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ ( undire7777452895879145676dges_a @ edges @ V ) )
=> ( member_a @ Y @ V ) ) ).
% induced_graph.wellformed_alt_snd
thf(fact_546_subset__emptyI,axiom,
! [A: set_a] :
( ! [X3: a] :
~ ( member_a @ X3 @ A )
=> ( ord_less_eq_set_a @ A @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_547_subset__emptyI,axiom,
! [A: set_set_a] :
( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A )
=> ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a ) ) ).
% subset_emptyI
thf(fact_548_all__edges__loops__ss_I1_J,axiom,
! [S: set_a] : ( ord_le3724670747650509150_set_a @ ( undire2918257014606996450dges_a @ S ) @ ( undire9065700607645037417oops_a @ S ) ) ).
% all_edges_loops_ss(1)
thf(fact_549_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A3: set_a] : ( A3 = bot_bot_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_550_Set_Ois__empty__def,axiom,
( is_empty_set_a
= ( ^ [A3: set_set_a] : ( A3 = bot_bot_set_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_551_comp__sgraph_Ois__isolated__vertex__edge,axiom,
! [S: set_a,V3: a,E3: set_a] :
( ( undire8931668460104145173rtex_a @ S @ ( undire2918257014606996450dges_a @ S ) @ V3 )
=> ( ( member_set_a @ E3 @ ( undire2918257014606996450dges_a @ S ) )
=> ~ ( undire1521409233611534436dent_a @ V3 @ E3 ) ) ) ).
% comp_sgraph.is_isolated_vertex_edge
thf(fact_552_Un__Pow__subset,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_le5722252365846178494_set_a @ ( sup_su2076012971530813682_set_a @ ( pow_set_a @ A ) @ ( pow_set_a @ B ) ) @ ( pow_set_a @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% Un_Pow_subset
thf(fact_553_Un__Pow__subset,axiom,
! [A: set_a,B: set_a] : ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ ( pow_a @ A ) @ ( pow_a @ B ) ) @ ( pow_a @ ( sup_sup_set_a @ A @ B ) ) ) ).
% Un_Pow_subset
thf(fact_554_insert__absorb2,axiom,
! [X2: a,A: set_a] :
( ( insert_a @ X2 @ ( insert_a @ X2 @ A ) )
= ( insert_a @ X2 @ A ) ) ).
% insert_absorb2
thf(fact_555_insert__iff,axiom,
! [A2: set_a,B2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ ( insert_set_a @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_set_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_556_insert__iff,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_557_insertCI,axiom,
! [A2: set_a,B: set_set_a,B2: set_a] :
( ( ~ ( member_set_a @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_set_a @ A2 @ ( insert_set_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_558_insertCI,axiom,
! [A2: a,B: set_a,B2: a] :
( ( ~ ( member_a @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_559_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_560_singletonI,axiom,
! [A2: set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_561_insert__subset,axiom,
! [X2: a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X2 @ A ) @ B )
= ( ( member_a @ X2 @ B )
& ( ord_less_eq_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_562_insert__subset,axiom,
! [X2: set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X2 @ A ) @ B )
= ( ( member_set_a @ X2 @ B )
& ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_563_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_564_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_565_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_566_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_567_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_568_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_569_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_570_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_571_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_572_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_573_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_574_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_575_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_576_PowI,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( member_set_a @ A @ ( pow_a @ B ) ) ) ).
% PowI
thf(fact_577_PowI,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( member_set_set_a @ A @ ( pow_set_a @ B ) ) ) ).
% PowI
thf(fact_578_Pow__iff,axiom,
! [A: set_a,B: set_a] :
( ( member_set_a @ A @ ( pow_a @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ).
% Pow_iff
thf(fact_579_Pow__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( member_set_set_a @ A @ ( pow_set_a @ B ) )
= ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% Pow_iff
thf(fact_580_singleton__insert__inj__eq,axiom,
! [B2: a,A2: a,A: set_a] :
( ( ( insert_a @ B2 @ bot_bot_set_a )
= ( insert_a @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_581_singleton__insert__inj__eq,axiom,
! [B2: set_a,A2: set_a,A: set_set_a] :
( ( ( insert_set_a @ B2 @ bot_bot_set_set_a )
= ( insert_set_a @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ B2 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_582_singleton__insert__inj__eq_H,axiom,
! [A2: a,A: set_a,B2: a] :
( ( ( insert_a @ A2 @ A )
= ( insert_a @ B2 @ bot_bot_set_a ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_583_singleton__insert__inj__eq_H,axiom,
! [A2: set_a,A: set_set_a,B2: set_a] :
( ( ( insert_set_a @ A2 @ A )
= ( insert_set_a @ B2 @ bot_bot_set_set_a ) )
= ( ( A2 = B2 )
& ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ B2 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_584_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_585_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_586_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_587_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_588_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_589_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_590_disjoint__insert_I2_J,axiom,
! [A: set_a,B2: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A @ ( insert_a @ B2 @ B ) ) )
= ( ~ ( member_a @ B2 @ A )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_591_disjoint__insert_I2_J,axiom,
! [A: set_set_a,B2: set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A @ ( insert_set_a @ B2 @ B ) ) )
= ( ~ ( member_set_a @ B2 @ A )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_592_Pow__Int__eq,axiom,
! [A: set_a,B: set_a] :
( ( pow_a @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_set_a @ ( pow_a @ A ) @ ( pow_a @ B ) ) ) ).
% Pow_Int_eq
thf(fact_593_Pow__not__empty,axiom,
! [A: set_a] :
( ( pow_a @ A )
!= bot_bot_set_set_a ) ).
% Pow_not_empty
thf(fact_594_insert__subsetI,axiom,
! [X2: a,A: set_a,X5: set_a] :
( ( member_a @ X2 @ A )
=> ( ( ord_less_eq_set_a @ X5 @ A )
=> ( ord_less_eq_set_a @ ( insert_a @ X2 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_595_insert__subsetI,axiom,
! [X2: set_a,A: set_set_a,X5: set_set_a] :
( ( member_set_a @ X2 @ A )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ A )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X2 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_596_mk__disjoint__insert,axiom,
! [A2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ? [B6: set_set_a] :
( ( A
= ( insert_set_a @ A2 @ B6 ) )
& ~ ( member_set_a @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_597_mk__disjoint__insert,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ? [B6: set_a] :
( ( A
= ( insert_a @ A2 @ B6 ) )
& ~ ( member_a @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_598_insert__commute,axiom,
! [X2: a,Y: a,A: set_a] :
( ( insert_a @ X2 @ ( insert_a @ Y @ A ) )
= ( insert_a @ Y @ ( insert_a @ X2 @ A ) ) ) ).
% insert_commute
thf(fact_599_insert__eq__iff,axiom,
! [A2: set_a,A: set_set_a,B2: set_a,B: set_set_a] :
( ~ ( member_set_a @ A2 @ A )
=> ( ~ ( member_set_a @ B2 @ B )
=> ( ( ( insert_set_a @ A2 @ A )
= ( insert_set_a @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_set_a] :
( ( A
= ( insert_set_a @ B2 @ C3 ) )
& ~ ( member_set_a @ B2 @ C3 )
& ( B
= ( insert_set_a @ A2 @ C3 ) )
& ~ ( member_set_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_600_insert__eq__iff,axiom,
! [A2: a,A: set_a,B2: a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ~ ( member_a @ B2 @ B )
=> ( ( ( insert_a @ A2 @ A )
= ( insert_a @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_a] :
( ( A
= ( insert_a @ B2 @ C3 ) )
& ~ ( member_a @ B2 @ C3 )
& ( B
= ( insert_a @ A2 @ C3 ) )
& ~ ( member_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_601_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_602_insert__absorb,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_603_insert__ident,axiom,
! [X2: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X2 @ A )
=> ( ~ ( member_set_a @ X2 @ B )
=> ( ( ( insert_set_a @ X2 @ A )
= ( insert_set_a @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_604_insert__ident,axiom,
! [X2: a,A: set_a,B: set_a] :
( ~ ( member_a @ X2 @ A )
=> ( ~ ( member_a @ X2 @ B )
=> ( ( ( insert_a @ X2 @ A )
= ( insert_a @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_605_Set_Oset__insert,axiom,
! [X2: set_a,A: set_set_a] :
( ( member_set_a @ X2 @ A )
=> ~ ! [B6: set_set_a] :
( ( A
= ( insert_set_a @ X2 @ B6 ) )
=> ( member_set_a @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_606_Set_Oset__insert,axiom,
! [X2: a,A: set_a] :
( ( member_a @ X2 @ A )
=> ~ ! [B6: set_a] :
( ( A
= ( insert_a @ X2 @ B6 ) )
=> ( member_a @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_607_insertI2,axiom,
! [A2: set_a,B: set_set_a,B2: set_a] :
( ( member_set_a @ A2 @ B )
=> ( member_set_a @ A2 @ ( insert_set_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_608_insertI2,axiom,
! [A2: a,B: set_a,B2: a] :
( ( member_a @ A2 @ B )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_609_insertI1,axiom,
! [A2: set_a,B: set_set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ B ) ) ).
% insertI1
thf(fact_610_insertI1,axiom,
! [A2: a,B: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B ) ) ).
% insertI1
thf(fact_611_insertE,axiom,
! [A2: set_a,B2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ ( insert_set_a @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_set_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_612_insertE,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_613_Pow__top,axiom,
! [A: set_a] : ( member_set_a @ A @ ( pow_a @ A ) ) ).
% Pow_top
thf(fact_614_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_615_bot__set__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a @ bot_bot_set_a_o ) ) ).
% bot_set_def
thf(fact_616_singleton__inject,axiom,
! [A2: a,B2: a] :
( ( ( insert_a @ A2 @ bot_bot_set_a )
= ( insert_a @ B2 @ bot_bot_set_a ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_617_singleton__inject,axiom,
! [A2: set_a,B2: set_a] :
( ( ( insert_set_a @ A2 @ bot_bot_set_set_a )
= ( insert_set_a @ B2 @ bot_bot_set_set_a ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_618_insert__not__empty,axiom,
! [A2: a,A: set_a] :
( ( insert_a @ A2 @ A )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_619_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_620_doubleton__eq__iff,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ( insert_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_621_doubleton__eq__iff,axiom,
! [A2: set_a,B2: set_a,C: set_a,D: set_a] :
( ( ( insert_set_a @ A2 @ ( insert_set_a @ B2 @ bot_bot_set_set_a ) )
= ( insert_set_a @ C @ ( insert_set_a @ D @ bot_bot_set_set_a ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_622_singleton__iff,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_623_singleton__iff,axiom,
! [B2: set_a,A2: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_624_singletonD,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_625_singletonD,axiom,
! [B2: set_a,A2: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_626_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_627_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_628_subset__insert,axiom,
! [X2: a,A: set_a,B: set_a] :
( ~ ( member_a @ X2 @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a @ X2 @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_629_subset__insert,axiom,
! [X2: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X2 @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X2 @ B ) )
= ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_630_subset__insertI,axiom,
! [B: set_a,A2: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A2 @ B ) ) ).
% subset_insertI
thf(fact_631_subset__insertI,axiom,
! [B: set_set_a,A2: set_a] : ( ord_le3724670747650509150_set_a @ B @ ( insert_set_a @ A2 @ B ) ) ).
% subset_insertI
thf(fact_632_subset__insertI2,axiom,
! [A: set_a,B: set_a,B2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_633_subset__insertI2,axiom,
! [A: set_set_a,B: set_set_a,B2: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_634_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_635_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_636_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_637_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_638_Pow__bottom,axiom,
! [B: set_a] : ( member_set_a @ bot_bot_set_a @ ( pow_a @ B ) ) ).
% Pow_bottom
thf(fact_639_Pow__bottom,axiom,
! [B: set_set_a] : ( member_set_set_a @ bot_bot_set_set_a @ ( pow_set_a @ B ) ) ).
% Pow_bottom
thf(fact_640_PowD,axiom,
! [A: set_a,B: set_a] :
( ( member_set_a @ A @ ( pow_a @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% PowD
thf(fact_641_PowD,axiom,
! [A: set_set_a,B: set_set_a] :
( ( member_set_set_a @ A @ ( pow_set_a @ B ) )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% PowD
thf(fact_642_subset__singletonD,axiom,
! [A: set_a,X2: a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ( A = bot_bot_set_a )
| ( A
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_643_subset__singletonD,axiom,
! [A: set_set_a,X2: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
=> ( ( A = bot_bot_set_set_a )
| ( A
= ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_644_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_645_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_646_insert__is__Un,axiom,
( insert_a
= ( ^ [A4: a] : ( sup_sup_set_a @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_647_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_648_Un__singleton__iff,axiom,
! [A: set_a,B: set_a,X2: a] :
( ( ( sup_sup_set_a @ A @ B )
= ( insert_a @ X2 @ bot_bot_set_a ) )
= ( ( ( A = bot_bot_set_a )
& ( B
= ( insert_a @ X2 @ bot_bot_set_a ) ) )
| ( ( A
= ( insert_a @ X2 @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A
= ( insert_a @ X2 @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_649_Un__singleton__iff,axiom,
! [A: set_set_a,B: set_set_a,X2: set_a] :
( ( ( sup_sup_set_set_a @ A @ B )
= ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
= ( ( ( A = bot_bot_set_set_a )
& ( B
= ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) )
| ( ( A
= ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
& ( B = bot_bot_set_set_a ) )
| ( ( A
= ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
& ( B
= ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_650_singleton__Un__iff,axiom,
! [X2: a,A: set_a,B: set_a] :
( ( ( insert_a @ X2 @ bot_bot_set_a )
= ( sup_sup_set_a @ A @ B ) )
= ( ( ( A = bot_bot_set_a )
& ( B
= ( insert_a @ X2 @ bot_bot_set_a ) ) )
| ( ( A
= ( insert_a @ X2 @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A
= ( insert_a @ X2 @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_651_singleton__Un__iff,axiom,
! [X2: set_a,A: set_set_a,B: set_set_a] :
( ( ( insert_set_a @ X2 @ bot_bot_set_set_a )
= ( sup_sup_set_set_a @ A @ B ) )
= ( ( ( A = bot_bot_set_set_a )
& ( B
= ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) )
| ( ( A
= ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
& ( B = bot_bot_set_set_a ) )
| ( ( A
= ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
& ( B
= ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_652_comp__sgraph_Osingleton__not__edge,axiom,
! [X2: a,S: set_a] :
~ ( member_set_a @ ( insert_a @ X2 @ bot_bot_set_a ) @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.singleton_not_edge
thf(fact_653_comp__sgraph_Osingleton__not__edge,axiom,
! [X2: set_a,S: set_set_a] :
~ ( member_set_set_a @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) @ ( undire8247866692393712962_set_a @ S ) ) ).
% comp_sgraph.singleton_not_edge
thf(fact_654_comp__sgraph_Owellformed__alt__fst,axiom,
! [X2: a,Y: a,S: set_a] :
( ( member_set_a @ ( insert_a @ X2 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ ( undire2918257014606996450dges_a @ S ) )
=> ( member_a @ X2 @ S ) ) ).
% comp_sgraph.wellformed_alt_fst
thf(fact_655_comp__sgraph_Owellformed__alt__fst,axiom,
! [X2: set_a,Y: set_a,S: set_set_a] :
( ( member_set_set_a @ ( insert_set_a @ X2 @ ( insert_set_a @ Y @ bot_bot_set_set_a ) ) @ ( undire8247866692393712962_set_a @ S ) )
=> ( member_set_a @ X2 @ S ) ) ).
% comp_sgraph.wellformed_alt_fst
thf(fact_656_comp__sgraph_Owellformed__alt__snd,axiom,
! [X2: a,Y: a,S: set_a] :
( ( member_set_a @ ( insert_a @ X2 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ ( undire2918257014606996450dges_a @ S ) )
=> ( member_a @ Y @ S ) ) ).
% comp_sgraph.wellformed_alt_snd
thf(fact_657_comp__sgraph_Owellformed__alt__snd,axiom,
! [X2: set_a,Y: set_a,S: set_set_a] :
( ( member_set_set_a @ ( insert_set_a @ X2 @ ( insert_set_a @ Y @ bot_bot_set_set_a ) ) @ ( undire8247866692393712962_set_a @ S ) )
=> ( member_set_a @ Y @ S ) ) ).
% comp_sgraph.wellformed_alt_snd
thf(fact_658_graph__system_Owellformed__alt__fst,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X2: set_a,Y: set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( member_set_set_a @ ( insert_set_a @ X2 @ ( insert_set_a @ Y @ bot_bot_set_set_a ) ) @ Edges )
=> ( member_set_a @ X2 @ Vertices ) ) ) ).
% graph_system.wellformed_alt_fst
thf(fact_659_graph__system_Owellformed__alt__fst,axiom,
! [Vertices: set_a,Edges: set_set_a,X2: a,Y: a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( member_set_a @ ( insert_a @ X2 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ Edges )
=> ( member_a @ X2 @ Vertices ) ) ) ).
% graph_system.wellformed_alt_fst
thf(fact_660_graph__system_Owellformed__alt__snd,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X2: set_a,Y: set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ( member_set_set_a @ ( insert_set_a @ X2 @ ( insert_set_a @ Y @ bot_bot_set_set_a ) ) @ Edges )
=> ( member_set_a @ Y @ Vertices ) ) ) ).
% graph_system.wellformed_alt_snd
thf(fact_661_graph__system_Owellformed__alt__snd,axiom,
! [Vertices: set_a,Edges: set_set_a,X2: a,Y: a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( member_set_a @ ( insert_a @ X2 @ ( insert_a @ Y @ bot_bot_set_a ) ) @ Edges )
=> ( member_a @ Y @ Vertices ) ) ) ).
% graph_system.wellformed_alt_snd
thf(fact_662_comp__sgraph_Oinduced__edges__alt,axiom,
! [S: set_a,V: set_a] :
( ( undire7777452895879145676dges_a @ ( undire2918257014606996450dges_a @ S ) @ V )
= ( inf_inf_set_set_a @ ( undire2918257014606996450dges_a @ S ) @ ( undire2918257014606996450dges_a @ V ) ) ) ).
% comp_sgraph.induced_edges_alt
thf(fact_663_Pow__mono,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ ( pow_a @ A ) @ ( pow_a @ B ) ) ) ).
% Pow_mono
thf(fact_664_Pow__mono,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_le5722252365846178494_set_a @ ( pow_set_a @ A ) @ ( pow_set_a @ B ) ) ) ).
% Pow_mono
thf(fact_665_all__edges__subset__Pow,axiom,
! [A: set_a] : ( ord_le3724670747650509150_set_a @ ( undire2918257014606996450dges_a @ A ) @ ( pow_a @ A ) ) ).
% all_edges_subset_Pow
thf(fact_666_incident__edges__empty,axiom,
! [V3: a] :
( ~ ( member_a @ V3 @ vertices )
=> ( ( undire3231912044278729248dges_a @ edges @ V3 )
= bot_bot_set_set_a ) ) ).
% incident_edges_empty
thf(fact_667_induced__graph_Oincident__edges__empty,axiom,
! [V3: a,V: set_a] :
( ~ ( member_a @ V3 @ V )
=> ( ( undire3231912044278729248dges_a @ ( undire7777452895879145676dges_a @ edges @ V ) @ V3 )
= bot_bot_set_set_a ) ) ).
% induced_graph.incident_edges_empty
thf(fact_668_comp__sgraph_Ois__edge__between__def,axiom,
( undire8544646567961481629ween_a
= ( ^ [X6: set_a,Y5: set_a,E5: set_a] :
? [X4: a,Y4: a] :
( ( E5
= ( insert_a @ X4 @ ( insert_a @ Y4 @ bot_bot_set_a ) ) )
& ( member_a @ X4 @ X6 )
& ( member_a @ Y4 @ Y5 ) ) ) ) ).
% comp_sgraph.is_edge_between_def
thf(fact_669_comp__sgraph_Ois__edge__between__def,axiom,
( undire2578756059399487229_set_a
= ( ^ [X6: set_set_a,Y5: set_set_a,E5: set_set_a] :
? [X4: set_a,Y4: set_a] :
( ( E5
= ( insert_set_a @ X4 @ ( insert_set_a @ Y4 @ bot_bot_set_set_a ) ) )
& ( member_set_a @ X4 @ X6 )
& ( member_set_a @ Y4 @ Y5 ) ) ) ) ).
% comp_sgraph.is_edge_between_def
thf(fact_670_the__elem__eq,axiom,
! [X2: a] :
( ( the_elem_a @ ( insert_a @ X2 @ bot_bot_set_a ) )
= X2 ) ).
% the_elem_eq
thf(fact_671_the__elem__eq,axiom,
! [X2: set_a] :
( ( the_elem_set_a @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
= X2 ) ).
% the_elem_eq
thf(fact_672_is__singletonI,axiom,
! [X2: a] : ( is_singleton_a @ ( insert_a @ X2 @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_673_is__singletonI,axiom,
! [X2: set_a] : ( is_singleton_set_a @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ).
% is_singletonI
thf(fact_674_comp__sgraph_Overt__adj__inc__edge__iff,axiom,
! [S: set_set_a,V1: set_a,V22: set_a] :
( ( undire3510646817838285160_set_a @ ( undire8247866692393712962_set_a @ S ) @ V1 @ V22 )
= ( ( undire2320338297334612420_set_a @ V1 @ ( insert_set_a @ V1 @ ( insert_set_a @ V22 @ bot_bot_set_set_a ) ) )
& ( undire2320338297334612420_set_a @ V22 @ ( insert_set_a @ V1 @ ( insert_set_a @ V22 @ bot_bot_set_set_a ) ) )
& ( member_set_set_a @ ( insert_set_a @ V1 @ ( insert_set_a @ V22 @ bot_bot_set_set_a ) ) @ ( undire8247866692393712962_set_a @ S ) ) ) ) ).
% comp_sgraph.vert_adj_inc_edge_iff
thf(fact_675_comp__sgraph_Overt__adj__inc__edge__iff,axiom,
! [S: set_a,V1: a,V22: a] :
( ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ V1 @ V22 )
= ( ( undire1521409233611534436dent_a @ V1 @ ( insert_a @ V1 @ ( insert_a @ V22 @ bot_bot_set_a ) ) )
& ( undire1521409233611534436dent_a @ V22 @ ( insert_a @ V1 @ ( insert_a @ V22 @ bot_bot_set_a ) ) )
& ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V22 @ bot_bot_set_a ) ) @ ( undire2918257014606996450dges_a @ S ) ) ) ) ).
% comp_sgraph.vert_adj_inc_edge_iff
thf(fact_676_Pow__singleton__iff,axiom,
! [X5: set_set_a,Y6: set_set_a] :
( ( ( pow_set_a @ X5 )
= ( insert_set_set_a @ Y6 @ bot_bo3380559777022489994_set_a ) )
= ( ( X5 = bot_bot_set_set_a )
& ( Y6 = bot_bot_set_set_a ) ) ) ).
% Pow_singleton_iff
thf(fact_677_Pow__singleton__iff,axiom,
! [X5: set_a,Y6: set_a] :
( ( ( pow_a @ X5 )
= ( insert_set_a @ Y6 @ bot_bot_set_set_a ) )
= ( ( X5 = bot_bot_set_a )
& ( Y6 = bot_bot_set_a ) ) ) ).
% Pow_singleton_iff
thf(fact_678_Pow__empty,axiom,
( ( pow_a @ bot_bot_set_a )
= ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) ) ).
% Pow_empty
thf(fact_679_Pow__empty,axiom,
( ( pow_set_a @ bot_bot_set_set_a )
= ( insert_set_set_a @ bot_bot_set_set_a @ bot_bo3380559777022489994_set_a ) ) ).
% Pow_empty
thf(fact_680_graph__system_Oincident__edges_Ocong,axiom,
undire3231912044278729248dges_a = undire3231912044278729248dges_a ).
% graph_system.incident_edges.cong
thf(fact_681_comp__sgraph_Overt__adj__imp__inV,axiom,
! [S: set_set_a,V1: set_a,V22: set_a] :
( ( undire3510646817838285160_set_a @ ( undire8247866692393712962_set_a @ S ) @ V1 @ V22 )
=> ( ( member_set_a @ V1 @ S )
& ( member_set_a @ V22 @ S ) ) ) ).
% comp_sgraph.vert_adj_imp_inV
thf(fact_682_comp__sgraph_Overt__adj__imp__inV,axiom,
! [S: set_a,V1: a,V22: a] :
( ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ V1 @ V22 )
=> ( ( member_a @ V1 @ S )
& ( member_a @ V22 @ S ) ) ) ).
% comp_sgraph.vert_adj_imp_inV
thf(fact_683_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_684_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_685_is__singletonI_H,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
=> ( ! [X3: a,Y3: a] :
( ( member_a @ X3 @ A )
=> ( ( member_a @ Y3 @ A )
=> ( X3 = Y3 ) ) )
=> ( is_singleton_a @ A ) ) ) ).
% is_singletonI'
thf(fact_686_is__singletonI_H,axiom,
! [A: set_set_a] :
( ( A != bot_bot_set_set_a )
=> ( ! [X3: set_a,Y3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( ( member_set_a @ Y3 @ A )
=> ( X3 = Y3 ) ) )
=> ( is_singleton_set_a @ A ) ) ) ).
% is_singletonI'
thf(fact_687_comp__sgraph_Oincident__edges__empty,axiom,
! [V3: set_a,S: set_set_a] :
( ~ ( member_set_a @ V3 @ S )
=> ( ( undire4631953023069350784_set_a @ ( undire8247866692393712962_set_a @ S ) @ V3 )
= bot_bo3380559777022489994_set_a ) ) ).
% comp_sgraph.incident_edges_empty
thf(fact_688_comp__sgraph_Oincident__edges__empty,axiom,
! [V3: a,S: set_a] :
( ~ ( member_a @ V3 @ S )
=> ( ( undire3231912044278729248dges_a @ ( undire2918257014606996450dges_a @ S ) @ V3 )
= bot_bot_set_set_a ) ) ).
% comp_sgraph.incident_edges_empty
thf(fact_689_comp__sgraph_Overt__adj__edge__iff2,axiom,
! [V1: a,V22: a,S: set_a] :
( ( V1 != V22 )
=> ( ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ V1 @ V22 )
= ( ? [X4: set_a] :
( ( member_set_a @ X4 @ ( undire2918257014606996450dges_a @ S ) )
& ( undire1521409233611534436dent_a @ V1 @ X4 )
& ( undire1521409233611534436dent_a @ V22 @ X4 ) ) ) ) ) ).
% comp_sgraph.vert_adj_edge_iff2
thf(fact_690_graph__system_Oincident__edges__empty,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V3: set_a] :
( ( undire7159349782766787846_set_a @ Vertices @ Edges )
=> ( ~ ( member_set_a @ V3 @ Vertices )
=> ( ( undire4631953023069350784_set_a @ Edges @ V3 )
= bot_bo3380559777022489994_set_a ) ) ) ).
% graph_system.incident_edges_empty
thf(fact_691_graph__system_Oincident__edges__empty,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ~ ( member_a @ V3 @ Vertices )
=> ( ( undire3231912044278729248dges_a @ Edges @ V3 )
= bot_bot_set_set_a ) ) ) ).
% graph_system.incident_edges_empty
thf(fact_692_comp__sgraph_Ois__isolated__vertex__def,axiom,
! [S: set_set_a,V3: set_a] :
( ( undire6879241558604981877_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ V3 )
= ( ( member_set_a @ V3 @ S )
& ! [X4: set_a] :
( ( member_set_a @ X4 @ S )
=> ~ ( undire3510646817838285160_set_a @ ( undire8247866692393712962_set_a @ S ) @ X4 @ V3 ) ) ) ) ).
% comp_sgraph.is_isolated_vertex_def
thf(fact_693_comp__sgraph_Ois__isolated__vertex__def,axiom,
! [S: set_a,V3: a] :
( ( undire8931668460104145173rtex_a @ S @ ( undire2918257014606996450dges_a @ S ) @ V3 )
= ( ( member_a @ V3 @ S )
& ! [X4: a] :
( ( member_a @ X4 @ S )
=> ~ ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ X4 @ V3 ) ) ) ) ).
% comp_sgraph.is_isolated_vertex_def
thf(fact_694_comp__sgraph_Overt__adj__def,axiom,
! [S: set_a,V1: a,V22: a] :
( ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ V1 @ V22 )
= ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V22 @ bot_bot_set_a ) ) @ ( undire2918257014606996450dges_a @ S ) ) ) ).
% comp_sgraph.vert_adj_def
thf(fact_695_comp__sgraph_Overt__adj__def,axiom,
! [S: set_set_a,V1: set_a,V22: set_a] :
( ( undire3510646817838285160_set_a @ ( undire8247866692393712962_set_a @ S ) @ V1 @ V22 )
= ( member_set_set_a @ ( insert_set_a @ V1 @ ( insert_set_a @ V22 @ bot_bot_set_set_a ) ) @ ( undire8247866692393712962_set_a @ S ) ) ) ).
% comp_sgraph.vert_adj_def
thf(fact_696_comp__sgraph_Onot__vert__adj,axiom,
! [S: set_a,V3: a,U: a] :
( ~ ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ V3 @ U )
=> ~ ( member_set_a @ ( insert_a @ V3 @ ( insert_a @ U @ bot_bot_set_a ) ) @ ( undire2918257014606996450dges_a @ S ) ) ) ).
% comp_sgraph.not_vert_adj
thf(fact_697_comp__sgraph_Onot__vert__adj,axiom,
! [S: set_set_a,V3: set_a,U: set_a] :
( ~ ( undire3510646817838285160_set_a @ ( undire8247866692393712962_set_a @ S ) @ V3 @ U )
=> ~ ( member_set_set_a @ ( insert_set_a @ V3 @ ( insert_set_a @ U @ bot_bot_set_set_a ) ) @ ( undire8247866692393712962_set_a @ S ) ) ) ).
% comp_sgraph.not_vert_adj
thf(fact_698_is__singletonE,axiom,
! [A: set_a] :
( ( is_singleton_a @ A )
=> ~ ! [X3: a] :
( A
!= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_699_is__singletonE,axiom,
! [A: set_set_a] :
( ( is_singleton_set_a @ A )
=> ~ ! [X3: set_a] :
( A
!= ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) ) ).
% is_singletonE
thf(fact_700_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A3: set_a] :
? [X4: a] :
( A3
= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_701_is__singleton__def,axiom,
( is_singleton_set_a
= ( ^ [A3: set_set_a] :
? [X4: set_a] :
( A3
= ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_702_induced__graph_Ofinite__incident__edges,axiom,
! [V: set_a,V3: a] :
( ( finite_finite_set_a @ ( undire7777452895879145676dges_a @ edges @ V ) )
=> ( finite_finite_set_a @ ( undire3231912044278729248dges_a @ ( undire7777452895879145676dges_a @ edges @ V ) @ V3 ) ) ) ).
% induced_graph.finite_incident_edges
thf(fact_703_finite__incident__edges,axiom,
! [V3: a] :
( ( finite_finite_set_a @ edges )
=> ( finite_finite_set_a @ ( undire3231912044278729248dges_a @ edges @ V3 ) ) ) ).
% finite_incident_edges
thf(fact_704_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : ( member_a @ X4 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_705_bot__empty__eq,axiom,
( bot_bot_set_a_o
= ( ^ [X4: set_a] : ( member_set_a @ X4 @ bot_bot_set_set_a ) ) ) ).
% bot_empty_eq
thf(fact_706_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_707_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_708_comp__sgraph_Oneighborhood__incident,axiom,
! [U: set_a,S: set_set_a,V3: set_a] :
( ( member_set_a @ U @ ( undire2074812191327625774_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ V3 ) )
= ( member_set_set_a @ ( insert_set_a @ U @ ( insert_set_a @ V3 @ bot_bot_set_set_a ) ) @ ( undire4631953023069350784_set_a @ ( undire8247866692393712962_set_a @ S ) @ V3 ) ) ) ).
% comp_sgraph.neighborhood_incident
thf(fact_709_comp__sgraph_Oneighborhood__incident,axiom,
! [U: a,S: set_a,V3: a] :
( ( member_a @ U @ ( undire8504279938402040014hood_a @ S @ ( undire2918257014606996450dges_a @ S ) @ V3 ) )
= ( member_set_a @ ( insert_a @ U @ ( insert_a @ V3 @ bot_bot_set_a ) ) @ ( undire3231912044278729248dges_a @ ( undire2918257014606996450dges_a @ S ) @ V3 ) ) ) ).
% comp_sgraph.neighborhood_incident
thf(fact_710_finite__all__edges,axiom,
! [S: set_a] :
( ( finite_finite_a @ S )
=> ( finite_finite_set_a @ ( undire2918257014606996450dges_a @ S ) ) ) ).
% finite_all_edges
thf(fact_711_finite__all__edges,axiom,
! [S: set_set_a] :
( ( finite_finite_set_a @ S )
=> ( finite7209287970140883943_set_a @ ( undire8247866692393712962_set_a @ S ) ) ) ).
% finite_all_edges
thf(fact_712_finite__all__edges__loops,axiom,
! [S: set_a] :
( ( finite_finite_a @ S )
=> ( finite_finite_set_a @ ( undire9065700607645037417oops_a @ S ) ) ) ).
% finite_all_edges_loops
thf(fact_713_finite__all__edges__loops,axiom,
! [S: set_set_a] :
( ( finite_finite_set_a @ S )
=> ( finite7209287970140883943_set_a @ ( undire3533089111843156169_set_a @ S ) ) ) ).
% finite_all_edges_loops
thf(fact_714_comp__sgraph_Ofinite__incident__edges,axiom,
! [S: set_a,V3: a] :
( ( finite_finite_set_a @ ( undire2918257014606996450dges_a @ S ) )
=> ( finite_finite_set_a @ ( undire3231912044278729248dges_a @ ( undire2918257014606996450dges_a @ S ) @ V3 ) ) ) ).
% comp_sgraph.finite_incident_edges
thf(fact_715_graph__system_Ofinite__incident__edges,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( finite_finite_set_a @ Edges )
=> ( finite_finite_set_a @ ( undire3231912044278729248dges_a @ Edges @ V3 ) ) ) ) ).
% graph_system.finite_incident_edges
thf(fact_716_comp__sgraph_Oiso__vertex__empty__neighborhood,axiom,
! [S: set_a,V3: a] :
( ( undire8931668460104145173rtex_a @ S @ ( undire2918257014606996450dges_a @ S ) @ V3 )
=> ( ( undire8504279938402040014hood_a @ S @ ( undire2918257014606996450dges_a @ S ) @ V3 )
= bot_bot_set_a ) ) ).
% comp_sgraph.iso_vertex_empty_neighborhood
thf(fact_717_comp__sgraph_Oiso__vertex__empty__neighborhood,axiom,
! [S: set_set_a,V3: set_a] :
( ( undire6879241558604981877_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ V3 )
=> ( ( undire2074812191327625774_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ V3 )
= bot_bot_set_set_a ) ) ).
% comp_sgraph.iso_vertex_empty_neighborhood
thf(fact_718_finite__Pow__iff,axiom,
! [A: set_set_a] :
( ( finite7209287970140883943_set_a @ ( pow_set_a @ A ) )
= ( finite_finite_set_a @ A ) ) ).
% finite_Pow_iff
thf(fact_719_finite__Pow__iff,axiom,
! [A: set_a] :
( ( finite_finite_set_a @ ( pow_a @ A ) )
= ( finite_finite_a @ A ) ) ).
% finite_Pow_iff
thf(fact_720_finite__Un,axiom,
! [F2: set_a,G: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G ) )
= ( ( finite_finite_a @ F2 )
& ( finite_finite_a @ G ) ) ) ).
% finite_Un
thf(fact_721_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_722_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_723_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_724_finite__insert,axiom,
! [A2: a,A: set_a] :
( ( finite_finite_a @ ( insert_a @ A2 @ A ) )
= ( finite_finite_a @ A ) ) ).
% finite_insert
thf(fact_725_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_726_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 )
=> ( ! [A6: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A6 @ A )
=> ( ( ord_less_eq_set_a @ F3 @ A )
=> ( ~ ( member_a @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A6 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_727_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 )
=> ( ! [A6: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A6 @ A )
=> ( ( ord_le3724670747650509150_set_a @ F3 @ A )
=> ( ~ ( member_set_a @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ A6 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_728_finite__has__minimal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A )
& ( ord_less_eq_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_729_finite__has__minimal2,axiom,
! [A: set_set_set_a,A2: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( member_set_set_a @ A2 @ A )
=> ? [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A )
& ( ord_le3724670747650509150_set_a @ X3 @ A2 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A )
=> ( ( ord_le3724670747650509150_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_730_finite__has__maximal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A )
& ( ord_less_eq_set_a @ A2 @ X3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_731_finite__has__maximal2,axiom,
! [A: set_set_set_a,A2: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( member_set_set_a @ A2 @ A )
=> ? [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A )
& ( ord_le3724670747650509150_set_a @ A2 @ X3 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_732_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_733_finite_OemptyI,axiom,
finite_finite_set_a @ bot_bot_set_set_a ).
% finite.emptyI
thf(fact_734_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_735_infinite__imp__nonempty,axiom,
! [S: set_set_a] :
( ~ ( finite_finite_set_a @ S )
=> ( S != bot_bot_set_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_736_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_737_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_738_infinite__super,axiom,
! [S: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_739_infinite__super,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ S @ T2 )
=> ( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ T2 ) ) ) ).
% infinite_super
thf(fact_740_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_741_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_742_finite_OinsertI,axiom,
! [A: set_a,A2: a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( insert_a @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_743_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_744_infinite__Un,axiom,
! [S: set_a,T2: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T2 ) ) )
= ( ~ ( finite_finite_a @ S )
| ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_Un
thf(fact_745_infinite__Un,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( ~ ( finite_finite_set_a @ ( sup_sup_set_set_a @ S @ T2 ) ) )
= ( ~ ( finite_finite_set_a @ S )
| ~ ( finite_finite_set_a @ T2 ) ) ) ).
% infinite_Un
thf(fact_746_Un__infinite,axiom,
! [S: set_a,T2: set_a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_747_Un__infinite,axiom,
! [S: set_set_a,T2: set_set_a] :
( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ ( sup_sup_set_set_a @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_748_finite__UnI,axiom,
! [F2: set_a,G: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( finite_finite_a @ G )
=> ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_749_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_750_finite__has__maximal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_751_finite__has__maximal,axiom,
! [A: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ? [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_752_finite__has__minimal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_753_finite__has__minimal,axiom,
! [A: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ? [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A )
=> ( ( ord_le3724670747650509150_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_754_finite_Ocases,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( A2 != bot_bot_set_a )
=> ~ ! [A7: set_a] :
( ? [A6: a] :
( A2
= ( insert_a @ A6 @ A7 ) )
=> ~ ( finite_finite_a @ A7 ) ) ) ) ).
% finite.cases
thf(fact_755_finite_Ocases,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ~ ! [A7: set_set_a] :
( ? [A6: set_a] :
( A2
= ( insert_set_a @ A6 @ A7 ) )
=> ~ ( finite_finite_set_a @ A7 ) ) ) ) ).
% finite.cases
thf(fact_756_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A4: set_a] :
( ( A4 = bot_bot_set_a )
| ? [A3: set_a,B4: a] :
( ( A4
= ( insert_a @ B4 @ A3 ) )
& ( finite_finite_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_757_finite_Osimps,axiom,
( finite_finite_set_a
= ( ^ [A4: set_set_a] :
( ( A4 = bot_bot_set_set_a )
| ? [A3: set_set_a,B4: set_a] :
( ( A4
= ( insert_set_a @ B4 @ A3 ) )
& ( finite_finite_set_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_758_finite__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_759_finite__induct,axiom,
! [F2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X3: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_760_finite__ne__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ! [X3: a] : ( P @ ( insert_a @ X3 @ bot_bot_set_a ) )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_761_finite__ne__induct,axiom,
! [F2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( F2 != bot_bot_set_set_a )
=> ( ! [X3: set_a] : ( P @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
=> ( ! [X3: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( F3 != bot_bot_set_set_a )
=> ( ~ ( member_set_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_762_infinite__finite__induct,axiom,
! [P: set_a > $o,A: set_a] :
( ! [A7: set_a] :
( ~ ( finite_finite_a @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_763_infinite__finite__induct,axiom,
! [P: set_set_a > $o,A: set_set_a] :
( ! [A7: set_set_a] :
( ~ ( finite_finite_set_a @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X3: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X3 @ F3 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_764_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 )
=> ( ! [A6: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A6 @ A )
=> ( ~ ( member_a @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A6 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_765_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 )
=> ( ! [A6: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A6 @ A )
=> ( ~ ( member_set_a @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ A6 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_766_finite__transitivity__chain,axiom,
! [A: set_a,R: a > a > $o] :
( ( finite_finite_a @ A )
=> ( ! [X3: a] :
~ ( R @ X3 @ X3 )
=> ( ! [X3: a,Y3: a,Z3: a] :
( ( R @ X3 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X3 @ Z3 ) ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ? [Y7: a] :
( ( member_a @ Y7 @ A )
& ( R @ X3 @ Y7 ) ) )
=> ( A = bot_bot_set_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_767_finite__transitivity__chain,axiom,
! [A: set_set_a,R: set_a > set_a > $o] :
( ( finite_finite_set_a @ A )
=> ( ! [X3: set_a] :
~ ( R @ X3 @ X3 )
=> ( ! [X3: set_a,Y3: set_a,Z3: set_a] :
( ( R @ X3 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X3 @ Z3 ) ) )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ? [Y7: set_a] :
( ( member_set_a @ Y7 @ A )
& ( R @ X3 @ Y7 ) ) )
=> ( A = bot_bot_set_set_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_768_chains__extend,axiom,
! [C: set_set_a,S: set_set_a,Z: set_a] :
( ( member_set_set_a @ C @ ( chains_a @ S ) )
=> ( ( member_set_a @ Z @ S )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ C )
=> ( ord_less_eq_set_a @ X3 @ Z ) )
=> ( member_set_set_a @ ( sup_sup_set_set_a @ ( insert_set_a @ Z @ bot_bot_set_set_a ) @ C ) @ ( chains_a @ S ) ) ) ) ) ).
% chains_extend
thf(fact_769_chains__extend,axiom,
! [C: set_set_set_a,S: set_set_set_a,Z: set_set_a] :
( ( member_set_set_set_a @ C @ ( chains_set_a @ S ) )
=> ( ( member_set_set_a @ Z @ S )
=> ( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ C )
=> ( ord_le3724670747650509150_set_a @ X3 @ Z ) )
=> ( member_set_set_set_a @ ( sup_su2076012971530813682_set_a @ ( insert_set_set_a @ Z @ bot_bo3380559777022489994_set_a ) @ C ) @ ( chains_set_a @ S ) ) ) ) ) ).
% chains_extend
thf(fact_770_fin__sgraph_Ofin__neighbourhood,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X2: set_a] :
( ( undire2910814396099768221_set_a @ Vertices @ Edges )
=> ( finite_finite_set_a @ ( undire2074812191327625774_set_a @ Vertices @ Edges @ X2 ) ) ) ).
% fin_sgraph.fin_neighbourhood
thf(fact_771_comp__sgraph_Oincident__sedges__empty,axiom,
! [V3: set_a,S: set_set_a] :
( ~ ( member_set_a @ V3 @ S )
=> ( ( undire5844230293943614535_set_a @ ( undire8247866692393712962_set_a @ S ) @ V3 )
= bot_bo3380559777022489994_set_a ) ) ).
% comp_sgraph.incident_sedges_empty
thf(fact_772_comp__sgraph_Oincident__sedges__empty,axiom,
! [V3: a,S: set_a] :
( ~ ( member_a @ V3 @ S )
=> ( ( undire1270416042309875431dges_a @ ( undire2918257014606996450dges_a @ S ) @ V3 )
= bot_bot_set_set_a ) ) ).
% comp_sgraph.incident_sedges_empty
thf(fact_773_Zorn__Lemma2,axiom,
! [A: set_set_a] :
( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ ( chains_a @ A ) )
=> ? [Xa: set_a] :
( ( member_set_a @ Xa @ A )
& ! [Xb: set_a] :
( ( member_set_a @ Xb @ X3 )
=> ( ord_less_eq_set_a @ Xb @ Xa ) ) ) )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( Xa = X3 ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_774_Zorn__Lemma2,axiom,
! [A: set_set_set_a] :
( ! [X3: set_set_set_a] :
( ( member_set_set_set_a @ X3 @ ( chains_set_a @ A ) )
=> ? [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A )
& ! [Xb: set_set_a] :
( ( member_set_set_a @ Xb @ X3 )
=> ( ord_le3724670747650509150_set_a @ Xb @ Xa ) ) ) )
=> ? [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Xa )
=> ( Xa = X3 ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_775_chainsD,axiom,
! [C: set_set_a,S: set_set_a,X2: set_a,Y: set_a] :
( ( member_set_set_a @ C @ ( chains_a @ S ) )
=> ( ( member_set_a @ X2 @ C )
=> ( ( member_set_a @ Y @ C )
=> ( ( ord_less_eq_set_a @ X2 @ Y )
| ( ord_less_eq_set_a @ Y @ X2 ) ) ) ) ) ).
% chainsD
thf(fact_776_chainsD,axiom,
! [C: set_set_set_a,S: set_set_set_a,X2: set_set_a,Y: set_set_a] :
( ( member_set_set_set_a @ C @ ( chains_set_a @ S ) )
=> ( ( member_set_set_a @ X2 @ C )
=> ( ( member_set_set_a @ Y @ C )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Y )
| ( ord_le3724670747650509150_set_a @ Y @ X2 ) ) ) ) ) ).
% chainsD
thf(fact_777_chainsD2,axiom,
! [C: set_set_a,S: set_set_a] :
( ( member_set_set_a @ C @ ( chains_a @ S ) )
=> ( ord_le3724670747650509150_set_a @ C @ S ) ) ).
% chainsD2
thf(fact_778_comp__sgraph_Ofinite__inc__sedges,axiom,
! [S: set_a,V3: a] :
( ( finite_finite_set_a @ ( undire2918257014606996450dges_a @ S ) )
=> ( finite_finite_set_a @ ( undire1270416042309875431dges_a @ ( undire2918257014606996450dges_a @ S ) @ V3 ) ) ) ).
% comp_sgraph.finite_inc_sedges
thf(fact_779_complete__fin__sgraph,axiom,
! [S: set_set_a] :
( ( finite_finite_set_a @ S )
=> ( undire2910814396099768221_set_a @ S @ ( undire8247866692393712962_set_a @ S ) ) ) ).
% complete_fin_sgraph
thf(fact_780_fin__sgraph_Ofin__all__edges,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire5670813279940215357raph_a @ Vertices @ Edges )
=> ( finite_finite_set_a @ ( undire2918257014606996450dges_a @ Vertices ) ) ) ).
% fin_sgraph.fin_all_edges
thf(fact_781_comp__sgraph_Oincident__edges__union,axiom,
! [S: set_a,V3: a] :
( ( undire3231912044278729248dges_a @ ( undire2918257014606996450dges_a @ S ) @ V3 )
= ( sup_sup_set_set_a @ ( undire1270416042309875431dges_a @ ( undire2918257014606996450dges_a @ S ) @ V3 ) @ ( undire4753905205749729249oops_a @ ( undire2918257014606996450dges_a @ S ) @ V3 ) ) ) ).
% comp_sgraph.incident_edges_union
thf(fact_782_comp__sgraph_Oincident__edges__sedges,axiom,
! [S: set_a,V3: a] :
( ~ ( undire3617971648856834880loop_a @ ( undire2918257014606996450dges_a @ S ) @ V3 )
=> ( ( undire3231912044278729248dges_a @ ( undire2918257014606996450dges_a @ S ) @ V3 )
= ( undire1270416042309875431dges_a @ ( undire2918257014606996450dges_a @ S ) @ V3 ) ) ) ).
% comp_sgraph.incident_edges_sedges
thf(fact_783_Sup__fin_Oinsert,axiom,
! [A: set_set_set_a,X2: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ( ( lattic338143333561554293_set_a @ ( insert_set_set_a @ X2 @ A ) )
= ( sup_sup_set_set_a @ X2 @ ( lattic338143333561554293_set_a @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_784_Sup__fin_Oinsert,axiom,
! [A: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( insert_set_a @ X2 @ A ) )
= ( sup_sup_set_a @ X2 @ ( lattic2918178356826803221_set_a @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_785_Inf__fin_Oinsert,axiom,
! [A: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X2 @ A ) )
= ( inf_inf_set_a @ X2 @ ( lattic8209813465164889211_set_a @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_786_Inf__fin_Osingleton,axiom,
! [X2: set_a] :
( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
= X2 ) ).
% Inf_fin.singleton
thf(fact_787_Sup__fin_Osingleton,axiom,
! [X2: set_a] :
( ( lattic2918178356826803221_set_a @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
= X2 ) ).
% Sup_fin.singleton
thf(fact_788_inf__Sup__absorb,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A2 @ ( lattic2918178356826803221_set_a @ A ) )
= A2 ) ) ) ).
% inf_Sup_absorb
thf(fact_789_sup__Inf__absorb,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ( ( sup_sup_set_a @ ( lattic8209813465164889211_set_a @ A ) @ A2 )
= A2 ) ) ) ).
% sup_Inf_absorb
thf(fact_790_sup__Inf__absorb,axiom,
! [A: set_set_set_a,A2: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( member_set_set_a @ A2 @ A )
=> ( ( sup_sup_set_set_a @ ( lattic4609221726464634331_set_a @ A ) @ A2 )
= A2 ) ) ) ).
% sup_Inf_absorb
thf(fact_791_comp__sgraph_Oincident__loops__simp_I2_J,axiom,
! [S: set_a,V3: a] :
( ~ ( undire3617971648856834880loop_a @ ( undire2918257014606996450dges_a @ S ) @ V3 )
=> ( ( undire4753905205749729249oops_a @ ( undire2918257014606996450dges_a @ S ) @ V3 )
= bot_bot_set_set_a ) ) ).
% comp_sgraph.incident_loops_simp(2)
thf(fact_792_Inf__fin__le__Sup__fin,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ A ) @ ( lattic2918178356826803221_set_a @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_793_Inf__fin__le__Sup__fin,axiom,
! [A: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ( ord_le3724670747650509150_set_a @ ( lattic4609221726464634331_set_a @ A ) @ ( lattic338143333561554293_set_a @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_794_comp__sgraph_Ohas__loop__in__verts,axiom,
! [S: set_set_a,V3: set_a] :
( ( undire5774735625301615776_set_a @ ( undire8247866692393712962_set_a @ S ) @ V3 )
=> ( member_set_a @ V3 @ S ) ) ).
% comp_sgraph.has_loop_in_verts
thf(fact_795_comp__sgraph_Ohas__loop__in__verts,axiom,
! [S: set_a,V3: a] :
( ( undire3617971648856834880loop_a @ ( undire2918257014606996450dges_a @ S ) @ V3 )
=> ( member_a @ V3 @ S ) ) ).
% comp_sgraph.has_loop_in_verts
thf(fact_796_comp__sgraph_Ono__loops,axiom,
! [V3: set_a,S: set_set_a] :
( ( member_set_a @ V3 @ S )
=> ~ ( undire5774735625301615776_set_a @ ( undire8247866692393712962_set_a @ S ) @ V3 ) ) ).
% comp_sgraph.no_loops
thf(fact_797_comp__sgraph_Ono__loops,axiom,
! [V3: a,S: set_a] :
( ( member_a @ V3 @ S )
=> ~ ( undire3617971648856834880loop_a @ ( undire2918257014606996450dges_a @ S ) @ V3 ) ) ).
% comp_sgraph.no_loops
thf(fact_798_Inf__fin_OcoboundedI,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_799_Inf__fin_OcoboundedI,axiom,
! [A: set_set_set_a,A2: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( member_set_set_a @ A2 @ A )
=> ( ord_le3724670747650509150_set_a @ ( lattic4609221726464634331_set_a @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_800_Sup__fin_OcoboundedI,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ( ord_less_eq_set_a @ A2 @ ( lattic2918178356826803221_set_a @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_801_Sup__fin_OcoboundedI,axiom,
! [A: set_set_set_a,A2: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( member_set_set_a @ A2 @ A )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( lattic338143333561554293_set_a @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_802_Inf__fin_Oin__idem,axiom,
! [A: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ X2 @ A )
=> ( ( inf_inf_set_a @ X2 @ ( lattic8209813465164889211_set_a @ A ) )
= ( lattic8209813465164889211_set_a @ A ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_803_Sup__fin_Oin__idem,axiom,
! [A: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ X2 @ A )
=> ( ( sup_sup_set_a @ X2 @ ( lattic2918178356826803221_set_a @ A ) )
= ( lattic2918178356826803221_set_a @ A ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_804_Sup__fin_Oin__idem,axiom,
! [A: set_set_set_a,X2: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( member_set_set_a @ X2 @ A )
=> ( ( sup_sup_set_set_a @ X2 @ ( lattic338143333561554293_set_a @ A ) )
= ( lattic338143333561554293_set_a @ A ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_805_comp__sgraph_Ofinite__incident__loops,axiom,
! [S: set_a,V3: a] : ( finite_finite_set_a @ ( undire4753905205749729249oops_a @ ( undire2918257014606996450dges_a @ S ) @ V3 ) ) ).
% comp_sgraph.finite_incident_loops
thf(fact_806_Inf__fin_OboundedE,axiom,
! [A: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ X2 @ ( lattic8209813465164889211_set_a @ A ) )
=> ! [A8: set_a] :
( ( member_set_a @ A8 @ A )
=> ( ord_less_eq_set_a @ X2 @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_807_Inf__fin_OboundedE,axiom,
! [A: set_set_set_a,X2: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ ( lattic4609221726464634331_set_a @ A ) )
=> ! [A8: set_set_a] :
( ( member_set_set_a @ A8 @ A )
=> ( ord_le3724670747650509150_set_a @ X2 @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_808_Inf__fin_OboundedI,axiom,
! [A: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ! [A6: set_a] :
( ( member_set_a @ A6 @ A )
=> ( ord_less_eq_set_a @ X2 @ A6 ) )
=> ( ord_less_eq_set_a @ X2 @ ( lattic8209813465164889211_set_a @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_809_Inf__fin_OboundedI,axiom,
! [A: set_set_set_a,X2: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ( ! [A6: set_set_a] :
( ( member_set_set_a @ A6 @ A )
=> ( ord_le3724670747650509150_set_a @ X2 @ A6 ) )
=> ( ord_le3724670747650509150_set_a @ X2 @ ( lattic4609221726464634331_set_a @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_810_Sup__fin_OboundedE,axiom,
! [A: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A ) @ X2 )
=> ! [A8: set_a] :
( ( member_set_a @ A8 @ A )
=> ( ord_less_eq_set_a @ A8 @ X2 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_811_Sup__fin_OboundedE,axiom,
! [A: set_set_set_a,X2: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ( ( ord_le3724670747650509150_set_a @ ( lattic338143333561554293_set_a @ A ) @ X2 )
=> ! [A8: set_set_a] :
( ( member_set_set_a @ A8 @ A )
=> ( ord_le3724670747650509150_set_a @ A8 @ X2 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_812_Sup__fin_OboundedI,axiom,
! [A: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ! [A6: set_a] :
( ( member_set_a @ A6 @ A )
=> ( ord_less_eq_set_a @ A6 @ X2 ) )
=> ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A ) @ X2 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_813_Sup__fin_OboundedI,axiom,
! [A: set_set_set_a,X2: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ( ! [A6: set_set_a] :
( ( member_set_set_a @ A6 @ A )
=> ( ord_le3724670747650509150_set_a @ A6 @ X2 ) )
=> ( ord_le3724670747650509150_set_a @ ( lattic338143333561554293_set_a @ A ) @ X2 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_814_Inf__fin_Obounded__iff,axiom,
! [A: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ X2 @ ( lattic8209813465164889211_set_a @ A ) )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ( ord_less_eq_set_a @ X2 @ X4 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_815_Inf__fin_Obounded__iff,axiom,
! [A: set_set_set_a,X2: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ ( lattic4609221726464634331_set_a @ A ) )
= ( ! [X4: set_set_a] :
( ( member_set_set_a @ X4 @ A )
=> ( ord_le3724670747650509150_set_a @ X2 @ X4 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_816_Sup__fin_Obounded__iff,axiom,
! [A: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A ) @ X2 )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ( ord_less_eq_set_a @ X4 @ X2 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_817_Sup__fin_Obounded__iff,axiom,
! [A: set_set_set_a,X2: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ( ( ord_le3724670747650509150_set_a @ ( lattic338143333561554293_set_a @ A ) @ X2 )
= ( ! [X4: set_set_a] :
( ( member_set_set_a @ X4 @ A )
=> ( ord_le3724670747650509150_set_a @ X4 @ X2 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_818_comp__sgraph_Oincident__loops__simp_I1_J,axiom,
! [S: set_a,V3: a] :
( ( undire3617971648856834880loop_a @ ( undire2918257014606996450dges_a @ S ) @ V3 )
=> ( ( undire4753905205749729249oops_a @ ( undire2918257014606996450dges_a @ S ) @ V3 )
= ( insert_set_a @ ( insert_a @ V3 @ bot_bot_set_a ) @ bot_bot_set_set_a ) ) ) ).
% comp_sgraph.incident_loops_simp(1)
thf(fact_819_comp__sgraph_Oincident__loops__simp_I1_J,axiom,
! [S: set_set_a,V3: set_a] :
( ( undire5774735625301615776_set_a @ ( undire8247866692393712962_set_a @ S ) @ V3 )
=> ( ( undire7215034953758041409_set_a @ ( undire8247866692393712962_set_a @ S ) @ V3 )
= ( insert_set_set_a @ ( insert_set_a @ V3 @ bot_bot_set_set_a ) @ bot_bo3380559777022489994_set_a ) ) ) ).
% comp_sgraph.incident_loops_simp(1)
thf(fact_820_comp__sgraph_Ohas__loop__def,axiom,
! [S: set_a,V3: a] :
( ( undire3617971648856834880loop_a @ ( undire2918257014606996450dges_a @ S ) @ V3 )
= ( member_set_a @ ( insert_a @ V3 @ bot_bot_set_a ) @ ( undire2918257014606996450dges_a @ S ) ) ) ).
% comp_sgraph.has_loop_def
thf(fact_821_comp__sgraph_Ohas__loop__def,axiom,
! [S: set_set_a,V3: set_a] :
( ( undire5774735625301615776_set_a @ ( undire8247866692393712962_set_a @ S ) @ V3 )
= ( member_set_set_a @ ( insert_set_a @ V3 @ bot_bot_set_set_a ) @ ( undire8247866692393712962_set_a @ S ) ) ) ).
% comp_sgraph.has_loop_def
thf(fact_822_Inf__fin_Osubset__imp,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ( ( finite7209287970140883943_set_a @ B )
=> ( ord_le3724670747650509150_set_a @ ( lattic4609221726464634331_set_a @ B ) @ ( lattic4609221726464634331_set_a @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_823_Inf__fin_Osubset__imp,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( A != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ B ) @ ( lattic8209813465164889211_set_a @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_824_Sup__fin_Osubset__imp,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ( ( finite7209287970140883943_set_a @ B )
=> ( ord_le3724670747650509150_set_a @ ( lattic338143333561554293_set_a @ A ) @ ( lattic338143333561554293_set_a @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_825_Sup__fin_Osubset__imp,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( A != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B )
=> ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A ) @ ( lattic2918178356826803221_set_a @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_826_Inf__fin_Osubset,axiom,
! [A: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( B != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( inf_inf_set_a @ ( lattic8209813465164889211_set_a @ B ) @ ( lattic8209813465164889211_set_a @ A ) )
= ( lattic8209813465164889211_set_a @ A ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_827_Sup__fin_Osubset,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( B != bot_bo3380559777022489994_set_a )
=> ( ( ord_le5722252365846178494_set_a @ B @ A )
=> ( ( sup_sup_set_set_a @ ( lattic338143333561554293_set_a @ B ) @ ( lattic338143333561554293_set_a @ A ) )
= ( lattic338143333561554293_set_a @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_828_Sup__fin_Osubset,axiom,
! [A: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( B != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( sup_sup_set_a @ ( lattic2918178356826803221_set_a @ B ) @ ( lattic2918178356826803221_set_a @ A ) )
= ( lattic2918178356826803221_set_a @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_829_Inf__fin_Oclosed,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ! [X3: set_a,Y3: set_a] : ( member_set_a @ ( inf_inf_set_a @ X3 @ Y3 ) @ ( insert_set_a @ X3 @ ( insert_set_a @ Y3 @ bot_bot_set_set_a ) ) )
=> ( member_set_a @ ( lattic8209813465164889211_set_a @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_830_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ~ ( member_set_a @ X2 @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X2 @ A ) )
= ( inf_inf_set_a @ X2 @ ( lattic8209813465164889211_set_a @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_831_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_set_set_a,X2: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ~ ( member_set_set_a @ X2 @ A )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ( ( lattic338143333561554293_set_a @ ( insert_set_set_a @ X2 @ A ) )
= ( sup_sup_set_set_a @ X2 @ ( lattic338143333561554293_set_a @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_832_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ~ ( member_set_a @ X2 @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( insert_set_a @ X2 @ A ) )
= ( sup_sup_set_a @ X2 @ ( lattic2918178356826803221_set_a @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_833_Sup__fin_Oclosed,axiom,
! [A: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ( ! [X3: set_set_a,Y3: set_set_a] : ( member_set_set_a @ ( sup_sup_set_set_a @ X3 @ Y3 ) @ ( insert_set_set_a @ X3 @ ( insert_set_set_a @ Y3 @ bot_bo3380559777022489994_set_a ) ) )
=> ( member_set_set_a @ ( lattic338143333561554293_set_a @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_834_Sup__fin_Oclosed,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ! [X3: set_a,Y3: set_a] : ( member_set_a @ ( sup_sup_set_a @ X3 @ Y3 ) @ ( insert_set_a @ X3 @ ( insert_set_a @ Y3 @ bot_bot_set_set_a ) ) )
=> ( member_set_a @ ( lattic2918178356826803221_set_a @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_835_Inf__fin_Ounion,axiom,
! [A: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B )
=> ( ( B != bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( sup_sup_set_set_a @ A @ B ) )
= ( inf_inf_set_a @ ( lattic8209813465164889211_set_a @ A ) @ ( lattic8209813465164889211_set_a @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_836_Sup__fin_Ounion,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ( ( finite7209287970140883943_set_a @ B )
=> ( ( B != bot_bo3380559777022489994_set_a )
=> ( ( lattic338143333561554293_set_a @ ( sup_su2076012971530813682_set_a @ A @ B ) )
= ( sup_sup_set_set_a @ ( lattic338143333561554293_set_a @ A ) @ ( lattic338143333561554293_set_a @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_837_Sup__fin_Ounion,axiom,
! [A: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B )
=> ( ( B != bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( sup_sup_set_set_a @ A @ B ) )
= ( sup_sup_set_a @ ( lattic2918178356826803221_set_a @ A ) @ ( lattic2918178356826803221_set_a @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_838_ulgraph_Oincident__loops__simp_I1_J,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3617971648856834880loop_a @ Edges @ V3 )
=> ( ( undire4753905205749729249oops_a @ Edges @ V3 )
= ( insert_set_a @ ( insert_a @ V3 @ bot_bot_set_a ) @ bot_bot_set_set_a ) ) ) ) ).
% ulgraph.incident_loops_simp(1)
thf(fact_839_ulgraph_Oincident__loops__simp_I1_J,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V3: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire5774735625301615776_set_a @ Edges @ V3 )
=> ( ( undire7215034953758041409_set_a @ Edges @ V3 )
= ( insert_set_set_a @ ( insert_set_a @ V3 @ bot_bot_set_set_a ) @ bot_bo3380559777022489994_set_a ) ) ) ) ).
% ulgraph.incident_loops_simp(1)
thf(fact_840_Sup__fin_Oinsert__remove,axiom,
! [A: set_set_set_a,X2: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( ( ( minus_3359197881701045381_set_a @ A @ ( insert_set_set_a @ X2 @ bot_bo3380559777022489994_set_a ) )
= bot_bo3380559777022489994_set_a )
=> ( ( lattic338143333561554293_set_a @ ( insert_set_set_a @ X2 @ A ) )
= X2 ) )
& ( ( ( minus_3359197881701045381_set_a @ A @ ( insert_set_set_a @ X2 @ bot_bo3380559777022489994_set_a ) )
!= bot_bo3380559777022489994_set_a )
=> ( ( lattic338143333561554293_set_a @ ( insert_set_set_a @ X2 @ A ) )
= ( sup_sup_set_set_a @ X2 @ ( lattic338143333561554293_set_a @ ( minus_3359197881701045381_set_a @ A @ ( insert_set_set_a @ X2 @ bot_bo3380559777022489994_set_a ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_841_Sup__fin_Oinsert__remove,axiom,
! [A: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( insert_set_a @ X2 @ A ) )
= X2 ) )
& ( ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( insert_set_a @ X2 @ A ) )
= ( sup_sup_set_a @ X2 @ ( lattic2918178356826803221_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_842_Sup__fin_Oremove,axiom,
! [A: set_set_set_a,X2: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( member_set_set_a @ X2 @ A )
=> ( ( ( ( minus_3359197881701045381_set_a @ A @ ( insert_set_set_a @ X2 @ bot_bo3380559777022489994_set_a ) )
= bot_bo3380559777022489994_set_a )
=> ( ( lattic338143333561554293_set_a @ A )
= X2 ) )
& ( ( ( minus_3359197881701045381_set_a @ A @ ( insert_set_set_a @ X2 @ bot_bo3380559777022489994_set_a ) )
!= bot_bo3380559777022489994_set_a )
=> ( ( lattic338143333561554293_set_a @ A )
= ( sup_sup_set_set_a @ X2 @ ( lattic338143333561554293_set_a @ ( minus_3359197881701045381_set_a @ A @ ( insert_set_set_a @ X2 @ bot_bo3380559777022489994_set_a ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_843_Sup__fin_Oremove,axiom,
! [A: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ X2 @ A )
=> ( ( ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ A )
= X2 ) )
& ( ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ A )
= ( sup_sup_set_a @ X2 @ ( lattic2918178356826803221_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_844_Inf__fin_Oremove,axiom,
! [A: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ X2 @ A )
=> ( ( ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ A )
= X2 ) )
& ( ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ A )
= ( inf_inf_set_a @ X2 @ ( lattic8209813465164889211_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_845_Diff__iff,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) )
= ( ( member_set_a @ C @ A )
& ~ ( member_set_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_846_Diff__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ~ ( member_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_847_DiffI,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 @ ( minus_5736297505244876581_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_848_DiffI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_849_Diff__cancel,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ A )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_850_Diff__cancel,axiom,
! [A: set_set_a] :
( ( minus_5736297505244876581_set_a @ A @ A )
= bot_bot_set_set_a ) ).
% Diff_cancel
thf(fact_851_empty__Diff,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_852_empty__Diff,axiom,
! [A: set_set_a] :
( ( minus_5736297505244876581_set_a @ bot_bot_set_set_a @ A )
= bot_bot_set_set_a ) ).
% empty_Diff
thf(fact_853_Diff__empty,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ bot_bot_set_a )
= A ) ).
% Diff_empty
thf(fact_854_Diff__empty,axiom,
! [A: set_set_a] :
( ( minus_5736297505244876581_set_a @ A @ bot_bot_set_set_a )
= A ) ).
% Diff_empty
thf(fact_855_finite__Diff2,axiom,
! [B: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) )
= ( finite_finite_set_a @ A ) ) ) ).
% finite_Diff2
thf(fact_856_finite__Diff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) ) ) ).
% finite_Diff
thf(fact_857_insert__Diff1,axiom,
! [X2: set_a,B: set_set_a,A: set_set_a] :
( ( member_set_a @ X2 @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X2 @ A ) @ B )
= ( minus_5736297505244876581_set_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_858_insert__Diff1,axiom,
! [X2: a,B: set_a,A: set_a] :
( ( member_a @ X2 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_859_Diff__insert0,axiom,
! [X2: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X2 @ A )
=> ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X2 @ B ) )
= ( minus_5736297505244876581_set_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_860_Diff__insert0,axiom,
! [X2: a,A: set_a,B: set_a] :
( ~ ( member_a @ X2 @ A )
=> ( ( minus_minus_set_a @ A @ ( insert_a @ X2 @ B ) )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_861_Un__Diff__cancel2,axiom,
! [B: set_a,A: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B @ A ) @ A )
= ( sup_sup_set_a @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_862_Un__Diff__cancel2,axiom,
! [B: set_set_a,A: set_set_a] :
( ( sup_sup_set_set_a @ ( minus_5736297505244876581_set_a @ B @ A ) @ A )
= ( sup_sup_set_set_a @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_863_Un__Diff__cancel,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= ( sup_sup_set_a @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_864_Un__Diff__cancel,axiom,
! [A: set_set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ A @ ( minus_5736297505244876581_set_a @ B @ A ) )
= ( sup_sup_set_set_a @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_865_Diff__eq__empty__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_866_Diff__eq__empty__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( minus_5736297505244876581_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_867_insert__Diff__single,axiom,
! [A2: a,A: set_a] :
( ( insert_a @ A2 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= ( insert_a @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_868_insert__Diff__single,axiom,
! [A2: set_a,A: set_set_a] :
( ( insert_set_a @ A2 @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) )
= ( insert_set_a @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_869_finite__Diff__insert,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_870_finite__Diff__insert,axiom,
! [A: set_set_a,A2: set_a,B: set_set_a] :
( ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ A2 @ B ) ) )
= ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_871_Diff__disjoint,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_872_Diff__disjoint,axiom,
! [A: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ A @ ( minus_5736297505244876581_set_a @ B @ A ) )
= bot_bot_set_set_a ) ).
% Diff_disjoint
thf(fact_873_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V3: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire5774735625301615776_set_a @ Edges @ V3 )
=> ( member_set_a @ V3 @ Vertices ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_874_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3617971648856834880loop_a @ Edges @ V3 )
=> ( member_a @ V3 @ Vertices ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_875_Diff__infinite__finite,axiom,
! [T2: set_set_a,S: set_set_a] :
( ( finite_finite_set_a @ T2 )
=> ( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_876_Un__Diff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( minus_minus_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A @ C2 ) @ ( minus_minus_set_a @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_877_Un__Diff,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( minus_5736297505244876581_set_a @ ( sup_sup_set_set_a @ A @ B ) @ C2 )
= ( sup_sup_set_set_a @ ( minus_5736297505244876581_set_a @ A @ C2 ) @ ( minus_5736297505244876581_set_a @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_878_ulgraph_Oempty__not__edge,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ~ ( member_set_a @ bot_bot_set_a @ Edges ) ) ).
% ulgraph.empty_not_edge
thf(fact_879_ulgraph_Oempty__not__edge,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ~ ( member_set_set_a @ bot_bot_set_set_a @ Edges ) ) ).
% ulgraph.empty_not_edge
thf(fact_880_DiffD2,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) )
=> ~ ( member_set_a @ C @ B ) ) ).
% DiffD2
thf(fact_881_DiffD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( member_a @ C @ B ) ) ).
% DiffD2
thf(fact_882_DiffD1,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) )
=> ( member_set_a @ C @ A ) ) ).
% DiffD1
thf(fact_883_DiffD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% DiffD1
thf(fact_884_DiffE,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) )
=> ~ ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% DiffE
thf(fact_885_DiffE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% DiffE
thf(fact_886_Int__Diff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_887_Diff__Int2,axiom,
! [A: set_a,C2: set_a,B: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C2 ) @ ( inf_inf_set_a @ B @ C2 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_888_Diff__Diff__Int,axiom,
! [A: set_a,B: set_a] :
( ( minus_minus_set_a @ A @ ( minus_minus_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_889_Diff__Int__distrib,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ C2 @ ( minus_minus_set_a @ A @ B ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C2 @ A ) @ ( inf_inf_set_a @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_890_Diff__Int__distrib2,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A @ B ) @ C2 )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C2 ) @ ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_891_ulgraph_Oaxioms_I1_J,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( undire2554140024507503526stem_a @ Vertices @ Edges ) ) ).
% ulgraph.axioms(1)
thf(fact_892_insert__Diff__if,axiom,
! [X2: set_a,B: set_set_a,A: set_set_a] :
( ( ( member_set_a @ X2 @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X2 @ A ) @ B )
= ( minus_5736297505244876581_set_a @ A @ B ) ) )
& ( ~ ( member_set_a @ X2 @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X2 @ A ) @ B )
= ( insert_set_a @ X2 @ ( minus_5736297505244876581_set_a @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_893_insert__Diff__if,axiom,
! [X2: a,B: set_a,A: set_a] :
( ( ( member_a @ X2 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) )
& ( ~ ( member_a @ X2 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A ) @ B )
= ( insert_a @ X2 @ ( minus_minus_set_a @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_894_double__diff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_895_double__diff,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 )
=> ( ( minus_5736297505244876581_set_a @ B @ ( minus_5736297505244876581_set_a @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_896_Diff__subset,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_897_Diff__subset,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_898_Diff__mono,axiom,
! [A: set_a,C2: set_a,D2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ D2 @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_899_Diff__mono,axiom,
! [A: set_set_a,C2: set_set_a,D2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ D2 @ B )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) @ ( minus_5736297505244876581_set_a @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_900_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V1: set_a,V22: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3510646817838285160_set_a @ Edges @ V1 @ V22 )
=> ( ( member_set_a @ V1 @ Vertices )
& ( member_set_a @ V22 @ Vertices ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_901_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V22: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V22 )
=> ( ( member_a @ V1 @ Vertices )
& ( member_a @ V22 @ Vertices ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_902_subgraph_Ois__subgraph__ulgraph,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ( undire7251896706689453996raph_a @ V_G @ E_G )
=> ( undire7251896706689453996raph_a @ V_H @ E_H ) ) ) ).
% subgraph.is_subgraph_ulgraph
thf(fact_903_diff__shunt__var,axiom,
! [X2: set_a,Y: set_a] :
( ( ( minus_minus_set_a @ X2 @ Y )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_904_diff__shunt__var,axiom,
! [X2: set_set_a,Y: set_set_a] :
( ( ( minus_5736297505244876581_set_a @ X2 @ Y )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_905_Diff__insert,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B ) @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_906_Diff__insert,axiom,
! [A: set_set_a,A2: set_a,B: set_set_a] :
( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( minus_5736297505244876581_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ) ).
% Diff_insert
thf(fact_907_insert__Diff,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= A ) ) ).
% insert_Diff
thf(fact_908_insert__Diff,axiom,
! [A2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ( ( insert_set_a @ A2 @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) )
= A ) ) ).
% insert_Diff
thf(fact_909_Diff__insert2,axiom,
! [A: set_a,A2: a,B: set_a] :
( ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_910_Diff__insert2,axiom,
! [A: set_set_a,A2: set_a,B: set_set_a] :
( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( minus_5736297505244876581_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_911_Diff__insert__absorb,axiom,
! [X2: a,A: set_a] :
( ~ ( member_a @ X2 @ A )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A ) @ ( insert_a @ X2 @ bot_bot_set_a ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_912_Diff__insert__absorb,axiom,
! [X2: set_a,A: set_set_a] :
( ~ ( member_set_a @ X2 @ A )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X2 @ A ) @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_913_subset__Diff__insert,axiom,
! [A: set_a,B: set_a,X2: a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ ( insert_a @ X2 @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ C2 ) )
& ~ ( member_a @ X2 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_914_subset__Diff__insert,axiom,
! [A: set_set_a,B: set_set_a,X2: set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( minus_5736297505244876581_set_a @ B @ ( insert_set_a @ X2 @ C2 ) ) )
= ( ( ord_le3724670747650509150_set_a @ A @ ( minus_5736297505244876581_set_a @ B @ C2 ) )
& ~ ( member_set_a @ X2 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_915_Int__Diff__disjoint,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ B ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_916_Int__Diff__disjoint,axiom,
! [A: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A @ B ) @ ( minus_5736297505244876581_set_a @ A @ B ) )
= bot_bot_set_set_a ) ).
% Int_Diff_disjoint
thf(fact_917_Diff__triv,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_918_Diff__triv,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a )
=> ( ( minus_5736297505244876581_set_a @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_919_Diff__partition,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ ( minus_minus_set_a @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_920_Diff__partition,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( sup_sup_set_set_a @ A @ ( minus_5736297505244876581_set_a @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_921_Diff__subset__conv,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ C2 )
= ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_922_Diff__subset__conv,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) @ C2 )
= ( ord_le3724670747650509150_set_a @ A @ ( sup_sup_set_set_a @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_923_Diff__Un,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( minus_minus_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) )
= ( inf_inf_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_924_Diff__Un,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( minus_5736297505244876581_set_a @ A @ ( sup_sup_set_set_a @ B @ C2 ) )
= ( inf_inf_set_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) @ ( minus_5736297505244876581_set_a @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_925_Diff__Int,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( minus_minus_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_926_Diff__Int,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( minus_5736297505244876581_set_a @ A @ ( inf_inf_set_set_a @ B @ C2 ) )
= ( sup_sup_set_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) @ ( minus_5736297505244876581_set_a @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_927_Int__Diff__Un,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B ) @ ( minus_minus_set_a @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_928_Int__Diff__Un,axiom,
! [A: set_set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ ( inf_inf_set_set_a @ A @ B ) @ ( minus_5736297505244876581_set_a @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_929_Un__Diff__Int,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ A @ B ) @ ( inf_inf_set_a @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_930_Un__Diff__Int,axiom,
! [A: set_set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) @ ( inf_inf_set_set_a @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_931_ulgraph_Ofinite__incident__loops,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( finite_finite_set_a @ ( undire4753905205749729249oops_a @ Edges @ V3 ) ) ) ).
% ulgraph.finite_incident_loops
thf(fact_932_ulgraph_Oincident__sedges__empty,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V3: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ~ ( member_set_a @ V3 @ Vertices )
=> ( ( undire5844230293943614535_set_a @ Edges @ V3 )
= bot_bo3380559777022489994_set_a ) ) ) ).
% ulgraph.incident_sedges_empty
thf(fact_933_ulgraph_Oincident__sedges__empty,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ~ ( member_a @ V3 @ Vertices )
=> ( ( undire1270416042309875431dges_a @ Edges @ V3 )
= bot_bot_set_set_a ) ) ) ).
% ulgraph.incident_sedges_empty
thf(fact_934_ulgraph_Ofinite__inc__sedges,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( finite_finite_set_a @ Edges )
=> ( finite_finite_set_a @ ( undire1270416042309875431dges_a @ Edges @ V3 ) ) ) ) ).
% ulgraph.finite_inc_sedges
thf(fact_935_ulgraph_Overt__adj__edge__iff2,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V22: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( V1 != V22 )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V22 )
= ( ? [X4: set_a] :
( ( member_set_a @ X4 @ Edges )
& ( undire1521409233611534436dent_a @ V1 @ X4 )
& ( undire1521409233611534436dent_a @ V22 @ X4 ) ) ) ) ) ) ).
% ulgraph.vert_adj_edge_iff2
thf(fact_936_ulgraph_Ois__isolated__vertex__edge,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a,E3: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V3 )
=> ( ( member_set_a @ E3 @ Edges )
=> ~ ( undire1521409233611534436dent_a @ V3 @ E3 ) ) ) ) ).
% ulgraph.is_isolated_vertex_edge
thf(fact_937_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V3: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire6879241558604981877_set_a @ Vertices @ Edges @ V3 )
= ( ( member_set_a @ V3 @ Vertices )
& ! [X4: set_a] :
( ( member_set_a @ X4 @ Vertices )
=> ~ ( undire3510646817838285160_set_a @ Edges @ X4 @ V3 ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_938_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V3 )
= ( ( member_a @ V3 @ Vertices )
& ! [X4: a] :
( ( member_a @ X4 @ Vertices )
=> ~ ( undire397441198561214472_adj_a @ Edges @ X4 @ V3 ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_939_ulgraph_Overt__adj__def,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V22: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V22 )
= ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V22 @ bot_bot_set_a ) ) @ Edges ) ) ) ).
% ulgraph.vert_adj_def
thf(fact_940_ulgraph_Overt__adj__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V1: set_a,V22: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3510646817838285160_set_a @ Edges @ V1 @ V22 )
= ( member_set_set_a @ ( insert_set_a @ V1 @ ( insert_set_a @ V22 @ bot_bot_set_set_a ) ) @ Edges ) ) ) ).
% ulgraph.vert_adj_def
thf(fact_941_ulgraph_Onot__vert__adj,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a,U: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ~ ( undire397441198561214472_adj_a @ Edges @ V3 @ U )
=> ~ ( member_set_a @ ( insert_a @ V3 @ ( insert_a @ U @ bot_bot_set_a ) ) @ Edges ) ) ) ).
% ulgraph.not_vert_adj
thf(fact_942_ulgraph_Onot__vert__adj,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V3: set_a,U: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ~ ( undire3510646817838285160_set_a @ Edges @ V3 @ U )
=> ~ ( member_set_set_a @ ( insert_set_a @ V3 @ ( insert_set_a @ U @ bot_bot_set_set_a ) ) @ Edges ) ) ) ).
% ulgraph.not_vert_adj
thf(fact_943_ulgraph_Ohas__loop__def,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3617971648856834880loop_a @ Edges @ V3 )
= ( member_set_a @ ( insert_a @ V3 @ bot_bot_set_a ) @ Edges ) ) ) ).
% ulgraph.has_loop_def
thf(fact_944_ulgraph_Ohas__loop__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V3: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire5774735625301615776_set_a @ Edges @ V3 )
= ( member_set_set_a @ ( insert_set_a @ V3 @ bot_bot_set_set_a ) @ Edges ) ) ) ).
% ulgraph.has_loop_def
thf(fact_945_finite__empty__induct,axiom,
! [A: set_a,P: set_a > $o] :
( ( finite_finite_a @ A )
=> ( ( P @ A )
=> ( ! [A6: a,A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( ( member_a @ A6 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_minus_set_a @ A7 @ ( insert_a @ A6 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_946_finite__empty__induct,axiom,
! [A: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ A )
=> ( ( P @ A )
=> ( ! [A6: set_a,A7: set_set_a] :
( ( finite_finite_set_a @ A7 )
=> ( ( member_set_a @ A6 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_5736297505244876581_set_a @ A7 @ ( insert_set_a @ A6 @ bot_bot_set_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_947_infinite__coinduct,axiom,
! [X5: set_a > $o,A: set_a] :
( ( X5 @ A )
=> ( ! [A7: set_a] :
( ( X5 @ A7 )
=> ? [X: a] :
( ( member_a @ X @ A7 )
& ( ( X5 @ ( minus_minus_set_a @ A7 @ ( insert_a @ X @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A7 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A ) ) ) ).
% infinite_coinduct
thf(fact_948_infinite__coinduct,axiom,
! [X5: set_set_a > $o,A: set_set_a] :
( ( X5 @ A )
=> ( ! [A7: set_set_a] :
( ( X5 @ A7 )
=> ? [X: set_a] :
( ( member_set_a @ X @ A7 )
& ( ( X5 @ ( minus_5736297505244876581_set_a @ A7 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) )
| ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A7 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) )
=> ~ ( finite_finite_set_a @ A ) ) ) ).
% infinite_coinduct
thf(fact_949_infinite__remove,axiom,
! [S: set_a,A2: a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_950_infinite__remove,axiom,
! [S: set_set_a,A2: set_a] :
( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ S @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ) ) ).
% infinite_remove
thf(fact_951_Diff__single__insert,axiom,
! [A: set_a,X2: a,B: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ X2 @ B ) ) ) ).
% Diff_single_insert
thf(fact_952_Diff__single__insert,axiom,
! [A: set_set_a,X2: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) @ B )
=> ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X2 @ B ) ) ) ).
% Diff_single_insert
thf(fact_953_subset__insert__iff,axiom,
! [A: set_a,X2: a,B: set_a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X2 @ B ) )
= ( ( ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_954_subset__insert__iff,axiom,
! [A: set_set_a,X2: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X2 @ B ) )
= ( ( ( member_set_a @ X2 @ A )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) @ B ) )
& ( ~ ( member_set_a @ X2 @ A )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_955_ulgraph_Oiso__vertex__empty__neighborhood,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V3 )
=> ( ( undire8504279938402040014hood_a @ Vertices @ Edges @ V3 )
= bot_bot_set_a ) ) ) ).
% ulgraph.iso_vertex_empty_neighborhood
thf(fact_956_ulgraph_Oiso__vertex__empty__neighborhood,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V3: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire6879241558604981877_set_a @ Vertices @ Edges @ V3 )
=> ( ( undire2074812191327625774_set_a @ Vertices @ Edges @ V3 )
= bot_bot_set_set_a ) ) ) ).
% ulgraph.iso_vertex_empty_neighborhood
thf(fact_957_ulgraph_Oincident__loops__simp_I2_J,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ~ ( undire3617971648856834880loop_a @ Edges @ V3 )
=> ( ( undire4753905205749729249oops_a @ Edges @ V3 )
= bot_bot_set_set_a ) ) ) ).
% ulgraph.incident_loops_simp(2)
thf(fact_958_ulgraph_Oincident__edges__sedges,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ~ ( undire3617971648856834880loop_a @ Edges @ V3 )
=> ( ( undire3231912044278729248dges_a @ Edges @ V3 )
= ( undire1270416042309875431dges_a @ Edges @ V3 ) ) ) ) ).
% ulgraph.incident_edges_sedges
thf(fact_959_ulgraph_Ois__edge__between__def,axiom,
! [Vertices: set_a,Edges: set_set_a,X5: set_a,Y6: set_a,E3: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8544646567961481629ween_a @ X5 @ Y6 @ E3 )
= ( ? [X4: a,Y4: a] :
( ( E3
= ( insert_a @ X4 @ ( insert_a @ Y4 @ bot_bot_set_a ) ) )
& ( member_a @ X4 @ X5 )
& ( member_a @ Y4 @ Y6 ) ) ) ) ) ).
% ulgraph.is_edge_between_def
thf(fact_960_ulgraph_Ois__edge__between__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X5: set_set_a,Y6: set_set_a,E3: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire2578756059399487229_set_a @ X5 @ Y6 @ E3 )
= ( ? [X4: set_a,Y4: set_a] :
( ( E3
= ( insert_set_a @ X4 @ ( insert_set_a @ Y4 @ bot_bot_set_set_a ) ) )
& ( member_set_a @ X4 @ X5 )
& ( member_set_a @ Y4 @ Y6 ) ) ) ) ) ).
% ulgraph.is_edge_between_def
thf(fact_961_finite__remove__induct,axiom,
! [B: set_a,P: set_a > $o] :
( ( finite_finite_a @ B )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( ( A7 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A7 @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A7 )
=> ( P @ ( minus_minus_set_a @ A7 @ ( insert_a @ X @ bot_bot_set_a ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_962_finite__remove__induct,axiom,
! [B: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ B )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A7: set_set_a] :
( ( finite_finite_set_a @ A7 )
=> ( ( A7 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A7 @ B )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ A7 )
=> ( P @ ( minus_5736297505244876581_set_a @ A7 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_963_remove__induct,axiom,
! [P: set_a > $o,B: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B )
=> ( P @ B ) )
=> ( ! [A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( ( A7 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A7 @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A7 )
=> ( P @ ( minus_minus_set_a @ A7 @ ( insert_a @ X @ bot_bot_set_a ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_964_remove__induct,axiom,
! [P: set_set_a > $o,B: set_set_a] :
( ( P @ bot_bot_set_set_a )
=> ( ( ~ ( finite_finite_set_a @ B )
=> ( P @ B ) )
=> ( ! [A7: set_set_a] :
( ( finite_finite_set_a @ A7 )
=> ( ( A7 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A7 @ B )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ A7 )
=> ( P @ ( minus_5736297505244876581_set_a @ A7 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_965_ulgraph_Overt__adj__inc__edge__iff,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V1: set_a,V22: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3510646817838285160_set_a @ Edges @ V1 @ V22 )
= ( ( undire2320338297334612420_set_a @ V1 @ ( insert_set_a @ V1 @ ( insert_set_a @ V22 @ bot_bot_set_set_a ) ) )
& ( undire2320338297334612420_set_a @ V22 @ ( insert_set_a @ V1 @ ( insert_set_a @ V22 @ bot_bot_set_set_a ) ) )
& ( member_set_set_a @ ( insert_set_a @ V1 @ ( insert_set_a @ V22 @ bot_bot_set_set_a ) ) @ Edges ) ) ) ) ).
% ulgraph.vert_adj_inc_edge_iff
thf(fact_966_ulgraph_Overt__adj__inc__edge__iff,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V22: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V22 )
= ( ( undire1521409233611534436dent_a @ V1 @ ( insert_a @ V1 @ ( insert_a @ V22 @ bot_bot_set_a ) ) )
& ( undire1521409233611534436dent_a @ V22 @ ( insert_a @ V1 @ ( insert_a @ V22 @ bot_bot_set_a ) ) )
& ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V22 @ bot_bot_set_a ) ) @ Edges ) ) ) ) ).
% ulgraph.vert_adj_inc_edge_iff
thf(fact_967_ulgraph_Oneighborhood__incident,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,U: set_a,V3: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( member_set_a @ U @ ( undire2074812191327625774_set_a @ Vertices @ Edges @ V3 ) )
= ( member_set_set_a @ ( insert_set_a @ U @ ( insert_set_a @ V3 @ bot_bot_set_set_a ) ) @ ( undire4631953023069350784_set_a @ Edges @ V3 ) ) ) ) ).
% ulgraph.neighborhood_incident
thf(fact_968_ulgraph_Oneighborhood__incident,axiom,
! [Vertices: set_a,Edges: set_set_a,U: a,V3: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( member_a @ U @ ( undire8504279938402040014hood_a @ Vertices @ Edges @ V3 ) )
= ( member_set_a @ ( insert_a @ U @ ( insert_a @ V3 @ bot_bot_set_a ) ) @ ( undire3231912044278729248dges_a @ Edges @ V3 ) ) ) ) ).
% ulgraph.neighborhood_incident
thf(fact_969_ulgraph_Oincident__edges__union,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3231912044278729248dges_a @ Edges @ V3 )
= ( sup_sup_set_set_a @ ( undire1270416042309875431dges_a @ Edges @ V3 ) @ ( undire4753905205749729249oops_a @ Edges @ V3 ) ) ) ) ).
% ulgraph.incident_edges_union
thf(fact_970_Inf__fin_Oinsert__remove,axiom,
! [A: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X2 @ A ) )
= X2 ) )
& ( ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X2 @ A ) )
= ( inf_inf_set_a @ X2 @ ( lattic8209813465164889211_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_971_remove__def,axiom,
( remove_a
= ( ^ [X4: a,A3: set_a] : ( minus_minus_set_a @ A3 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ).
% remove_def
thf(fact_972_remove__def,axiom,
( remove_set_a
= ( ^ [X4: set_a,A3: set_set_a] : ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) ) ) ).
% remove_def
thf(fact_973_ulgraph_Ointro,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( undire2177556672586781897ioms_a @ Edges )
=> ( undire7251896706689453996raph_a @ Vertices @ Edges ) ) ) ).
% ulgraph.intro
thf(fact_974_ulgraph__def,axiom,
( undire7251896706689453996raph_a
= ( ^ [Vertices2: set_a,Edges2: set_set_a] :
( ( undire2554140024507503526stem_a @ Vertices2 @ Edges2 )
& ( undire2177556672586781897ioms_a @ Edges2 ) ) ) ) ).
% ulgraph_def
thf(fact_975_member__remove,axiom,
! [X2: set_a,Y: set_a,A: set_set_a] :
( ( member_set_a @ X2 @ ( remove_set_a @ Y @ A ) )
= ( ( member_set_a @ X2 @ A )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_976_member__remove,axiom,
! [X2: a,Y: a,A: set_a] :
( ( member_a @ X2 @ ( remove_a @ Y @ A ) )
= ( ( member_a @ X2 @ A )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_977_sgraph_Oinduced__edges__alt,axiom,
! [Vertices: set_a,Edges: set_set_a,V: set_a] :
( ( undire3507641187627840796raph_a @ Vertices @ Edges )
=> ( ( undire7777452895879145676dges_a @ Edges @ V )
= ( inf_inf_set_set_a @ Edges @ ( undire2918257014606996450dges_a @ V ) ) ) ) ).
% sgraph.induced_edges_alt
thf(fact_978_Sup__fin_Oeq__fold,axiom,
! [A: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( lattic2918178356826803221_set_a @ ( insert_set_a @ X2 @ A ) )
= ( finite5985231929012247624_set_a @ sup_sup_set_a @ X2 @ A ) ) ) ).
% Sup_fin.eq_fold
thf(fact_979_Sup__fin_Oeq__fold,axiom,
! [A: set_set_set_a,X2: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( lattic338143333561554293_set_a @ ( insert_set_set_a @ X2 @ A ) )
= ( finite1111689594665117768_set_a @ sup_sup_set_set_a @ X2 @ A ) ) ) ).
% Sup_fin.eq_fold
thf(fact_980_Inf__fin_Oeq__fold,axiom,
! [A: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( lattic8209813465164889211_set_a @ ( insert_set_a @ X2 @ A ) )
= ( finite5985231929012247624_set_a @ inf_inf_set_a @ X2 @ A ) ) ) ).
% Inf_fin.eq_fold
thf(fact_981_sgraph_Oaxioms_I1_J,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire3507641187627840796raph_a @ Vertices @ Edges )
=> ( undire2554140024507503526stem_a @ Vertices @ Edges ) ) ).
% sgraph.axioms(1)
thf(fact_982_sgraph_Oinduced__edges__self,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire3507641187627840796raph_a @ Vertices @ Edges )
=> ( ( undire7777452895879145676dges_a @ Edges @ Vertices )
= Edges ) ) ).
% sgraph.induced_edges_self
thf(fact_983_sgraph_Oe__in__all__edges,axiom,
! [Vertices: set_a,Edges: set_set_a,E3: set_a] :
( ( undire3507641187627840796raph_a @ Vertices @ Edges )
=> ( ( member_set_a @ E3 @ Edges )
=> ( member_set_a @ E3 @ ( undire2918257014606996450dges_a @ Vertices ) ) ) ) ).
% sgraph.e_in_all_edges
thf(fact_984_subgraph_Ois__simp__subgraph,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ( undire3507641187627840796raph_a @ V_G @ E_G )
=> ( undire3507641187627840796raph_a @ V_H @ E_H ) ) ) ).
% subgraph.is_simp_subgraph
thf(fact_985_sgraph_Ono__loops,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V3: set_a] :
( ( undire6035205377725458044_set_a @ Vertices @ Edges )
=> ( ( member_set_a @ V3 @ Vertices )
=> ~ ( undire5774735625301615776_set_a @ Edges @ V3 ) ) ) ).
% sgraph.no_loops
thf(fact_986_sgraph_Ono__loops,axiom,
! [Vertices: set_a,Edges: set_set_a,V3: a] :
( ( undire3507641187627840796raph_a @ Vertices @ Edges )
=> ( ( member_a @ V3 @ Vertices )
=> ~ ( undire3617971648856834880loop_a @ Edges @ V3 ) ) ) ).
% sgraph.no_loops
thf(fact_987_sgraph_Osingleton__not__edge,axiom,
! [Vertices: set_a,Edges: set_set_a,X2: a] :
( ( undire3507641187627840796raph_a @ Vertices @ Edges )
=> ~ ( member_set_a @ ( insert_a @ X2 @ bot_bot_set_a ) @ Edges ) ) ).
% sgraph.singleton_not_edge
thf(fact_988_sgraph_Osingleton__not__edge,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X2: set_a] :
( ( undire6035205377725458044_set_a @ Vertices @ Edges )
=> ~ ( member_set_set_a @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) @ Edges ) ) ).
% sgraph.singleton_not_edge
thf(fact_989_sgraph_Oe__in__all__edges__ss,axiom,
! [Vertices: set_a,Edges: set_set_a,E3: set_a,V: set_a] :
( ( undire3507641187627840796raph_a @ Vertices @ Edges )
=> ( ( member_set_a @ E3 @ Edges )
=> ( ( ord_less_eq_set_a @ E3 @ V )
=> ( ( ord_less_eq_set_a @ V @ Vertices )
=> ( member_set_a @ E3 @ ( undire2918257014606996450dges_a @ V ) ) ) ) ) ) ).
% sgraph.e_in_all_edges_ss
thf(fact_990_sgraph_Oe__in__all__edges__ss,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,E3: set_set_a,V: set_set_a] :
( ( undire6035205377725458044_set_a @ Vertices @ Edges )
=> ( ( member_set_set_a @ E3 @ Edges )
=> ( ( ord_le3724670747650509150_set_a @ E3 @ V )
=> ( ( ord_le3724670747650509150_set_a @ V @ Vertices )
=> ( member_set_set_a @ E3 @ ( undire8247866692393712962_set_a @ V ) ) ) ) ) ) ).
% sgraph.e_in_all_edges_ss
thf(fact_991_sgraph_Owellformed__all__edges,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire3507641187627840796raph_a @ Vertices @ Edges )
=> ( ord_le3724670747650509150_set_a @ Edges @ ( undire2918257014606996450dges_a @ Vertices ) ) ) ).
% sgraph.wellformed_all_edges
thf(fact_992_union__fold__insert,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( sup_sup_set_a @ A @ B )
= ( finite_fold_a_set_a @ insert_a @ B @ A ) ) ) ).
% union_fold_insert
thf(fact_993_union__fold__insert,axiom,
! [A: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( sup_sup_set_set_a @ A @ B )
= ( finite8981829120792779176_set_a @ insert_set_a @ B @ A ) ) ) ).
% union_fold_insert
thf(fact_994_sgraph_Osubgraph__complete,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire3507641187627840796raph_a @ Vertices @ Edges )
=> ( undire7103218114511261257raph_a @ Vertices @ Edges @ Vertices @ ( undire2918257014606996450dges_a @ Vertices ) ) ) ).
% sgraph.subgraph_complete
thf(fact_995_minus__fold__remove,axiom,
! [A: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( minus_5736297505244876581_set_a @ B @ A )
= ( finite8981829120792779176_set_a @ remove_set_a @ B @ A ) ) ) ).
% minus_fold_remove
thf(fact_996_sgraph_Ointro,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( undire3875311282895952441ioms_a @ Edges )
=> ( undire3507641187627840796raph_a @ Vertices @ Edges ) ) ) ).
% sgraph.intro
thf(fact_997_sgraph__def,axiom,
( undire3507641187627840796raph_a
= ( ^ [Vertices2: set_a,Edges2: set_set_a] :
( ( undire2554140024507503526stem_a @ Vertices2 @ Edges2 )
& ( undire3875311282895952441ioms_a @ Edges2 ) ) ) ) ).
% sgraph_def
thf(fact_998_pairwise__alt,axiom,
( pairwise_a
= ( ^ [R2: a > a > $o,S2: set_a] :
! [X4: a] :
( ( member_a @ X4 @ S2 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ ( minus_minus_set_a @ S2 @ ( insert_a @ X4 @ bot_bot_set_a ) ) )
=> ( R2 @ X4 @ Y4 ) ) ) ) ) ).
% pairwise_alt
thf(fact_999_pairwise__alt,axiom,
( pairwise_set_a
= ( ^ [R2: set_a > set_a > $o,S2: set_set_a] :
! [X4: set_a] :
( ( member_set_a @ X4 @ S2 )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ ( minus_5736297505244876581_set_a @ S2 @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) )
=> ( R2 @ X4 @ Y4 ) ) ) ) ) ).
% pairwise_alt
thf(fact_1000_pairwiseD,axiom,
! [R: set_a > set_a > $o,S: set_set_a,X2: set_a,Y: set_a] :
( ( pairwise_set_a @ R @ S )
=> ( ( member_set_a @ X2 @ S )
=> ( ( member_set_a @ Y @ S )
=> ( ( X2 != Y )
=> ( R @ X2 @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_1001_pairwiseD,axiom,
! [R: a > a > $o,S: set_a,X2: a,Y: a] :
( ( pairwise_a @ R @ S )
=> ( ( member_a @ X2 @ S )
=> ( ( member_a @ Y @ S )
=> ( ( X2 != Y )
=> ( R @ X2 @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_1002_pairwiseI,axiom,
! [S: set_set_a,R: set_a > set_a > $o] :
( ! [X3: set_a,Y3: set_a] :
( ( member_set_a @ X3 @ S )
=> ( ( member_set_a @ Y3 @ S )
=> ( ( X3 != Y3 )
=> ( R @ X3 @ Y3 ) ) ) )
=> ( pairwise_set_a @ R @ S ) ) ).
% pairwiseI
thf(fact_1003_pairwiseI,axiom,
! [S: set_a,R: a > a > $o] :
( ! [X3: a,Y3: a] :
( ( member_a @ X3 @ S )
=> ( ( member_a @ Y3 @ S )
=> ( ( X3 != Y3 )
=> ( R @ X3 @ Y3 ) ) ) )
=> ( pairwise_a @ R @ S ) ) ).
% pairwiseI
thf(fact_1004_pairwise__insert,axiom,
! [R3: set_a > set_a > $o,X2: set_a,S3: set_set_a] :
( ( pairwise_set_a @ R3 @ ( insert_set_a @ X2 @ S3 ) )
= ( ! [Y4: set_a] :
( ( ( member_set_a @ Y4 @ S3 )
& ( Y4 != X2 ) )
=> ( ( R3 @ X2 @ Y4 )
& ( R3 @ Y4 @ X2 ) ) )
& ( pairwise_set_a @ R3 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_1005_pairwise__insert,axiom,
! [R3: a > a > $o,X2: a,S3: set_a] :
( ( pairwise_a @ R3 @ ( insert_a @ X2 @ S3 ) )
= ( ! [Y4: a] :
( ( ( member_a @ Y4 @ S3 )
& ( Y4 != X2 ) )
=> ( ( R3 @ X2 @ Y4 )
& ( R3 @ Y4 @ X2 ) ) )
& ( pairwise_a @ R3 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_1006_pairwise__mono,axiom,
! [P: a > a > $o,A: set_a,Q: a > a > $o,B: set_a] :
( ( pairwise_a @ P @ A )
=> ( ! [X3: a,Y3: a] :
( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( pairwise_a @ Q @ B ) ) ) ) ).
% pairwise_mono
thf(fact_1007_pairwise__mono,axiom,
! [P: set_a > set_a > $o,A: set_set_a,Q: set_a > set_a > $o,B: set_set_a] :
( ( pairwise_set_a @ P @ A )
=> ( ! [X3: set_a,Y3: set_a] :
( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( pairwise_set_a @ Q @ B ) ) ) ) ).
% pairwise_mono
thf(fact_1008_pairwise__subset,axiom,
! [P: a > a > $o,S: set_a,T2: set_a] :
( ( pairwise_a @ P @ S )
=> ( ( ord_less_eq_set_a @ T2 @ S )
=> ( pairwise_a @ P @ T2 ) ) ) ).
% pairwise_subset
thf(fact_1009_pairwise__subset,axiom,
! [P: set_a > set_a > $o,S: set_set_a,T2: set_set_a] :
( ( pairwise_set_a @ P @ S )
=> ( ( ord_le3724670747650509150_set_a @ T2 @ S )
=> ( pairwise_set_a @ P @ T2 ) ) ) ).
% pairwise_subset
thf(fact_1010_pairwise__empty,axiom,
! [P: a > a > $o] : ( pairwise_a @ P @ bot_bot_set_a ) ).
% pairwise_empty
thf(fact_1011_pairwise__empty,axiom,
! [P: set_a > set_a > $o] : ( pairwise_set_a @ P @ bot_bot_set_set_a ) ).
% pairwise_empty
thf(fact_1012_fin__graph__system_OfinV,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a] :
( ( undire1228946095791996325_set_a @ Vertices @ Edges )
=> ( finite_finite_set_a @ Vertices ) ) ).
% fin_graph_system.finV
thf(fact_1013_fin__graph__system_Oaxioms_I1_J,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire945497512398942277stem_a @ Vertices @ Edges )
=> ( undire2554140024507503526stem_a @ Vertices @ Edges ) ) ).
% fin_graph_system.axioms(1)
thf(fact_1014_fin__graph__system_Ofin__edges,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire945497512398942277stem_a @ Vertices @ Edges )
=> ( finite_finite_set_a @ Edges ) ) ).
% fin_graph_system.fin_edges
thf(fact_1015_subgraph_Ois__finite__subgraph,axiom,
! [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a] :
( ( undire7103218114511261257raph_a @ V_H @ E_H @ V_G @ E_G )
=> ( ( undire945497512398942277stem_a @ V_G @ E_G )
=> ( undire945497512398942277stem_a @ V_H @ E_H ) ) ) ).
% subgraph.is_finite_subgraph
thf(fact_1016_pairwise__singleton,axiom,
! [P: a > a > $o,A: a] : ( pairwise_a @ P @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% pairwise_singleton
thf(fact_1017_pairwise__singleton,axiom,
! [P: set_a > set_a > $o,A: set_a] : ( pairwise_set_a @ P @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% pairwise_singleton
thf(fact_1018_fin__graph__system_Ointro,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire2554140024507503526stem_a @ Vertices @ Edges )
=> ( ( undire5875954478249984866ioms_a @ Vertices )
=> ( undire945497512398942277stem_a @ Vertices @ Edges ) ) ) ).
% fin_graph_system.intro
thf(fact_1019_fin__graph__system__def,axiom,
( undire945497512398942277stem_a
= ( ^ [Vertices2: set_a,Edges2: set_set_a] :
( ( undire2554140024507503526stem_a @ Vertices2 @ Edges2 )
& ( undire5875954478249984866ioms_a @ Vertices2 ) ) ) ) ).
% fin_graph_system_def
thf(fact_1020_fin__graph__system__axioms_Ointro,axiom,
! [Vertices: set_set_a] :
( ( finite_finite_set_a @ Vertices )
=> ( undire819665811519888578_set_a @ Vertices ) ) ).
% fin_graph_system_axioms.intro
thf(fact_1021_fin__graph__system__axioms__def,axiom,
undire819665811519888578_set_a = finite_finite_set_a ).
% fin_graph_system_axioms_def
thf(fact_1022_psubset__insert__iff,axiom,
! [A: set_a,X2: a,B: set_a] :
( ( ord_less_set_a @ A @ ( insert_a @ X2 @ B ) )
= ( ( ( member_a @ X2 @ B )
=> ( ord_less_set_a @ A @ B ) )
& ( ~ ( member_a @ X2 @ B )
=> ( ( ( member_a @ X2 @ A )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ A @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1023_psubset__insert__iff,axiom,
! [A: set_set_a,X2: set_a,B: set_set_a] :
( ( ord_less_set_set_a @ A @ ( insert_set_a @ X2 @ B ) )
= ( ( ( member_set_a @ X2 @ B )
=> ( ord_less_set_set_a @ A @ B ) )
& ( ~ ( member_set_a @ X2 @ B )
=> ( ( ( member_set_a @ X2 @ A )
=> ( ord_less_set_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) @ B ) )
& ( ~ ( member_set_a @ X2 @ A )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1024_finite__induct__select,axiom,
! [S: set_a,P: set_a > $o] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [T3: set_a] :
( ( ord_less_set_a @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X: a] :
( ( member_a @ X @ ( minus_minus_set_a @ S @ T3 ) )
& ( P @ ( insert_a @ X @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1025_finite__induct__select,axiom,
! [S: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ S )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [T3: set_set_a] :
( ( ord_less_set_set_a @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X: set_a] :
( ( member_set_a @ X @ ( minus_5736297505244876581_set_a @ S @ T3 ) )
& ( P @ ( insert_set_a @ X @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1026_Fpow__subset__Pow,axiom,
! [A: set_a] : ( ord_le3724670747650509150_set_a @ ( finite_Fpow_a @ A ) @ ( pow_a @ A ) ) ).
% Fpow_subset_Pow
thf(fact_1027_psubsetI,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% psubsetI
thf(fact_1028_psubsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_set_a @ A @ B ) ) ) ).
% psubsetI
thf(fact_1029_psubset__imp__ex__mem,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ? [B7: set_a] : ( member_set_a @ B7 @ ( minus_5736297505244876581_set_a @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1030_psubset__imp__ex__mem,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ? [B7: a] : ( member_a @ B7 @ ( minus_minus_set_a @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1031_bot_Oextremum__strict,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_1032_bot_Oextremum__strict,axiom,
! [A2: set_set_a] :
~ ( ord_less_set_set_a @ A2 @ bot_bot_set_set_a ) ).
% bot.extremum_strict
thf(fact_1033_bot_Onot__eq__extremum,axiom,
! [A2: set_a] :
( ( A2 != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_1034_bot_Onot__eq__extremum,axiom,
! [A2: set_set_a] :
( ( A2 != bot_bot_set_set_a )
= ( ord_less_set_set_a @ bot_bot_set_set_a @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_1035_not__psubset__empty,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_1036_not__psubset__empty,axiom,
! [A: set_set_a] :
~ ( ord_less_set_set_a @ A @ bot_bot_set_set_a ) ).
% not_psubset_empty
thf(fact_1037_psubsetE,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ B @ A ) ) ) ).
% psubsetE
thf(fact_1038_psubsetE,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ~ ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ B @ A ) ) ) ).
% psubsetE
thf(fact_1039_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_1040_psubset__eq,axiom,
( ord_less_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_1041_psubset__imp__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_1042_psubset__imp__subset,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_1043_psubset__subset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_1044_psubset__subset__trans,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ord_less_set_set_a @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_1045_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ~ ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1046_subset__not__subset__eq,axiom,
( ord_less_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
& ~ ( ord_le3724670747650509150_set_a @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1047_subset__psubset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_1048_subset__psubset__trans,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_less_set_set_a @ B @ C2 )
=> ( ord_less_set_set_a @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_1049_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1050_subset__iff__psubset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_less_set_set_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1051_inf_Ostrict__coboundedI2,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_1052_inf_Ostrict__coboundedI1,axiom,
! [A2: set_a,C: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ C )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_1053_inf_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( A4
= ( inf_inf_set_a @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_1054_inf_Ostrict__boundedE,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C ) )
=> ~ ( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ord_less_set_a @ A2 @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_1055_inf_Oabsorb4,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_1056_inf_Oabsorb3,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb3
thf(fact_1057_less__infI2,axiom,
! [B2: set_a,X2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ X2 )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X2 ) ) ).
% less_infI2
thf(fact_1058_less__infI1,axiom,
! [A2: set_a,X2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ X2 )
=> ( ord_less_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ X2 ) ) ).
% less_infI1
thf(fact_1059_sup_Ostrict__coboundedI2,axiom,
! [C: set_a,B2: set_a,A2: set_a] :
( ( ord_less_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_1060_sup_Ostrict__coboundedI2,axiom,
! [C: set_set_a,B2: set_set_a,A2: set_set_a] :
( ( ord_less_set_set_a @ C @ B2 )
=> ( ord_less_set_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_1061_sup_Ostrict__coboundedI1,axiom,
! [C: set_a,A2: set_a,B2: set_a] :
( ( ord_less_set_a @ C @ A2 )
=> ( ord_less_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_1062_sup_Ostrict__coboundedI1,axiom,
! [C: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ C @ A2 )
=> ( ord_less_set_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_1063_sup_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( A4
= ( sup_sup_set_a @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1064_sup_Ostrict__order__iff,axiom,
( ord_less_set_set_a
= ( ^ [B4: set_set_a,A4: set_set_a] :
( ( A4
= ( sup_sup_set_set_a @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1065_sup_Ostrict__boundedE,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_set_a @ B2 @ A2 )
=> ~ ( ord_less_set_a @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_1066_sup_Ostrict__boundedE,axiom,
! [B2: set_set_a,C: set_set_a,A2: set_set_a] :
( ( ord_less_set_set_a @ ( sup_sup_set_set_a @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_set_set_a @ B2 @ A2 )
=> ~ ( ord_less_set_set_a @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_1067_sup_Oabsorb4,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_1068_sup_Oabsorb4,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_set_a @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_1069_sup_Oabsorb3,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_1070_sup_Oabsorb3,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_less_set_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_set_a @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_1071_less__supI2,axiom,
! [X2: set_a,B2: set_a,A2: set_a] :
( ( ord_less_set_a @ X2 @ B2 )
=> ( ord_less_set_a @ X2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_1072_less__supI2,axiom,
! [X2: set_set_a,B2: set_set_a,A2: set_set_a] :
( ( ord_less_set_set_a @ X2 @ B2 )
=> ( ord_less_set_set_a @ X2 @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_1073_less__supI1,axiom,
! [X2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_set_a @ X2 @ A2 )
=> ( ord_less_set_a @ X2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_1074_less__supI1,axiom,
! [X2: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ X2 @ A2 )
=> ( ord_less_set_set_a @ X2 @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_1075_psubsetD,axiom,
! [A: set_set_a,B: set_set_a,C: set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% psubsetD
thf(fact_1076_psubsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% psubsetD
thf(fact_1077_leD,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ~ ( ord_less_set_a @ X2 @ Y ) ) ).
% leD
thf(fact_1078_leD,axiom,
! [Y: set_set_a,X2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X2 )
=> ~ ( ord_less_set_set_a @ X2 @ Y ) ) ).
% leD
thf(fact_1079_nless__le,axiom,
! [A2: set_a,B2: set_a] :
( ( ~ ( ord_less_set_a @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_set_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_1080_nless__le,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ~ ( ord_less_set_set_a @ A2 @ B2 ) )
= ( ~ ( ord_le3724670747650509150_set_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_1081_antisym__conv1,axiom,
! [X2: set_a,Y: set_a] :
( ~ ( ord_less_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_1082_antisym__conv1,axiom,
! [X2: set_set_a,Y: set_set_a] :
( ~ ( ord_less_set_set_a @ X2 @ Y )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_1083_antisym__conv2,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ~ ( ord_less_set_a @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_1084_antisym__conv2,axiom,
! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ( ~ ( ord_less_set_set_a @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_1085_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
& ~ ( ord_less_eq_set_a @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_1086_less__le__not__le,axiom,
( ord_less_set_set_a
= ( ^ [X4: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y4 )
& ~ ( ord_le3724670747650509150_set_a @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_1087_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_set_a @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1088_order_Oorder__iff__strict,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_less_set_set_a @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1089_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1090_order_Ostrict__iff__order,axiom,
( ord_less_set_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1091_order_Ostrict__trans1,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_1092_order_Ostrict__trans1,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_less_set_set_a @ B2 @ C )
=> ( ord_less_set_set_a @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_1093_order_Ostrict__trans2,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_1094_order_Ostrict__trans2,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_less_set_set_a @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_1095_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ~ ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1096_order_Ostrict__iff__not,axiom,
( ord_less_set_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B4 )
& ~ ( ord_le3724670747650509150_set_a @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1097_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( ord_less_set_a @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1098_dual__order_Oorder__iff__strict,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B4: set_set_a,A4: set_set_a] :
( ( ord_less_set_set_a @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1099_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1100_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_set_a
= ( ^ [B4: set_set_a,A4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1101_dual__order_Ostrict__trans1,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1102_dual__order_Ostrict__trans1,axiom,
! [B2: set_set_a,A2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( ord_less_set_set_a @ C @ B2 )
=> ( ord_less_set_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1103_dual__order_Ostrict__trans2,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1104_dual__order_Ostrict__trans2,axiom,
! [B2: set_set_a,A2: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ B2 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ C @ B2 )
=> ( ord_less_set_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1105_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ~ ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1106_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_set_a
= ( ^ [B4: set_set_a,A4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A4 )
& ~ ( ord_le3724670747650509150_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1107_order_Ostrict__implies__order,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_1108_order_Ostrict__implies__order,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_1109_dual__order_Ostrict__implies__order,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1110_dual__order_Ostrict__implies__order,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_less_set_set_a @ B2 @ A2 )
=> ( ord_le3724670747650509150_set_a @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1111_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X4: set_a,Y4: set_a] :
( ( ord_less_set_a @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_1112_order__le__less,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X4: set_set_a,Y4: set_set_a] :
( ( ord_less_set_set_a @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_1113_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_1114_order__less__le,axiom,
( ord_less_set_set_a
= ( ^ [X4: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_1115_order__less__imp__le,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_1116_order__less__imp__le,axiom,
! [X2: set_set_a,Y: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y )
=> ( ord_le3724670747650509150_set_a @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_1117_order__le__neq__trans,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1118_order__le__neq__trans,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_set_a @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1119_order__neq__le__trans,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 != B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1120_order__neq__le__trans,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 != B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ord_less_set_set_a @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1121_order__le__less__trans,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_set_a @ Y @ Z )
=> ( ord_less_set_a @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1122_order__le__less__trans,axiom,
! [X2: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ( ord_less_set_set_a @ Y @ Z )
=> ( ord_less_set_set_a @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1123_order__less__le__trans,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_set_a @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1124_order__less__le__trans,axiom,
! [X2: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_less_set_set_a @ X2 @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ Z )
=> ( ord_less_set_set_a @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1125_order__le__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1126_order__le__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_set_a,C: set_set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_set_a @ ( F @ B2 ) @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1127_order__le__less__subst2,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1128_order__le__less__subst2,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_set_a > set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_less_set_set_a @ ( F @ B2 ) @ C )
=> ( ! [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1129_order__less__le__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1130_order__less__le__subst1,axiom,
! [A2: set_set_a,F: set_a > set_set_a,B2: set_a,C: set_a] :
( ( ord_less_set_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1131_order__less__le__subst1,axiom,
! [A2: set_a,F: set_set_a > set_a,B2: set_set_a,C: set_set_a] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1132_order__less__le__subst1,axiom,
! [A2: set_set_a,F: set_set_a > set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_less_set_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1133_order__le__imp__less__or__eq,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_set_a @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1134_order__le__imp__less__or__eq,axiom,
! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ( ord_less_set_set_a @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1135_finite__psubset__induct,axiom,
! [A: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ A )
=> ( ! [A7: set_set_a] :
( ( finite_finite_set_a @ A7 )
=> ( ! [B8: set_set_a] :
( ( ord_less_set_set_a @ B8 @ A7 )
=> ( P @ B8 ) )
=> ( P @ A7 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_1136_empty__in__Fpow,axiom,
! [A: set_a] : ( member_set_a @ bot_bot_set_a @ ( finite_Fpow_a @ A ) ) ).
% empty_in_Fpow
thf(fact_1137_empty__in__Fpow,axiom,
! [A: set_set_a] : ( member_set_set_a @ bot_bot_set_set_a @ ( finite_Fpow_set_a @ A ) ) ).
% empty_in_Fpow
thf(fact_1138_ex__min__if__finite,axiom,
! [S: set_set_a] :
( ( finite_finite_set_a @ S )
=> ( ( S != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ S )
& ~ ? [Xa: set_a] :
( ( member_set_a @ Xa @ S )
& ( ord_less_set_a @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1139_Fpow__not__empty,axiom,
! [A: set_a] :
( ( finite_Fpow_a @ A )
!= bot_bot_set_set_a ) ).
% Fpow_not_empty
thf(fact_1140_Fpow__mono,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ ( finite_Fpow_a @ A ) @ ( finite_Fpow_a @ B ) ) ) ).
% Fpow_mono
thf(fact_1141_Fpow__mono,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_le5722252365846178494_set_a @ ( finite_Fpow_set_a @ A ) @ ( finite_Fpow_set_a @ B ) ) ) ).
% Fpow_mono
thf(fact_1142_verit__comp__simplify1_I2_J,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_1143_verit__comp__simplify1_I2_J,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_1144_Inf__fin_Osemilattice__order__set__axioms,axiom,
lattic8986249270076014136_set_a @ inf_inf_set_a @ ord_less_eq_set_a @ ord_less_set_a ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_1145_Inf__fin_Osemilattice__order__set__axioms,axiom,
lattic4037270620248062872_set_a @ inf_inf_set_set_a @ ord_le3724670747650509150_set_a @ ord_less_set_set_a ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_1146_subset_Ochain__extend,axiom,
! [A: set_set_a,C2: set_set_a,Z: set_a] :
( ( pred_chain_set_a @ A @ ord_less_set_a @ C2 )
=> ( ( member_set_a @ Z @ A )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ C2 )
=> ( sup_su8131220024717662786et_a_o @ ord_less_set_a
@ ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 )
@ X3
@ Z ) )
=> ( pred_chain_set_a @ A @ ord_less_set_a @ ( sup_sup_set_set_a @ ( insert_set_a @ Z @ bot_bot_set_set_a ) @ C2 ) ) ) ) ) ).
% subset.chain_extend
thf(fact_1147_subset__Zorn,axiom,
! [A: set_set_a] :
( ! [C4: set_set_a] :
( ( pred_chain_set_a @ A @ ord_less_set_a @ C4 )
=> ? [X: set_a] :
( ( member_set_a @ X @ A )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ C4 )
=> ( ord_less_eq_set_a @ Xa2 @ X ) ) ) )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( Xa = X3 ) ) ) ) ) ).
% subset_Zorn
thf(fact_1148_subset__Zorn,axiom,
! [A: set_set_set_a] :
( ! [C4: set_set_set_a] :
( ( pred_chain_set_set_a @ A @ ord_less_set_set_a @ C4 )
=> ? [X: set_set_a] :
( ( member_set_set_a @ X @ A )
& ! [Xa2: set_set_a] :
( ( member_set_set_a @ Xa2 @ C4 )
=> ( ord_le3724670747650509150_set_a @ Xa2 @ X ) ) ) )
=> ? [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Xa )
=> ( Xa = X3 ) ) ) ) ) ).
% subset_Zorn
thf(fact_1149_subset_OchainI,axiom,
! [C2: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ A )
=> ( ! [X3: set_a,Y3: set_a] :
( ( member_set_a @ X3 @ C2 )
=> ( ( member_set_a @ Y3 @ C2 )
=> ( ( sup_su8131220024717662786et_a_o @ ord_less_set_a
@ ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 )
@ X3
@ Y3 )
| ( sup_su8131220024717662786et_a_o @ ord_less_set_a
@ ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 )
@ Y3
@ X3 ) ) ) )
=> ( pred_chain_set_a @ A @ ord_less_set_a @ C2 ) ) ) ).
% subset.chainI
thf(fact_1150_subset_Ochain__def,axiom,
! [A: set_set_a,C2: set_set_a] :
( ( pred_chain_set_a @ A @ ord_less_set_a @ C2 )
= ( ( ord_le3724670747650509150_set_a @ C2 @ A )
& ! [X4: set_a] :
( ( member_set_a @ X4 @ C2 )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ C2 )
=> ( ( sup_su8131220024717662786et_a_o @ ord_less_set_a
@ ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 )
@ X4
@ Y4 )
| ( sup_su8131220024717662786et_a_o @ ord_less_set_a
@ ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 )
@ Y4
@ X4 ) ) ) ) ) ) ).
% subset.chain_def
thf(fact_1151_subset_Ochain__empty,axiom,
! [A: set_set_a] : ( pred_chain_set_a @ A @ ord_less_set_a @ bot_bot_set_set_a ) ).
% subset.chain_empty
thf(fact_1152_pred__on_Ochain__empty,axiom,
! [A: set_a,P: a > a > $o] : ( pred_chain_a @ A @ P @ bot_bot_set_a ) ).
% pred_on.chain_empty
thf(fact_1153_pred__on_Ochain__empty,axiom,
! [A: set_set_a,P: set_a > set_a > $o] : ( pred_chain_set_a @ A @ P @ bot_bot_set_set_a ) ).
% pred_on.chain_empty
thf(fact_1154_pred__on_Ochain__def,axiom,
( pred_chain_a
= ( ^ [A3: set_a,P2: a > a > $o,C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A3 )
& ! [X4: a] :
( ( member_a @ X4 @ C3 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ C3 )
=> ( ( sup_sup_a_a_o @ P2
@ ^ [Y2: a,Z2: a] : ( Y2 = Z2 )
@ X4
@ Y4 )
| ( sup_sup_a_a_o @ P2
@ ^ [Y2: a,Z2: a] : ( Y2 = Z2 )
@ Y4
@ X4 ) ) ) ) ) ) ) ).
% pred_on.chain_def
thf(fact_1155_pred__on_Ochain__def,axiom,
( pred_chain_set_a
= ( ^ [A3: set_set_a,P2: set_a > set_a > $o,C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C3 @ A3 )
& ! [X4: set_a] :
( ( member_set_a @ X4 @ C3 )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ C3 )
=> ( ( sup_su8131220024717662786et_a_o @ P2
@ ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 )
@ X4
@ Y4 )
| ( sup_su8131220024717662786et_a_o @ P2
@ ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 )
@ Y4
@ X4 ) ) ) ) ) ) ) ).
% pred_on.chain_def
thf(fact_1156_pred__on_OchainI,axiom,
! [C2: set_a,A: set_a,P: a > a > $o] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ! [X3: a,Y3: a] :
( ( member_a @ X3 @ C2 )
=> ( ( member_a @ Y3 @ C2 )
=> ( ( sup_sup_a_a_o @ P
@ ^ [Y2: a,Z2: a] : ( Y2 = Z2 )
@ X3
@ Y3 )
| ( sup_sup_a_a_o @ P
@ ^ [Y2: a,Z2: a] : ( Y2 = Z2 )
@ Y3
@ X3 ) ) ) )
=> ( pred_chain_a @ A @ P @ C2 ) ) ) ).
% pred_on.chainI
thf(fact_1157_pred__on_OchainI,axiom,
! [C2: set_set_a,A: set_set_a,P: set_a > set_a > $o] :
( ( ord_le3724670747650509150_set_a @ C2 @ A )
=> ( ! [X3: set_a,Y3: set_a] :
( ( member_set_a @ X3 @ C2 )
=> ( ( member_set_a @ Y3 @ C2 )
=> ( ( sup_su8131220024717662786et_a_o @ P
@ ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 )
@ X3
@ Y3 )
| ( sup_su8131220024717662786et_a_o @ P
@ ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 )
@ Y3
@ X3 ) ) ) )
=> ( pred_chain_set_a @ A @ P @ C2 ) ) ) ).
% pred_on.chainI
thf(fact_1158_subset__chain__def,axiom,
! [A9: set_set_set_a,C5: set_set_set_a] :
( ( pred_chain_set_set_a @ A9 @ ord_less_set_set_a @ C5 )
= ( ( ord_le5722252365846178494_set_a @ C5 @ A9 )
& ! [X4: set_set_a] :
( ( member_set_set_a @ X4 @ C5 )
=> ! [Y4: set_set_a] :
( ( member_set_set_a @ Y4 @ C5 )
=> ( ( ord_le3724670747650509150_set_a @ X4 @ Y4 )
| ( ord_le3724670747650509150_set_a @ Y4 @ X4 ) ) ) ) ) ) ).
% subset_chain_def
thf(fact_1159_subset__chain__def,axiom,
! [A9: set_set_a,C5: set_set_a] :
( ( pred_chain_set_a @ A9 @ ord_less_set_a @ C5 )
= ( ( ord_le3724670747650509150_set_a @ C5 @ A9 )
& ! [X4: set_a] :
( ( member_set_a @ X4 @ C5 )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ C5 )
=> ( ( ord_less_eq_set_a @ X4 @ Y4 )
| ( ord_less_eq_set_a @ Y4 @ X4 ) ) ) ) ) ) ).
% subset_chain_def
thf(fact_1160_subset__chain__insert,axiom,
! [A9: set_set_a,B: set_a,B9: set_set_a] :
( ( pred_chain_set_a @ A9 @ ord_less_set_a @ ( insert_set_a @ B @ B9 ) )
= ( ( member_set_a @ B @ A9 )
& ! [X4: set_a] :
( ( member_set_a @ X4 @ B9 )
=> ( ( ord_less_eq_set_a @ X4 @ B )
| ( ord_less_eq_set_a @ B @ X4 ) ) )
& ( pred_chain_set_a @ A9 @ ord_less_set_a @ B9 ) ) ) ).
% subset_chain_insert
thf(fact_1161_subset__chain__insert,axiom,
! [A9: set_set_set_a,B: set_set_a,B9: set_set_set_a] :
( ( pred_chain_set_set_a @ A9 @ ord_less_set_set_a @ ( insert_set_set_a @ B @ B9 ) )
= ( ( member_set_set_a @ B @ A9 )
& ! [X4: set_set_a] :
( ( member_set_set_a @ X4 @ B9 )
=> ( ( ord_le3724670747650509150_set_a @ X4 @ B )
| ( ord_le3724670747650509150_set_a @ B @ X4 ) ) )
& ( pred_chain_set_set_a @ A9 @ ord_less_set_set_a @ B9 ) ) ) ).
% subset_chain_insert
thf(fact_1162_pred__on_Ochain__extend,axiom,
! [A: set_a,P: a > a > $o,C2: set_a,Z: a] :
( ( pred_chain_a @ A @ P @ C2 )
=> ( ( member_a @ Z @ A )
=> ( ! [X3: a] :
( ( member_a @ X3 @ C2 )
=> ( sup_sup_a_a_o @ P
@ ^ [Y2: a,Z2: a] : ( Y2 = Z2 )
@ X3
@ Z ) )
=> ( pred_chain_a @ A @ P @ ( sup_sup_set_a @ ( insert_a @ Z @ bot_bot_set_a ) @ C2 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_1163_pred__on_Ochain__extend,axiom,
! [A: set_set_a,P: set_a > set_a > $o,C2: set_set_a,Z: set_a] :
( ( pred_chain_set_a @ A @ P @ C2 )
=> ( ( member_set_a @ Z @ A )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ C2 )
=> ( sup_su8131220024717662786et_a_o @ P
@ ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 )
@ X3
@ Z ) )
=> ( pred_chain_set_a @ A @ P @ ( sup_sup_set_set_a @ ( insert_set_a @ Z @ bot_bot_set_set_a ) @ C2 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_1164_finite__subset__Union__chain,axiom,
! [A: set_a,B9: set_set_a,A9: set_set_a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ A @ ( comple2307003609928055243_set_a @ B9 ) )
=> ( ( B9 != bot_bot_set_set_a )
=> ( ( pred_chain_set_a @ A9 @ ord_less_set_a @ B9 )
=> ~ ! [B6: set_a] :
( ( member_set_a @ B6 @ B9 )
=> ~ ( ord_less_eq_set_a @ A @ B6 ) ) ) ) ) ) ).
% finite_subset_Union_chain
thf(fact_1165_finite__subset__Union__chain,axiom,
! [A: set_set_a,B9: set_set_set_a,A9: set_set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ ( comple3958522678809307947_set_a @ B9 ) )
=> ( ( B9 != bot_bo3380559777022489994_set_a )
=> ( ( pred_chain_set_set_a @ A9 @ ord_less_set_set_a @ B9 )
=> ~ ! [B6: set_set_a] :
( ( member_set_set_a @ B6 @ B9 )
=> ~ ( ord_le3724670747650509150_set_a @ A @ B6 ) ) ) ) ) ) ).
% finite_subset_Union_chain
thf(fact_1166_semilattice__order__set_Osubset__imp,axiom,
! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: set_a,B: set_a] :
( ( lattic5078705180708912344_set_a @ F @ Less_eq @ Less )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != bot_bot_set_a )
=> ( ( finite_finite_a @ B )
=> ( Less_eq @ ( lattic5116578512385870296ce_F_a @ F @ B ) @ ( lattic5116578512385870296ce_F_a @ F @ A ) ) ) ) ) ) ).
% semilattice_order_set.subset_imp
thf(fact_1167_semilattice__order__set_Osubset__imp,axiom,
! [F: set_a > set_a > set_a,Less_eq: set_a > set_a > $o,Less: set_a > set_a > $o,A: set_set_a,B: set_set_a] :
( ( lattic8986249270076014136_set_a @ F @ Less_eq @ Less )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( A != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B )
=> ( Less_eq @ ( lattic2714821017709792056_set_a @ F @ B ) @ ( lattic2714821017709792056_set_a @ F @ A ) ) ) ) ) ) ).
% semilattice_order_set.subset_imp
thf(fact_1168_cSup__singleton,axiom,
! [X2: set_a] :
( ( comple2307003609928055243_set_a @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
= X2 ) ).
% cSup_singleton
thf(fact_1169_finite__Union,axiom,
! [A: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ! [M: set_set_a] :
( ( member_set_set_a @ M @ A )
=> ( finite_finite_set_a @ M ) )
=> ( finite_finite_set_a @ ( comple3958522678809307947_set_a @ A ) ) ) ) ).
% finite_Union
thf(fact_1170_finite__Union,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ! [M: set_a] :
( ( member_set_a @ M @ A )
=> ( finite_finite_a @ M ) )
=> ( finite_finite_a @ ( comple2307003609928055243_set_a @ A ) ) ) ) ).
% finite_Union
thf(fact_1171_finite__UnionD,axiom,
! [A: set_set_a] :
( ( finite_finite_a @ ( comple2307003609928055243_set_a @ A ) )
=> ( finite_finite_set_a @ A ) ) ).
% finite_UnionD
thf(fact_1172_finite__UnionD,axiom,
! [A: set_set_set_a] :
( ( finite_finite_set_a @ ( comple3958522678809307947_set_a @ A ) )
=> ( finite7209287970140883943_set_a @ A ) ) ).
% finite_UnionD
thf(fact_1173_cSup__eq__non__empty,axiom,
! [X5: set_set_a,A2: set_a] :
( ( X5 != bot_bot_set_set_a )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ X5 )
=> ( ord_less_eq_set_a @ X3 @ A2 ) )
=> ( ! [Y3: set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ X5 )
=> ( ord_less_eq_set_a @ X @ Y3 ) )
=> ( ord_less_eq_set_a @ A2 @ Y3 ) )
=> ( ( comple2307003609928055243_set_a @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1174_cSup__eq__non__empty,axiom,
! [X5: set_set_set_a,A2: set_set_a] :
( ( X5 != bot_bo3380559777022489994_set_a )
=> ( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ X5 )
=> ( ord_le3724670747650509150_set_a @ X3 @ A2 ) )
=> ( ! [Y3: set_set_a] :
( ! [X: set_set_a] :
( ( member_set_set_a @ X @ X5 )
=> ( ord_le3724670747650509150_set_a @ X @ Y3 ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ Y3 ) )
=> ( ( comple3958522678809307947_set_a @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1175_cSup__least,axiom,
! [X5: set_set_a,Z: set_a] :
( ( X5 != bot_bot_set_set_a )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ X5 )
=> ( ord_less_eq_set_a @ X3 @ Z ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1176_cSup__least,axiom,
! [X5: set_set_set_a,Z: set_set_a] :
( ( X5 != bot_bo3380559777022489994_set_a )
=> ( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ X5 )
=> ( ord_le3724670747650509150_set_a @ X3 @ Z ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1177_le__cSup__finite,axiom,
! [X5: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ X5 )
=> ( ( member_set_a @ X2 @ X5 )
=> ( ord_less_eq_set_a @ X2 @ ( comple2307003609928055243_set_a @ X5 ) ) ) ) ).
% le_cSup_finite
thf(fact_1178_le__cSup__finite,axiom,
! [X5: set_set_set_a,X2: set_set_a] :
( ( finite7209287970140883943_set_a @ X5 )
=> ( ( member_set_set_a @ X2 @ X5 )
=> ( ord_le3724670747650509150_set_a @ X2 @ ( comple3958522678809307947_set_a @ X5 ) ) ) ) ).
% le_cSup_finite
thf(fact_1179_cSup__eq__maximum,axiom,
! [Z: set_a,X5: set_set_a] :
( ( member_set_a @ Z @ X5 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ X5 )
=> ( ord_less_eq_set_a @ X3 @ Z ) )
=> ( ( comple2307003609928055243_set_a @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_1180_cSup__eq__maximum,axiom,
! [Z: set_set_a,X5: set_set_set_a] :
( ( member_set_set_a @ Z @ X5 )
=> ( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ X5 )
=> ( ord_le3724670747650509150_set_a @ X3 @ Z ) )
=> ( ( comple3958522678809307947_set_a @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_1181_Inf__fin__def,axiom,
( lattic8209813465164889211_set_a
= ( lattic2714821017709792056_set_a @ inf_inf_set_a ) ) ).
% Inf_fin_def
thf(fact_1182_Sup__fin__def,axiom,
( lattic2918178356826803221_set_a
= ( lattic2714821017709792056_set_a @ sup_sup_set_a ) ) ).
% Sup_fin_def
thf(fact_1183_Sup__fin__def,axiom,
( lattic338143333561554293_set_a
= ( lattic2919170233852397720_set_a @ sup_sup_set_set_a ) ) ).
% Sup_fin_def
thf(fact_1184_Zorn__Lemma,axiom,
! [A: set_set_a] :
( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ ( chains_a @ A ) )
=> ( member_set_a @ ( comple2307003609928055243_set_a @ X3 ) @ A ) )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( Xa = X3 ) ) ) ) ) ).
% Zorn_Lemma
thf(fact_1185_Zorn__Lemma,axiom,
! [A: set_set_set_a] :
( ! [X3: set_set_set_a] :
( ( member_set_set_set_a @ X3 @ ( chains_set_a @ A ) )
=> ( member_set_set_a @ ( comple3958522678809307947_set_a @ X3 ) @ A ) )
=> ? [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Xa )
=> ( Xa = X3 ) ) ) ) ) ).
% Zorn_Lemma
thf(fact_1186_semilattice__order__set_OcoboundedI,axiom,
! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: set_a,A2: a] :
( ( lattic5078705180708912344_set_a @ F @ Less_eq @ Less )
=> ( ( finite_finite_a @ A )
=> ( ( member_a @ A2 @ A )
=> ( Less_eq @ ( lattic5116578512385870296ce_F_a @ F @ A ) @ A2 ) ) ) ) ).
% semilattice_order_set.coboundedI
thf(fact_1187_semilattice__order__set_OcoboundedI,axiom,
! [F: set_a > set_a > set_a,Less_eq: set_a > set_a > $o,Less: set_a > set_a > $o,A: set_set_a,A2: set_a] :
( ( lattic8986249270076014136_set_a @ F @ Less_eq @ Less )
=> ( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ( Less_eq @ ( lattic2714821017709792056_set_a @ F @ A ) @ A2 ) ) ) ) ).
% semilattice_order_set.coboundedI
thf(fact_1188_finite__Sup__in,axiom,
! [A: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ( ! [X3: set_set_a,Y3: set_set_a] :
( ( member_set_set_a @ X3 @ A )
=> ( ( member_set_set_a @ Y3 @ A )
=> ( member_set_set_a @ ( sup_sup_set_set_a @ X3 @ Y3 ) @ A ) ) )
=> ( member_set_set_a @ ( comple3958522678809307947_set_a @ A ) @ A ) ) ) ) ).
% finite_Sup_in
thf(fact_1189_finite__Sup__in,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ! [X3: set_a,Y3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( ( member_set_a @ Y3 @ A )
=> ( member_set_a @ ( sup_sup_set_a @ X3 @ Y3 ) @ A ) ) )
=> ( member_set_a @ ( comple2307003609928055243_set_a @ A ) @ A ) ) ) ) ).
% finite_Sup_in
thf(fact_1190_sup__Sup__fold__sup,axiom,
! [A: set_set_a,B: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ A ) @ B )
= ( finite5985231929012247624_set_a @ sup_sup_set_a @ B @ A ) ) ) ).
% sup_Sup_fold_sup
thf(fact_1191_sup__Sup__fold__sup,axiom,
! [A: set_set_set_a,B: set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( sup_sup_set_set_a @ ( comple3958522678809307947_set_a @ A ) @ B )
= ( finite1111689594665117768_set_a @ sup_sup_set_set_a @ B @ A ) ) ) ).
% sup_Sup_fold_sup
thf(fact_1192_cSup__eq__Sup__fin,axiom,
! [X5: set_set_a] :
( ( finite_finite_set_a @ X5 )
=> ( ( X5 != bot_bot_set_set_a )
=> ( ( comple2307003609928055243_set_a @ X5 )
= ( lattic2918178356826803221_set_a @ X5 ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_1193_Sup__fin__Sup,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ A )
= ( comple2307003609928055243_set_a @ A ) ) ) ) ).
% Sup_fin_Sup
thf(fact_1194_insert__partition,axiom,
! [X2: set_a,F2: set_set_a] :
( ~ ( member_set_a @ X2 @ F2 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ ( insert_set_a @ X2 @ F2 ) )
=> ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ ( insert_set_a @ X2 @ F2 ) )
=> ( ( X3 != Xa2 )
=> ( ( inf_inf_set_a @ X3 @ Xa2 )
= bot_bot_set_a ) ) ) )
=> ( ( inf_inf_set_a @ X2 @ ( comple2307003609928055243_set_a @ F2 ) )
= bot_bot_set_a ) ) ) ).
% insert_partition
thf(fact_1195_insert__partition,axiom,
! [X2: set_set_a,F2: set_set_set_a] :
( ~ ( member_set_set_a @ X2 @ F2 )
=> ( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ ( insert_set_set_a @ X2 @ F2 ) )
=> ! [Xa2: set_set_a] :
( ( member_set_set_a @ Xa2 @ ( insert_set_set_a @ X2 @ F2 ) )
=> ( ( X3 != Xa2 )
=> ( ( inf_inf_set_set_a @ X3 @ Xa2 )
= bot_bot_set_set_a ) ) ) )
=> ( ( inf_inf_set_set_a @ X2 @ ( comple3958522678809307947_set_a @ F2 ) )
= bot_bot_set_set_a ) ) ) ).
% insert_partition
thf(fact_1196_subset__Zorn_H,axiom,
! [A: set_set_a] :
( ! [C4: set_set_a] :
( ( pred_chain_set_a @ A @ ord_less_set_a @ C4 )
=> ( member_set_a @ ( comple2307003609928055243_set_a @ C4 ) @ A ) )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( Xa = X3 ) ) ) ) ) ).
% subset_Zorn'
thf(fact_1197_subset__Zorn_H,axiom,
! [A: set_set_set_a] :
( ! [C4: set_set_set_a] :
( ( pred_chain_set_set_a @ A @ ord_less_set_set_a @ C4 )
=> ( member_set_set_a @ ( comple3958522678809307947_set_a @ C4 ) @ A ) )
=> ? [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Xa )
=> ( Xa = X3 ) ) ) ) ) ).
% subset_Zorn'
thf(fact_1198_semilattice__order__set_OboundedE,axiom,
! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: set_a,X2: a] :
( ( lattic5078705180708912344_set_a @ F @ Less_eq @ Less )
=> ( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ( ( Less_eq @ X2 @ ( lattic5116578512385870296ce_F_a @ F @ A ) )
=> ! [A8: a] :
( ( member_a @ A8 @ A )
=> ( Less_eq @ X2 @ A8 ) ) ) ) ) ) ).
% semilattice_order_set.boundedE
thf(fact_1199_semilattice__order__set_OboundedE,axiom,
! [F: set_a > set_a > set_a,Less_eq: set_a > set_a > $o,Less: set_a > set_a > $o,A: set_set_a,X2: set_a] :
( ( lattic8986249270076014136_set_a @ F @ Less_eq @ Less )
=> ( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( Less_eq @ X2 @ ( lattic2714821017709792056_set_a @ F @ A ) )
=> ! [A8: set_a] :
( ( member_set_a @ A8 @ A )
=> ( Less_eq @ X2 @ A8 ) ) ) ) ) ) ).
% semilattice_order_set.boundedE
thf(fact_1200_semilattice__order__set_OboundedI,axiom,
! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: set_a,X2: a] :
( ( lattic5078705180708912344_set_a @ F @ Less_eq @ Less )
=> ( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ( ! [A6: a] :
( ( member_a @ A6 @ A )
=> ( Less_eq @ X2 @ A6 ) )
=> ( Less_eq @ X2 @ ( lattic5116578512385870296ce_F_a @ F @ A ) ) ) ) ) ) ).
% semilattice_order_set.boundedI
thf(fact_1201_semilattice__order__set_OboundedI,axiom,
! [F: set_a > set_a > set_a,Less_eq: set_a > set_a > $o,Less: set_a > set_a > $o,A: set_set_a,X2: set_a] :
( ( lattic8986249270076014136_set_a @ F @ Less_eq @ Less )
=> ( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ! [A6: set_a] :
( ( member_set_a @ A6 @ A )
=> ( Less_eq @ X2 @ A6 ) )
=> ( Less_eq @ X2 @ ( lattic2714821017709792056_set_a @ F @ A ) ) ) ) ) ) ).
% semilattice_order_set.boundedI
thf(fact_1202_semilattice__order__set_Obounded__iff,axiom,
! [F: a > a > a,Less_eq: a > a > $o,Less: a > a > $o,A: set_a,X2: a] :
( ( lattic5078705180708912344_set_a @ F @ Less_eq @ Less )
=> ( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ( ( Less_eq @ X2 @ ( lattic5116578512385870296ce_F_a @ F @ A ) )
= ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( Less_eq @ X2 @ X4 ) ) ) ) ) ) ) ).
% semilattice_order_set.bounded_iff
thf(fact_1203_semilattice__order__set_Obounded__iff,axiom,
! [F: set_a > set_a > set_a,Less_eq: set_a > set_a > $o,Less: set_a > set_a > $o,A: set_set_a,X2: set_a] :
( ( lattic8986249270076014136_set_a @ F @ Less_eq @ Less )
=> ( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ( ( Less_eq @ X2 @ ( lattic2714821017709792056_set_a @ F @ A ) )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ( Less_eq @ X2 @ X4 ) ) ) ) ) ) ) ).
% semilattice_order_set.bounded_iff
thf(fact_1204_Sup__fold__sup,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( comple2307003609928055243_set_a @ A )
= ( finite5985231929012247624_set_a @ sup_sup_set_a @ bot_bot_set_a @ A ) ) ) ).
% Sup_fold_sup
thf(fact_1205_Sup__fold__sup,axiom,
! [A: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A )
=> ( ( comple3958522678809307947_set_a @ A )
= ( finite1111689594665117768_set_a @ sup_sup_set_set_a @ bot_bot_set_set_a @ A ) ) ) ).
% Sup_fold_sup
thf(fact_1206_subset__Zorn__nonempty,axiom,
! [A9: set_set_a] :
( ( A9 != bot_bot_set_set_a )
=> ( ! [C6: set_set_a] :
( ( C6 != bot_bot_set_set_a )
=> ( ( pred_chain_set_a @ A9 @ ord_less_set_a @ C6 )
=> ( member_set_a @ ( comple2307003609928055243_set_a @ C6 ) @ A9 ) ) )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A9 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A9 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( Xa = X3 ) ) ) ) ) ) ).
% subset_Zorn_nonempty
thf(fact_1207_subset__Zorn__nonempty,axiom,
! [A9: set_set_set_a] :
( ( A9 != bot_bo3380559777022489994_set_a )
=> ( ! [C6: set_set_set_a] :
( ( C6 != bot_bo3380559777022489994_set_a )
=> ( ( pred_chain_set_set_a @ A9 @ ord_less_set_set_a @ C6 )
=> ( member_set_set_a @ ( comple3958522678809307947_set_a @ C6 ) @ A9 ) ) )
=> ? [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A9 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A9 )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Xa )
=> ( Xa = X3 ) ) ) ) ) ) ).
% subset_Zorn_nonempty
thf(fact_1208_Union__in__chain,axiom,
! [B9: set_set_a,A9: set_set_a] :
( ( finite_finite_set_a @ B9 )
=> ( ( B9 != bot_bot_set_set_a )
=> ( ( pred_chain_set_a @ A9 @ ord_less_set_a @ B9 )
=> ( member_set_a @ ( comple2307003609928055243_set_a @ B9 ) @ B9 ) ) ) ) ).
% Union_in_chain
thf(fact_1209_Union__Un__distrib,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ( comple3958522678809307947_set_a @ ( sup_su2076012971530813682_set_a @ A @ B ) )
= ( sup_sup_set_set_a @ ( comple3958522678809307947_set_a @ A ) @ ( comple3958522678809307947_set_a @ B ) ) ) ).
% Union_Un_distrib
thf(fact_1210_Union__Un__distrib,axiom,
! [A: set_set_a,B: set_set_a] :
( ( comple2307003609928055243_set_a @ ( sup_sup_set_set_a @ A @ B ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_Un_distrib
thf(fact_1211_Sup__insert,axiom,
! [A2: set_a,A: set_set_a] :
( ( comple2307003609928055243_set_a @ ( insert_set_a @ A2 @ A ) )
= ( sup_sup_set_a @ A2 @ ( comple2307003609928055243_set_a @ A ) ) ) ).
% Sup_insert
thf(fact_1212_Sup__insert,axiom,
! [A2: set_set_a,A: set_set_set_a] :
( ( comple3958522678809307947_set_a @ ( insert_set_set_a @ A2 @ A ) )
= ( sup_sup_set_set_a @ A2 @ ( comple3958522678809307947_set_a @ A ) ) ) ).
% Sup_insert
thf(fact_1213_Sup__bot__conv_I1_J,axiom,
! [A: set_set_a] :
( ( ( comple2307003609928055243_set_a @ A )
= bot_bot_set_a )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ( X4 = bot_bot_set_a ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_1214_Sup__bot__conv_I1_J,axiom,
! [A: set_set_set_a] :
( ( ( comple3958522678809307947_set_a @ A )
= bot_bot_set_set_a )
= ( ! [X4: set_set_a] :
( ( member_set_set_a @ X4 @ A )
=> ( X4 = bot_bot_set_set_a ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_1215_Sup__bot__conv_I2_J,axiom,
! [A: set_set_a] :
( ( bot_bot_set_a
= ( comple2307003609928055243_set_a @ A ) )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ( X4 = bot_bot_set_a ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_1216_Sup__bot__conv_I2_J,axiom,
! [A: set_set_set_a] :
( ( bot_bot_set_set_a
= ( comple3958522678809307947_set_a @ A ) )
= ( ! [X4: set_set_a] :
( ( member_set_set_a @ X4 @ A )
=> ( X4 = bot_bot_set_set_a ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_1217_Sup__empty,axiom,
( ( comple3958522678809307947_set_a @ bot_bo3380559777022489994_set_a )
= bot_bot_set_set_a ) ).
% Sup_empty
thf(fact_1218_Sup__empty,axiom,
( ( comple2307003609928055243_set_a @ bot_bot_set_set_a )
= bot_bot_set_a ) ).
% Sup_empty
thf(fact_1219_Sup__upper2,axiom,
! [U: set_a,A: set_set_a,V3: set_a] :
( ( member_set_a @ U @ A )
=> ( ( ord_less_eq_set_a @ V3 @ U )
=> ( ord_less_eq_set_a @ V3 @ ( comple2307003609928055243_set_a @ A ) ) ) ) ).
% Sup_upper2
thf(fact_1220_Sup__upper2,axiom,
! [U: set_set_a,A: set_set_set_a,V3: set_set_a] :
( ( member_set_set_a @ U @ A )
=> ( ( ord_le3724670747650509150_set_a @ V3 @ U )
=> ( ord_le3724670747650509150_set_a @ V3 @ ( comple3958522678809307947_set_a @ A ) ) ) ) ).
% Sup_upper2
thf(fact_1221_Sup__le__iff,axiom,
! [A: set_set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ B2 )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ( ord_less_eq_set_a @ X4 @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_1222_Sup__le__iff,axiom,
! [A: set_set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A ) @ B2 )
= ( ! [X4: set_set_a] :
( ( member_set_set_a @ X4 @ A )
=> ( ord_le3724670747650509150_set_a @ X4 @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_1223_Sup__upper,axiom,
! [X2: set_a,A: set_set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ X2 @ ( comple2307003609928055243_set_a @ A ) ) ) ).
% Sup_upper
thf(fact_1224_Sup__upper,axiom,
! [X2: set_set_a,A: set_set_set_a] :
( ( member_set_set_a @ X2 @ A )
=> ( ord_le3724670747650509150_set_a @ X2 @ ( comple3958522678809307947_set_a @ A ) ) ) ).
% Sup_upper
thf(fact_1225_Sup__least,axiom,
! [A: set_set_a,Z: set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( ord_less_eq_set_a @ X3 @ Z ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ Z ) ) ).
% Sup_least
thf(fact_1226_Sup__least,axiom,
! [A: set_set_set_a,Z: set_set_a] :
( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A )
=> ( ord_le3724670747650509150_set_a @ X3 @ Z ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A ) @ Z ) ) ).
% Sup_least
thf(fact_1227_Sup__mono,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [A6: set_a] :
( ( member_set_a @ A6 @ A )
=> ? [X: set_a] :
( ( member_set_a @ X @ B )
& ( ord_less_eq_set_a @ A6 @ X ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Sup_mono
thf(fact_1228_Sup__mono,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ! [A6: set_set_a] :
( ( member_set_set_a @ A6 @ A )
=> ? [X: set_set_a] :
( ( member_set_set_a @ X @ B )
& ( ord_le3724670747650509150_set_a @ A6 @ X ) ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A ) @ ( comple3958522678809307947_set_a @ B ) ) ) ).
% Sup_mono
thf(fact_1229_Sup__eqI,axiom,
! [A: set_set_a,X2: set_a] :
( ! [Y3: set_a] :
( ( member_set_a @ Y3 @ A )
=> ( ord_less_eq_set_a @ Y3 @ X2 ) )
=> ( ! [Y3: set_a] :
( ! [Z4: set_a] :
( ( member_set_a @ Z4 @ A )
=> ( ord_less_eq_set_a @ Z4 @ Y3 ) )
=> ( ord_less_eq_set_a @ X2 @ Y3 ) )
=> ( ( comple2307003609928055243_set_a @ A )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_1230_Sup__eqI,axiom,
! [A: set_set_set_a,X2: set_set_a] :
( ! [Y3: set_set_a] :
( ( member_set_set_a @ Y3 @ A )
=> ( ord_le3724670747650509150_set_a @ Y3 @ X2 ) )
=> ( ! [Y3: set_set_a] :
( ! [Z4: set_set_a] :
( ( member_set_set_a @ Z4 @ A )
=> ( ord_le3724670747650509150_set_a @ Z4 @ Y3 ) )
=> ( ord_le3724670747650509150_set_a @ X2 @ Y3 ) )
=> ( ( comple3958522678809307947_set_a @ A )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_1231_Union__empty__conv,axiom,
! [A: set_set_a] :
( ( ( comple2307003609928055243_set_a @ A )
= bot_bot_set_a )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ( X4 = bot_bot_set_a ) ) ) ) ).
% Union_empty_conv
thf(fact_1232_Union__empty__conv,axiom,
! [A: set_set_set_a] :
( ( ( comple3958522678809307947_set_a @ A )
= bot_bot_set_set_a )
= ( ! [X4: set_set_a] :
( ( member_set_set_a @ X4 @ A )
=> ( X4 = bot_bot_set_set_a ) ) ) ) ).
% Union_empty_conv
thf(fact_1233_empty__Union__conv,axiom,
! [A: set_set_a] :
( ( bot_bot_set_a
= ( comple2307003609928055243_set_a @ A ) )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ( X4 = bot_bot_set_a ) ) ) ) ).
% empty_Union_conv
thf(fact_1234_empty__Union__conv,axiom,
! [A: set_set_set_a] :
( ( bot_bot_set_set_a
= ( comple3958522678809307947_set_a @ A ) )
= ( ! [X4: set_set_a] :
( ( member_set_set_a @ X4 @ A )
=> ( X4 = bot_bot_set_set_a ) ) ) ) ).
% empty_Union_conv
thf(fact_1235_Union__subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ? [Y7: set_a] :
( ( member_set_a @ Y7 @ B )
& ( ord_less_eq_set_a @ X3 @ Y7 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_subsetI
thf(fact_1236_Union__subsetI,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A )
=> ? [Y7: set_set_a] :
( ( member_set_set_a @ Y7 @ B )
& ( ord_le3724670747650509150_set_a @ X3 @ Y7 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A ) @ ( comple3958522678809307947_set_a @ B ) ) ) ).
% Union_subsetI
thf(fact_1237_Union__upper,axiom,
! [B: set_a,A: set_set_a] :
( ( member_set_a @ B @ A )
=> ( ord_less_eq_set_a @ B @ ( comple2307003609928055243_set_a @ A ) ) ) ).
% Union_upper
thf(fact_1238_Union__upper,axiom,
! [B: set_set_a,A: set_set_set_a] :
( ( member_set_set_a @ B @ A )
=> ( ord_le3724670747650509150_set_a @ B @ ( comple3958522678809307947_set_a @ A ) ) ) ).
% Union_upper
thf(fact_1239_Union__least,axiom,
! [A: set_set_a,C2: set_a] :
( ! [X7: set_a] :
( ( member_set_a @ X7 @ A )
=> ( ord_less_eq_set_a @ X7 @ C2 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ C2 ) ) ).
% Union_least
thf(fact_1240_Union__least,axiom,
! [A: set_set_set_a,C2: set_set_a] :
( ! [X7: set_set_a] :
( ( member_set_set_a @ X7 @ A )
=> ( ord_le3724670747650509150_set_a @ X7 @ C2 ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A ) @ C2 ) ) ).
% Union_least
thf(fact_1241_less__eq__Sup,axiom,
! [A: set_set_a,U: set_a] :
( ! [V5: set_a] :
( ( member_set_a @ V5 @ A )
=> ( ord_less_eq_set_a @ U @ V5 ) )
=> ( ( A != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_1242_less__eq__Sup,axiom,
! [A: set_set_set_a,U: set_set_a] :
( ! [V5: set_set_a] :
( ( member_set_set_a @ V5 @ A )
=> ( ord_le3724670747650509150_set_a @ U @ V5 ) )
=> ( ( A != bot_bo3380559777022489994_set_a )
=> ( ord_le3724670747650509150_set_a @ U @ ( comple3958522678809307947_set_a @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_1243_Sup__subset__mono,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A ) @ ( comple3958522678809307947_set_a @ B ) ) ) ).
% Sup_subset_mono
thf(fact_1244_Sup__subset__mono,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Sup_subset_mono
thf(fact_1245_Union__disjoint,axiom,
! [C2: set_set_a,A: set_a] :
( ( ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ C2 ) @ A )
= bot_bot_set_a )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ C2 )
=> ( ( inf_inf_set_a @ X4 @ A )
= bot_bot_set_a ) ) ) ) ).
% Union_disjoint
thf(fact_1246_Union__disjoint,axiom,
! [C2: set_set_set_a,A: set_set_a] :
( ( ( inf_inf_set_set_a @ ( comple3958522678809307947_set_a @ C2 ) @ A )
= bot_bot_set_set_a )
= ( ! [X4: set_set_a] :
( ( member_set_set_a @ X4 @ C2 )
=> ( ( inf_inf_set_set_a @ X4 @ A )
= bot_bot_set_set_a ) ) ) ) ).
% Union_disjoint
thf(fact_1247_Sup__union__distrib,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ( comple3958522678809307947_set_a @ ( sup_su2076012971530813682_set_a @ A @ B ) )
= ( sup_sup_set_set_a @ ( comple3958522678809307947_set_a @ A ) @ ( comple3958522678809307947_set_a @ B ) ) ) ).
% Sup_union_distrib
thf(fact_1248_Sup__union__distrib,axiom,
! [A: set_set_a,B: set_set_a] :
( ( comple2307003609928055243_set_a @ ( sup_sup_set_set_a @ A @ B ) )
= ( sup_sup_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Sup_union_distrib
thf(fact_1249_Union__mono,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A ) @ ( comple3958522678809307947_set_a @ B ) ) ) ).
% Union_mono
thf(fact_1250_Union__mono,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_mono
thf(fact_1251_Union__empty,axiom,
( ( comple3958522678809307947_set_a @ bot_bo3380559777022489994_set_a )
= bot_bot_set_set_a ) ).
% Union_empty
thf(fact_1252_Union__empty,axiom,
( ( comple2307003609928055243_set_a @ bot_bot_set_set_a )
= bot_bot_set_a ) ).
% Union_empty
thf(fact_1253_Union__insert,axiom,
! [A2: set_a,B: set_set_a] :
( ( comple2307003609928055243_set_a @ ( insert_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ A2 @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_insert
thf(fact_1254_Union__insert,axiom,
! [A2: set_set_a,B: set_set_set_a] :
( ( comple3958522678809307947_set_a @ ( insert_set_set_a @ A2 @ B ) )
= ( sup_sup_set_set_a @ A2 @ ( comple3958522678809307947_set_a @ B ) ) ) ).
% Union_insert
thf(fact_1255_subset__Pow__Union,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ ( pow_a @ ( comple2307003609928055243_set_a @ A ) ) ) ).
% subset_Pow_Union
thf(fact_1256_Sup__inter__less__eq,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A @ B ) ) @ ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_1257_Sup__inter__less__eq,axiom,
! [A: set_set_set_a,B: set_set_set_a] : ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ ( inf_in1205276777018777868_set_a @ A @ B ) ) @ ( inf_inf_set_set_a @ ( comple3958522678809307947_set_a @ A ) @ ( comple3958522678809307947_set_a @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_1258_Union__Int__subset,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A @ B ) ) @ ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_Int_subset
thf(fact_1259_Union__Int__subset,axiom,
! [A: set_set_set_a,B: set_set_set_a] : ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ ( inf_in1205276777018777868_set_a @ A @ B ) ) @ ( inf_inf_set_set_a @ ( comple3958522678809307947_set_a @ A ) @ ( comple3958522678809307947_set_a @ B ) ) ) ).
% Union_Int_subset
thf(fact_1260_ccpo__Sup__singleton,axiom,
! [X2: set_a] :
( ( comple2307003609928055243_set_a @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
= X2 ) ).
% ccpo_Sup_singleton
thf(fact_1261_ccSup__empty,axiom,
( ( comple3958522678809307947_set_a @ bot_bo3380559777022489994_set_a )
= bot_bot_set_set_a ) ).
% ccSup_empty
thf(fact_1262_ccSup__empty,axiom,
( ( comple2307003609928055243_set_a @ bot_bot_set_set_a )
= bot_bot_set_a ) ).
% ccSup_empty
thf(fact_1263_finite__subset__Union,axiom,
! [A: set_a,B9: set_set_a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ A @ ( comple2307003609928055243_set_a @ B9 ) )
=> ~ ! [F4: set_set_a] :
( ( finite_finite_set_a @ F4 )
=> ( ( ord_le3724670747650509150_set_a @ F4 @ B9 )
=> ~ ( ord_less_eq_set_a @ A @ ( comple2307003609928055243_set_a @ F4 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_1264_finite__subset__Union,axiom,
! [A: set_set_a,B9: set_set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ ( comple3958522678809307947_set_a @ B9 ) )
=> ~ ! [F4: set_set_set_a] :
( ( finite7209287970140883943_set_a @ F4 )
=> ( ( ord_le5722252365846178494_set_a @ F4 @ B9 )
=> ~ ( ord_le3724670747650509150_set_a @ A @ ( comple3958522678809307947_set_a @ F4 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_1265_Sup__inf__eq__bot__iff,axiom,
! [B: set_set_a,A2: set_a] :
( ( ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ B ) @ A2 )
= bot_bot_set_a )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ B )
=> ( ( inf_inf_set_a @ X4 @ A2 )
= bot_bot_set_a ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_1266_Sup__inf__eq__bot__iff,axiom,
! [B: set_set_set_a,A2: set_set_a] :
( ( ( inf_inf_set_set_a @ ( comple3958522678809307947_set_a @ B ) @ A2 )
= bot_bot_set_set_a )
= ( ! [X4: set_set_a] :
( ( member_set_set_a @ X4 @ B )
=> ( ( inf_inf_set_set_a @ X4 @ A2 )
= bot_bot_set_set_a ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_1267_semilattice__set_Oremove,axiom,
! [F: a > a > a,A: set_a,X2: a] :
( ( lattic5961991414251573132_set_a @ F )
=> ( ( finite_finite_a @ A )
=> ( ( member_a @ X2 @ A )
=> ( ( ( ( minus_minus_set_a @ A @ ( insert_a @ X2 @ bot_bot_set_a ) )
= bot_bot_set_a )
=> ( ( lattic5116578512385870296ce_F_a @ F @ A )
= X2 ) )
& ( ( ( minus_minus_set_a @ A @ ( insert_a @ X2 @ bot_bot_set_a ) )
!= bot_bot_set_a )
=> ( ( lattic5116578512385870296ce_F_a @ F @ A )
= ( F @ X2 @ ( lattic5116578512385870296ce_F_a @ F @ ( minus_minus_set_a @ A @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.remove
thf(fact_1268_semilattice__set_Oremove,axiom,
! [F: set_a > set_a > set_a,A: set_set_a,X2: set_a] :
( ( lattic1258622339881844972_set_a @ F )
=> ( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ X2 @ A )
=> ( ( ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic2714821017709792056_set_a @ F @ A )
= X2 ) )
& ( ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic2714821017709792056_set_a @ F @ A )
= ( F @ X2 @ ( lattic2714821017709792056_set_a @ F @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.remove
thf(fact_1269_semilattice__set_Oinsert__remove,axiom,
! [F: a > a > a,A: set_a,X2: a] :
( ( lattic5961991414251573132_set_a @ F )
=> ( ( finite_finite_a @ A )
=> ( ( ( ( minus_minus_set_a @ A @ ( insert_a @ X2 @ bot_bot_set_a ) )
= bot_bot_set_a )
=> ( ( lattic5116578512385870296ce_F_a @ F @ ( insert_a @ X2 @ A ) )
= X2 ) )
& ( ( ( minus_minus_set_a @ A @ ( insert_a @ X2 @ bot_bot_set_a ) )
!= bot_bot_set_a )
=> ( ( lattic5116578512385870296ce_F_a @ F @ ( insert_a @ X2 @ A ) )
= ( F @ X2 @ ( lattic5116578512385870296ce_F_a @ F @ ( minus_minus_set_a @ A @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ) ) ) ) ).
% semilattice_set.insert_remove
thf(fact_1270_semilattice__set_Oinsert__remove,axiom,
! [F: set_a > set_a > set_a,A: set_set_a,X2: set_a] :
( ( lattic1258622339881844972_set_a @ F )
=> ( ( finite_finite_set_a @ A )
=> ( ( ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic2714821017709792056_set_a @ F @ ( insert_set_a @ X2 @ A ) )
= X2 ) )
& ( ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic2714821017709792056_set_a @ F @ ( insert_set_a @ X2 @ A ) )
= ( F @ X2 @ ( lattic2714821017709792056_set_a @ F @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ) ).
% semilattice_set.insert_remove
thf(fact_1271_Inf__fin_Osemilattice__set__axioms,axiom,
lattic1258622339881844972_set_a @ inf_inf_set_a ).
% Inf_fin.semilattice_set_axioms
thf(fact_1272_Sup__fin_Osemilattice__set__axioms,axiom,
lattic1258622339881844972_set_a @ sup_sup_set_a ).
% Sup_fin.semilattice_set_axioms
thf(fact_1273_Sup__fin_Osemilattice__set__axioms,axiom,
lattic5230888003190500940_set_a @ sup_sup_set_set_a ).
% Sup_fin.semilattice_set_axioms
thf(fact_1274_semilattice__set_Oin__idem,axiom,
! [F: a > a > a,A: set_a,X2: a] :
( ( lattic5961991414251573132_set_a @ F )
=> ( ( finite_finite_a @ A )
=> ( ( member_a @ X2 @ A )
=> ( ( F @ X2 @ ( lattic5116578512385870296ce_F_a @ F @ A ) )
= ( lattic5116578512385870296ce_F_a @ F @ A ) ) ) ) ) ).
% semilattice_set.in_idem
thf(fact_1275_semilattice__set_Oin__idem,axiom,
! [F: set_a > set_a > set_a,A: set_set_a,X2: set_a] :
( ( lattic1258622339881844972_set_a @ F )
=> ( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ X2 @ A )
=> ( ( F @ X2 @ ( lattic2714821017709792056_set_a @ F @ A ) )
= ( lattic2714821017709792056_set_a @ F @ A ) ) ) ) ) ).
% semilattice_set.in_idem
thf(fact_1276_semilattice__set_Osingleton,axiom,
! [F: a > a > a,X2: a] :
( ( lattic5961991414251573132_set_a @ F )
=> ( ( lattic5116578512385870296ce_F_a @ F @ ( insert_a @ X2 @ bot_bot_set_a ) )
= X2 ) ) ).
% semilattice_set.singleton
thf(fact_1277_semilattice__set_Osingleton,axiom,
! [F: set_a > set_a > set_a,X2: set_a] :
( ( lattic1258622339881844972_set_a @ F )
=> ( ( lattic2714821017709792056_set_a @ F @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
= X2 ) ) ).
% semilattice_set.singleton
% Conjectures (1)
thf(conj_0,conjecture,
undire7103218114511261257raph_a @ ( sup_sup_set_a @ vH1 @ vH2 ) @ ( sup_sup_set_set_a @ eH1 @ eH2 ) @ ( sup_sup_set_a @ s @ t ) @ ( undire7777452895879145676dges_a @ edges @ ( sup_sup_set_a @ s @ t ) ) ).
%------------------------------------------------------------------------------