TPTP Problem File: SLH0136^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Undirected_Graph_Theory/0015_Graph_Triangles/prob_00126_004631__13145406_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1430 ( 633 unt; 152 typ; 0 def)
% Number of atoms : 3442 (1194 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10228 ( 378 ~; 35 |; 256 &;8070 @)
% ( 0 <=>;1489 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 401 ( 401 >; 0 *; 0 +; 0 <<)
% Number of symbols : 138 ( 137 usr; 18 con; 0-4 aty)
% Number of variables : 3339 ( 224 ^;3015 !; 100 ?;3339 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:33:04.204
%------------------------------------------------------------------------------
% Could-be-implicit typings (15)
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% Explicit typings (137)
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undire3617971648856834880loop_a: set_set_a > a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oincident__loops_001tf__a,type,
undire4753905205749729249oops_a: set_set_a > a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oincident__sedges_001tf__a,type,
undire1270416042309875431dges_a: set_set_a > a > set_set_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__edge__between_001t__Nat__Onat,type,
undire6814325412647357297en_nat: set_nat > set_nat > set_nat > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__edge__between_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
undire7011261089604658374od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__edge__between_001t__Real__Oreal,type,
undire3488164626074856909n_real: set_real > set_real > set_real > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__edge__between_001t__Set__Oset_Itf__a_J,type,
undire2578756059399487229_set_a: set_set_a > set_set_a > set_set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__edge__between_001tf__a,type,
undire8544646567961481629ween_a: set_a > set_a > set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__isolated__vertex_001tf__a,type,
undire8931668460104145173rtex_a: set_a > set_set_a > a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__loop_001tf__a,type,
undire2905028936066782638loop_a: set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Ois__sedge_001tf__a,type,
undire4917966558017083288edge_a: set_a > $o ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Oneighborhood_001tf__a,type,
undire8504279938402040014hood_a: set_a > set_set_a > a > set_a ).
thf(sy_c_Undirected__Graph__Basics_Oulgraph_Overt__adj_001tf__a,type,
undire397441198561214472_adj_a: set_set_a > a > a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
member268004040519299248_set_a: produc7943277765024757383_set_a > set_Pr4256959165342167655_set_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member1816616512716248880od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_E_H,type,
e: set_set_a ).
thf(sy_v_edges,type,
edges: set_set_a ).
thf(sy_v_vertices,type,
vertices: set_a ).
thf(sy_v_x,type,
x: a ).
thf(sy_v_y,type,
y: a ).
thf(sy_v_z,type,
z: a ).
% Relevant facts (1277)
thf(fact_0_assms_I1_J,axiom,
ord_le3724670747650509150_set_a @ e @ edges ).
% assms(1)
thf(fact_1_gnew_Otriangle__in__graph__def,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ e @ X @ Y @ Z )
= ( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ e )
& ( member_set_a @ ( insert_a @ Y @ ( insert_a @ Z @ bot_bot_set_a ) ) @ e )
& ( member_set_a @ ( insert_a @ X @ ( insert_a @ Z @ bot_bot_set_a ) ) @ e ) ) ) ).
% gnew.triangle_in_graph_def
thf(fact_2_empty__not__edge,axiom,
~ ( member_set_a @ bot_bot_set_a @ edges ) ).
% empty_not_edge
thf(fact_3_triangle__commu1,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z )
=> ( graph_4582152751571636272raph_a @ edges @ Y @ X @ Z ) ) ).
% triangle_commu1
thf(fact_4_triangle__vertices__distinct1,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z )
=> ( X != Y ) ) ).
% triangle_vertices_distinct1
thf(fact_5_triangle__vertices__distinct2,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z )
=> ( Y != Z ) ) ).
% triangle_vertices_distinct2
thf(fact_6_triangle__vertices__distinct3,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z )
=> ( Z != X ) ) ).
% triangle_vertices_distinct3
thf(fact_7_gnew_Oempty__not__edge,axiom,
~ ( member_set_a @ bot_bot_set_a @ e ) ).
% gnew.empty_not_edge
thf(fact_8_edge__vertices__not__equal,axiom,
! [X: a,Y: a] :
( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges )
=> ( X != Y ) ) ).
% edge_vertices_not_equal
thf(fact_9_singleton__not__edge,axiom,
! [X: a] :
~ ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ edges ) ).
% singleton_not_edge
thf(fact_10_is__edge__between__def,axiom,
( undire8544646567961481629ween_a
= ( ^ [X2: set_a,Y2: set_a,E: set_a] :
? [X3: a,Y3: a] :
( ( E
= ( insert_a @ X3 @ ( insert_a @ Y3 @ bot_bot_set_a ) ) )
& ( member_a @ X3 @ X2 )
& ( member_a @ Y3 @ Y2 ) ) ) ) ).
% is_edge_between_def
thf(fact_11_assms_I2_J,axiom,
graph_4582152751571636272raph_a @ e @ x @ y @ z ).
% assms(2)
thf(fact_12_gnew_Oedge__vertices__not__equal,axiom,
! [X: a,Y: a] :
( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ e )
=> ( X != Y ) ) ).
% gnew.edge_vertices_not_equal
thf(fact_13_gnew_Osingleton__not__edge,axiom,
! [X: a] :
~ ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ e ) ).
% gnew.singleton_not_edge
thf(fact_14_triangle__in__graph__def,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z )
= ( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges )
& ( member_set_a @ ( insert_a @ Y @ ( insert_a @ Z @ bot_bot_set_a ) ) @ edges )
& ( member_set_a @ ( insert_a @ X @ ( insert_a @ Z @ bot_bot_set_a ) ) @ edges ) ) ) ).
% triangle_in_graph_def
thf(fact_15_edge__adj__inE,axiom,
! [E1: set_a,E2: set_a] :
( ( undire4022703626023482010_adj_a @ edges @ E1 @ E2 )
=> ( ( member_set_a @ E1 @ edges )
& ( member_set_a @ E2 @ edges ) ) ) ).
% edge_adj_inE
thf(fact_16_gnew_Otriangle__commu1,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ e @ X @ Y @ Z )
=> ( graph_4582152751571636272raph_a @ e @ Y @ X @ Z ) ) ).
% gnew.triangle_commu1
thf(fact_17_gnew_Otriangle__vertices__distinct1,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ e @ X @ Y @ Z )
=> ( X != Y ) ) ).
% gnew.triangle_vertices_distinct1
thf(fact_18_gnew_Otriangle__vertices__distinct2,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ e @ X @ Y @ Z )
=> ( Y != Z ) ) ).
% gnew.triangle_vertices_distinct2
thf(fact_19_gnew_Otriangle__vertices__distinct3,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ e @ X @ Y @ Z )
=> ( Z != X ) ) ).
% gnew.triangle_vertices_distinct3
thf(fact_20_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_21_singletonI,axiom,
! [A: real] : ( member_real @ A @ ( insert_real @ A @ bot_bot_set_real ) ) ).
% singletonI
thf(fact_22_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_23_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_24_singletonI,axiom,
! [A: product_prod_a_a] : ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ) ).
% singletonI
thf(fact_25_has__loop__def,axiom,
! [V: a] :
( ( undire3617971648856834880loop_a @ edges @ V )
= ( member_set_a @ ( insert_a @ V @ bot_bot_set_a ) @ edges ) ) ).
% has_loop_def
thf(fact_26_not__vert__adj,axiom,
! [V: a,U: a] :
( ~ ( undire397441198561214472_adj_a @ edges @ V @ U )
=> ~ ( member_set_a @ ( insert_a @ V @ ( insert_a @ U @ bot_bot_set_a ) ) @ edges ) ) ).
% not_vert_adj
thf(fact_27_vert__adj__def,axiom,
! [V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
= ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) @ edges ) ) ).
% vert_adj_def
thf(fact_28_insertCI,axiom,
! [A: set_a,B: set_set_a,B2: set_a] :
( ( ~ ( member_set_a @ A @ B )
=> ( A = B2 ) )
=> ( member_set_a @ A @ ( insert_set_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_29_insertCI,axiom,
! [A: a,B: set_a,B2: a] :
( ( ~ ( member_a @ A @ B )
=> ( A = B2 ) )
=> ( member_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_30_insertCI,axiom,
! [A: product_prod_a_a,B: set_Product_prod_a_a,B2: product_prod_a_a] :
( ( ~ ( member1426531477525435216od_a_a @ A @ B )
=> ( A = B2 ) )
=> ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_31_insertCI,axiom,
! [A: nat,B: set_nat,B2: nat] :
( ( ~ ( member_nat @ A @ B )
=> ( A = B2 ) )
=> ( member_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_32_insertCI,axiom,
! [A: real,B: set_real,B2: real] :
( ( ~ ( member_real @ A @ B )
=> ( A = B2 ) )
=> ( member_real @ A @ ( insert_real @ B2 @ B ) ) ) ).
% insertCI
thf(fact_33_insert__iff,axiom,
! [A: set_a,B2: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_set_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_34_insert__iff,axiom,
! [A: a,B2: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_35_insert__iff,axiom,
! [A: product_prod_a_a,B2: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member1426531477525435216od_a_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_36_insert__iff,axiom,
! [A: nat,B2: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_37_insert__iff,axiom,
! [A: real,B2: real,A2: set_real] :
( ( member_real @ A @ ( insert_real @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_real @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_38_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_39_insert__absorb2,axiom,
! [X: set_a,A2: set_set_a] :
( ( insert_set_a @ X @ ( insert_set_a @ X @ A2 ) )
= ( insert_set_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_40_gnew_Otriangle__in__graph__edge__empty,axiom,
! [X: a,Y: a,Z: a] :
( ( e = bot_bot_set_set_a )
=> ~ ( graph_4582152751571636272raph_a @ e @ X @ Y @ Z ) ) ).
% gnew.triangle_in_graph_edge_empty
thf(fact_41_triangle__in__graph__edge__empty,axiom,
! [X: a,Y: a,Z: a] :
( ( edges = bot_bot_set_set_a )
=> ~ ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z ) ) ).
% triangle_in_graph_edge_empty
thf(fact_42_gnew_Overt__adj__sym,axiom,
! [V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ e @ V1 @ V2 )
= ( undire397441198561214472_adj_a @ e @ V2 @ V1 ) ) ).
% gnew.vert_adj_sym
thf(fact_43_vert__adj__sym,axiom,
! [V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
= ( undire397441198561214472_adj_a @ edges @ V2 @ V1 ) ) ).
% vert_adj_sym
thf(fact_44_gnew_Oedge__adj__inE,axiom,
! [E1: set_a,E2: set_a] :
( ( undire4022703626023482010_adj_a @ e @ E1 @ E2 )
=> ( ( member_set_a @ E1 @ e )
& ( member_set_a @ E2 @ e ) ) ) ).
% gnew.edge_adj_inE
thf(fact_45_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_46_empty__Collect__eq,axiom,
! [P: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P ) )
= ( ! [X3: set_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_47_empty__Collect__eq,axiom,
! [P: product_prod_a_a > $o] :
( ( bot_bo3357376287454694259od_a_a
= ( collec3336397797384452498od_a_a @ P ) )
= ( ! [X3: product_prod_a_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_48_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_49_Collect__empty__eq,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_50_Collect__empty__eq,axiom,
! [P: product_prod_a_a > $o] :
( ( ( collec3336397797384452498od_a_a @ P )
= bot_bo3357376287454694259od_a_a )
= ( ! [X3: product_prod_a_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_51_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_52_all__not__in__conv,axiom,
! [A2: set_real] :
( ( ! [X3: real] :
~ ( member_real @ X3 @ A2 ) )
= ( A2 = bot_bot_set_real ) ) ).
% all_not_in_conv
thf(fact_53_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_54_all__not__in__conv,axiom,
! [A2: set_set_a] :
( ( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_55_all__not__in__conv,axiom,
! [A2: set_Product_prod_a_a] :
( ( ! [X3: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X3 @ A2 ) )
= ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% all_not_in_conv
thf(fact_56_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_57_empty__iff,axiom,
! [C: real] :
~ ( member_real @ C @ bot_bot_set_real ) ).
% empty_iff
thf(fact_58_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_59_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_60_empty__iff,axiom,
! [C: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ C @ bot_bo3357376287454694259od_a_a ) ).
% empty_iff
thf(fact_61_subsetI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat @ X4 @ B ) )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_62_subsetI,axiom,
! [A2: set_real,B: set_real] :
( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( member_real @ X4 @ B ) )
=> ( ord_less_eq_set_real @ A2 @ B ) ) ).
% subsetI
thf(fact_63_subsetI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ( member_set_a @ X4 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_64_subsetI,axiom,
! [A2: set_a,B: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( member_a @ X4 @ B ) )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_65_subsetI,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A2 )
=> ( member1426531477525435216od_a_a @ X4 @ B ) )
=> ( ord_le746702958409616551od_a_a @ A2 @ B ) ) ).
% subsetI
thf(fact_66_subset__antisym,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_67_subset__antisym,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_68_subset__antisym,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( ord_le746702958409616551od_a_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_69_gnew_Overt__adj__def,axiom,
! [V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ e @ V1 @ V2 )
= ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) @ e ) ) ).
% gnew.vert_adj_def
thf(fact_70_gnew_Onot__vert__adj,axiom,
! [V: a,U: a] :
( ~ ( undire397441198561214472_adj_a @ e @ V @ U )
=> ~ ( member_set_a @ ( insert_a @ V @ ( insert_a @ U @ bot_bot_set_a ) ) @ e ) ) ).
% gnew.not_vert_adj
thf(fact_71_gnew_Ohas__loop__def,axiom,
! [V: a] :
( ( undire3617971648856834880loop_a @ e @ V )
= ( member_set_a @ ( insert_a @ V @ bot_bot_set_a ) @ e ) ) ).
% gnew.has_loop_def
thf(fact_72_gnew_Otriangle__in__graph__edge__point,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ e @ X @ Y @ Z )
= ( ( member_set_a @ ( insert_a @ Y @ ( insert_a @ Z @ bot_bot_set_a ) ) @ e )
& ( undire397441198561214472_adj_a @ e @ X @ Y )
& ( undire397441198561214472_adj_a @ e @ X @ Z ) ) ) ).
% gnew.triangle_in_graph_edge_point
thf(fact_73_triangle__in__graph__edge__point,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z )
= ( ( member_set_a @ ( insert_a @ Y @ ( insert_a @ Z @ bot_bot_set_a ) ) @ edges )
& ( undire397441198561214472_adj_a @ edges @ X @ Y )
& ( undire397441198561214472_adj_a @ edges @ X @ Z ) ) ) ).
% triangle_in_graph_edge_point
thf(fact_74_empty__subsetI,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A2 ) ).
% empty_subsetI
thf(fact_75_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_76_empty__subsetI,axiom,
! [A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ bot_bo3357376287454694259od_a_a @ A2 ) ).
% empty_subsetI
thf(fact_77_subset__empty,axiom,
! [A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a )
= ( A2 = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_78_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_79_subset__empty,axiom,
! [A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ bot_bo3357376287454694259od_a_a )
= ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% subset_empty
thf(fact_80_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_81_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_82_mem__Collect__eq,axiom,
! [A: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A @ ( collec3336397797384452498od_a_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_83_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_84_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_85_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_86_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_87_Collect__mem__eq,axiom,
! [A2: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_88_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_89_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X3: real] : ( member_real @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_90_insert__subset,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A2 ) @ B )
= ( ( member_nat @ X @ B )
& ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_91_insert__subset,axiom,
! [X: real,A2: set_real,B: set_real] :
( ( ord_less_eq_set_real @ ( insert_real @ X @ A2 ) @ B )
= ( ( member_real @ X @ B )
& ( ord_less_eq_set_real @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_92_insert__subset,axiom,
! [X: set_a,A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A2 ) @ B )
= ( ( member_set_a @ X @ B )
& ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_93_insert__subset,axiom,
! [X: a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( ( member_a @ X @ B )
& ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_94_insert__subset,axiom,
! [X: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ X @ A2 ) @ B )
= ( ( member1426531477525435216od_a_a @ X @ B )
& ( ord_le746702958409616551od_a_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_95_singleton__insert__inj__eq_H,axiom,
! [A: set_a,A2: set_set_a,B2: set_a] :
( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B2 @ bot_bot_set_set_a ) )
= ( ( A = B2 )
& ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B2 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_96_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B2: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B2 @ bot_bot_set_a ) )
= ( ( A = B2 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_97_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a,B2: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A @ A2 )
= ( insert4534936382041156343od_a_a @ B2 @ bot_bo3357376287454694259od_a_a ) )
= ( ( A = B2 )
& ( ord_le746702958409616551od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B2 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_98_singleton__insert__inj__eq,axiom,
! [B2: set_a,A: set_a,A2: set_set_a] :
( ( ( insert_set_a @ B2 @ bot_bot_set_set_a )
= ( insert_set_a @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B2 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_99_singleton__insert__inj__eq,axiom,
! [B2: a,A: a,A2: set_a] :
( ( ( insert_a @ B2 @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_100_singleton__insert__inj__eq,axiom,
! [B2: product_prod_a_a,A: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ B2 @ bot_bo3357376287454694259od_a_a )
= ( insert4534936382041156343od_a_a @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le746702958409616551od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B2 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_101_in__mono,axiom,
! [A2: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_102_in__mono,axiom,
! [A2: set_real,B: set_real,X: real] :
( ( ord_less_eq_set_real @ A2 @ B )
=> ( ( member_real @ X @ A2 )
=> ( member_real @ X @ B ) ) ) ).
% in_mono
thf(fact_103_in__mono,axiom,
! [A2: set_set_a,B: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_a @ X @ A2 )
=> ( member_set_a @ X @ B ) ) ) ).
% in_mono
thf(fact_104_in__mono,axiom,
! [A2: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_105_in__mono,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,X: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( member1426531477525435216od_a_a @ X @ A2 )
=> ( member1426531477525435216od_a_a @ X @ B ) ) ) ).
% in_mono
thf(fact_106_subsetD,axiom,
! [A2: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_107_subsetD,axiom,
! [A2: set_real,B: set_real,C: real] :
( ( ord_less_eq_set_real @ A2 @ B )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B ) ) ) ).
% subsetD
thf(fact_108_subsetD,axiom,
! [A2: set_set_a,B: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B ) ) ) ).
% subsetD
thf(fact_109_subsetD,axiom,
! [A2: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_110_subsetD,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( member1426531477525435216od_a_a @ C @ A2 )
=> ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% subsetD
thf(fact_111_equalityE,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( A2 = B )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ~ ( ord_le3724670747650509150_set_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_112_equalityE,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B )
=> ~ ( ord_less_eq_set_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_113_equalityE,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A2 = B )
=> ~ ( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ~ ( ord_le746702958409616551od_a_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_114_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_115_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B3: set_real] :
! [X3: real] :
( ( member_real @ X3 @ A3 )
=> ( member_real @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_116_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
=> ( member_set_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_117_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_118_subset__eq,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A3 )
=> ( member1426531477525435216od_a_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_119_equalityD1,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( A2 = B )
=> ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_120_equalityD1,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_121_equalityD1,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A2 = B )
=> ( ord_le746702958409616551od_a_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_122_equalityD2,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( A2 = B )
=> ( ord_le3724670747650509150_set_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_123_equalityD2,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_124_equalityD2,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( A2 = B )
=> ( ord_le746702958409616551od_a_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_125_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_126_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B3: set_real] :
! [T: real] :
( ( member_real @ T @ A3 )
=> ( member_real @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_127_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_128_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_129_subset__iff,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
! [T: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T @ A3 )
=> ( member1426531477525435216od_a_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_130_subset__refl,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_131_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_132_subset__refl,axiom,
! [A2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_133_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X4: set_a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_134_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_135_Collect__mono,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ! [X4: product_prod_a_a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P ) @ ( collec3336397797384452498od_a_a @ Q ) ) ) ).
% Collect_mono
thf(fact_136_subset__trans,axiom,
! [A2: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_137_subset__trans,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_138_subset__trans,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( ord_le746702958409616551od_a_a @ B @ C2 )
=> ( ord_le746702958409616551od_a_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_139_set__eq__subset,axiom,
( ( ^ [Y4: set_set_a,Z2: set_set_a] : ( Y4 = 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_140_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = 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_141_set__eq__subset,axiom,
( ( ^ [Y4: set_Product_prod_a_a,Z2: set_Product_prod_a_a] : ( Y4 = Z2 ) )
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A3 @ B3 )
& ( ord_le746702958409616551od_a_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_142_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X3: set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_143_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_144_Collect__mono__iff,axiom,
! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ P ) @ ( collec3336397797384452498od_a_a @ Q ) )
= ( ! [X3: product_prod_a_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_145_sgraph_Otriangle__in__graph_Ocong,axiom,
graph_4582152751571636272raph_a = graph_4582152751571636272raph_a ).
% sgraph.triangle_in_graph.cong
thf(fact_146_subset__insertI2,axiom,
! [A2: set_set_a,B: set_set_a,B2: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_147_subset__insertI2,axiom,
! [A2: set_a,B: set_a,B2: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_148_subset__insertI2,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,B2: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ord_le746702958409616551od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_149_subset__insertI,axiom,
! [B: set_set_a,A: set_a] : ( ord_le3724670747650509150_set_a @ B @ ( insert_set_a @ A @ B ) ) ).
% subset_insertI
thf(fact_150_subset__insertI,axiom,
! [B: set_a,A: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A @ B ) ) ).
% subset_insertI
thf(fact_151_subset__insertI,axiom,
! [B: set_Product_prod_a_a,A: product_prod_a_a] : ( ord_le746702958409616551od_a_a @ B @ ( insert4534936382041156343od_a_a @ A @ B ) ) ).
% subset_insertI
thf(fact_152_subset__insert,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B ) )
= ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_153_subset__insert,axiom,
! [X: real,A2: set_real,B: set_real] :
( ~ ( member_real @ X @ A2 )
=> ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B ) )
= ( ord_less_eq_set_real @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_154_subset__insert,axiom,
! [X: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B ) )
= ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_155_subset__insert,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_156_subset__insert,axiom,
! [X: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ X @ B ) )
= ( ord_le746702958409616551od_a_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_157_insert__mono,axiom,
! [C2: set_set_a,D: set_set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ D )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ A @ C2 ) @ ( insert_set_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_158_insert__mono,axiom,
! [C2: set_a,D: set_a,A: a] :
( ( ord_less_eq_set_a @ C2 @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C2 ) @ ( insert_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_159_insert__mono,axiom,
! [C2: set_Product_prod_a_a,D: set_Product_prod_a_a,A: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C2 @ D )
=> ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ A @ C2 ) @ ( insert4534936382041156343od_a_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_160_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_161_ex__in__conv,axiom,
! [A2: set_real] :
( ( ? [X3: real] : ( member_real @ X3 @ A2 ) )
= ( A2 != bot_bot_set_real ) ) ).
% ex_in_conv
thf(fact_162_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_163_ex__in__conv,axiom,
! [A2: set_set_a] :
( ( ? [X3: set_a] : ( member_set_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_164_ex__in__conv,axiom,
! [A2: set_Product_prod_a_a] :
( ( ? [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ A2 ) )
= ( A2 != bot_bo3357376287454694259od_a_a ) ) ).
% ex_in_conv
thf(fact_165_equals0I,axiom,
! [A2: set_nat] :
( ! [Y5: nat] :
~ ( member_nat @ Y5 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_166_equals0I,axiom,
! [A2: set_real] :
( ! [Y5: real] :
~ ( member_real @ Y5 @ A2 )
=> ( A2 = bot_bot_set_real ) ) ).
% equals0I
thf(fact_167_equals0I,axiom,
! [A2: set_a] :
( ! [Y5: a] :
~ ( member_a @ Y5 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_168_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y5: set_a] :
~ ( member_set_a @ Y5 @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_169_equals0I,axiom,
! [A2: set_Product_prod_a_a] :
( ! [Y5: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ Y5 @ A2 )
=> ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% equals0I
thf(fact_170_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_171_equals0D,axiom,
! [A2: set_real,A: real] :
( ( A2 = bot_bot_set_real )
=> ~ ( member_real @ A @ A2 ) ) ).
% equals0D
thf(fact_172_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_173_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_174_equals0D,axiom,
! [A2: set_Product_prod_a_a,A: product_prod_a_a] :
( ( A2 = bot_bo3357376287454694259od_a_a )
=> ~ ( member1426531477525435216od_a_a @ A @ A2 ) ) ).
% equals0D
thf(fact_175_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_176_emptyE,axiom,
! [A: real] :
~ ( member_real @ A @ bot_bot_set_real ) ).
% emptyE
thf(fact_177_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_178_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_179_emptyE,axiom,
! [A: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ).
% emptyE
thf(fact_180_mk__disjoint__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ? [B4: set_set_a] :
( ( A2
= ( insert_set_a @ A @ B4 ) )
& ~ ( member_set_a @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_181_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B4: set_a] :
( ( A2
= ( insert_a @ A @ B4 ) )
& ~ ( member_a @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_182_mk__disjoint__insert,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ A2 )
=> ? [B4: set_Product_prod_a_a] :
( ( A2
= ( insert4534936382041156343od_a_a @ A @ B4 ) )
& ~ ( member1426531477525435216od_a_a @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_183_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B4: set_nat] :
( ( A2
= ( insert_nat @ A @ B4 ) )
& ~ ( member_nat @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_184_mk__disjoint__insert,axiom,
! [A: real,A2: set_real] :
( ( member_real @ A @ A2 )
=> ? [B4: set_real] :
( ( A2
= ( insert_real @ A @ B4 ) )
& ~ ( member_real @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_185_insert__commute,axiom,
! [X: a,Y: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ Y @ A2 ) )
= ( insert_a @ Y @ ( insert_a @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_186_insert__commute,axiom,
! [X: set_a,Y: set_a,A2: set_set_a] :
( ( insert_set_a @ X @ ( insert_set_a @ Y @ A2 ) )
= ( insert_set_a @ Y @ ( insert_set_a @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_187_insert__eq__iff,axiom,
! [A: set_a,A2: set_set_a,B2: set_a,B: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ~ ( member_set_a @ B2 @ B )
=> ( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_set_a] :
( ( A2
= ( insert_set_a @ B2 @ C3 ) )
& ~ ( member_set_a @ B2 @ C3 )
& ( B
= ( insert_set_a @ A @ C3 ) )
& ~ ( member_set_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_188_insert__eq__iff,axiom,
! [A: a,A2: set_a,B2: a,B: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B2 @ B )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_a] :
( ( A2
= ( insert_a @ B2 @ C3 ) )
& ~ ( member_a @ B2 @ C3 )
& ( B
= ( insert_a @ A @ C3 ) )
& ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_189_insert__eq__iff,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a,B2: product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A @ A2 )
=> ( ~ ( member1426531477525435216od_a_a @ B2 @ B )
=> ( ( ( insert4534936382041156343od_a_a @ A @ A2 )
= ( insert4534936382041156343od_a_a @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_Product_prod_a_a] :
( ( A2
= ( insert4534936382041156343od_a_a @ B2 @ C3 ) )
& ~ ( member1426531477525435216od_a_a @ B2 @ C3 )
& ( B
= ( insert4534936382041156343od_a_a @ A @ C3 ) )
& ~ ( member1426531477525435216od_a_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_190_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B2: nat,B: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B2 @ B )
=> ( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_nat] :
( ( A2
= ( insert_nat @ B2 @ C3 ) )
& ~ ( member_nat @ B2 @ C3 )
& ( B
= ( insert_nat @ A @ C3 ) )
& ~ ( member_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_191_insert__eq__iff,axiom,
! [A: real,A2: set_real,B2: real,B: set_real] :
( ~ ( member_real @ A @ A2 )
=> ( ~ ( member_real @ B2 @ B )
=> ( ( ( insert_real @ A @ A2 )
= ( insert_real @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_real] :
( ( A2
= ( insert_real @ B2 @ C3 ) )
& ~ ( member_real @ B2 @ C3 )
& ( B
= ( insert_real @ A @ C3 ) )
& ~ ( member_real @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_192_insert__absorb,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( insert_set_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_193_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_194_insert__absorb,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ A2 )
=> ( ( insert4534936382041156343od_a_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_195_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_196_insert__absorb,axiom,
! [A: real,A2: set_real] :
( ( member_real @ A @ A2 )
=> ( ( insert_real @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_197_insert__ident,axiom,
! [X: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ~ ( member_set_a @ X @ B )
=> ( ( ( insert_set_a @ X @ A2 )
= ( insert_set_a @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_198_insert__ident,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_199_insert__ident,axiom,
! [X: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X @ A2 )
=> ( ~ ( member1426531477525435216od_a_a @ X @ B )
=> ( ( ( insert4534936382041156343od_a_a @ X @ A2 )
= ( insert4534936382041156343od_a_a @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_200_insert__ident,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ~ ( member_nat @ X @ B )
=> ( ( ( insert_nat @ X @ A2 )
= ( insert_nat @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_201_insert__ident,axiom,
! [X: real,A2: set_real,B: set_real] :
( ~ ( member_real @ X @ A2 )
=> ( ~ ( member_real @ X @ B )
=> ( ( ( insert_real @ X @ A2 )
= ( insert_real @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_202_Set_Oset__insert,axiom,
! [X: set_a,A2: set_set_a] :
( ( member_set_a @ X @ A2 )
=> ~ ! [B4: set_set_a] :
( ( A2
= ( insert_set_a @ X @ B4 ) )
=> ( member_set_a @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_203_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B4: set_a] :
( ( A2
= ( insert_a @ X @ B4 ) )
=> ( member_a @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_204_Set_Oset__insert,axiom,
! [X: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A2 )
=> ~ ! [B4: set_Product_prod_a_a] :
( ( A2
= ( insert4534936382041156343od_a_a @ X @ B4 ) )
=> ( member1426531477525435216od_a_a @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_205_Set_Oset__insert,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat @ X @ A2 )
=> ~ ! [B4: set_nat] :
( ( A2
= ( insert_nat @ X @ B4 ) )
=> ( member_nat @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_206_Set_Oset__insert,axiom,
! [X: real,A2: set_real] :
( ( member_real @ X @ A2 )
=> ~ ! [B4: set_real] :
( ( A2
= ( insert_real @ X @ B4 ) )
=> ( member_real @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_207_insertI2,axiom,
! [A: set_a,B: set_set_a,B2: set_a] :
( ( member_set_a @ A @ B )
=> ( member_set_a @ A @ ( insert_set_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_208_insertI2,axiom,
! [A: a,B: set_a,B2: a] :
( ( member_a @ A @ B )
=> ( member_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_209_insertI2,axiom,
! [A: product_prod_a_a,B: set_Product_prod_a_a,B2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ B )
=> ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_210_insertI2,axiom,
! [A: nat,B: set_nat,B2: nat] :
( ( member_nat @ A @ B )
=> ( member_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_211_insertI2,axiom,
! [A: real,B: set_real,B2: real] :
( ( member_real @ A @ B )
=> ( member_real @ A @ ( insert_real @ B2 @ B ) ) ) ).
% insertI2
thf(fact_212_insertI1,axiom,
! [A: set_a,B: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B ) ) ).
% insertI1
thf(fact_213_insertI1,axiom,
! [A: a,B: set_a] : ( member_a @ A @ ( insert_a @ A @ B ) ) ).
% insertI1
thf(fact_214_insertI1,axiom,
! [A: product_prod_a_a,B: set_Product_prod_a_a] : ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ A @ B ) ) ).
% insertI1
thf(fact_215_insertI1,axiom,
! [A: nat,B: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B ) ) ).
% insertI1
thf(fact_216_insertI1,axiom,
! [A: real,B: set_real] : ( member_real @ A @ ( insert_real @ A @ B ) ) ).
% insertI1
thf(fact_217_insertE,axiom,
! [A: set_a,B2: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_set_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_218_insertE,axiom,
! [A: a,B2: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_219_insertE,axiom,
! [A: product_prod_a_a,B2: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ ( insert4534936382041156343od_a_a @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member1426531477525435216od_a_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_220_insertE,axiom,
! [A: nat,B2: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_221_insertE,axiom,
! [A: real,B2: real,A2: set_real] :
( ( member_real @ A @ ( insert_real @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_real @ A @ A2 ) ) ) ).
% insertE
thf(fact_222_subset__singleton__iff,axiom,
! [X5: set_set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ X5 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( ( X5 = bot_bot_set_set_a )
| ( X5
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_223_subset__singleton__iff,axiom,
! [X5: set_a,A: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_224_subset__singleton__iff,axiom,
! [X5: set_Product_prod_a_a,A: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X5 @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) )
= ( ( X5 = bot_bo3357376287454694259od_a_a )
| ( X5
= ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_225_subset__singletonD,axiom,
! [A2: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
=> ( ( A2 = bot_bot_set_set_a )
| ( A2
= ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_226_subset__singletonD,axiom,
! [A2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A2 = bot_bot_set_a )
| ( A2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_227_subset__singletonD,axiom,
! [A2: set_Product_prod_a_a,X: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) )
=> ( ( A2 = bot_bo3357376287454694259od_a_a )
| ( A2
= ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% subset_singletonD
thf(fact_228_singleton__inject,axiom,
! [A: a,B2: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B2 @ bot_bot_set_a ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_229_singleton__inject,axiom,
! [A: set_a,B2: set_a] :
( ( ( insert_set_a @ A @ bot_bot_set_set_a )
= ( insert_set_a @ B2 @ bot_bot_set_set_a ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_230_singleton__inject,axiom,
! [A: product_prod_a_a,B2: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a )
= ( insert4534936382041156343od_a_a @ B2 @ bot_bo3357376287454694259od_a_a ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_231_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_232_insert__not__empty,axiom,
! [A: set_a,A2: set_set_a] :
( ( insert_set_a @ A @ A2 )
!= bot_bot_set_set_a ) ).
% insert_not_empty
thf(fact_233_insert__not__empty,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( insert4534936382041156343od_a_a @ A @ A2 )
!= bot_bo3357376287454694259od_a_a ) ).
% insert_not_empty
thf(fact_234_doubleton__eq__iff,axiom,
! [A: a,B2: a,C: a,D2: a] :
( ( ( insert_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B2 = D2 ) )
| ( ( A = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_235_doubleton__eq__iff,axiom,
! [A: set_a,B2: set_a,C: set_a,D2: set_a] :
( ( ( insert_set_a @ A @ ( insert_set_a @ B2 @ bot_bot_set_set_a ) )
= ( insert_set_a @ C @ ( insert_set_a @ D2 @ bot_bot_set_set_a ) ) )
= ( ( ( A = C )
& ( B2 = D2 ) )
| ( ( A = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_236_doubleton__eq__iff,axiom,
! [A: product_prod_a_a,B2: product_prod_a_a,C: product_prod_a_a,D2: product_prod_a_a] :
( ( ( insert4534936382041156343od_a_a @ A @ ( insert4534936382041156343od_a_a @ B2 @ bot_bo3357376287454694259od_a_a ) )
= ( insert4534936382041156343od_a_a @ C @ ( insert4534936382041156343od_a_a @ D2 @ bot_bo3357376287454694259od_a_a ) ) )
= ( ( ( A = C )
& ( B2 = D2 ) )
| ( ( A = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_237_singleton__iff,axiom,
! [B2: nat,A: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_238_singleton__iff,axiom,
! [B2: real,A: real] :
( ( member_real @ B2 @ ( insert_real @ A @ bot_bot_set_real ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_239_singleton__iff,axiom,
! [B2: a,A: a] :
( ( member_a @ B2 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_240_singleton__iff,axiom,
! [B2: set_a,A: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_241_singleton__iff,axiom,
! [B2: product_prod_a_a,A: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B2 @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_242_singletonD,axiom,
! [B2: nat,A: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_243_singletonD,axiom,
! [B2: real,A: real] :
( ( member_real @ B2 @ ( insert_real @ A @ bot_bot_set_real ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_244_singletonD,axiom,
! [B2: a,A: a] :
( ( member_a @ B2 @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_245_singletonD,axiom,
! [B2: set_a,A: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_246_singletonD,axiom,
! [B2: product_prod_a_a,A: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B2 @ ( insert4534936382041156343od_a_a @ A @ bot_bo3357376287454694259od_a_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_247_incident__loops__simp_I1_J,axiom,
! [V: a] :
( ( undire3617971648856834880loop_a @ edges @ V )
=> ( ( undire4753905205749729249oops_a @ edges @ V )
= ( insert_set_a @ ( insert_a @ V @ bot_bot_set_a ) @ bot_bot_set_set_a ) ) ) ).
% incident_loops_simp(1)
thf(fact_248_gnew_Oincident__loops__simp_I1_J,axiom,
! [V: a] :
( ( undire3617971648856834880loop_a @ e @ V )
=> ( ( undire4753905205749729249oops_a @ e @ V )
= ( insert_set_a @ ( insert_a @ V @ bot_bot_set_a ) @ bot_bot_set_set_a ) ) ) ).
% gnew.incident_loops_simp(1)
thf(fact_249_vert__adj__inc__edge__iff,axiom,
! [V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
= ( ( undire1521409233611534436dent_a @ V1 @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) )
& ( undire1521409233611534436dent_a @ V2 @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) )
& ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) @ edges ) ) ) ).
% vert_adj_inc_edge_iff
thf(fact_250_gnew_Overt__adj__inc__edge__iff,axiom,
! [V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ e @ V1 @ V2 )
= ( ( undire1521409233611534436dent_a @ V1 @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) )
& ( undire1521409233611534436dent_a @ V2 @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) )
& ( member_set_a @ ( insert_a @ V1 @ ( insert_a @ V2 @ bot_bot_set_a ) ) @ e ) ) ) ).
% gnew.vert_adj_inc_edge_iff
thf(fact_251_comp__sgraph_Ois__edge__between__def,axiom,
( undire6814325412647357297en_nat
= ( ^ [X2: set_nat,Y2: set_nat,E: set_nat] :
? [X3: nat,Y3: nat] :
( ( E
= ( insert_nat @ X3 @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) )
& ( member_nat @ X3 @ X2 )
& ( member_nat @ Y3 @ Y2 ) ) ) ) ).
% comp_sgraph.is_edge_between_def
thf(fact_252_comp__sgraph_Ois__edge__between__def,axiom,
( undire3488164626074856909n_real
= ( ^ [X2: set_real,Y2: set_real,E: set_real] :
? [X3: real,Y3: real] :
( ( E
= ( insert_real @ X3 @ ( insert_real @ Y3 @ bot_bot_set_real ) ) )
& ( member_real @ X3 @ X2 )
& ( member_real @ Y3 @ Y2 ) ) ) ) ).
% comp_sgraph.is_edge_between_def
thf(fact_253_comp__sgraph_Ois__edge__between__def,axiom,
( undire2578756059399487229_set_a
= ( ^ [X2: set_set_a,Y2: set_set_a,E: set_set_a] :
? [X3: set_a,Y3: set_a] :
( ( E
= ( insert_set_a @ X3 @ ( insert_set_a @ Y3 @ bot_bot_set_set_a ) ) )
& ( member_set_a @ X3 @ X2 )
& ( member_set_a @ Y3 @ Y2 ) ) ) ) ).
% comp_sgraph.is_edge_between_def
thf(fact_254_comp__sgraph_Ois__edge__between__def,axiom,
( undire7011261089604658374od_a_a
= ( ^ [X2: set_Product_prod_a_a,Y2: set_Product_prod_a_a,E: set_Product_prod_a_a] :
? [X3: product_prod_a_a,Y3: product_prod_a_a] :
( ( E
= ( insert4534936382041156343od_a_a @ X3 @ ( insert4534936382041156343od_a_a @ Y3 @ bot_bo3357376287454694259od_a_a ) ) )
& ( member1426531477525435216od_a_a @ X3 @ X2 )
& ( member1426531477525435216od_a_a @ Y3 @ Y2 ) ) ) ) ).
% comp_sgraph.is_edge_between_def
thf(fact_255_comp__sgraph_Ois__edge__between__def,axiom,
( undire8544646567961481629ween_a
= ( ^ [X2: set_a,Y2: set_a,E: set_a] :
? [X3: a,Y3: a] :
( ( E
= ( insert_a @ X3 @ ( insert_a @ Y3 @ bot_bot_set_a ) ) )
& ( member_a @ X3 @ X2 )
& ( member_a @ Y3 @ Y2 ) ) ) ) ).
% comp_sgraph.is_edge_between_def
thf(fact_256_incident__loops__simp_I2_J,axiom,
! [V: a] :
( ~ ( undire3617971648856834880loop_a @ edges @ V )
=> ( ( undire4753905205749729249oops_a @ edges @ V )
= bot_bot_set_set_a ) ) ).
% incident_loops_simp(2)
thf(fact_257_gnew_Oincident__loops__simp_I2_J,axiom,
! [V: a] :
( ~ ( undire3617971648856834880loop_a @ e @ V )
=> ( ( undire4753905205749729249oops_a @ e @ V )
= bot_bot_set_set_a ) ) ).
% gnew.incident_loops_simp(2)
thf(fact_258_edge__adj__def,axiom,
! [E1: set_a,E2: set_a] :
( ( undire4022703626023482010_adj_a @ edges @ E1 @ E2 )
= ( ( ( inf_inf_set_a @ E1 @ E2 )
!= bot_bot_set_a )
& ( member_set_a @ E1 @ edges )
& ( member_set_a @ E2 @ edges ) ) ) ).
% edge_adj_def
thf(fact_259_gnew_Oedge__adj__def,axiom,
! [E1: set_a,E2: set_a] :
( ( undire4022703626023482010_adj_a @ e @ E1 @ E2 )
= ( ( ( inf_inf_set_a @ E1 @ E2 )
!= bot_bot_set_a )
& ( member_set_a @ E1 @ e )
& ( member_set_a @ E2 @ e ) ) ) ).
% gnew.edge_adj_def
thf(fact_260_vert__adj__edge__iff2,axiom,
! [V1: a,V2: a] :
( ( V1 != V2 )
=> ( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
= ( ? [X3: set_a] :
( ( member_set_a @ X3 @ edges )
& ( undire1521409233611534436dent_a @ V1 @ X3 )
& ( undire1521409233611534436dent_a @ V2 @ X3 ) ) ) ) ) ).
% vert_adj_edge_iff2
thf(fact_261_gnew_Overt__adj__edge__iff2,axiom,
! [V1: a,V2: a] :
( ( V1 != V2 )
=> ( ( undire397441198561214472_adj_a @ e @ V1 @ V2 )
= ( ? [X3: set_a] :
( ( member_set_a @ X3 @ e )
& ( undire1521409233611534436dent_a @ V1 @ X3 )
& ( undire1521409233611534436dent_a @ V2 @ X3 ) ) ) ) ) ).
% gnew.vert_adj_edge_iff2
thf(fact_262_incident__def,axiom,
undire1521409233611534436dent_a = member_a ).
% incident_def
thf(fact_263_IntI,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A2 )
=> ( ( member1426531477525435216od_a_a @ C @ B )
=> ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_264_IntI,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_265_IntI,axiom,
! [C: real,A2: set_real,B: set_real] :
( ( member_real @ C @ A2 )
=> ( ( member_real @ C @ B )
=> ( member_real @ C @ ( inf_inf_set_real @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_266_IntI,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ A2 )
=> ( ( member_a @ C @ B )
=> ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_267_IntI,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A2 )
=> ( ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_268_Int__iff,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) )
= ( ( member1426531477525435216od_a_a @ C @ A2 )
& ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_269_Int__iff,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B ) )
= ( ( member_nat @ C @ A2 )
& ( member_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_270_Int__iff,axiom,
! [C: real,A2: set_real,B: set_real] :
( ( member_real @ C @ ( inf_inf_set_real @ A2 @ B ) )
= ( ( member_real @ C @ A2 )
& ( member_real @ C @ B ) ) ) ).
% Int_iff
thf(fact_271_Int__iff,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( member_a @ C @ A2 )
& ( member_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_272_Int__iff,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B ) )
= ( ( member_set_a @ C @ A2 )
& ( member_set_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_273_Int__subset__iff,axiom,
! [C2: set_set_a,A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) )
= ( ( ord_le3724670747650509150_set_a @ C2 @ A2 )
& ( ord_le3724670747650509150_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_274_Int__subset__iff,axiom,
! [C2: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( ord_less_eq_set_a @ C2 @ A2 )
& ( ord_less_eq_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_275_Int__subset__iff,axiom,
! [C2: set_Product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C2 @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) )
= ( ( ord_le746702958409616551od_a_a @ C2 @ A2 )
& ( ord_le746702958409616551od_a_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_276_Int__insert__left__if0,axiom,
! [A: product_prod_a_a,C2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ B ) @ C2 )
= ( inf_in8905007599844390133od_a_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_277_Int__insert__left__if0,axiom,
! [A: nat,C2: set_nat,B: set_nat] :
( ~ ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_278_Int__insert__left__if0,axiom,
! [A: real,C2: set_real,B: set_real] :
( ~ ( member_real @ A @ C2 )
=> ( ( inf_inf_set_real @ ( insert_real @ A @ B ) @ C2 )
= ( inf_inf_set_real @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_279_Int__insert__left__if0,axiom,
! [A: a,C2: set_a,B: set_a] :
( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_280_Int__insert__left__if0,axiom,
! [A: set_a,C2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_281_Int__insert__left__if1,axiom,
! [A: product_prod_a_a,C2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ B ) @ C2 )
= ( insert4534936382041156343od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_282_Int__insert__left__if1,axiom,
! [A: nat,C2: set_nat,B: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B ) @ C2 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_283_Int__insert__left__if1,axiom,
! [A: real,C2: set_real,B: set_real] :
( ( member_real @ A @ C2 )
=> ( ( inf_inf_set_real @ ( insert_real @ A @ B ) @ C2 )
= ( insert_real @ A @ ( inf_inf_set_real @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_284_Int__insert__left__if1,axiom,
! [A: a,C2: set_a,B: set_a] :
( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_285_Int__insert__left__if1,axiom,
! [A: set_a,C2: set_set_a,B: set_set_a] :
( ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C2 )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_286_insert__inter__insert,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_287_insert__inter__insert,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ ( insert_set_a @ A @ B ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_288_Int__insert__right__if0,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A @ A2 )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ A @ B ) )
= ( inf_in8905007599844390133od_a_a @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_289_Int__insert__right__if0,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( inf_inf_set_nat @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_290_Int__insert__right__if0,axiom,
! [A: real,A2: set_real,B: set_real] :
( ~ ( member_real @ A @ A2 )
=> ( ( inf_inf_set_real @ A2 @ ( insert_real @ A @ B ) )
= ( inf_inf_set_real @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_291_Int__insert__right__if0,axiom,
! [A: a,A2: set_a,B: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_292_Int__insert__right__if0,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_293_Int__insert__right__if1,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ A2 )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ A @ B ) )
= ( insert4534936382041156343od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_294_Int__insert__right__if1,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_295_Int__insert__right__if1,axiom,
! [A: real,A2: set_real,B: set_real] :
( ( member_real @ A @ A2 )
=> ( ( inf_inf_set_real @ A2 @ ( insert_real @ A @ B ) )
= ( insert_real @ A @ ( inf_inf_set_real @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_296_Int__insert__right__if1,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_297_Int__insert__right__if1,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_298_disjoint__insert_I2_J,axiom,
! [A2: set_nat,B2: nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ ( insert_nat @ B2 @ B ) ) )
= ( ~ ( member_nat @ B2 @ A2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_299_disjoint__insert_I2_J,axiom,
! [A2: set_real,B2: real,B: set_real] :
( ( bot_bot_set_real
= ( inf_inf_set_real @ A2 @ ( insert_real @ B2 @ B ) ) )
= ( ~ ( member_real @ B2 @ A2 )
& ( bot_bot_set_real
= ( inf_inf_set_real @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_300_disjoint__insert_I2_J,axiom,
! [A2: set_a,B2: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ ( insert_a @ B2 @ B ) ) )
= ( ~ ( member_a @ B2 @ A2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_301_disjoint__insert_I2_J,axiom,
! [A2: set_set_a,B2: set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ B2 @ B ) ) )
= ( ~ ( member_set_a @ B2 @ A2 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_302_disjoint__insert_I2_J,axiom,
! [A2: set_Product_prod_a_a,B2: product_prod_a_a,B: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B2 @ B ) ) )
= ( ~ ( member1426531477525435216od_a_a @ B2 @ A2 )
& ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_303_disjoint__insert_I1_J,axiom,
! [B: set_nat,A: nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ B @ ( insert_nat @ A @ A2 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B )
& ( ( inf_inf_set_nat @ B @ A2 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_304_disjoint__insert_I1_J,axiom,
! [B: set_real,A: real,A2: set_real] :
( ( ( inf_inf_set_real @ B @ ( insert_real @ A @ A2 ) )
= bot_bot_set_real )
= ( ~ ( member_real @ A @ B )
& ( ( inf_inf_set_real @ B @ A2 )
= bot_bot_set_real ) ) ) ).
% disjoint_insert(1)
thf(fact_305_disjoint__insert_I1_J,axiom,
! [B: set_a,A: a,A2: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A @ A2 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B )
& ( ( inf_inf_set_a @ B @ A2 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_306_disjoint__insert_I1_J,axiom,
! [B: set_set_a,A: set_a,A2: set_set_a] :
( ( ( inf_inf_set_set_a @ B @ ( insert_set_a @ A @ A2 ) )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B )
& ( ( inf_inf_set_set_a @ B @ A2 )
= bot_bot_set_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_307_disjoint__insert_I1_J,axiom,
! [B: set_Product_prod_a_a,A: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ B @ ( insert4534936382041156343od_a_a @ A @ A2 ) )
= bot_bo3357376287454694259od_a_a )
= ( ~ ( member1426531477525435216od_a_a @ A @ B )
& ( ( inf_in8905007599844390133od_a_a @ B @ A2 )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% disjoint_insert(1)
thf(fact_308_insert__disjoint_I2_J,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B ) )
= ( ~ ( member_nat @ A @ B )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_309_insert__disjoint_I2_J,axiom,
! [A: real,A2: set_real,B: set_real] :
( ( bot_bot_set_real
= ( inf_inf_set_real @ ( insert_real @ A @ A2 ) @ B ) )
= ( ~ ( member_real @ A @ B )
& ( bot_bot_set_real
= ( inf_inf_set_real @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_310_insert__disjoint_I2_J,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B ) )
= ( ~ ( member_a @ A @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_311_insert__disjoint_I2_J,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ B ) )
= ( ~ ( member_set_a @ A @ B )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_312_insert__disjoint_I2_J,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ A2 ) @ B ) )
= ( ~ ( member1426531477525435216od_a_a @ A @ B )
& ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_313_insert__disjoint_I1_J,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B )
& ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_314_insert__disjoint_I1_J,axiom,
! [A: real,A2: set_real,B: set_real] :
( ( ( inf_inf_set_real @ ( insert_real @ A @ A2 ) @ B )
= bot_bot_set_real )
= ( ~ ( member_real @ A @ B )
& ( ( inf_inf_set_real @ A2 @ B )
= bot_bot_set_real ) ) ) ).
% insert_disjoint(1)
thf(fact_315_insert__disjoint_I1_J,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B )
& ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_316_insert__disjoint_I1_J,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ B )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B )
& ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_317_insert__disjoint_I1_J,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ A2 ) @ B )
= bot_bo3357376287454694259od_a_a )
= ( ~ ( member1426531477525435216od_a_a @ A @ B )
& ( ( inf_in8905007599844390133od_a_a @ A2 @ B )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% insert_disjoint(1)
thf(fact_318_gnew_Ounique__triangles__def,axiom,
( ( graph_6144490306505338871gles_a @ e )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ e )
=> ? [Y3: set_a] :
( ? [Z3: a,Aa: a,Ab: a] :
( ( Y3
= ( insert_a @ Z3 @ ( insert_a @ Aa @ ( insert_a @ Ab @ bot_bot_set_a ) ) ) )
& ( graph_4582152751571636272raph_a @ e @ Z3 @ Aa @ Ab )
& ( ord_less_eq_set_a @ X3 @ Y3 ) )
& ! [Z3: set_a] :
( ? [Aa: a,Ab: a,Ac: a] :
( ( Z3
= ( insert_a @ Aa @ ( insert_a @ Ab @ ( insert_a @ Ac @ bot_bot_set_a ) ) ) )
& ( graph_4582152751571636272raph_a @ e @ Aa @ Ab @ Ac )
& ( ord_less_eq_set_a @ X3 @ Z3 ) )
=> ( Z3 = Y3 ) ) ) ) ) ) ).
% gnew.unique_triangles_def
thf(fact_319_unique__triangles__def,axiom,
( ( graph_6144490306505338871gles_a @ edges )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ edges )
=> ? [Y3: set_a] :
( ? [Z3: a,Aa: a,Ab: a] :
( ( Y3
= ( insert_a @ Z3 @ ( insert_a @ Aa @ ( insert_a @ Ab @ bot_bot_set_a ) ) ) )
& ( graph_4582152751571636272raph_a @ edges @ Z3 @ Aa @ Ab )
& ( ord_less_eq_set_a @ X3 @ Y3 ) )
& ! [Z3: set_a] :
( ? [Aa: a,Ab: a,Ac: a] :
( ( Z3
= ( insert_a @ Aa @ ( insert_a @ Ab @ ( insert_a @ Ac @ bot_bot_set_a ) ) ) )
& ( graph_4582152751571636272raph_a @ edges @ Aa @ Ab @ Ac )
& ( ord_less_eq_set_a @ X3 @ Z3 ) )
=> ( Z3 = Y3 ) ) ) ) ) ) ).
% unique_triangles_def
thf(fact_320_IntE,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) )
=> ~ ( ( member1426531477525435216od_a_a @ C @ A2 )
=> ~ ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% IntE
thf(fact_321_IntE,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ~ ( member_nat @ C @ B ) ) ) ).
% IntE
thf(fact_322_IntE,axiom,
! [C: real,A2: set_real,B: set_real] :
( ( member_real @ C @ ( inf_inf_set_real @ A2 @ B ) )
=> ~ ( ( member_real @ C @ A2 )
=> ~ ( member_real @ C @ B ) ) ) ).
% IntE
thf(fact_323_IntE,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
=> ~ ( ( member_a @ C @ A2 )
=> ~ ( member_a @ C @ B ) ) ) ).
% IntE
thf(fact_324_IntE,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ~ ( ( member_set_a @ C @ A2 )
=> ~ ( member_set_a @ C @ B ) ) ) ).
% IntE
thf(fact_325_IntD1,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) )
=> ( member1426531477525435216od_a_a @ C @ A2 ) ) ).
% IntD1
thf(fact_326_IntD1,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B ) )
=> ( member_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_327_IntD1,axiom,
! [C: real,A2: set_real,B: set_real] :
( ( member_real @ C @ ( inf_inf_set_real @ A2 @ B ) )
=> ( member_real @ C @ A2 ) ) ).
% IntD1
thf(fact_328_IntD1,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C @ A2 ) ) ).
% IntD1
thf(fact_329_IntD1,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ( member_set_a @ C @ A2 ) ) ).
% IntD1
thf(fact_330_IntD2,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) )
=> ( member1426531477525435216od_a_a @ C @ B ) ) ).
% IntD2
thf(fact_331_IntD2,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B ) )
=> ( member_nat @ C @ B ) ) ).
% IntD2
thf(fact_332_IntD2,axiom,
! [C: real,A2: set_real,B: set_real] :
( ( member_real @ C @ ( inf_inf_set_real @ A2 @ B ) )
=> ( member_real @ C @ B ) ) ).
% IntD2
thf(fact_333_IntD2,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C @ B ) ) ).
% IntD2
thf(fact_334_IntD2,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ( member_set_a @ C @ B ) ) ).
% IntD2
thf(fact_335_Int__assoc,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_336_Int__assoc,axiom,
! [A2: set_set_a,B: set_set_a,C2: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_337_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_338_Int__absorb,axiom,
! [A2: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_339_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A3: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_340_Int__commute,axiom,
( inf_inf_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] : ( inf_inf_set_set_a @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_341_Int__left__absorb,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ).
% Int_left_absorb
thf(fact_342_Int__left__absorb,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ ( inf_inf_set_set_a @ A2 @ B ) )
= ( inf_inf_set_set_a @ A2 @ B ) ) ).
% Int_left_absorb
thf(fact_343_Int__left__commute,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_344_Int__left__commute,axiom,
! [A2: set_set_a,B: set_set_a,C2: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C2 ) )
= ( inf_inf_set_set_a @ B @ ( inf_inf_set_set_a @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_345_ulgraph_Oincident__loops_Ocong,axiom,
undire4753905205749729249oops_a = undire4753905205749729249oops_a ).
% ulgraph.incident_loops.cong
thf(fact_346_comp__sgraph_Oincident__def,axiom,
undire2320338297334612420_set_a = member_set_a ).
% comp_sgraph.incident_def
thf(fact_347_comp__sgraph_Oincident__def,axiom,
undire3369688177417741453od_a_a = member1426531477525435216od_a_a ).
% comp_sgraph.incident_def
thf(fact_348_comp__sgraph_Oincident__def,axiom,
undire7858122600432113898nt_nat = member_nat ).
% comp_sgraph.incident_def
thf(fact_349_comp__sgraph_Oincident__def,axiom,
undire4230802696203636422t_real = member_real ).
% comp_sgraph.incident_def
thf(fact_350_comp__sgraph_Oincident__def,axiom,
undire1521409233611534436dent_a = member_a ).
% comp_sgraph.incident_def
thf(fact_351_Int__emptyI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ~ ( member_nat @ X4 @ B ) )
=> ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_352_Int__emptyI,axiom,
! [A2: set_real,B: set_real] :
( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ~ ( member_real @ X4 @ B ) )
=> ( ( inf_inf_set_real @ A2 @ B )
= bot_bot_set_real ) ) ).
% Int_emptyI
thf(fact_353_Int__emptyI,axiom,
! [A2: set_a,B: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ~ ( member_a @ X4 @ B ) )
=> ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_354_Int__emptyI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ~ ( member_set_a @ X4 @ B ) )
=> ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_355_Int__emptyI,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A2 )
=> ~ ( member1426531477525435216od_a_a @ X4 @ B ) )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ B )
= bot_bo3357376287454694259od_a_a ) ) ).
% Int_emptyI
thf(fact_356_disjoint__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ~ ( member_nat @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_357_disjoint__iff,axiom,
! [A2: set_real,B: set_real] :
( ( ( inf_inf_set_real @ A2 @ B )
= bot_bot_set_real )
= ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ~ ( member_real @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_358_disjoint__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_a @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_359_disjoint__iff,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ~ ( member_set_a @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_360_disjoint__iff,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ A2 @ B )
= bot_bo3357376287454694259od_a_a )
= ( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A2 )
=> ~ ( member1426531477525435216od_a_a @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_361_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_362_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_363_Int__empty__left,axiom,
! [B: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ B )
= bot_bo3357376287454694259od_a_a ) ).
% Int_empty_left
thf(fact_364_Int__empty__right,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_365_Int__empty__right,axiom,
! [A2: set_set_a] :
( ( inf_inf_set_set_a @ A2 @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% Int_empty_right
thf(fact_366_Int__empty__right,axiom,
! [A2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A2 @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% Int_empty_right
thf(fact_367_disjoint__iff__not__equal,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ! [Y3: a] :
( ( member_a @ Y3 @ B )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_368_disjoint__iff__not__equal,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ! [Y3: set_a] :
( ( member_set_a @ Y3 @ B )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_369_disjoint__iff__not__equal,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ A2 @ B )
= bot_bo3357376287454694259od_a_a )
= ( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A2 )
=> ! [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ B )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_370_Int__Collect__mono,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_371_Int__Collect__mono,axiom,
! [A2: set_real,B: set_real,P: real > $o,Q: real > $o] :
( ( ord_less_eq_set_real @ A2 @ B )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_real @ ( inf_inf_set_real @ A2 @ ( collect_real @ P ) ) @ ( inf_inf_set_real @ B @ ( collect_real @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_372_Int__Collect__mono,axiom,
! [A2: set_set_a,B: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_373_Int__Collect__mono,axiom,
! [A2: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_374_Int__Collect__mono,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ ( collec3336397797384452498od_a_a @ P ) ) @ ( inf_in8905007599844390133od_a_a @ B @ ( collec3336397797384452498od_a_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_375_Int__greatest,axiom,
! [C2: set_set_a,A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ C2 @ B )
=> ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_376_Int__greatest,axiom,
! [C2: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ B )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_377_Int__greatest,axiom,
! [C2: set_Product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C2 @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ C2 @ B )
=> ( ord_le746702958409616551od_a_a @ C2 @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_378_Int__absorb2,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( inf_inf_set_set_a @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_379_Int__absorb2,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( inf_inf_set_a @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_380_Int__absorb2,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_381_Int__absorb1,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_382_Int__absorb1,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_383_Int__absorb1,axiom,
! [B: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B @ A2 )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_384_Int__lower2,axiom,
! [A2: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_385_Int__lower2,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_386_Int__lower2,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_387_Int__lower1,axiom,
! [A2: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_388_Int__lower1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_389_Int__lower1,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_390_Int__mono,axiom,
! [A2: set_set_a,C2: set_set_a,B: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B @ D )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ B ) @ ( inf_inf_set_set_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_391_Int__mono,axiom,
! [A2: set_a,C2: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_392_Int__mono,axiom,
! [A2: set_Product_prod_a_a,C2: set_Product_prod_a_a,B: set_Product_prod_a_a,D: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ C2 )
=> ( ( ord_le746702958409616551od_a_a @ B @ D )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) @ ( inf_in8905007599844390133od_a_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_393_Int__insert__left,axiom,
! [A: product_prod_a_a,C2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ A @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ B ) @ C2 )
= ( insert4534936382041156343od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ B @ C2 ) ) ) )
& ( ~ ( member1426531477525435216od_a_a @ A @ C2 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A @ B ) @ C2 )
= ( inf_in8905007599844390133od_a_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_394_Int__insert__left,axiom,
! [A: nat,C2: set_nat,B: set_nat] :
( ( ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B ) @ C2 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) ) ) )
& ( ~ ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_395_Int__insert__left,axiom,
! [A: real,C2: set_real,B: set_real] :
( ( ( member_real @ A @ C2 )
=> ( ( inf_inf_set_real @ ( insert_real @ A @ B ) @ C2 )
= ( insert_real @ A @ ( inf_inf_set_real @ B @ C2 ) ) ) )
& ( ~ ( member_real @ A @ C2 )
=> ( ( inf_inf_set_real @ ( insert_real @ A @ B ) @ C2 )
= ( inf_inf_set_real @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_396_Int__insert__left,axiom,
! [A: a,C2: set_a,B: set_a] :
( ( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_397_Int__insert__left,axiom,
! [A: set_a,C2: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C2 )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_set_a @ A @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_398_Int__insert__right,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ A @ A2 )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ A @ B ) )
= ( insert4534936382041156343od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ A2 @ B ) ) ) )
& ( ~ ( member1426531477525435216od_a_a @ A @ A2 )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ A @ B ) )
= ( inf_in8905007599844390133od_a_a @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_399_Int__insert__right,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B ) ) ) )
& ( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_400_Int__insert__right,axiom,
! [A: real,A2: set_real,B: set_real] :
( ( ( member_real @ A @ A2 )
=> ( ( inf_inf_set_real @ A2 @ ( insert_real @ A @ B ) )
= ( insert_real @ A @ ( inf_inf_set_real @ A2 @ B ) ) ) )
& ( ~ ( member_real @ A @ A2 )
=> ( ( inf_inf_set_real @ A2 @ ( insert_real @ A @ B ) )
= ( inf_inf_set_real @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_401_Int__insert__right,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_402_Int__insert__right,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B ) ) ) )
& ( ~ ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_403_ulgraph_Overt__adj_Ocong,axiom,
undire397441198561214472_adj_a = undire397441198561214472_adj_a ).
% ulgraph.vert_adj.cong
thf(fact_404_ulgraph_Ohas__loop_Ocong,axiom,
undire3617971648856834880loop_a = undire3617971648856834880loop_a ).
% ulgraph.has_loop.cong
thf(fact_405_graph__system_Oedge__adj_Ocong,axiom,
undire4022703626023482010_adj_a = undire4022703626023482010_adj_a ).
% graph_system.edge_adj.cong
thf(fact_406_inf__bot__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_407_inf__bot__left,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X )
= bot_bot_set_set_a ) ).
% inf_bot_left
thf(fact_408_inf__bot__left,axiom,
! [X: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ X )
= bot_bo3357376287454694259od_a_a ) ).
% inf_bot_left
thf(fact_409_inf__bot__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_410_inf__bot__right,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ X @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% inf_bot_right
thf(fact_411_inf__bot__right,axiom,
! [X: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% inf_bot_right
thf(fact_412_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_413_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_414_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ X )
= bot_bo3357376287454694259od_a_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_415_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_416_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ X @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_417_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_418_inf_Obounded__iff,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( inf_inf_set_set_a @ B2 @ C ) )
= ( ( ord_le3724670747650509150_set_a @ A @ B2 )
& ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_419_inf_Obounded__iff,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B2 @ C ) )
= ( ( ord_less_eq_set_a @ A @ B2 )
& ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_420_inf_Obounded__iff,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ B2 @ C ) )
= ( ( ord_le746702958409616551od_a_a @ A @ B2 )
& ( ord_le746702958409616551od_a_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_421_inf_Obounded__iff,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B2 @ C ) )
= ( ( ord_less_eq_nat @ A @ B2 )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_422_inf_Obounded__iff,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ ( inf_inf_real @ B2 @ C ) )
= ( ( ord_less_eq_real @ A @ B2 )
& ( ord_less_eq_real @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_423_le__inf__iff,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) )
= ( ( ord_le3724670747650509150_set_a @ X @ Y )
& ( ord_le3724670747650509150_set_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_424_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_425_le__inf__iff,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ Y @ Z ) )
= ( ( ord_le746702958409616551od_a_a @ X @ Y )
& ( ord_le746702958409616551od_a_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_426_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_427_le__inf__iff,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ ( inf_inf_real @ Y @ Z ) )
= ( ( ord_less_eq_real @ X @ Y )
& ( ord_less_eq_real @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_428_inf__right__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_429_inf__right__idem,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Y )
= ( inf_inf_set_set_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_430_inf_Oright__idem,axiom,
! [A: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B2 ) @ B2 )
= ( inf_inf_set_a @ A @ B2 ) ) ).
% inf.right_idem
thf(fact_431_inf_Oright__idem,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ B2 )
= ( inf_inf_set_set_a @ A @ B2 ) ) ).
% inf.right_idem
thf(fact_432_inf__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_left_idem
thf(fact_433_inf__left__idem,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ X @ Y ) )
= ( inf_inf_set_set_a @ X @ Y ) ) ).
% inf_left_idem
thf(fact_434_inf_Oidem,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_435_inf_Oidem,axiom,
! [A: set_set_a] :
( ( inf_inf_set_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_436_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_437_inf__idem,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_438_inf_Oleft__idem,axiom,
! [A: set_a,B2: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B2 ) )
= ( inf_inf_set_a @ A @ B2 ) ) ).
% inf.left_idem
thf(fact_439_inf_Oleft__idem,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ A @ B2 ) )
= ( inf_inf_set_set_a @ A @ B2 ) ) ).
% inf.left_idem
thf(fact_440_sgraph_Ounique__triangles_Ocong,axiom,
graph_6144490306505338871gles_a = graph_6144490306505338871gles_a ).
% sgraph.unique_triangles.cong
thf(fact_441_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_442_bot__set__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a @ bot_bot_set_a_o ) ) ).
% bot_set_def
thf(fact_443_bot__set__def,axiom,
( bot_bo3357376287454694259od_a_a
= ( collec3336397797384452498od_a_a @ bot_bo4160289986317612842_a_a_o ) ) ).
% bot_set_def
thf(fact_444_inf__sup__aci_I4_J,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_445_inf__sup__aci_I4_J,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ X @ Y ) )
= ( inf_inf_set_set_a @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_446_inf__sup__aci_I3_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_447_inf__sup__aci_I3_J,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) )
= ( inf_inf_set_set_a @ Y @ ( inf_inf_set_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_448_inf__sup__aci_I2_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_449_inf__sup__aci_I2_J,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_450_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X3: set_a,Y3: set_a] : ( inf_inf_set_a @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_451_inf__sup__aci_I1_J,axiom,
( inf_inf_set_set_a
= ( ^ [X3: set_set_a,Y3: set_set_a] : ( inf_inf_set_set_a @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_452_inf_Oassoc,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B2 ) @ C )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_453_inf_Oassoc,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ C )
= ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_454_inf__assoc,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_455_inf__assoc,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_456_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B5: set_a] : ( inf_inf_set_a @ B5 @ A4 ) ) ) ).
% inf.commute
thf(fact_457_inf_Ocommute,axiom,
( inf_inf_set_set_a
= ( ^ [A4: set_set_a,B5: set_set_a] : ( inf_inf_set_set_a @ B5 @ A4 ) ) ) ).
% inf.commute
thf(fact_458_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X3: set_a,Y3: set_a] : ( inf_inf_set_a @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_459_inf__commute,axiom,
( inf_inf_set_set_a
= ( ^ [X3: set_set_a,Y3: set_set_a] : ( inf_inf_set_set_a @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_460_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_a,K: set_a,A: set_a,B2: set_a] :
( ( A2
= ( inf_inf_set_a @ K @ A ) )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_461_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_set_a,K: set_set_a,A: set_set_a,B2: set_set_a] :
( ( A2
= ( inf_inf_set_set_a @ K @ A ) )
=> ( ( inf_inf_set_set_a @ A2 @ B2 )
= ( inf_inf_set_set_a @ K @ ( inf_inf_set_set_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_462_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_a,K: set_a,B2: set_a,A: set_a] :
( ( B
= ( inf_inf_set_a @ K @ B2 ) )
=> ( ( inf_inf_set_a @ A @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_463_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_set_a,K: set_set_a,B2: set_set_a,A: set_set_a] :
( ( B
= ( inf_inf_set_set_a @ K @ B2 ) )
=> ( ( inf_inf_set_set_a @ A @ B )
= ( inf_inf_set_set_a @ K @ ( inf_inf_set_set_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_464_inf_Oleft__commute,axiom,
! [B2: set_a,A: set_a,C: set_a] :
( ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A @ C ) )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_465_inf_Oleft__commute,axiom,
! [B2: set_set_a,A: set_set_a,C: set_set_a] :
( ( inf_inf_set_set_a @ B2 @ ( inf_inf_set_set_a @ A @ C ) )
= ( inf_inf_set_set_a @ A @ ( inf_inf_set_set_a @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_466_inf__left__commute,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_467_inf__left__commute,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) )
= ( inf_inf_set_set_a @ Y @ ( inf_inf_set_set_a @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_468_inf__sup__ord_I2_J,axiom,
! [X: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_469_inf__sup__ord_I2_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_470_inf__sup__ord_I2_J,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_471_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_472_inf__sup__ord_I2_J,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_473_inf__sup__ord_I1_J,axiom,
! [X: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_474_inf__sup__ord_I1_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_475_inf__sup__ord_I1_J,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_476_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_477_inf__sup__ord_I1_J,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_478_inf__le1,axiom,
! [X: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_479_inf__le1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_480_inf__le1,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_481_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_482_inf__le1,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_483_inf__le2,axiom,
! [X: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_484_inf__le2,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_485_inf__le2,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_486_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_487_inf__le2,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_488_le__infE,axiom,
! [X: set_set_a,A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ A @ B2 ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ X @ A )
=> ~ ( ord_le3724670747650509150_set_a @ X @ B2 ) ) ) ).
% le_infE
thf(fact_489_le__infE,axiom,
! [X: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B2 ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A )
=> ~ ( ord_less_eq_set_a @ X @ B2 ) ) ) ).
% le_infE
thf(fact_490_le__infE,axiom,
! [X: set_Product_prod_a_a,A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ A @ B2 ) )
=> ~ ( ( ord_le746702958409616551od_a_a @ X @ A )
=> ~ ( ord_le746702958409616551od_a_a @ X @ B2 ) ) ) ).
% le_infE
thf(fact_491_le__infE,axiom,
! [X: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X @ A )
=> ~ ( ord_less_eq_nat @ X @ B2 ) ) ) ).
% le_infE
thf(fact_492_le__infE,axiom,
! [X: real,A: real,B2: real] :
( ( ord_less_eq_real @ X @ ( inf_inf_real @ A @ B2 ) )
=> ~ ( ( ord_less_eq_real @ X @ A )
=> ~ ( ord_less_eq_real @ X @ B2 ) ) ) ).
% le_infE
thf(fact_493_le__infI,axiom,
! [X: set_set_a,A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ A )
=> ( ( ord_le3724670747650509150_set_a @ X @ B2 )
=> ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_494_le__infI,axiom,
! [X: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X @ B2 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_495_le__infI,axiom,
! [X: set_Product_prod_a_a,A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ A )
=> ( ( ord_le746702958409616551od_a_a @ X @ B2 )
=> ( ord_le746702958409616551od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_496_le__infI,axiom,
! [X: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_497_le__infI,axiom,
! [X: real,A: real,B2: real] :
( ( ord_less_eq_real @ X @ A )
=> ( ( ord_less_eq_real @ X @ B2 )
=> ( ord_less_eq_real @ X @ ( inf_inf_real @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_498_inf__mono,axiom,
! [A: set_set_a,C: set_set_a,B2: set_set_a,D2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ ( inf_inf_set_set_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_499_inf__mono,axiom,
! [A: set_a,C: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_500_inf__mono,axiom,
! [A: set_Product_prod_a_a,C: set_Product_prod_a_a,B2: set_Product_prod_a_a,D2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ C )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ D2 )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B2 ) @ ( inf_in8905007599844390133od_a_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_501_inf__mono,axiom,
! [A: nat,C: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_502_inf__mono,axiom,
! [A: real,C: real,B2: real,D2: real] :
( ( ord_less_eq_real @ A @ C )
=> ( ( ord_less_eq_real @ B2 @ D2 )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B2 ) @ ( inf_inf_real @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_503_le__infI1,axiom,
! [A: set_set_a,X: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ X )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_504_le__infI1,axiom,
! [A: set_a,X: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_505_le__infI1,axiom,
! [A: set_Product_prod_a_a,X: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ X )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_506_le__infI1,axiom,
! [A: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_507_le__infI1,axiom,
! [A: real,X: real,B2: real] :
( ( ord_less_eq_real @ A @ X )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_508_le__infI2,axiom,
! [B2: set_set_a,X: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ X )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_509_le__infI2,axiom,
! [B2: set_a,X: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_510_le__infI2,axiom,
! [B2: set_Product_prod_a_a,X: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B2 @ X )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_511_le__infI2,axiom,
! [B2: nat,X: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_512_le__infI2,axiom,
! [B2: real,X: real,A: real] :
( ( ord_less_eq_real @ B2 @ X )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_513_inf_OorderE,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( A
= ( inf_inf_set_set_a @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_514_inf_OorderE,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( A
= ( inf_inf_set_a @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_515_inf_OorderE,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B2 )
=> ( A
= ( inf_in8905007599844390133od_a_a @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_516_inf_OorderE,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( A
= ( inf_inf_nat @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_517_inf_OorderE,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( A
= ( inf_inf_real @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_518_inf_OorderI,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( A
= ( inf_inf_set_set_a @ A @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A @ B2 ) ) ).
% inf.orderI
thf(fact_519_inf_OorderI,axiom,
! [A: set_a,B2: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B2 ) )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% inf.orderI
thf(fact_520_inf_OorderI,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( A
= ( inf_in8905007599844390133od_a_a @ A @ B2 ) )
=> ( ord_le746702958409616551od_a_a @ A @ B2 ) ) ).
% inf.orderI
thf(fact_521_inf_OorderI,axiom,
! [A: nat,B2: nat] :
( ( A
= ( inf_inf_nat @ A @ B2 ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% inf.orderI
thf(fact_522_inf_OorderI,axiom,
! [A: real,B2: real] :
( ( A
= ( inf_inf_real @ A @ B2 ) )
=> ( ord_less_eq_real @ A @ B2 ) ) ).
% inf.orderI
thf(fact_523_inf__unique,axiom,
! [F: set_set_a > set_set_a > set_set_a,X: set_set_a,Y: set_set_a] :
( ! [X4: set_set_a,Y5: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F @ X4 @ Y5 ) @ X4 )
=> ( ! [X4: set_set_a,Y5: set_set_a] : ( ord_le3724670747650509150_set_a @ ( F @ X4 @ Y5 ) @ Y5 )
=> ( ! [X4: set_set_a,Y5: set_set_a,Z4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y5 )
=> ( ( ord_le3724670747650509150_set_a @ X4 @ Z4 )
=> ( ord_le3724670747650509150_set_a @ X4 @ ( F @ Y5 @ Z4 ) ) ) )
=> ( ( inf_inf_set_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_524_inf__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X4: set_a,Y5: set_a] : ( ord_less_eq_set_a @ ( F @ X4 @ Y5 ) @ X4 )
=> ( ! [X4: set_a,Y5: set_a] : ( ord_less_eq_set_a @ ( F @ X4 @ Y5 ) @ Y5 )
=> ( ! [X4: set_a,Y5: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y5 )
=> ( ( ord_less_eq_set_a @ X4 @ Z4 )
=> ( ord_less_eq_set_a @ X4 @ ( F @ Y5 @ Z4 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_525_inf__unique,axiom,
! [F: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a,X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ! [X4: set_Product_prod_a_a,Y5: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( F @ X4 @ Y5 ) @ X4 )
=> ( ! [X4: set_Product_prod_a_a,Y5: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( F @ X4 @ Y5 ) @ Y5 )
=> ( ! [X4: set_Product_prod_a_a,Y5: set_Product_prod_a_a,Z4: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X4 @ Y5 )
=> ( ( ord_le746702958409616551od_a_a @ X4 @ Z4 )
=> ( ord_le746702958409616551od_a_a @ X4 @ ( F @ Y5 @ Z4 ) ) ) )
=> ( ( inf_in8905007599844390133od_a_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_526_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X4: nat,Y5: nat] : ( ord_less_eq_nat @ ( F @ X4 @ Y5 ) @ X4 )
=> ( ! [X4: nat,Y5: nat] : ( ord_less_eq_nat @ ( F @ X4 @ Y5 ) @ Y5 )
=> ( ! [X4: nat,Y5: nat,Z4: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ( ord_less_eq_nat @ X4 @ Z4 )
=> ( ord_less_eq_nat @ X4 @ ( F @ Y5 @ Z4 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_527_inf__unique,axiom,
! [F: real > real > real,X: real,Y: real] :
( ! [X4: real,Y5: real] : ( ord_less_eq_real @ ( F @ X4 @ Y5 ) @ X4 )
=> ( ! [X4: real,Y5: real] : ( ord_less_eq_real @ ( F @ X4 @ Y5 ) @ Y5 )
=> ( ! [X4: real,Y5: real,Z4: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
=> ( ( ord_less_eq_real @ X4 @ Z4 )
=> ( ord_less_eq_real @ X4 @ ( F @ Y5 @ Z4 ) ) ) )
=> ( ( inf_inf_real @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_528_le__iff__inf,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X3: set_set_a,Y3: set_set_a] :
( ( inf_inf_set_set_a @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_529_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_530_le__iff__inf,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [X3: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_531_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y3: nat] :
( ( inf_inf_nat @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_532_le__iff__inf,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y3: real] :
( ( inf_inf_real @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_533_inf_Oabsorb1,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( inf_inf_set_set_a @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_534_inf_Oabsorb1,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( inf_inf_set_a @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_535_inf_Oabsorb1,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B2 )
=> ( ( inf_in8905007599844390133od_a_a @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_536_inf_Oabsorb1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( inf_inf_nat @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_537_inf_Oabsorb1,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( inf_inf_real @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_538_inf_Oabsorb2,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( ( inf_inf_set_set_a @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_539_inf_Oabsorb2,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( inf_inf_set_a @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_540_inf_Oabsorb2,axiom,
! [B2: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B2 @ A )
=> ( ( inf_in8905007599844390133od_a_a @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_541_inf_Oabsorb2,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( inf_inf_nat @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_542_inf_Oabsorb2,axiom,
! [B2: real,A: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( inf_inf_real @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_543_inf__absorb1,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ( inf_inf_set_set_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_544_inf__absorb1,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( inf_inf_set_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_545_inf__absorb1,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ Y )
=> ( ( inf_in8905007599844390133od_a_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_546_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_547_inf__absorb1,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( inf_inf_real @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_548_inf__absorb2,axiom,
! [Y: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X )
=> ( ( inf_inf_set_set_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_549_inf__absorb2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( inf_inf_set_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_550_inf__absorb2,axiom,
! [Y: set_Product_prod_a_a,X: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y @ X )
=> ( ( inf_in8905007599844390133od_a_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_551_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_552_inf__absorb2,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( inf_inf_real @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_553_inf_OboundedE,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( inf_inf_set_set_a @ B2 @ C ) )
=> ~ ( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ~ ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_554_inf_OboundedE,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B2 )
=> ~ ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_555_inf_OboundedE,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ B2 @ C ) )
=> ~ ( ( ord_le746702958409616551od_a_a @ A @ B2 )
=> ~ ( ord_le746702958409616551od_a_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_556_inf_OboundedE,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B2 )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_557_inf_OboundedE,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ ( inf_inf_real @ B2 @ C ) )
=> ~ ( ( ord_less_eq_real @ A @ B2 )
=> ~ ( ord_less_eq_real @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_558_inf_OboundedI,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A @ C )
=> ( ord_le3724670747650509150_set_a @ A @ ( inf_inf_set_set_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_559_inf_OboundedI,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_560_inf_OboundedI,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B2 )
=> ( ( ord_le746702958409616551od_a_a @ A @ C )
=> ( ord_le746702958409616551od_a_a @ A @ ( inf_in8905007599844390133od_a_a @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_561_inf_OboundedI,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_562_inf_OboundedI,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ A @ C )
=> ( ord_less_eq_real @ A @ ( inf_inf_real @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_563_inf__greatest,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ( ord_le3724670747650509150_set_a @ X @ Z )
=> ( ord_le3724670747650509150_set_a @ X @ ( inf_inf_set_set_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_564_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_565_inf__greatest,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ Y )
=> ( ( ord_le746702958409616551od_a_a @ X @ Z )
=> ( ord_le746702958409616551od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_566_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_567_inf__greatest,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Z )
=> ( ord_less_eq_real @ X @ ( inf_inf_real @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_568_inf_Oorder__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B5: set_set_a] :
( A4
= ( inf_inf_set_set_a @ A4 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_569_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B5: set_a] :
( A4
= ( inf_inf_set_a @ A4 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_570_inf_Oorder__iff,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A4: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( A4
= ( inf_in8905007599844390133od_a_a @ A4 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_571_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] :
( A4
= ( inf_inf_nat @ A4 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_572_inf_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B5: real] :
( A4
= ( inf_inf_real @ A4 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_573_inf_Ocobounded1,axiom,
! [A: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_574_inf_Ocobounded1,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_575_inf_Ocobounded1,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_576_inf_Ocobounded1,axiom,
! [A: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_577_inf_Ocobounded1,axiom,
! [A: real,B2: real] : ( ord_less_eq_real @ ( inf_inf_real @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_578_inf_Ocobounded2,axiom,
! [A: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_579_inf_Ocobounded2,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_580_inf_Ocobounded2,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_581_inf_Ocobounded2,axiom,
! [A: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_582_inf_Ocobounded2,axiom,
! [A: real,B2: real] : ( ord_less_eq_real @ ( inf_inf_real @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_583_inf_Oabsorb__iff1,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B5: set_set_a] :
( ( inf_inf_set_set_a @ A4 @ B5 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_584_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B5: set_a] :
( ( inf_inf_set_a @ A4 @ B5 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_585_inf_Oabsorb__iff1,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A4: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A4 @ B5 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_586_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] :
( ( inf_inf_nat @ A4 @ B5 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_587_inf_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B5: real] :
( ( inf_inf_real @ A4 @ B5 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_588_inf_Oabsorb__iff2,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B5: set_set_a,A4: set_set_a] :
( ( inf_inf_set_set_a @ A4 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_589_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B5: set_a,A4: set_a] :
( ( inf_inf_set_a @ A4 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_590_inf_Oabsorb__iff2,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [B5: set_Product_prod_a_a,A4: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A4 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_591_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_592_inf_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [B5: real,A4: real] :
( ( inf_inf_real @ A4 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_593_inf_OcoboundedI1,axiom,
! [A: set_set_a,C: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_594_inf_OcoboundedI1,axiom,
! [A: set_a,C: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_595_inf_OcoboundedI1,axiom,
! [A: set_Product_prod_a_a,C: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ C )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_596_inf_OcoboundedI1,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_597_inf_OcoboundedI1,axiom,
! [A: real,C: real,B2: real] :
( ( ord_less_eq_real @ A @ C )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_598_inf_OcoboundedI2,axiom,
! [B2: set_set_a,C: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_599_inf_OcoboundedI2,axiom,
! [B2: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_600_inf_OcoboundedI2,axiom,
! [B2: set_Product_prod_a_a,C: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B2 @ C )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_601_inf_OcoboundedI2,axiom,
! [B2: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_602_inf_OcoboundedI2,axiom,
! [B2: real,C: real,A: real] :
( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_603_the__elem__eq,axiom,
! [X: a] :
( ( the_elem_a @ ( insert_a @ X @ bot_bot_set_a ) )
= X ) ).
% the_elem_eq
thf(fact_604_the__elem__eq,axiom,
! [X: set_a] :
( ( the_elem_set_a @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= X ) ).
% the_elem_eq
thf(fact_605_the__elem__eq,axiom,
! [X: product_prod_a_a] :
( ( the_el8589169208993665564od_a_a @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) )
= X ) ).
% the_elem_eq
thf(fact_606_finite__incident__loops,axiom,
! [V: a] : ( finite_finite_set_a @ ( undire4753905205749729249oops_a @ edges @ V ) ) ).
% finite_incident_loops
thf(fact_607_gnew_Ofinite__incident__loops,axiom,
! [V: a] : ( finite_finite_set_a @ ( undire4753905205749729249oops_a @ e @ V ) ) ).
% gnew.finite_incident_loops
thf(fact_608_order__refl,axiom,
! [X: set_set_a] : ( ord_le3724670747650509150_set_a @ X @ X ) ).
% order_refl
thf(fact_609_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_610_order__refl,axiom,
! [X: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ X @ X ) ).
% order_refl
thf(fact_611_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_612_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_613_dual__order_Orefl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_614_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_615_dual__order_Orefl,axiom,
! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A @ A ) ).
% dual_order.refl
thf(fact_616_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_617_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_618_is__singletonI,axiom,
! [X: a] : ( is_singleton_a @ ( insert_a @ X @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_619_is__singletonI,axiom,
! [X: set_a] : ( is_singleton_set_a @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ).
% is_singletonI
thf(fact_620_is__singletonI,axiom,
! [X: product_prod_a_a] : ( is_sin3171834905898671131od_a_a @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) ) ).
% is_singletonI
thf(fact_621_insert__subsetI,axiom,
! [X: nat,A2: set_nat,X5: set_nat] :
( ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ X5 @ A2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_622_insert__subsetI,axiom,
! [X: real,A2: set_real,X5: set_real] :
( ( member_real @ X @ A2 )
=> ( ( ord_less_eq_set_real @ X5 @ A2 )
=> ( ord_less_eq_set_real @ ( insert_real @ X @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_623_insert__subsetI,axiom,
! [X: set_a,A2: set_set_a,X5: set_set_a] :
( ( member_set_a @ X @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_624_insert__subsetI,axiom,
! [X: a,A2: set_a,X5: set_a] :
( ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ X5 @ A2 )
=> ( ord_less_eq_set_a @ ( insert_a @ X @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_625_insert__subsetI,axiom,
! [X: product_prod_a_a,A2: set_Product_prod_a_a,X5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ X5 @ A2 )
=> ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ X @ X5 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_626_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X4: nat] :
~ ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_627_subset__emptyI,axiom,
! [A2: set_real] :
( ! [X4: real] :
~ ( member_real @ X4 @ A2 )
=> ( ord_less_eq_set_real @ A2 @ bot_bot_set_real ) ) ).
% subset_emptyI
thf(fact_628_subset__emptyI,axiom,
! [A2: set_set_a] :
( ! [X4: set_a] :
~ ( member_set_a @ X4 @ A2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a ) ) ).
% subset_emptyI
thf(fact_629_subset__emptyI,axiom,
! [A2: set_a] :
( ! [X4: a] :
~ ( member_a @ X4 @ A2 )
=> ( ord_less_eq_set_a @ A2 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_630_subset__emptyI,axiom,
! [A2: set_Product_prod_a_a] :
( ! [X4: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X4 @ A2 )
=> ( ord_le746702958409616551od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) ).
% subset_emptyI
thf(fact_631_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_632_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_633_is__singleton__the__elem,axiom,
( is_sin3171834905898671131od_a_a
= ( ^ [A3: set_Product_prod_a_a] :
( A3
= ( insert4534936382041156343od_a_a @ ( the_el8589169208993665564od_a_a @ A3 ) @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_634_is__singletonI_H,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X4: nat,Y5: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( member_nat @ Y5 @ A2 )
=> ( X4 = Y5 ) ) )
=> ( is_singleton_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_635_is__singletonI_H,axiom,
! [A2: set_real] :
( ( A2 != bot_bot_set_real )
=> ( ! [X4: real,Y5: real] :
( ( member_real @ X4 @ A2 )
=> ( ( member_real @ Y5 @ A2 )
=> ( X4 = Y5 ) ) )
=> ( is_singleton_real @ A2 ) ) ) ).
% is_singletonI'
thf(fact_636_is__singletonI_H,axiom,
! [A2: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ! [X4: a,Y5: a] :
( ( member_a @ X4 @ A2 )
=> ( ( member_a @ Y5 @ A2 )
=> ( X4 = Y5 ) ) )
=> ( is_singleton_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_637_is__singletonI_H,axiom,
! [A2: set_set_a] :
( ( A2 != bot_bot_set_set_a )
=> ( ! [X4: set_a,Y5: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ( ( member_set_a @ Y5 @ A2 )
=> ( X4 = Y5 ) ) )
=> ( is_singleton_set_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_638_is__singletonI_H,axiom,
! [A2: set_Product_prod_a_a] :
( ( A2 != bot_bo3357376287454694259od_a_a )
=> ( ! [X4: product_prod_a_a,Y5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A2 )
=> ( ( member1426531477525435216od_a_a @ Y5 @ A2 )
=> ( X4 = Y5 ) ) )
=> ( is_sin3171834905898671131od_a_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_639_order__antisym__conv,axiom,
! [Y: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_640_order__antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_641_order__antisym__conv,axiom,
! [Y: set_Product_prod_a_a,X: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y @ X )
=> ( ( ord_le746702958409616551od_a_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_642_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_643_order__antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_644_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_645_linorder__le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_646_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_647_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_648_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_649_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_650_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_651_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > real,C: real] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_652_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_653_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > set_a,C: set_a] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_654_ord__le__eq__subst,axiom,
! [A: set_set_a,B2: set_set_a,F: set_set_a > nat,C: nat] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_set_a,Y5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_655_ord__le__eq__subst,axiom,
! [A: set_set_a,B2: set_set_a,F: set_set_a > real,C: real] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_set_a,Y5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_656_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_657_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_658_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_659_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_660_ord__eq__le__subst,axiom,
! [A: nat,F: set_a > nat,B2: set_a,C: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_661_ord__eq__le__subst,axiom,
! [A: real,F: set_a > real,B2: set_a,C: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_662_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_663_ord__eq__le__subst,axiom,
! [A: set_a,F: real > set_a,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_664_ord__eq__le__subst,axiom,
! [A: nat,F: set_set_a > nat,B2: set_set_a,C: set_set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X4: set_set_a,Y5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_665_ord__eq__le__subst,axiom,
! [A: real,F: set_set_a > real,B2: set_set_a,C: set_set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X4: set_set_a,Y5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_666_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_667_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_668_order__eq__refl,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( X = Y )
=> ( ord_le3724670747650509150_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_669_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_670_order__eq__refl,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( X = Y )
=> ( ord_le746702958409616551od_a_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_671_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_672_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_673_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_674_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_675_order__subst2,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_676_order__subst2,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_677_order__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_678_order__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > real,C: real] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_679_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_680_order__subst2,axiom,
! [A: real,B2: real,F: real > set_a,C: set_a] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_681_order__subst2,axiom,
! [A: set_set_a,B2: set_set_a,F: set_set_a > nat,C: nat] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: set_set_a,Y5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_682_order__subst2,axiom,
! [A: set_set_a,B2: set_set_a,F: set_set_a > real,C: real] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X4: set_set_a,Y5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_683_order__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_684_order__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_685_order__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_686_order__subst1,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_687_order__subst1,axiom,
! [A: set_a,F: nat > set_a,B2: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_688_order__subst1,axiom,
! [A: set_a,F: real > set_a,B2: real,C: real] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_689_order__subst1,axiom,
! [A: nat,F: set_a > nat,B2: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_690_order__subst1,axiom,
! [A: real,F: set_a > real,B2: set_a,C: set_a] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X4: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y5 )
=> ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_691_order__subst1,axiom,
! [A: set_set_a,F: nat > set_set_a,B2: nat,C: nat] :
( ( ord_le3724670747650509150_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_692_order__subst1,axiom,
! [A: set_set_a,F: real > set_set_a,B2: real,C: real] :
( ( ord_le3724670747650509150_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_693_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_a,Z2: set_set_a] : ( Y4 = Z2 ) )
= ( ^ [A4: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_694_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [A4: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_695_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_a_a,Z2: set_Product_prod_a_a] : ( Y4 = Z2 ) )
= ( ^ [A4: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A4 @ B5 )
& ( ord_le746702958409616551od_a_a @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_696_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
& ( ord_less_eq_nat @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_697_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z2: real] : ( Y4 = Z2 ) )
= ( ^ [A4: real,B5: real] :
( ( ord_less_eq_real @ A4 @ B5 )
& ( ord_less_eq_real @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_698_antisym,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_699_antisym,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_700_antisym,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_701_antisym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_702_antisym,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_703_dual__order_Otrans,axiom,
! [B2: set_set_a,A: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( ( ord_le3724670747650509150_set_a @ C @ B2 )
=> ( ord_le3724670747650509150_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_704_dual__order_Otrans,axiom,
! [B2: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_705_dual__order_Otrans,axiom,
! [B2: set_Product_prod_a_a,A: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B2 @ A )
=> ( ( ord_le746702958409616551od_a_a @ C @ B2 )
=> ( ord_le746702958409616551od_a_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_706_dual__order_Otrans,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_707_dual__order_Otrans,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_708_dual__order_Oantisym,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_709_dual__order_Oantisym,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_710_dual__order_Oantisym,axiom,
! [B2: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B2 @ A )
=> ( ( ord_le746702958409616551od_a_a @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_711_dual__order_Oantisym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_712_dual__order_Oantisym,axiom,
! [B2: real,A: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_713_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_set_a,Z2: set_set_a] : ( Y4 = Z2 ) )
= ( ^ [A4: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ A4 )
& ( ord_le3724670747650509150_set_a @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_714_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [A4: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_715_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Product_prod_a_a,Z2: set_Product_prod_a_a] : ( Y4 = Z2 ) )
= ( ^ [A4: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ B5 @ A4 )
& ( ord_le746702958409616551od_a_a @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_716_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A4 )
& ( ord_less_eq_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_717_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: real,Z2: real] : ( Y4 = Z2 ) )
= ( ^ [A4: real,B5: real] :
( ( ord_less_eq_real @ B5 @ A4 )
& ( ord_less_eq_real @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_718_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: nat,B6: nat] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_719_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B2: real] :
( ! [A5: real,B6: real] :
( ( ord_less_eq_real @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: real,B6: real] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_720_order__trans,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ Z )
=> ( ord_le3724670747650509150_set_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_721_order__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_722_order__trans,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ Y )
=> ( ( ord_le746702958409616551od_a_a @ Y @ Z )
=> ( ord_le746702958409616551od_a_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_723_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_724_order__trans,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_eq_real @ X @ Z ) ) ) ).
% order_trans
thf(fact_725_order_Otrans,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_726_order_Otrans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_727_order_Otrans,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ C )
=> ( ord_le746702958409616551od_a_a @ A @ C ) ) ) ).
% order.trans
thf(fact_728_order_Otrans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_729_order_Otrans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_730_order__antisym,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_731_order__antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_732_order__antisym,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X @ Y )
=> ( ( ord_le746702958409616551od_a_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_733_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_734_order__antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_735_ord__le__eq__trans,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_736_ord__le__eq__trans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_737_ord__le__eq__trans,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le746702958409616551od_a_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_738_ord__le__eq__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_739_ord__le__eq__trans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_740_ord__eq__le__trans,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( A = B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_741_ord__eq__le__trans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( A = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_742_ord__eq__le__trans,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( A = B2 )
=> ( ( ord_le746702958409616551od_a_a @ B2 @ C )
=> ( ord_le746702958409616551od_a_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_743_ord__eq__le__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_744_ord__eq__le__trans,axiom,
! [A: real,B2: real,C: real] :
( ( A = B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_745_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_a,Z2: set_set_a] : ( Y4 = Z2 ) )
= ( ^ [X3: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y3 )
& ( ord_le3724670747650509150_set_a @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_746_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
& ( ord_less_eq_set_a @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_747_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_a_a,Z2: set_Product_prod_a_a] : ( Y4 = Z2 ) )
= ( ^ [X3: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ X3 @ Y3 )
& ( ord_le746702958409616551od_a_a @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_748_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_749_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z2: real] : ( Y4 = Z2 ) )
= ( ^ [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
& ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_750_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_751_le__cases3,axiom,
! [X: real,Y: real,Z: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z ) )
=> ( ( ( ord_less_eq_real @ X @ Z )
=> ~ ( ord_less_eq_real @ Z @ Y ) )
=> ( ( ( ord_less_eq_real @ Z @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z )
=> ~ ( ord_less_eq_real @ Z @ X ) )
=> ~ ( ( ord_less_eq_real @ Z @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_752_nle__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_753_nle__le,axiom,
! [A: real,B2: real] :
( ( ~ ( ord_less_eq_real @ A @ B2 ) )
= ( ( ord_less_eq_real @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_754_is__singletonE,axiom,
! [A2: set_a] :
( ( is_singleton_a @ A2 )
=> ~ ! [X4: a] :
( A2
!= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_755_is__singletonE,axiom,
! [A2: set_set_a] :
( ( is_singleton_set_a @ A2 )
=> ~ ! [X4: set_a] :
( A2
!= ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) ) ).
% is_singletonE
thf(fact_756_is__singletonE,axiom,
! [A2: set_Product_prod_a_a] :
( ( is_sin3171834905898671131od_a_a @ A2 )
=> ~ ! [X4: product_prod_a_a] :
( A2
!= ( insert4534936382041156343od_a_a @ X4 @ bot_bo3357376287454694259od_a_a ) ) ) ).
% is_singletonE
thf(fact_757_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A3: set_a] :
? [X3: a] :
( A3
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_758_is__singleton__def,axiom,
( is_singleton_set_a
= ( ^ [A3: set_set_a] :
? [X3: set_a] :
( A3
= ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_759_is__singleton__def,axiom,
( is_sin3171834905898671131od_a_a
= ( ^ [A3: set_Product_prod_a_a] :
? [X3: product_prod_a_a] :
( A3
= ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) ) ) ) ).
% is_singleton_def
thf(fact_760_bot_Oextremum,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% bot.extremum
thf(fact_761_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_762_bot_Oextremum,axiom,
! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ bot_bo3357376287454694259od_a_a @ A ) ).
% bot.extremum
thf(fact_763_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_764_bot_Oextremum__unique,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% bot.extremum_unique
thf(fact_765_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_766_bot_Oextremum__unique,axiom,
! [A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ bot_bo3357376287454694259od_a_a )
= ( A = bot_bo3357376287454694259od_a_a ) ) ).
% bot.extremum_unique
thf(fact_767_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_768_bot_Oextremum__uniqueI,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
=> ( A = bot_bot_set_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_769_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_770_bot_Oextremum__uniqueI,axiom,
! [A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A @ bot_bo3357376287454694259od_a_a )
=> ( A = bot_bo3357376287454694259od_a_a ) ) ).
% bot.extremum_uniqueI
thf(fact_771_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_772_finite__Int,axiom,
! [F2: set_Product_prod_a_a,G: set_Product_prod_a_a] :
( ( ( finite6544458595007987280od_a_a @ F2 )
| ( finite6544458595007987280od_a_a @ G ) )
=> ( finite6544458595007987280od_a_a @ ( inf_in8905007599844390133od_a_a @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_773_finite__Int,axiom,
! [F2: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_774_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_775_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_776_finite__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( finite_finite_set_a @ ( insert_set_a @ A @ A2 ) )
= ( finite_finite_set_a @ A2 ) ) ).
% finite_insert
thf(fact_777_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_778_finite__insert,axiom,
! [A: product_prod_a_a,A2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ ( insert4534936382041156343od_a_a @ A @ A2 ) )
= ( finite6544458595007987280od_a_a @ A2 ) ) ).
% finite_insert
thf(fact_779_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_780_finite__inc__sedges,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( finite_finite_set_a @ ( undire1270416042309875431dges_a @ edges @ V ) ) ) ).
% finite_inc_sedges
thf(fact_781_gnew_Ofinite__inc__sedges,axiom,
! [V: a] :
( ( finite_finite_set_a @ e )
=> ( finite_finite_set_a @ ( undire1270416042309875431dges_a @ e @ V ) ) ) ).
% gnew.finite_inc_sedges
thf(fact_782_finite__subset__induct,axiom,
! [F2: set_real,A2: set_real,P: set_real > $o] :
( ( finite_finite_real @ F2 )
=> ( ( ord_less_eq_set_real @ F2 @ A2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [A5: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ( member_real @ A5 @ A2 )
=> ( ~ ( member_real @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_783_finite__subset__induct,axiom,
! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A5: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A5 @ A2 )
=> ( ~ ( member_nat @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_784_finite__subset__induct,axiom,
! [F2: set_set_a,A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A5: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A5 @ A2 )
=> ( ~ ( member_set_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_785_finite__subset__induct,axiom,
! [F2: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A5 @ A2 )
=> ( ~ ( member_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_786_finite__subset__induct,axiom,
! [F2: set_Product_prod_a_a,A2: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ F2 )
=> ( ( ord_le746702958409616551od_a_a @ F2 @ A2 )
=> ( ( P @ bot_bo3357376287454694259od_a_a )
=> ( ! [A5: product_prod_a_a,F3: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F3 )
=> ( ( member1426531477525435216od_a_a @ A5 @ A2 )
=> ( ~ ( member1426531477525435216od_a_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert4534936382041156343od_a_a @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_787_finite__subset__induct_H,axiom,
! [F2: set_real,A2: set_real,P: set_real > $o] :
( ( finite_finite_real @ F2 )
=> ( ( ord_less_eq_set_real @ F2 @ A2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [A5: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ( member_real @ A5 @ A2 )
=> ( ( ord_less_eq_set_real @ F3 @ A2 )
=> ( ~ ( member_real @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_788_finite__subset__induct_H,axiom,
! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A5: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A5 @ A2 )
=> ( ( ord_less_eq_set_nat @ F3 @ A2 )
=> ( ~ ( member_nat @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_789_finite__subset__induct_H,axiom,
! [F2: set_set_a,A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A5: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A5 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ F3 @ A2 )
=> ( ~ ( member_set_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_790_finite__subset__induct_H,axiom,
! [F2: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A5 @ A2 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ~ ( member_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_791_finite__subset__induct_H,axiom,
! [F2: set_Product_prod_a_a,A2: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ F2 )
=> ( ( ord_le746702958409616551od_a_a @ F2 @ A2 )
=> ( ( P @ bot_bo3357376287454694259od_a_a )
=> ( ! [A5: product_prod_a_a,F3: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F3 )
=> ( ( member1426531477525435216od_a_a @ A5 @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ F3 @ A2 )
=> ( ~ ( member1426531477525435216od_a_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert4534936382041156343od_a_a @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_792_finite__ranking__induct,axiom,
! [S: set_real,P: set_real > $o,F: real > nat] :
( ( finite_finite_real @ S )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X4: real,S2: set_real] :
( ( finite_finite_real @ S2 )
=> ( ! [Y6: real] :
( ( member_real @ Y6 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_real @ X4 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_793_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X4: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y6: nat] :
( ( member_nat @ Y6 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat @ X4 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_794_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X4: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y6: a] :
( ( member_a @ Y6 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X4 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_795_finite__ranking__induct,axiom,
! [S: set_set_a,P: set_set_a > $o,F: set_a > nat] :
( ( finite_finite_set_a @ S )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X4: set_a,S2: set_set_a] :
( ( finite_finite_set_a @ S2 )
=> ( ! [Y6: set_a] :
( ( member_set_a @ Y6 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_set_a @ X4 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_796_finite__ranking__induct,axiom,
! [S: set_Product_prod_a_a,P: set_Product_prod_a_a > $o,F: product_prod_a_a > nat] :
( ( finite6544458595007987280od_a_a @ S )
=> ( ( P @ bot_bo3357376287454694259od_a_a )
=> ( ! [X4: product_prod_a_a,S2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ S2 )
=> ( ! [Y6: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y6 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert4534936382041156343od_a_a @ X4 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_797_finite__ranking__induct,axiom,
! [S: set_real,P: set_real > $o,F: real > real] :
( ( finite_finite_real @ S )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X4: real,S2: set_real] :
( ( finite_finite_real @ S2 )
=> ( ! [Y6: real] :
( ( member_real @ Y6 @ S2 )
=> ( ord_less_eq_real @ ( F @ Y6 ) @ ( F @ X4 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_real @ X4 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_798_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X4: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y6: nat] :
( ( member_nat @ Y6 @ S2 )
=> ( ord_less_eq_real @ ( F @ Y6 ) @ ( F @ X4 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat @ X4 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_799_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F: a > real] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X4: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y6: a] :
( ( member_a @ Y6 @ S2 )
=> ( ord_less_eq_real @ ( F @ Y6 ) @ ( F @ X4 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X4 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_800_finite__ranking__induct,axiom,
! [S: set_set_a,P: set_set_a > $o,F: set_a > real] :
( ( finite_finite_set_a @ S )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X4: set_a,S2: set_set_a] :
( ( finite_finite_set_a @ S2 )
=> ( ! [Y6: set_a] :
( ( member_set_a @ Y6 @ S2 )
=> ( ord_less_eq_real @ ( F @ Y6 ) @ ( F @ X4 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_set_a @ X4 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_801_finite__ranking__induct,axiom,
! [S: set_Product_prod_a_a,P: set_Product_prod_a_a > $o,F: product_prod_a_a > real] :
( ( finite6544458595007987280od_a_a @ S )
=> ( ( P @ bot_bo3357376287454694259od_a_a )
=> ( ! [X4: product_prod_a_a,S2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ S2 )
=> ( ! [Y6: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y6 @ S2 )
=> ( ord_less_eq_real @ ( F @ Y6 ) @ ( F @ X4 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert4534936382041156343od_a_a @ X4 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_802_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A6: set_nat] :
( ? [A5: nat] :
( A
= ( insert_nat @ A5 @ A6 ) )
=> ~ ( finite_finite_nat @ A6 ) ) ) ) ).
% finite.cases
thf(fact_803_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A6: set_a] :
( ? [A5: a] :
( A
= ( insert_a @ A5 @ A6 ) )
=> ~ ( finite_finite_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_804_finite_Ocases,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ~ ! [A6: set_set_a] :
( ? [A5: set_a] :
( A
= ( insert_set_a @ A5 @ A6 ) )
=> ~ ( finite_finite_set_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_805_finite_Ocases,axiom,
! [A: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A )
=> ( ( A != bot_bo3357376287454694259od_a_a )
=> ~ ! [A6: set_Product_prod_a_a] :
( ? [A5: product_prod_a_a] :
( A
= ( insert4534936382041156343od_a_a @ A5 @ A6 ) )
=> ~ ( finite6544458595007987280od_a_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_806_ulgraph_Oincident__sedges_Ocong,axiom,
undire1270416042309875431dges_a = undire1270416042309875431dges_a ).
% ulgraph.incident_sedges.cong
thf(fact_807_finite__has__minimal2,axiom,
! [A2: set_set_set_a,A: set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( member_set_set_a @ A @ A2 )
=> ? [X4: set_set_a] :
( ( member_set_set_a @ X4 @ A2 )
& ( ord_le3724670747650509150_set_a @ X4 @ A )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_808_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
& ( ord_less_eq_set_a @ X4 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_809_finite__has__minimal2,axiom,
! [A2: set_se5735800977113168103od_a_a,A: set_Product_prod_a_a] :
( ( finite8717734299975451184od_a_a @ A2 )
=> ( ( member1816616512716248880od_a_a @ A @ A2 )
=> ? [X4: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X4 @ A2 )
& ( ord_le746702958409616551od_a_a @ X4 @ A )
& ! [Xa: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ Xa @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_810_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_nat @ X4 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_811_finite__has__minimal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ord_less_eq_real @ X4 @ A )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_812_finite__has__maximal2,axiom,
! [A2: set_set_set_a,A: set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( member_set_set_a @ A @ A2 )
=> ? [X4: set_set_a] :
( ( member_set_set_a @ X4 @ A2 )
& ( ord_le3724670747650509150_set_a @ A @ X4 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_813_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
& ( ord_less_eq_set_a @ A @ X4 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_814_finite__has__maximal2,axiom,
! [A2: set_se5735800977113168103od_a_a,A: set_Product_prod_a_a] :
( ( finite8717734299975451184od_a_a @ A2 )
=> ( ( member1816616512716248880od_a_a @ A @ A2 )
=> ? [X4: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X4 @ A2 )
& ( ord_le746702958409616551od_a_a @ A @ X4 )
& ! [Xa: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ Xa @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_815_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_nat @ A @ X4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_816_finite__has__maximal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ord_less_eq_real @ A @ X4 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_817_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_818_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_819_finite_OemptyI,axiom,
finite_finite_set_a @ bot_bot_set_set_a ).
% finite.emptyI
thf(fact_820_finite_OemptyI,axiom,
finite6544458595007987280od_a_a @ bot_bo3357376287454694259od_a_a ).
% finite.emptyI
thf(fact_821_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_822_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_823_infinite__imp__nonempty,axiom,
! [S: set_set_a] :
( ~ ( finite_finite_set_a @ S )
=> ( S != bot_bot_set_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_824_infinite__imp__nonempty,axiom,
! [S: set_Product_prod_a_a] :
( ~ ( finite6544458595007987280od_a_a @ S )
=> ( S != bot_bo3357376287454694259od_a_a ) ) ).
% infinite_imp_nonempty
thf(fact_825_rev__finite__subset,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_826_rev__finite__subset,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_827_rev__finite__subset,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_828_rev__finite__subset,axiom,
! [B: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ B )
=> ( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( finite6544458595007987280od_a_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_829_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_830_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_831_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_832_infinite__super,axiom,
! [S: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ S @ T2 )
=> ( ~ ( finite6544458595007987280od_a_a @ S )
=> ~ ( finite6544458595007987280od_a_a @ T2 ) ) ) ).
% infinite_super
thf(fact_833_finite__subset,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_834_finite__subset,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( finite_finite_set_a @ B )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% finite_subset
thf(fact_835_finite__subset,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_836_finite__subset,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B )
=> ( ( finite6544458595007987280od_a_a @ B )
=> ( finite6544458595007987280od_a_a @ A2 ) ) ) ).
% finite_subset
thf(fact_837_finite_OinsertI,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( finite_finite_set_a @ ( insert_set_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_838_finite_OinsertI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_839_finite_OinsertI,axiom,
! [A2: set_Product_prod_a_a,A: product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A2 )
=> ( finite6544458595007987280od_a_a @ ( insert4534936382041156343od_a_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_840_finite_OinsertI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_841_finite__has__minimal,axiom,
! [A2: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( A2 != bot_bo3380559777022489994_set_a )
=> ? [X4: set_set_a] :
( ( member_set_set_a @ X4 @ A2 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_842_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_843_finite__has__minimal,axiom,
! [A2: set_se5735800977113168103od_a_a] :
( ( finite8717734299975451184od_a_a @ A2 )
=> ( ( A2 != bot_bo777872063958040403od_a_a )
=> ? [X4: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X4 @ A2 )
& ! [Xa: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ Xa @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_844_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_845_finite__has__minimal,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_846_finite__has__maximal,axiom,
! [A2: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( A2 != bot_bo3380559777022489994_set_a )
=> ? [X4: set_set_a] :
( ( member_set_set_a @ X4 @ A2 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_847_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_848_finite__has__maximal,axiom,
! [A2: set_se5735800977113168103od_a_a] :
( ( finite8717734299975451184od_a_a @ A2 )
=> ( ( A2 != bot_bo777872063958040403od_a_a )
=> ? [X4: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X4 @ A2 )
& ! [Xa: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ Xa @ A2 )
=> ( ( ord_le746702958409616551od_a_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_849_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_850_finite__has__maximal,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_851_infinite__finite__induct,axiom,
! [P: set_real > $o,A2: set_real] :
( ! [A6: set_real] :
( ~ ( finite_finite_real @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X4: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ~ ( member_real @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ X4 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_852_infinite__finite__induct,axiom,
! [P: set_nat > $o,A2: set_nat] :
( ! [A6: set_nat] :
( ~ ( finite_finite_nat @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X4 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_853_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A6: set_a] :
( ~ ( finite_finite_a @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X4: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X4 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_854_infinite__finite__induct,axiom,
! [P: set_set_a > $o,A2: set_set_a] :
( ! [A6: set_set_a] :
( ~ ( finite_finite_set_a @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X4: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X4 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_855_infinite__finite__induct,axiom,
! [P: set_Product_prod_a_a > $o,A2: set_Product_prod_a_a] :
( ! [A6: set_Product_prod_a_a] :
( ~ ( finite6544458595007987280od_a_a @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bo3357376287454694259od_a_a )
=> ( ! [X4: product_prod_a_a,F3: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F3 )
=> ( ~ ( member1426531477525435216od_a_a @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert4534936382041156343od_a_a @ X4 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_856_finite__ne__induct,axiom,
! [F2: set_real,P: set_real > $o] :
( ( finite_finite_real @ F2 )
=> ( ( F2 != bot_bot_set_real )
=> ( ! [X4: real] : ( P @ ( insert_real @ X4 @ bot_bot_set_real ) )
=> ( ! [X4: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ( F3 != bot_bot_set_real )
=> ( ~ ( member_real @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ X4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_857_finite__ne__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X4: nat] : ( P @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
=> ( ! [X4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_858_finite__ne__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ! [X4: a] : ( P @ ( insert_a @ X4 @ bot_bot_set_a ) )
=> ( ! [X4: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_859_finite__ne__induct,axiom,
! [F2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( F2 != bot_bot_set_set_a )
=> ( ! [X4: set_a] : ( P @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) )
=> ( ! [X4: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( F3 != bot_bot_set_set_a )
=> ( ~ ( member_set_a @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_860_finite__ne__induct,axiom,
! [F2: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ F2 )
=> ( ( F2 != bot_bo3357376287454694259od_a_a )
=> ( ! [X4: product_prod_a_a] : ( P @ ( insert4534936382041156343od_a_a @ X4 @ bot_bo3357376287454694259od_a_a ) )
=> ( ! [X4: product_prod_a_a,F3: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F3 )
=> ( ( F3 != bot_bo3357376287454694259od_a_a )
=> ( ~ ( member1426531477525435216od_a_a @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert4534936382041156343od_a_a @ X4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_861_finite__induct,axiom,
! [F2: set_real,P: set_real > $o] :
( ( finite_finite_real @ F2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [X4: real,F3: set_real] :
( ( finite_finite_real @ F3 )
=> ( ~ ( member_real @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_real @ X4 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_862_finite__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X4 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_863_finite__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X4: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X4 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_864_finite__induct,axiom,
! [F2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X4: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X4 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_865_finite__induct,axiom,
! [F2: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ F2 )
=> ( ( P @ bot_bo3357376287454694259od_a_a )
=> ( ! [X4: product_prod_a_a,F3: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F3 )
=> ( ~ ( member1426531477525435216od_a_a @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert4534936382041156343od_a_a @ X4 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_866_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A4: set_nat] :
( ( A4 = bot_bot_set_nat )
| ? [A3: set_nat,B5: nat] :
( ( A4
= ( insert_nat @ B5 @ A3 ) )
& ( finite_finite_nat @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_867_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A4: set_a] :
( ( A4 = bot_bot_set_a )
| ? [A3: set_a,B5: a] :
( ( A4
= ( insert_a @ B5 @ A3 ) )
& ( finite_finite_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_868_finite_Osimps,axiom,
( finite_finite_set_a
= ( ^ [A4: set_set_a] :
( ( A4 = bot_bot_set_set_a )
| ? [A3: set_set_a,B5: set_a] :
( ( A4
= ( insert_set_a @ B5 @ A3 ) )
& ( finite_finite_set_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_869_finite_Osimps,axiom,
( finite6544458595007987280od_a_a
= ( ^ [A4: set_Product_prod_a_a] :
( ( A4 = bot_bo3357376287454694259od_a_a )
| ? [A3: set_Product_prod_a_a,B5: product_prod_a_a] :
( ( A4
= ( insert4534936382041156343od_a_a @ B5 @ A3 ) )
& ( finite6544458595007987280od_a_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_870_finite__incident__edges,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( finite_finite_set_a @ ( undire3231912044278729248dges_a @ edges @ V ) ) ) ).
% finite_incident_edges
thf(fact_871_gnew_Ofinite__incident__edges,axiom,
! [V: a] :
( ( finite_finite_set_a @ e )
=> ( finite_finite_set_a @ ( undire3231912044278729248dges_a @ e @ V ) ) ) ).
% gnew.finite_incident_edges
thf(fact_872_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_873_bot__empty__eq,axiom,
( bot_bot_real_o
= ( ^ [X3: real] : ( member_real @ X3 @ bot_bot_set_real ) ) ) ).
% bot_empty_eq
thf(fact_874_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_875_bot__empty__eq,axiom,
( bot_bot_set_a_o
= ( ^ [X3: set_a] : ( member_set_a @ X3 @ bot_bot_set_set_a ) ) ) ).
% bot_empty_eq
thf(fact_876_bot__empty__eq,axiom,
( bot_bo4160289986317612842_a_a_o
= ( ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) ) ) ).
% bot_empty_eq
thf(fact_877_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_878_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_879_Collect__empty__eq__bot,axiom,
! [P: product_prod_a_a > $o] :
( ( ( collec3336397797384452498od_a_a @ P )
= bot_bo3357376287454694259od_a_a )
= ( P = bot_bo4160289986317612842_a_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_880_arg__min__least,axiom,
! [S: set_real,Y: real,F: real > nat] :
( ( finite_finite_real @ S )
=> ( ( S != bot_bot_set_real )
=> ( ( member_real @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic5055836439445974935al_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_881_arg__min__least,axiom,
! [S: set_nat,Y: nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_882_arg__min__least,axiom,
! [S: set_a,Y: a,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ( ( member_a @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic6340287419671400565_a_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_883_arg__min__least,axiom,
! [S: set_set_a,Y: set_a,F: set_a > nat] :
( ( finite_finite_set_a @ S )
=> ( ( S != bot_bot_set_set_a )
=> ( ( member_set_a @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic4678118661306933717_a_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_884_arg__min__least,axiom,
! [S: set_Product_prod_a_a,Y: product_prod_a_a,F: product_prod_a_a > nat] :
( ( finite6544458595007987280od_a_a @ S )
=> ( ( S != bot_bo3357376287454694259od_a_a )
=> ( ( member1426531477525435216od_a_a @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic806887198133436574_a_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_885_arg__min__least,axiom,
! [S: set_real,Y: real,F: real > real] :
( ( finite_finite_real @ S )
=> ( ( S != bot_bot_set_real )
=> ( ( member_real @ Y @ S )
=> ( ord_less_eq_real @ ( F @ ( lattic8440615504127631091l_real @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_886_arg__min__least,axiom,
! [S: set_nat,Y: nat,F: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S )
=> ( ord_less_eq_real @ ( F @ ( lattic488527866317076247t_real @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_887_arg__min__least,axiom,
! [S: set_a,Y: a,F: a > real] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ( ( member_a @ Y @ S )
=> ( ord_less_eq_real @ ( F @ ( lattic7288945864786915537a_real @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_888_arg__min__least,axiom,
! [S: set_set_a,Y: set_a,F: set_a > real] :
( ( finite_finite_set_a @ S )
=> ( ( S != bot_bot_set_set_a )
=> ( ( member_set_a @ Y @ S )
=> ( ord_less_eq_real @ ( F @ ( lattic8696895084918575665a_real @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_889_arg__min__least,axiom,
! [S: set_Product_prod_a_a,Y: product_prod_a_a,F: product_prod_a_a > real] :
( ( finite6544458595007987280od_a_a @ S )
=> ( ( S != bot_bo3357376287454694259od_a_a )
=> ( ( member1426531477525435216od_a_a @ Y @ S )
=> ( ord_less_eq_real @ ( F @ ( lattic4338785915121900666a_real @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_890_finite__transitivity__chain,axiom,
! [A2: set_real,R: real > real > $o] :
( ( finite_finite_real @ A2 )
=> ( ! [X4: real] :
~ ( R @ X4 @ X4 )
=> ( ! [X4: real,Y5: real,Z4: real] :
( ( R @ X4 @ Y5 )
=> ( ( R @ Y5 @ Z4 )
=> ( R @ X4 @ Z4 ) ) )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ? [Y6: real] :
( ( member_real @ Y6 @ A2 )
& ( R @ X4 @ Y6 ) ) )
=> ( A2 = bot_bot_set_real ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_891_finite__transitivity__chain,axiom,
! [A2: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [X4: nat] :
~ ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y5: nat,Z4: nat] :
( ( R @ X4 @ Y5 )
=> ( ( R @ Y5 @ Z4 )
=> ( R @ X4 @ Z4 ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ? [Y6: nat] :
( ( member_nat @ Y6 @ A2 )
& ( R @ X4 @ Y6 ) ) )
=> ( A2 = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_892_finite__transitivity__chain,axiom,
! [A2: set_a,R: a > a > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [X4: a] :
~ ( R @ X4 @ X4 )
=> ( ! [X4: a,Y5: a,Z4: a] :
( ( R @ X4 @ Y5 )
=> ( ( R @ Y5 @ Z4 )
=> ( R @ X4 @ Z4 ) ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ? [Y6: a] :
( ( member_a @ Y6 @ A2 )
& ( R @ X4 @ Y6 ) ) )
=> ( A2 = bot_bot_set_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_893_finite__transitivity__chain,axiom,
! [A2: set_set_a,R: set_a > set_a > $o] :
( ( finite_finite_set_a @ A2 )
=> ( ! [X4: set_a] :
~ ( R @ X4 @ X4 )
=> ( ! [X4: set_a,Y5: set_a,Z4: set_a] :
( ( R @ X4 @ Y5 )
=> ( ( R @ Y5 @ Z4 )
=> ( R @ X4 @ Z4 ) ) )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ? [Y6: set_a] :
( ( member_set_a @ Y6 @ A2 )
& ( R @ X4 @ Y6 ) ) )
=> ( A2 = bot_bot_set_set_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_894_finite__transitivity__chain,axiom,
! [A2: set_Product_prod_a_a,R: product_prod_a_a > product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ A2 )
=> ( ! [X4: product_prod_a_a] :
~ ( R @ X4 @ X4 )
=> ( ! [X4: product_prod_a_a,Y5: product_prod_a_a,Z4: product_prod_a_a] :
( ( R @ X4 @ Y5 )
=> ( ( R @ Y5 @ Z4 )
=> ( R @ X4 @ Z4 ) ) )
=> ( ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A2 )
=> ? [Y6: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y6 @ A2 )
& ( R @ X4 @ Y6 ) ) )
=> ( A2 = bot_bo3357376287454694259od_a_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_895_all__edges__between__mono2,axiom,
! [Y7: set_a,Z5: set_a,X5: set_a] :
( ( ord_less_eq_set_a @ Y7 @ Z5 )
=> ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y7 ) @ ( undire8383842906760478443ween_a @ edges @ X5 @ Z5 ) ) ) ).
% all_edges_between_mono2
thf(fact_896_gnew_Ofinite__all__edges__between,axiom,
! [X5: set_a,Y7: set_a] :
( ( finite_finite_a @ X5 )
=> ( ( finite_finite_a @ Y7 )
=> ( finite6544458595007987280od_a_a @ ( undire8383842906760478443ween_a @ e @ X5 @ Y7 ) ) ) ) ).
% gnew.finite_all_edges_between
thf(fact_897_finite__all__edges__between,axiom,
! [X5: set_a,Y7: set_a] :
( ( finite_finite_a @ X5 )
=> ( ( finite_finite_a @ Y7 )
=> ( finite6544458595007987280od_a_a @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y7 ) ) ) ) ).
% finite_all_edges_between
thf(fact_898_gnew_Oall__edges__between__mono2,axiom,
! [Y7: set_a,Z5: set_a,X5: set_a] :
( ( ord_less_eq_set_a @ Y7 @ Z5 )
=> ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ e @ X5 @ Y7 ) @ ( undire8383842906760478443ween_a @ e @ X5 @ Z5 ) ) ) ).
% gnew.all_edges_between_mono2
thf(fact_899_gnew_Oall__edges__between__mono1,axiom,
! [Y7: set_a,Z5: set_a,X5: set_a] :
( ( ord_less_eq_set_a @ Y7 @ Z5 )
=> ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ e @ Y7 @ X5 ) @ ( undire8383842906760478443ween_a @ e @ Z5 @ X5 ) ) ) ).
% gnew.all_edges_between_mono1
thf(fact_900_all__edges__between__mono1,axiom,
! [Y7: set_a,Z5: set_a,X5: set_a] :
( ( ord_less_eq_set_a @ Y7 @ Z5 )
=> ( ord_le746702958409616551od_a_a @ ( undire8383842906760478443ween_a @ edges @ Y7 @ X5 ) @ ( undire8383842906760478443ween_a @ edges @ Z5 @ X5 ) ) ) ).
% all_edges_between_mono1
thf(fact_901_gnew_Oall__edges__between__empty_I2_J,axiom,
! [Z5: set_a] :
( ( undire8383842906760478443ween_a @ e @ Z5 @ bot_bot_set_a )
= bot_bo3357376287454694259od_a_a ) ).
% gnew.all_edges_between_empty(2)
thf(fact_902_gnew_Oall__edges__between__empty_I1_J,axiom,
! [Z5: set_a] :
( ( undire8383842906760478443ween_a @ e @ bot_bot_set_a @ Z5 )
= bot_bo3357376287454694259od_a_a ) ).
% gnew.all_edges_between_empty(1)
thf(fact_903_all__edges__between__empty_I2_J,axiom,
! [Z5: set_a] :
( ( undire8383842906760478443ween_a @ edges @ Z5 @ bot_bot_set_a )
= bot_bo3357376287454694259od_a_a ) ).
% all_edges_between_empty(2)
thf(fact_904_all__edges__between__empty_I1_J,axiom,
! [Z5: set_a] :
( ( undire8383842906760478443ween_a @ edges @ bot_bot_set_a @ Z5 )
= bot_bo3357376287454694259od_a_a ) ).
% all_edges_between_empty(1)
thf(fact_905_gnew_Oincident__edges__sedges,axiom,
! [V: a] :
( ~ ( undire3617971648856834880loop_a @ e @ V )
=> ( ( undire3231912044278729248dges_a @ e @ V )
= ( undire1270416042309875431dges_a @ e @ V ) ) ) ).
% gnew.incident_edges_sedges
thf(fact_906_incident__edges__sedges,axiom,
! [V: a] :
( ~ ( undire3617971648856834880loop_a @ edges @ V )
=> ( ( undire3231912044278729248dges_a @ edges @ V )
= ( undire1270416042309875431dges_a @ edges @ V ) ) ) ).
% incident_edges_sedges
thf(fact_907_ulgraph_Oall__edges__between_Ocong,axiom,
undire8383842906760478443ween_a = undire8383842906760478443ween_a ).
% ulgraph.all_edges_between.cong
thf(fact_908_graph__system_Oincident__edges_Ocong,axiom,
undire3231912044278729248dges_a = undire3231912044278729248dges_a ).
% graph_system.incident_edges.cong
thf(fact_909_incident__edges__union,axiom,
! [V: a] :
( ( undire3231912044278729248dges_a @ edges @ V )
= ( sup_sup_set_set_a @ ( undire1270416042309875431dges_a @ edges @ V ) @ ( undire4753905205749729249oops_a @ edges @ V ) ) ) ).
% incident_edges_union
thf(fact_910_gnew_Oincident__edges__union,axiom,
! [V: a] :
( ( undire3231912044278729248dges_a @ e @ V )
= ( sup_sup_set_set_a @ ( undire1270416042309875431dges_a @ e @ V ) @ ( undire4753905205749729249oops_a @ e @ V ) ) ) ).
% gnew.incident_edges_union
thf(fact_911_all__edges__betw__D3,axiom,
! [X: a,Y: a,X5: set_a,Y7: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y7 ) )
=> ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges ) ) ).
% all_edges_betw_D3
thf(fact_912_all__edges__betw__I,axiom,
! [X: a,X5: set_a,Y: a,Y7: set_a] :
( ( member_a @ X @ X5 )
=> ( ( member_a @ Y @ Y7 )
=> ( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y7 ) ) ) ) ) ).
% all_edges_betw_I
thf(fact_913_gnew_Oall__edges__betw__D3,axiom,
! [X: a,Y: a,X5: set_a,Y7: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( undire8383842906760478443ween_a @ e @ X5 @ Y7 ) )
=> ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ e ) ) ).
% gnew.all_edges_betw_D3
thf(fact_914_gnew_Oall__edges__betw__I,axiom,
! [X: a,X5: set_a,Y: a,Y7: set_a] :
( ( member_a @ X @ X5 )
=> ( ( member_a @ Y @ Y7 )
=> ( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ e )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( undire8383842906760478443ween_a @ e @ X5 @ Y7 ) ) ) ) ) ).
% gnew.all_edges_betw_I
thf(fact_915_mk__triangle__from__ss__edges,axiom,
! [X: a,Y: a,X5: set_a,Y7: set_a,Z: a,Z5: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y7 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z ) @ ( undire8383842906760478443ween_a @ edges @ X5 @ Z5 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ ( undire8383842906760478443ween_a @ edges @ Y7 @ Z5 ) )
=> ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z ) ) ) ) ).
% mk_triangle_from_ss_edges
thf(fact_916_gnew_Oedge__btw__vertices__not__equal,axiom,
! [X: a,Y: a,X5: set_a,Y7: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( undire8383842906760478443ween_a @ e @ X5 @ Y7 ) )
=> ( X != Y ) ) ).
% gnew.edge_btw_vertices_not_equal
thf(fact_917_edge__btw__vertices__not__equal,axiom,
! [X: a,Y: a,X5: set_a,Y7: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y7 ) )
=> ( X != Y ) ) ).
% edge_btw_vertices_not_equal
thf(fact_918_sup_Oright__idem,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A @ B2 ) @ B2 )
= ( sup_sup_set_set_a @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_919_sup_Oright__idem,axiom,
! [A: set_a,B2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B2 ) @ B2 )
= ( sup_sup_set_a @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_920_sup_Oright__idem,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ A @ B2 ) @ B2 )
= ( sup_su3048258781599657691od_a_a @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_921_sup__left__idem,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( sup_sup_set_set_a @ X @ ( sup_sup_set_set_a @ X @ Y ) )
= ( sup_sup_set_set_a @ X @ Y ) ) ).
% sup_left_idem
thf(fact_922_sup__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% sup_left_idem
thf(fact_923_sup__left__idem,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ X @ Y ) )
= ( sup_su3048258781599657691od_a_a @ X @ Y ) ) ).
% sup_left_idem
thf(fact_924_sup_Oleft__idem,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( sup_sup_set_set_a @ A @ ( sup_sup_set_set_a @ A @ B2 ) )
= ( sup_sup_set_set_a @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_925_sup_Oleft__idem,axiom,
! [A: set_a,B2: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B2 ) )
= ( sup_sup_set_a @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_926_sup_Oleft__idem,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ A @ B2 ) )
= ( sup_su3048258781599657691od_a_a @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_927_sup__idem,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_928_sup__idem,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_929_sup__idem,axiom,
! [X: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_930_sup_Oidem,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_931_sup_Oidem,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_932_sup_Oidem,axiom,
! [A: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_933_Un__iff,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) )
= ( ( member_nat @ C @ A2 )
| ( member_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_934_Un__iff,axiom,
! [C: real,A2: set_real,B: set_real] :
( ( member_real @ C @ ( sup_sup_set_real @ A2 @ B ) )
= ( ( member_real @ C @ A2 )
| ( member_real @ C @ B ) ) ) ).
% Un_iff
thf(fact_935_Un__iff,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B ) )
= ( ( member_set_a @ C @ A2 )
| ( member_set_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_936_Un__iff,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) )
= ( ( member_a @ C @ A2 )
| ( member_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_937_Un__iff,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) )
= ( ( member1426531477525435216od_a_a @ C @ A2 )
| ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_938_UnCI,axiom,
! [C: nat,B: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ A2 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_939_UnCI,axiom,
! [C: real,B: set_real,A2: set_real] :
( ( ~ ( member_real @ C @ B )
=> ( member_real @ C @ A2 ) )
=> ( member_real @ C @ ( sup_sup_set_real @ A2 @ B ) ) ) ).
% UnCI
thf(fact_940_UnCI,axiom,
! [C: set_a,B: set_set_a,A2: set_set_a] :
( ( ~ ( member_set_a @ C @ B )
=> ( member_set_a @ C @ A2 ) )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B ) ) ) ).
% UnCI
thf(fact_941_UnCI,axiom,
! [C: a,B: set_a,A2: set_a] :
( ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ A2 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnCI
thf(fact_942_UnCI,axiom,
! [C: product_prod_a_a,B: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( ~ ( member1426531477525435216od_a_a @ C @ B )
=> ( member1426531477525435216od_a_a @ C @ A2 ) )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ) ).
% UnCI
thf(fact_943_gnew_Omk__triangle__from__ss__edges,axiom,
! [X: a,Y: a,X5: set_a,Y7: set_a,Z: a,Z5: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( undire8383842906760478443ween_a @ e @ X5 @ Y7 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z ) @ ( undire8383842906760478443ween_a @ e @ X5 @ Z5 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ ( undire8383842906760478443ween_a @ e @ Y7 @ Z5 ) )
=> ( graph_4582152751571636272raph_a @ e @ X @ Y @ Z ) ) ) ) ).
% gnew.mk_triangle_from_ss_edges
thf(fact_944_le__sup__iff,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ X @ Y ) @ Z )
= ( ( ord_le3724670747650509150_set_a @ X @ Z )
& ( ord_le3724670747650509150_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_945_le__sup__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( ( ord_less_eq_set_a @ X @ Z )
& ( ord_less_eq_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_946_le__sup__iff,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ X @ Y ) @ Z )
= ( ( ord_le746702958409616551od_a_a @ X @ Z )
& ( ord_le746702958409616551od_a_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_947_le__sup__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_948_le__sup__iff,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ X @ Y ) @ Z )
= ( ( ord_less_eq_real @ X @ Z )
& ( ord_less_eq_real @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_949_sup_Obounded__iff,axiom,
! [B2: set_set_a,C: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B2 @ C ) @ A )
= ( ( ord_le3724670747650509150_set_a @ B2 @ A )
& ( ord_le3724670747650509150_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_950_sup_Obounded__iff,axiom,
! [B2: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A )
= ( ( ord_less_eq_set_a @ B2 @ A )
& ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_951_sup_Obounded__iff,axiom,
! [B2: set_Product_prod_a_a,C: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ B2 @ C ) @ A )
= ( ( ord_le746702958409616551od_a_a @ B2 @ A )
& ( ord_le746702958409616551od_a_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_952_sup_Obounded__iff,axiom,
! [B2: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_953_sup_Obounded__iff,axiom,
! [B2: real,C: real,A: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ B2 @ C ) @ A )
= ( ( ord_less_eq_real @ B2 @ A )
& ( ord_less_eq_real @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_954_sup__bot_Oright__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_955_sup__bot_Oright__neutral,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ A @ bot_bot_set_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_956_sup__bot_Oright__neutral,axiom,
! [A: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A @ bot_bo3357376287454694259od_a_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_957_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A @ B2 ) )
= ( ( A = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_958_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( bot_bot_set_set_a
= ( sup_sup_set_set_a @ A @ B2 ) )
= ( ( A = bot_bot_set_set_a )
& ( B2 = bot_bot_set_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_959_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( sup_su3048258781599657691od_a_a @ A @ B2 ) )
= ( ( A = bot_bo3357376287454694259od_a_a )
& ( B2 = bot_bo3357376287454694259od_a_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_960_sup__bot_Oleft__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_961_sup__bot_Oleft__neutral,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_962_sup__bot_Oleft__neutral,axiom,
! [A: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ bot_bo3357376287454694259od_a_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_963_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A @ B2 )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_964_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ( sup_sup_set_set_a @ A @ B2 )
= bot_bot_set_set_a )
= ( ( A = bot_bot_set_set_a )
& ( B2 = bot_bot_set_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_965_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ( sup_su3048258781599657691od_a_a @ A @ B2 )
= bot_bo3357376287454694259od_a_a )
= ( ( A = bot_bo3357376287454694259od_a_a )
& ( B2 = bot_bo3357376287454694259od_a_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_966_sup__eq__bot__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X @ Y )
= bot_bot_set_a )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_967_sup__eq__bot__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ( sup_sup_set_set_a @ X @ Y )
= bot_bot_set_set_a )
= ( ( X = bot_bot_set_set_a )
& ( Y = bot_bot_set_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_968_sup__eq__bot__iff,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ( sup_su3048258781599657691od_a_a @ X @ Y )
= bot_bo3357376287454694259od_a_a )
= ( ( X = bot_bo3357376287454694259od_a_a )
& ( Y = bot_bo3357376287454694259od_a_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_969_bot__eq__sup__iff,axiom,
! [X: set_a,Y: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X @ Y ) )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_970_bot__eq__sup__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( bot_bot_set_set_a
= ( sup_sup_set_set_a @ X @ Y ) )
= ( ( X = bot_bot_set_set_a )
& ( Y = bot_bot_set_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_971_bot__eq__sup__iff,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( sup_su3048258781599657691od_a_a @ X @ Y ) )
= ( ( X = bot_bo3357376287454694259od_a_a )
& ( Y = bot_bo3357376287454694259od_a_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_972_sup__bot__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% sup_bot_right
thf(fact_973_sup__bot__right,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ X @ bot_bot_set_set_a )
= X ) ).
% sup_bot_right
thf(fact_974_sup__bot__right,axiom,
! [X: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ bot_bo3357376287454694259od_a_a )
= X ) ).
% sup_bot_right
thf(fact_975_sup__bot__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_976_sup__bot__left,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_977_sup__bot__left,axiom,
! [X: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ bot_bo3357376287454694259od_a_a @ X )
= X ) ).
% sup_bot_left
thf(fact_978_inf__sup__absorb,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( sup_sup_set_set_a @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_979_inf__sup__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_980_inf__sup__absorb,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_981_sup__inf__absorb,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( sup_sup_set_set_a @ X @ ( inf_inf_set_set_a @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_982_sup__inf__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_983_sup__inf__absorb,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ ( inf_in8905007599844390133od_a_a @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_984_Un__empty,axiom,
! [A2: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_985_Un__empty,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ( sup_sup_set_set_a @ A2 @ B )
= bot_bot_set_set_a )
= ( ( A2 = bot_bot_set_set_a )
& ( B = bot_bot_set_set_a ) ) ) ).
% Un_empty
thf(fact_986_Un__empty,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( ( sup_su3048258781599657691od_a_a @ A2 @ B )
= bot_bo3357376287454694259od_a_a )
= ( ( A2 = bot_bo3357376287454694259od_a_a )
& ( B = bot_bo3357376287454694259od_a_a ) ) ) ).
% Un_empty
thf(fact_987_finite__Un,axiom,
! [F2: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) )
= ( ( finite_finite_nat @ F2 )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_988_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_989_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_990_finite__Un,axiom,
! [F2: set_Product_prod_a_a,G: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ ( sup_su3048258781599657691od_a_a @ F2 @ G ) )
= ( ( finite6544458595007987280od_a_a @ F2 )
& ( finite6544458595007987280od_a_a @ G ) ) ) ).
% finite_Un
thf(fact_991_Un__subset__iff,axiom,
! [A2: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B ) @ C2 )
= ( ( ord_le3724670747650509150_set_a @ A2 @ C2 )
& ( ord_le3724670747650509150_set_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_992_Un__subset__iff,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C2 )
= ( ( ord_less_eq_set_a @ A2 @ C2 )
& ( ord_less_eq_set_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_993_Un__subset__iff,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) @ C2 )
= ( ( ord_le746702958409616551od_a_a @ A2 @ C2 )
& ( ord_le746702958409616551od_a_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_994_Un__insert__right,axiom,
! [A2: set_set_a,A: set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( insert_set_a @ A @ ( sup_sup_set_set_a @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_995_Un__insert__right,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_996_Un__insert__right,axiom,
! [A2: set_Product_prod_a_a,A: product_prod_a_a,B: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ A @ B ) )
= ( insert4534936382041156343od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_997_Un__insert__left,axiom,
! [A: set_a,B: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ ( insert_set_a @ A @ B ) @ C2 )
= ( insert_set_a @ A @ ( sup_sup_set_set_a @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_998_Un__insert__left,axiom,
! [A: a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A @ B ) @ C2 )
= ( insert_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_999_Un__insert__left,axiom,
! [A: product_prod_a_a,B: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( insert4534936382041156343od_a_a @ A @ B ) @ C2 )
= ( insert4534936382041156343od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_1000_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_1001_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_1002_Un__Int__eq_I1_J,axiom,
! [S: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_1003_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_1004_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_1005_Un__Int__eq_I2_J,axiom,
! [S: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( sup_su3048258781599657691od_a_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_1006_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_1007_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_1008_Un__Int__eq_I3_J,axiom,
! [S: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ S @ ( sup_su3048258781599657691od_a_a @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_1009_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_1010_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_1011_Un__Int__eq_I4_J,axiom,
! [T2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ T2 @ ( sup_su3048258781599657691od_a_a @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_1012_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_1013_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_1014_Int__Un__eq_I1_J,axiom,
! [S: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_1015_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_1016_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_1017_Int__Un__eq_I2_J,axiom,
! [S: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( inf_in8905007599844390133od_a_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_1018_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_1019_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_1020_Int__Un__eq_I3_J,axiom,
! [S: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ S @ ( inf_in8905007599844390133od_a_a @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_1021_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_1022_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_1023_Int__Un__eq_I4_J,axiom,
! [T2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ T2 @ ( inf_in8905007599844390133od_a_a @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_1024_ssubst__Pair__rhs,axiom,
! [R2: a,S3: a,R: set_Product_prod_a_a,S4: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ R2 @ S3 ) @ R )
=> ( ( S4 = S3 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ R2 @ S4 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_1025_ssubst__Pair__rhs,axiom,
! [R2: set_a,S3: set_set_a,R: set_Pr4256959165342167655_set_a,S4: set_set_a] :
( ( member268004040519299248_set_a @ ( produc2116933609460601975_set_a @ R2 @ S3 ) @ R )
=> ( ( S4 = S3 )
=> ( member268004040519299248_set_a @ ( produc2116933609460601975_set_a @ R2 @ S4 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_1026_mk__triangle__set_Ocases,axiom,
! [X: produc4044097585999906000od_a_a] :
~ ! [X4: a,Y5: a,Z4: a] :
( X
!= ( produc431845341423274048od_a_a @ X4 @ ( product_Pair_a_a @ Y5 @ Z4 ) ) ) ).
% mk_triangle_set.cases
thf(fact_1027_mk__edge_Ocases,axiom,
! [X: product_prod_a_a] :
~ ! [U2: a,V3: a] :
( X
!= ( product_Pair_a_a @ U2 @ V3 ) ) ).
% mk_edge.cases
thf(fact_1028_Un__left__commute,axiom,
! [A2: set_set_a,B: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ ( sup_sup_set_set_a @ B @ C2 ) )
= ( sup_sup_set_set_a @ B @ ( sup_sup_set_set_a @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_1029_Un__left__commute,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B @ C2 ) )
= ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_1030_Un__left__commute,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A2 @ ( sup_su3048258781599657691od_a_a @ B @ C2 ) )
= ( sup_su3048258781599657691od_a_a @ B @ ( sup_su3048258781599657691od_a_a @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_1031_Un__left__absorb,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ ( sup_sup_set_set_a @ A2 @ B ) )
= ( sup_sup_set_set_a @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_1032_Un__left__absorb,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_1033_Un__left__absorb,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A2 @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) )
= ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_1034_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_1035_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_1036_Un__commute,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_1037_Un__absorb,axiom,
! [A2: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_1038_Un__absorb,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_1039_Un__absorb,axiom,
! [A2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_1040_Un__assoc,axiom,
! [A2: set_set_a,B: set_set_a,C2: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A2 @ B ) @ C2 )
= ( sup_sup_set_set_a @ A2 @ ( sup_sup_set_set_a @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_1041_Un__assoc,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C2 )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_1042_Un__assoc,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) @ C2 )
= ( sup_su3048258781599657691od_a_a @ A2 @ ( sup_su3048258781599657691od_a_a @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_1043_ball__Un,axiom,
! [A2: set_set_a,B: set_set_a,P: set_a > $o] :
( ( ! [X3: set_a] :
( ( member_set_a @ X3 @ ( sup_sup_set_set_a @ A2 @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( P @ X3 ) )
& ! [X3: set_a] :
( ( member_set_a @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_1044_ball__Un,axiom,
! [A2: set_a,B: set_a,P: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A2 @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( P @ X3 ) )
& ! [X3: a] :
( ( member_a @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_1045_ball__Un,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A2 )
=> ( P @ X3 ) )
& ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_1046_bex__Un,axiom,
! [A2: set_set_a,B: set_set_a,P: set_a > $o] :
( ( ? [X3: set_a] :
( ( member_set_a @ X3 @ ( sup_sup_set_set_a @ A2 @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( P @ X3 ) )
| ? [X3: set_a] :
( ( member_set_a @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_1047_bex__Un,axiom,
! [A2: set_a,B: set_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A2 @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ X3 ) )
| ? [X3: a] :
( ( member_a @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_1048_bex__Un,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ( ? [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A2 )
& ( P @ X3 ) )
| ? [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_1049_UnI2,axiom,
! [C: nat,B: set_nat,A2: set_nat] :
( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_1050_UnI2,axiom,
! [C: real,B: set_real,A2: set_real] :
( ( member_real @ C @ B )
=> ( member_real @ C @ ( sup_sup_set_real @ A2 @ B ) ) ) ).
% UnI2
thf(fact_1051_UnI2,axiom,
! [C: set_a,B: set_set_a,A2: set_set_a] :
( ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B ) ) ) ).
% UnI2
thf(fact_1052_UnI2,axiom,
! [C: a,B: set_a,A2: set_a] :
( ( member_a @ C @ B )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnI2
thf(fact_1053_UnI2,axiom,
! [C: product_prod_a_a,B: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ B )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ) ).
% UnI2
thf(fact_1054_UnI1,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_1055_UnI1,axiom,
! [C: real,A2: set_real,B: set_real] :
( ( member_real @ C @ A2 )
=> ( member_real @ C @ ( sup_sup_set_real @ A2 @ B ) ) ) ).
% UnI1
thf(fact_1056_UnI1,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B ) ) ) ).
% UnI1
thf(fact_1057_UnI1,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ A2 )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnI1
thf(fact_1058_UnI1,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A2 )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) ) ) ).
% UnI1
thf(fact_1059_UnE,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) )
=> ( ~ ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B ) ) ) ).
% UnE
thf(fact_1060_UnE,axiom,
! [C: real,A2: set_real,B: set_real] :
( ( member_real @ C @ ( sup_sup_set_real @ A2 @ B ) )
=> ( ~ ( member_real @ C @ A2 )
=> ( member_real @ C @ B ) ) ) ).
% UnE
thf(fact_1061_UnE,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B ) )
=> ( ~ ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B ) ) ) ).
% UnE
thf(fact_1062_UnE,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A2 @ B ) )
=> ( ~ ( member_a @ C @ A2 )
=> ( member_a @ C @ B ) ) ) ).
% UnE
thf(fact_1063_UnE,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A2 @ B ) )
=> ( ~ ( member1426531477525435216od_a_a @ C @ A2 )
=> ( member1426531477525435216od_a_a @ C @ B ) ) ) ).
% UnE
thf(fact_1064_sup__left__commute,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ X @ ( sup_sup_set_set_a @ Y @ Z ) )
= ( sup_sup_set_set_a @ Y @ ( sup_sup_set_set_a @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_1065_sup__left__commute,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_1066_sup__left__commute,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ Y @ Z ) )
= ( sup_su3048258781599657691od_a_a @ Y @ ( sup_su3048258781599657691od_a_a @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_1067_sup_Oleft__commute,axiom,
! [B2: set_set_a,A: set_set_a,C: set_set_a] :
( ( sup_sup_set_set_a @ B2 @ ( sup_sup_set_set_a @ A @ C ) )
= ( sup_sup_set_set_a @ A @ ( sup_sup_set_set_a @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_1068_sup_Oleft__commute,axiom,
! [B2: set_a,A: set_a,C: set_a] :
( ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A @ C ) )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_1069_sup_Oleft__commute,axiom,
! [B2: set_Product_prod_a_a,A: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ B2 @ ( sup_su3048258781599657691od_a_a @ A @ C ) )
= ( sup_su3048258781599657691od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_1070_boolean__algebra__cancel_Osup2,axiom,
! [B: set_set_a,K: set_set_a,B2: set_set_a,A: set_set_a] :
( ( B
= ( sup_sup_set_set_a @ K @ B2 ) )
=> ( ( sup_sup_set_set_a @ A @ B )
= ( sup_sup_set_set_a @ K @ ( sup_sup_set_set_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_1071_boolean__algebra__cancel_Osup2,axiom,
! [B: set_a,K: set_a,B2: set_a,A: set_a] :
( ( B
= ( sup_sup_set_a @ K @ B2 ) )
=> ( ( sup_sup_set_a @ A @ B )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_1072_boolean__algebra__cancel_Osup2,axiom,
! [B: set_Product_prod_a_a,K: set_Product_prod_a_a,B2: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( B
= ( sup_su3048258781599657691od_a_a @ K @ B2 ) )
=> ( ( sup_su3048258781599657691od_a_a @ A @ B )
= ( sup_su3048258781599657691od_a_a @ K @ ( sup_su3048258781599657691od_a_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_1073_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_set_a,K: set_set_a,A: set_set_a,B2: set_set_a] :
( ( A2
= ( sup_sup_set_set_a @ K @ A ) )
=> ( ( sup_sup_set_set_a @ A2 @ B2 )
= ( sup_sup_set_set_a @ K @ ( sup_sup_set_set_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_1074_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_a,K: set_a,A: set_a,B2: set_a] :
( ( A2
= ( sup_sup_set_a @ K @ A ) )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_1075_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_Product_prod_a_a,K: set_Product_prod_a_a,A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( A2
= ( sup_su3048258781599657691od_a_a @ K @ A ) )
=> ( ( sup_su3048258781599657691od_a_a @ A2 @ B2 )
= ( sup_su3048258781599657691od_a_a @ K @ ( sup_su3048258781599657691od_a_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_1076_sup__commute,axiom,
( sup_sup_set_set_a
= ( ^ [X3: set_set_a,Y3: set_set_a] : ( sup_sup_set_set_a @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_1077_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X3: set_a,Y3: set_a] : ( sup_sup_set_a @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_1078_sup__commute,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [X3: set_Product_prod_a_a,Y3: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_1079_sup_Ocommute,axiom,
( sup_sup_set_set_a
= ( ^ [A4: set_set_a,B5: set_set_a] : ( sup_sup_set_set_a @ B5 @ A4 ) ) ) ).
% sup.commute
thf(fact_1080_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B5: set_a] : ( sup_sup_set_a @ B5 @ A4 ) ) ) ).
% sup.commute
thf(fact_1081_sup_Ocommute,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [A4: set_Product_prod_a_a,B5: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ B5 @ A4 ) ) ) ).
% sup.commute
thf(fact_1082_sup__assoc,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ X @ Y ) @ Z )
= ( sup_sup_set_set_a @ X @ ( sup_sup_set_set_a @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_1083_sup__assoc,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_1084_sup__assoc,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ X @ Y ) @ Z )
= ( sup_su3048258781599657691od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_1085_sup_Oassoc,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ A @ B2 ) @ C )
= ( sup_sup_set_set_a @ A @ ( sup_sup_set_set_a @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_1086_sup_Oassoc,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B2 ) @ C )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_1087_sup_Oassoc,axiom,
! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ A @ B2 ) @ C )
= ( sup_su3048258781599657691od_a_a @ A @ ( sup_su3048258781599657691od_a_a @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_1088_inf__sup__aci_I5_J,axiom,
( sup_sup_set_set_a
= ( ^ [X3: set_set_a,Y3: set_set_a] : ( sup_sup_set_set_a @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_1089_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X3: set_a,Y3: set_a] : ( sup_sup_set_a @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_1090_inf__sup__aci_I5_J,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [X3: set_Product_prod_a_a,Y3: set_Product_prod_a_a] : ( sup_su3048258781599657691od_a_a @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_1091_inf__sup__aci_I6_J,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ X @ Y ) @ Z )
= ( sup_sup_set_set_a @ X @ ( sup_sup_set_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_1092_inf__sup__aci_I6_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_1093_inf__sup__aci_I6_J,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ ( sup_su3048258781599657691od_a_a @ X @ Y ) @ Z )
= ( sup_su3048258781599657691od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_1094_inf__sup__aci_I7_J,axiom,
! [X: set_set_a,Y: set_set_a,Z: set_set_a] :
( ( sup_sup_set_set_a @ X @ ( sup_sup_set_set_a @ Y @ Z ) )
= ( sup_sup_set_set_a @ Y @ ( sup_sup_set_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_1095_inf__sup__aci_I7_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_1096_inf__sup__aci_I7_J,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ Y @ Z ) )
= ( sup_su3048258781599657691od_a_a @ Y @ ( sup_su3048258781599657691od_a_a @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_1097_inf__sup__aci_I8_J,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( sup_sup_set_set_a @ X @ ( sup_sup_set_set_a @ X @ Y ) )
= ( sup_sup_set_set_a @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_1098_inf__sup__aci_I8_J,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_1099_inf__sup__aci_I8_J,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ X @ ( sup_su3048258781599657691od_a_a @ X @ Y ) )
= ( sup_su3048258781599657691od_a_a @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_1100_sup_OcoboundedI2,axiom,
! [C: set_set_a,B2: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ B2 )
=> ( ord_le3724670747650509150_set_a @ C @ ( sup_sup_set_set_a @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_1101_sup_OcoboundedI2,axiom,
! [C: set_a,B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_1102_sup_OcoboundedI2,axiom,
! [C: set_Product_prod_a_a,B2: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C @ B2 )
=> ( ord_le746702958409616551od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_1103_sup_OcoboundedI2,axiom,
! [C: nat,B2: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_1104_sup_OcoboundedI2,axiom,
! [C: real,B2: real,A: real] :
( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_eq_real @ C @ ( sup_sup_real @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_1105_sup_OcoboundedI1,axiom,
! [C: set_set_a,A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ A )
=> ( ord_le3724670747650509150_set_a @ C @ ( sup_sup_set_set_a @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_1106_sup_OcoboundedI1,axiom,
! [C: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_1107_sup_OcoboundedI1,axiom,
! [C: set_Product_prod_a_a,A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ C @ A )
=> ( ord_le746702958409616551od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_1108_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_1109_sup_OcoboundedI1,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_eq_real @ C @ A )
=> ( ord_less_eq_real @ C @ ( sup_sup_real @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_1110_sup_Oabsorb__iff2,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B5: set_set_a] :
( ( sup_sup_set_set_a @ A4 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_1111_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B5: set_a] :
( ( sup_sup_set_a @ A4 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_1112_sup_Oabsorb__iff2,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A4: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( sup_su3048258781599657691od_a_a @ A4 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_1113_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] :
( ( sup_sup_nat @ A4 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_1114_sup_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B5: real] :
( ( sup_sup_real @ A4 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_1115_sup_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [B5: real,A4: real] :
( ( sup_sup_real @ A4 @ B5 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_1116_card__all__edges__between__commute,axiom,
! [X5: set_a,Y7: set_a] :
( ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y7 ) )
= ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ Y7 @ X5 ) ) ) ).
% card_all_edges_between_commute
thf(fact_1117_gnew_Ocard__all__edges__between__commute,axiom,
! [X5: set_a,Y7: set_a] :
( ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ e @ X5 @ Y7 ) )
= ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ e @ Y7 @ X5 ) ) ) ).
% gnew.card_all_edges_between_commute
thf(fact_1118_gnew_Oall__edges__between__Un1,axiom,
! [X5: set_a,Y7: set_a,Z5: set_a] :
( ( undire8383842906760478443ween_a @ e @ ( sup_sup_set_a @ X5 @ Y7 ) @ Z5 )
= ( sup_su3048258781599657691od_a_a @ ( undire8383842906760478443ween_a @ e @ X5 @ Z5 ) @ ( undire8383842906760478443ween_a @ e @ Y7 @ Z5 ) ) ) ).
% gnew.all_edges_between_Un1
thf(fact_1119_gnew_Oall__edges__between__Un2,axiom,
! [X5: set_a,Y7: set_a,Z5: set_a] :
( ( undire8383842906760478443ween_a @ e @ X5 @ ( sup_sup_set_a @ Y7 @ Z5 ) )
= ( sup_su3048258781599657691od_a_a @ ( undire8383842906760478443ween_a @ e @ X5 @ Y7 ) @ ( undire8383842906760478443ween_a @ e @ X5 @ Z5 ) ) ) ).
% gnew.all_edges_between_Un2
thf(fact_1120_all__edges__between__Un2,axiom,
! [X5: set_a,Y7: set_a,Z5: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X5 @ ( sup_sup_set_a @ Y7 @ Z5 ) )
= ( sup_su3048258781599657691od_a_a @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y7 ) @ ( undire8383842906760478443ween_a @ edges @ X5 @ Z5 ) ) ) ).
% all_edges_between_Un2
thf(fact_1121_all__edges__between__Un1,axiom,
! [X5: set_a,Y7: set_a,Z5: set_a] :
( ( undire8383842906760478443ween_a @ edges @ ( sup_sup_set_a @ X5 @ Y7 ) @ Z5 )
= ( sup_su3048258781599657691od_a_a @ ( undire8383842906760478443ween_a @ edges @ X5 @ Z5 ) @ ( undire8383842906760478443ween_a @ edges @ Y7 @ Z5 ) ) ) ).
% all_edges_between_Un1
thf(fact_1122_edge__density__commute,axiom,
! [X5: set_a,Y7: set_a] :
( ( undire297304480579013331sity_a @ edges @ X5 @ Y7 )
= ( undire297304480579013331sity_a @ edges @ Y7 @ X5 ) ) ).
% edge_density_commute
thf(fact_1123_gnew_Oedge__density__commute,axiom,
! [X5: set_a,Y7: set_a] :
( ( undire297304480579013331sity_a @ e @ X5 @ Y7 )
= ( undire297304480579013331sity_a @ e @ Y7 @ X5 ) ) ).
% gnew.edge_density_commute
thf(fact_1124_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M: nat] :
? [N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( member_nat @ N @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_1125_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N2: set_nat] :
? [M: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N2 )
=> ( ord_less_eq_nat @ X3 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1126_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M2: nat] :
( ( P @ X )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M2 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1127_card__triangle__triples__rotate,axiom,
! [X5: set_a,Y7: set_a,Z5: set_a] :
( ( finite6893194910719049976od_a_a @ ( graph_4774508486909600516ples_a @ edges @ X5 @ Y7 @ Z5 ) )
= ( finite6893194910719049976od_a_a @ ( graph_4774508486909600516ples_a @ edges @ Y7 @ Z5 @ X5 ) ) ) ).
% card_triangle_triples_rotate
thf(fact_1128_gnew_Ocard__triangle__triples__rotate,axiom,
! [X5: set_a,Y7: set_a,Z5: set_a] :
( ( finite6893194910719049976od_a_a @ ( graph_4774508486909600516ples_a @ e @ X5 @ Y7 @ Z5 ) )
= ( finite6893194910719049976od_a_a @ ( graph_4774508486909600516ples_a @ e @ Y7 @ Z5 @ X5 ) ) ) ).
% gnew.card_triangle_triples_rotate
thf(fact_1129_edge__density__eq0,axiom,
! [A2: set_a,B: set_a,X5: set_a,Y7: set_a] :
( ( ( undire8383842906760478443ween_a @ edges @ A2 @ B )
= bot_bo3357376287454694259od_a_a )
=> ( ( ord_less_eq_set_a @ X5 @ A2 )
=> ( ( ord_less_eq_set_a @ Y7 @ B )
=> ( ( undire297304480579013331sity_a @ edges @ X5 @ Y7 )
= zero_zero_real ) ) ) ) ).
% edge_density_eq0
thf(fact_1130_gnew_Oedge__density__eq0,axiom,
! [A2: set_a,B: set_a,X5: set_a,Y7: set_a] :
( ( ( undire8383842906760478443ween_a @ e @ A2 @ B )
= bot_bo3357376287454694259od_a_a )
=> ( ( ord_less_eq_set_a @ X5 @ A2 )
=> ( ( ord_less_eq_set_a @ Y7 @ B )
=> ( ( undire297304480579013331sity_a @ e @ X5 @ Y7 )
= zero_zero_real ) ) ) ) ).
% gnew.edge_density_eq0
thf(fact_1131_gnew_Oedge__density__ge0,axiom,
! [X5: set_a,Y7: set_a] : ( ord_less_eq_real @ zero_zero_real @ ( undire297304480579013331sity_a @ e @ X5 @ Y7 ) ) ).
% gnew.edge_density_ge0
thf(fact_1132_edge__density__ge0,axiom,
! [X5: set_a,Y7: set_a] : ( ord_less_eq_real @ zero_zero_real @ ( undire297304480579013331sity_a @ edges @ X5 @ Y7 ) ) ).
% edge_density_ge0
thf(fact_1133_gnew_Oedge__density__zero,axiom,
! [Y7: set_a,X5: set_a] :
( ( Y7 = bot_bot_set_a )
=> ( ( undire297304480579013331sity_a @ e @ X5 @ Y7 )
= zero_zero_real ) ) ).
% gnew.edge_density_zero
thf(fact_1134_edge__density__zero,axiom,
! [Y7: set_a,X5: set_a] :
( ( Y7 = bot_bot_set_a )
=> ( ( undire297304480579013331sity_a @ edges @ X5 @ Y7 )
= zero_zero_real ) ) ).
% edge_density_zero
thf(fact_1135_max__all__edges__between,axiom,
! [X5: set_a,Y7: set_a] :
( ( finite_finite_a @ X5 )
=> ( ( finite_finite_a @ Y7 )
=> ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y7 ) ) @ ( times_times_nat @ ( finite_card_a @ X5 ) @ ( finite_card_a @ Y7 ) ) ) ) ) ).
% max_all_edges_between
thf(fact_1136_gnew_Omax__all__edges__between,axiom,
! [X5: set_a,Y7: set_a] :
( ( finite_finite_a @ X5 )
=> ( ( finite_finite_a @ Y7 )
=> ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ e @ X5 @ Y7 ) ) @ ( times_times_nat @ ( finite_card_a @ X5 ) @ ( finite_card_a @ Y7 ) ) ) ) ) ).
% gnew.max_all_edges_between
thf(fact_1137_incident__loops__card,axiom,
! [V: a] : ( ord_less_eq_nat @ ( finite_card_set_a @ ( undire4753905205749729249oops_a @ edges @ V ) ) @ one_one_nat ) ).
% incident_loops_card
thf(fact_1138_card1__incident__imp__vert,axiom,
! [V: a,E3: set_a] :
( ( ( undire1521409233611534436dent_a @ V @ E3 )
& ( ( finite_card_a @ E3 )
= one_one_nat ) )
=> ( E3
= ( insert_a @ V @ bot_bot_set_a ) ) ) ).
% card1_incident_imp_vert
thf(fact_1139_gnew_Oincident__loops__card,axiom,
! [V: a] : ( ord_less_eq_nat @ ( finite_card_set_a @ ( undire4753905205749729249oops_a @ e @ V ) ) @ one_one_nat ) ).
% gnew.incident_loops_card
thf(fact_1140_degree0__inc__edges__empt__iff,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( ( ( undire8867928226783802224gree_a @ edges @ V )
= zero_zero_nat )
= ( ( undire3231912044278729248dges_a @ edges @ V )
= bot_bot_set_set_a ) ) ) ).
% degree0_inc_edges_empt_iff
thf(fact_1141_gnew_Odegree0__inc__edges__empt__iff,axiom,
! [V: a] :
( ( finite_finite_set_a @ e )
=> ( ( ( undire8867928226783802224gree_a @ e @ V )
= zero_zero_nat )
= ( ( undire3231912044278729248dges_a @ e @ V )
= bot_bot_set_set_a ) ) ) ).
% gnew.degree0_inc_edges_empt_iff
thf(fact_1142_is__loop__def,axiom,
( undire2905028936066782638loop_a
= ( ^ [E: set_a] :
( ( finite_card_a @ E )
= one_one_nat ) ) ) ).
% is_loop_def
thf(fact_1143_gnew_Oedge__density__le1,axiom,
! [X5: set_a,Y7: set_a] : ( ord_less_eq_real @ ( undire297304480579013331sity_a @ e @ X5 @ Y7 ) @ one_one_real ) ).
% gnew.edge_density_le1
thf(fact_1144_edge__density__le1,axiom,
! [X5: set_a,Y7: set_a] : ( ord_less_eq_real @ ( undire297304480579013331sity_a @ edges @ X5 @ Y7 ) @ one_one_real ) ).
% edge_density_le1
thf(fact_1145_gnew_Oalt__degree__def,axiom,
! [V: a] :
( ( undire8867928226783802224gree_a @ e @ V )
= ( finite_card_set_a @ ( undire3231912044278729248dges_a @ e @ V ) ) ) ).
% gnew.alt_degree_def
thf(fact_1146_alt__degree__def,axiom,
! [V: a] :
( ( undire8867928226783802224gree_a @ edges @ V )
= ( finite_card_set_a @ ( undire3231912044278729248dges_a @ edges @ V ) ) ) ).
% alt_degree_def
thf(fact_1147_gnew_Odegree__no__loops,axiom,
! [V: a] :
( ~ ( undire3617971648856834880loop_a @ e @ V )
=> ( ( undire8867928226783802224gree_a @ e @ V )
= ( finite_card_set_a @ ( undire3231912044278729248dges_a @ e @ V ) ) ) ) ).
% gnew.degree_no_loops
thf(fact_1148_degree__no__loops,axiom,
! [V: a] :
( ~ ( undire3617971648856834880loop_a @ edges @ V )
=> ( ( undire8867928226783802224gree_a @ edges @ V )
= ( finite_card_set_a @ ( undire3231912044278729248dges_a @ edges @ V ) ) ) ) ).
% degree_no_loops
thf(fact_1149_is__edge__or__loop,axiom,
! [E3: set_a] :
( ( member_set_a @ E3 @ edges )
=> ( ( undire2905028936066782638loop_a @ E3 )
| ( undire4917966558017083288edge_a @ E3 ) ) ) ).
% is_edge_or_loop
thf(fact_1150_gnew_Ois__edge__or__loop,axiom,
! [E3: set_a] :
( ( member_set_a @ E3 @ e )
=> ( ( undire2905028936066782638loop_a @ E3 )
| ( undire4917966558017083288edge_a @ E3 ) ) ) ).
% gnew.is_edge_or_loop
thf(fact_1151_le0,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% le0
thf(fact_1152_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1153_le__refl,axiom,
! [N3: nat] : ( ord_less_eq_nat @ N3 @ N3 ) ).
% le_refl
thf(fact_1154_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_1155_eq__imp__le,axiom,
! [M4: nat,N3: nat] :
( ( M4 = N3 )
=> ( ord_less_eq_nat @ M4 @ N3 ) ) ).
% eq_imp_le
thf(fact_1156_le__antisym,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ M4 )
=> ( M4 = N3 ) ) ) ).
% le_antisym
thf(fact_1157_nat__le__linear,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
| ( ord_less_eq_nat @ N3 @ M4 ) ) ).
% nat_le_linear
thf(fact_1158_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ B2 ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1159_less__eq__nat_Osimps_I1_J,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% less_eq_nat.simps(1)
thf(fact_1160_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1161_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1162_le__0__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1163_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1164_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1165_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1166_le__square,axiom,
! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ).
% le_square
thf(fact_1167_le__cube,axiom,
! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ) ).
% le_cube
thf(fact_1168_degree0__neighborhood__empt__iff,axiom,
! [V: a] :
( ( finite_finite_set_a @ edges )
=> ( ( ( undire8867928226783802224gree_a @ edges @ V )
= zero_zero_nat )
= ( ( undire8504279938402040014hood_a @ vertices @ edges @ V )
= bot_bot_set_a ) ) ) ).
% degree0_neighborhood_empt_iff
thf(fact_1169_gnew_Odegree0__neighborhood__empt__iff,axiom,
! [V: a] :
( ( finite_finite_set_a @ e )
=> ( ( ( undire8867928226783802224gree_a @ e @ V )
= zero_zero_nat )
= ( ( undire8504279938402040014hood_a @ vertices @ e @ V )
= bot_bot_set_a ) ) ) ).
% gnew.degree0_neighborhood_empt_iff
thf(fact_1170_sgraph__axioms,axiom,
undire3507641187627840796raph_a @ vertices @ edges ).
% sgraph_axioms
thf(fact_1171_gnew_Oe__in__all__edges,axiom,
! [E3: set_a] :
( ( member_set_a @ E3 @ e )
=> ( member_set_a @ E3 @ ( undire2918257014606996450dges_a @ vertices ) ) ) ).
% gnew.e_in_all_edges
thf(fact_1172_gnew_Osgraph__axioms,axiom,
undire3507641187627840796raph_a @ vertices @ e ).
% gnew.sgraph_axioms
thf(fact_1173_e__in__all__edges,axiom,
! [E3: set_a] :
( ( member_set_a @ E3 @ edges )
=> ( member_set_a @ E3 @ ( undire2918257014606996450dges_a @ vertices ) ) ) ).
% e_in_all_edges
thf(fact_1174_gnew_Owellformed,axiom,
! [E3: set_a] :
( ( member_set_a @ E3 @ e )
=> ( ord_less_eq_set_a @ E3 @ vertices ) ) ).
% gnew.wellformed
thf(fact_1175_gnew_Oe__in__all__edges__ss,axiom,
! [E3: set_a,V4: set_a] :
( ( member_set_a @ E3 @ e )
=> ( ( ord_less_eq_set_a @ E3 @ V4 )
=> ( ( ord_less_eq_set_a @ V4 @ vertices )
=> ( member_set_a @ E3 @ ( undire2918257014606996450dges_a @ V4 ) ) ) ) ) ).
% gnew.e_in_all_edges_ss
thf(fact_1176_wellformed,axiom,
! [E3: set_a] :
( ( member_set_a @ E3 @ edges )
=> ( ord_less_eq_set_a @ E3 @ vertices ) ) ).
% wellformed
thf(fact_1177_e__in__all__edges__ss,axiom,
! [E3: set_a,V4: set_a] :
( ( member_set_a @ E3 @ edges )
=> ( ( ord_less_eq_set_a @ E3 @ V4 )
=> ( ( ord_less_eq_set_a @ V4 @ vertices )
=> ( member_set_a @ E3 @ ( undire2918257014606996450dges_a @ V4 ) ) ) ) ) ).
% e_in_all_edges_ss
thf(fact_1178_gnew_Otriangle__in__graph__verts_I3_J,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ e @ X @ Y @ Z )
=> ( member_a @ Z @ vertices ) ) ).
% gnew.triangle_in_graph_verts(3)
thf(fact_1179_gnew_Otriangle__in__graph__verts_I2_J,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ e @ X @ Y @ Z )
=> ( member_a @ Y @ vertices ) ) ).
% gnew.triangle_in_graph_verts(2)
thf(fact_1180_gnew_Otriangle__in__graph__verts_I1_J,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ e @ X @ Y @ Z )
=> ( member_a @ X @ vertices ) ) ).
% gnew.triangle_in_graph_verts(1)
thf(fact_1181_triangle__in__graph__verts_I1_J,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z )
=> ( member_a @ X @ vertices ) ) ).
% triangle_in_graph_verts(1)
thf(fact_1182_triangle__in__graph__verts_I2_J,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z )
=> ( member_a @ Y @ vertices ) ) ).
% triangle_in_graph_verts(2)
thf(fact_1183_triangle__in__graph__verts_I3_J,axiom,
! [X: a,Y: a,Z: a] :
( ( graph_4582152751571636272raph_a @ edges @ X @ Y @ Z )
=> ( member_a @ Z @ vertices ) ) ).
% triangle_in_graph_verts(3)
thf(fact_1184_gnew_Overt__adj__imp__inV,axiom,
! [V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ e @ V1 @ V2 )
=> ( ( member_a @ V1 @ vertices )
& ( member_a @ V2 @ vertices ) ) ) ).
% gnew.vert_adj_imp_inV
thf(fact_1185_vert__adj__imp__inV,axiom,
! [V1: a,V2: a] :
( ( undire397441198561214472_adj_a @ edges @ V1 @ V2 )
=> ( ( member_a @ V1 @ vertices )
& ( member_a @ V2 @ vertices ) ) ) ).
% vert_adj_imp_inV
thf(fact_1186_gnew_Oincident__edge__in__wf,axiom,
! [E3: set_a,V: a] :
( ( member_set_a @ E3 @ e )
=> ( ( undire1521409233611534436dent_a @ V @ E3 )
=> ( member_a @ V @ vertices ) ) ) ).
% gnew.incident_edge_in_wf
thf(fact_1187_incident__edge__in__wf,axiom,
! [E3: set_a,V: a] :
( ( member_set_a @ E3 @ edges )
=> ( ( undire1521409233611534436dent_a @ V @ E3 )
=> ( member_a @ V @ vertices ) ) ) ).
% incident_edge_in_wf
thf(fact_1188_gnew_Ohas__loop__in__verts,axiom,
! [V: a] :
( ( undire3617971648856834880loop_a @ e @ V )
=> ( member_a @ V @ vertices ) ) ).
% gnew.has_loop_in_verts
thf(fact_1189_gnew_Ono__loops,axiom,
! [V: a] :
( ( member_a @ V @ vertices )
=> ~ ( undire3617971648856834880loop_a @ e @ V ) ) ).
% gnew.no_loops
thf(fact_1190_no__loops,axiom,
! [V: a] :
( ( member_a @ V @ vertices )
=> ~ ( undire3617971648856834880loop_a @ edges @ V ) ) ).
% no_loops
thf(fact_1191_has__loop__in__verts,axiom,
! [V: a] :
( ( undire3617971648856834880loop_a @ edges @ V )
=> ( member_a @ V @ vertices ) ) ).
% has_loop_in_verts
thf(fact_1192_gnew_Owellformed__all__edges,axiom,
ord_le3724670747650509150_set_a @ e @ ( undire2918257014606996450dges_a @ vertices ) ).
% gnew.wellformed_all_edges
thf(fact_1193_wellformed__all__edges,axiom,
ord_le3724670747650509150_set_a @ edges @ ( undire2918257014606996450dges_a @ vertices ) ).
% wellformed_all_edges
thf(fact_1194_gnew_Oedge__adjacent__alt__def,axiom,
! [E1: set_a,E2: set_a] :
( ( member_set_a @ E1 @ e )
=> ( ( member_set_a @ E2 @ e )
=> ( ? [X6: a] :
( ( member_a @ X6 @ vertices )
& ( member_a @ X6 @ E1 )
& ( member_a @ X6 @ E2 ) )
=> ( undire4022703626023482010_adj_a @ e @ E1 @ E2 ) ) ) ) ).
% gnew.edge_adjacent_alt_def
thf(fact_1195_edge__adjacent__alt__def,axiom,
! [E1: set_a,E2: set_a] :
( ( member_set_a @ E1 @ edges )
=> ( ( member_set_a @ E2 @ edges )
=> ( ? [X6: a] :
( ( member_a @ X6 @ vertices )
& ( member_a @ X6 @ E1 )
& ( member_a @ X6 @ E2 ) )
=> ( undire4022703626023482010_adj_a @ edges @ E1 @ E2 ) ) ) ) ).
% edge_adjacent_alt_def
thf(fact_1196_gnew_Owellformed__alt__snd,axiom,
! [X: a,Y: a] :
( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ e )
=> ( member_a @ Y @ vertices ) ) ).
% gnew.wellformed_alt_snd
thf(fact_1197_gnew_Owellformed__alt__fst,axiom,
! [X: a,Y: a] :
( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ e )
=> ( member_a @ X @ vertices ) ) ).
% gnew.wellformed_alt_fst
thf(fact_1198_wellformed__alt__snd,axiom,
! [X: a,Y: a] :
( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges )
=> ( member_a @ Y @ vertices ) ) ).
% wellformed_alt_snd
thf(fact_1199_wellformed__alt__fst,axiom,
! [X: a,Y: a] :
( ( member_set_a @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) @ edges )
=> ( member_a @ X @ vertices ) ) ).
% wellformed_alt_fst
thf(fact_1200_gnew_Oall__edges__between__rem__wf,axiom,
! [X5: set_a,Y7: set_a] :
( ( undire8383842906760478443ween_a @ e @ X5 @ Y7 )
= ( undire8383842906760478443ween_a @ e @ ( inf_inf_set_a @ X5 @ vertices ) @ ( inf_inf_set_a @ Y7 @ vertices ) ) ) ).
% gnew.all_edges_between_rem_wf
thf(fact_1201_all__edges__between__rem__wf,axiom,
! [X5: set_a,Y7: set_a] :
( ( undire8383842906760478443ween_a @ edges @ X5 @ Y7 )
= ( undire8383842906760478443ween_a @ edges @ ( inf_inf_set_a @ X5 @ vertices ) @ ( inf_inf_set_a @ Y7 @ vertices ) ) ) ).
% all_edges_between_rem_wf
thf(fact_1202_gnew_Oincident__edges__empty,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire3231912044278729248dges_a @ e @ V )
= bot_bot_set_set_a ) ) ).
% gnew.incident_edges_empty
thf(fact_1203_incident__edges__empty,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire3231912044278729248dges_a @ edges @ V )
= bot_bot_set_set_a ) ) ).
% incident_edges_empty
thf(fact_1204_gnew_Oalt__deg__neighborhood,axiom,
! [V: a] :
( ( undire8867928226783802224gree_a @ e @ V )
= ( finite_card_a @ ( undire8504279938402040014hood_a @ vertices @ e @ V ) ) ) ).
% gnew.alt_deg_neighborhood
thf(fact_1205_alt__deg__neighborhood,axiom,
! [V: a] :
( ( undire8867928226783802224gree_a @ edges @ V )
= ( finite_card_a @ ( undire8504279938402040014hood_a @ vertices @ edges @ V ) ) ) ).
% alt_deg_neighborhood
thf(fact_1206_gnew_Oneighborhood__incident,axiom,
! [U: a,V: a] :
( ( member_a @ U @ ( undire8504279938402040014hood_a @ vertices @ e @ V ) )
= ( member_set_a @ ( insert_a @ U @ ( insert_a @ V @ bot_bot_set_a ) ) @ ( undire3231912044278729248dges_a @ e @ V ) ) ) ).
% gnew.neighborhood_incident
thf(fact_1207_neighborhood__incident,axiom,
! [U: a,V: a] :
( ( member_a @ U @ ( undire8504279938402040014hood_a @ vertices @ edges @ V ) )
= ( member_set_a @ ( insert_a @ U @ ( insert_a @ V @ bot_bot_set_a ) ) @ ( undire3231912044278729248dges_a @ edges @ V ) ) ) ).
% neighborhood_incident
thf(fact_1208_gnew_Ocard__incident__sedges__neighborhood,axiom,
! [V: a] :
( ( finite_card_set_a @ ( undire3231912044278729248dges_a @ e @ V ) )
= ( finite_card_a @ ( undire8504279938402040014hood_a @ vertices @ e @ V ) ) ) ).
% gnew.card_incident_sedges_neighborhood
thf(fact_1209_card__incident__sedges__neighborhood,axiom,
! [V: a] :
( ( finite_card_set_a @ ( undire3231912044278729248dges_a @ edges @ V ) )
= ( finite_card_a @ ( undire8504279938402040014hood_a @ vertices @ edges @ V ) ) ) ).
% card_incident_sedges_neighborhood
thf(fact_1210_gnew_Ois__complete__n__graph__def,axiom,
! [N3: nat] :
( ( undire6087271738840788937raph_a @ vertices @ e @ N3 )
= ( ( ( finite_card_a @ vertices )
= N3 )
& ( e
= ( undire2918257014606996450dges_a @ vertices ) ) ) ) ).
% gnew.is_complete_n_graph_def
thf(fact_1211_is__complete__n__graph__def,axiom,
! [N3: nat] :
( ( undire6087271738840788937raph_a @ vertices @ edges @ N3 )
= ( ( ( finite_card_a @ vertices )
= N3 )
& ( edges
= ( undire2918257014606996450dges_a @ vertices ) ) ) ) ).
% is_complete_n_graph_def
thf(fact_1212_gnew_Odegree__none,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire8867928226783802224gree_a @ e @ V )
= zero_zero_nat ) ) ).
% gnew.degree_none
thf(fact_1213_degree__none,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire8867928226783802224gree_a @ edges @ V )
= zero_zero_nat ) ) ).
% degree_none
thf(fact_1214_gnew_Oincident__sedges__empty,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire1270416042309875431dges_a @ e @ V )
= bot_bot_set_set_a ) ) ).
% gnew.incident_sedges_empty
thf(fact_1215_incident__sedges__empty,axiom,
! [V: a] :
( ~ ( member_a @ V @ vertices )
=> ( ( undire1270416042309875431dges_a @ edges @ V )
= bot_bot_set_set_a ) ) ).
% incident_sedges_empty
thf(fact_1216_iso__vertex__empty__neighborhood,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
=> ( ( undire8504279938402040014hood_a @ vertices @ edges @ V )
= bot_bot_set_a ) ) ).
% iso_vertex_empty_neighborhood
thf(fact_1217_gnew_Oiso__vertex__empty__neighborhood,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ e @ V )
=> ( ( undire8504279938402040014hood_a @ vertices @ e @ V )
= bot_bot_set_a ) ) ).
% gnew.iso_vertex_empty_neighborhood
thf(fact_1218_is__isolated__vertex__degree0,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
=> ( ( undire8867928226783802224gree_a @ edges @ V )
= zero_zero_nat ) ) ).
% is_isolated_vertex_degree0
thf(fact_1219_gnew_Ois__isolated__vertex__def,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ e @ V )
= ( ( member_a @ V @ vertices )
& ! [X3: a] :
( ( member_a @ X3 @ vertices )
=> ~ ( undire397441198561214472_adj_a @ e @ X3 @ V ) ) ) ) ).
% gnew.is_isolated_vertex_def
thf(fact_1220_is__isolated__vertex__def,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
= ( ( member_a @ V @ vertices )
& ! [X3: a] :
( ( member_a @ X3 @ vertices )
=> ~ ( undire397441198561214472_adj_a @ edges @ X3 @ V ) ) ) ) ).
% is_isolated_vertex_def
thf(fact_1221_gnew_Ois__isolated__vertex__edge,axiom,
! [V: a,E3: set_a] :
( ( undire8931668460104145173rtex_a @ vertices @ e @ V )
=> ( ( member_set_a @ E3 @ e )
=> ~ ( undire1521409233611534436dent_a @ V @ E3 ) ) ) ).
% gnew.is_isolated_vertex_edge
thf(fact_1222_is__isolated__vertex__edge,axiom,
! [V: a,E3: set_a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
=> ( ( member_set_a @ E3 @ edges )
=> ~ ( undire1521409233611534436dent_a @ V @ E3 ) ) ) ).
% is_isolated_vertex_edge
thf(fact_1223_gnew_Ois__isolated__vertex__no__loop,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ e @ V )
=> ~ ( undire3617971648856834880loop_a @ e @ V ) ) ).
% gnew.is_isolated_vertex_no_loop
thf(fact_1224_is__isolated__vertex__no__loop,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ edges @ V )
=> ~ ( undire3617971648856834880loop_a @ edges @ V ) ) ).
% is_isolated_vertex_no_loop
thf(fact_1225_gnew_Ois__isolated__vertex__degree0,axiom,
! [V: a] :
( ( undire8931668460104145173rtex_a @ vertices @ e @ V )
=> ( ( undire8867928226783802224gree_a @ e @ V )
= zero_zero_nat ) ) ).
% gnew.is_isolated_vertex_degree0
thf(fact_1226_induced__edges__ss,axiom,
! [V4: set_a] :
( ( ord_less_eq_set_a @ V4 @ vertices )
=> ( ord_le3724670747650509150_set_a @ ( undire7777452895879145676dges_a @ edges @ V4 ) @ edges ) ) ).
% induced_edges_ss
thf(fact_1227_gnew_Oinduced__edges__ss,axiom,
! [V4: set_a] :
( ( ord_less_eq_set_a @ V4 @ vertices )
=> ( ord_le3724670747650509150_set_a @ ( undire7777452895879145676dges_a @ e @ V4 ) @ e ) ) ).
% gnew.induced_edges_ss
thf(fact_1228_subgraph__complete,axiom,
undire7103218114511261257raph_a @ vertices @ edges @ vertices @ ( undire2918257014606996450dges_a @ vertices ) ).
% subgraph_complete
thf(fact_1229_gnew_Oinduced__edges__alt,axiom,
! [V4: set_a] :
( ( undire7777452895879145676dges_a @ e @ V4 )
= ( inf_inf_set_set_a @ e @ ( undire2918257014606996450dges_a @ V4 ) ) ) ).
% gnew.induced_edges_alt
thf(fact_1230_induced__edges__alt,axiom,
! [V4: set_a] :
( ( undire7777452895879145676dges_a @ edges @ V4 )
= ( inf_inf_set_set_a @ edges @ ( undire2918257014606996450dges_a @ V4 ) ) ) ).
% induced_edges_alt
thf(fact_1231_gnew_Oinduced__edges__self,axiom,
( ( undire7777452895879145676dges_a @ e @ vertices )
= e ) ).
% gnew.induced_edges_self
thf(fact_1232_induced__edges__self,axiom,
( ( undire7777452895879145676dges_a @ edges @ vertices )
= edges ) ).
% induced_edges_self
thf(fact_1233_gnew_Osubgraph__refl,axiom,
undire7103218114511261257raph_a @ vertices @ e @ vertices @ e ).
% gnew.subgraph_refl
thf(fact_1234_subgraph__refl,axiom,
undire7103218114511261257raph_a @ vertices @ edges @ vertices @ edges ).
% subgraph_refl
thf(fact_1235_gnew_Osubgraph__complete,axiom,
undire7103218114511261257raph_a @ vertices @ e @ vertices @ ( undire2918257014606996450dges_a @ vertices ) ).
% gnew.subgraph_complete
thf(fact_1236_gnew_Oinduced__is__subgraph,axiom,
! [V4: set_a] :
( ( ord_less_eq_set_a @ V4 @ vertices )
=> ( undire7103218114511261257raph_a @ V4 @ ( undire7777452895879145676dges_a @ e @ V4 ) @ vertices @ e ) ) ).
% gnew.induced_is_subgraph
thf(fact_1237_induced__is__subgraph,axiom,
! [V4: set_a] :
( ( ord_less_eq_set_a @ V4 @ vertices )
=> ( undire7103218114511261257raph_a @ V4 @ ( undire7777452895879145676dges_a @ edges @ V4 ) @ vertices @ edges ) ) ).
% induced_is_subgraph
thf(fact_1238_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_1239_gnew_Oinduced__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 @ e @ ( sup_sup_set_a @ S @ T2 ) ) )
=> ( ord_le3724670747650509150_set_a @ EH1 @ ( undire7777452895879145676dges_a @ e @ S ) ) ) ) ) ) ) ).
% gnew.induced_edges_union
thf(fact_1240_induced__union__subgraph,axiom,
! [VH1: set_a,S: set_a,VH2: set_a,T2: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T2 )
=> ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
=> ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
=> ( ( ( undire7103218114511261257raph_a @ VH1 @ EH1 @ S @ ( undire7777452895879145676dges_a @ edges @ S ) )
& ( undire7103218114511261257raph_a @ VH2 @ EH2 @ T2 @ ( undire7777452895879145676dges_a @ edges @ T2 ) ) )
= ( undire7103218114511261257raph_a @ ( sup_sup_set_a @ VH1 @ VH2 ) @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( sup_sup_set_a @ S @ T2 ) @ ( undire7777452895879145676dges_a @ edges @ ( sup_sup_set_a @ S @ T2 ) ) ) ) ) ) ) ) ).
% induced_union_subgraph
thf(fact_1241_gnew_Oinduced__is__graph__sys,axiom,
! [V4: set_a] : ( undire2554140024507503526stem_a @ V4 @ ( undire7777452895879145676dges_a @ e @ V4 ) ) ).
% gnew.induced_is_graph_sys
thf(fact_1242_induced__is__graph__sys,axiom,
! [V4: set_a] : ( undire2554140024507503526stem_a @ V4 @ ( undire7777452895879145676dges_a @ edges @ V4 ) ) ).
% induced_is_graph_sys
thf(fact_1243_gnew_Ograph__system__axioms,axiom,
undire2554140024507503526stem_a @ vertices @ e ).
% gnew.graph_system_axioms
thf(fact_1244_graph__system__axioms,axiom,
undire2554140024507503526stem_a @ vertices @ edges ).
% graph_system_axioms
thf(fact_1245_gnew_Oinduced__edges__union__subgraph__single,axiom,
! [VH1: set_a,S: set_a,VH2: set_a,T2: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T2 )
=> ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
=> ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
=> ( ( undire7103218114511261257raph_a @ ( sup_sup_set_a @ VH1 @ VH2 ) @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( sup_sup_set_a @ S @ T2 ) @ ( undire7777452895879145676dges_a @ e @ ( sup_sup_set_a @ S @ T2 ) ) )
=> ( undire7103218114511261257raph_a @ VH1 @ EH1 @ S @ ( undire7777452895879145676dges_a @ e @ S ) ) ) ) ) ) ) ).
% gnew.induced_edges_union_subgraph_single
thf(fact_1246_gnew_Oinduced__union__subgraph,axiom,
! [VH1: set_a,S: set_a,VH2: set_a,T2: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T2 )
=> ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
=> ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
=> ( ( ( undire7103218114511261257raph_a @ VH1 @ EH1 @ S @ ( undire7777452895879145676dges_a @ e @ S ) )
& ( undire7103218114511261257raph_a @ VH2 @ EH2 @ T2 @ ( undire7777452895879145676dges_a @ e @ T2 ) ) )
= ( undire7103218114511261257raph_a @ ( sup_sup_set_a @ VH1 @ VH2 ) @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( sup_sup_set_a @ S @ T2 ) @ ( undire7777452895879145676dges_a @ e @ ( sup_sup_set_a @ S @ T2 ) ) ) ) ) ) ) ) ).
% gnew.induced_union_subgraph
thf(fact_1247_induced__edges__union__subgraph__single,axiom,
! [VH1: set_a,S: set_a,VH2: set_a,T2: set_a,EH1: set_set_a,EH2: set_set_a] :
( ( ord_less_eq_set_a @ VH1 @ S )
=> ( ( ord_less_eq_set_a @ VH2 @ T2 )
=> ( ( undire2554140024507503526stem_a @ VH1 @ EH1 )
=> ( ( undire2554140024507503526stem_a @ VH2 @ EH2 )
=> ( ( undire7103218114511261257raph_a @ ( sup_sup_set_a @ VH1 @ VH2 ) @ ( sup_sup_set_set_a @ EH1 @ EH2 ) @ ( sup_sup_set_a @ S @ T2 ) @ ( undire7777452895879145676dges_a @ edges @ ( sup_sup_set_a @ S @ T2 ) ) )
=> ( undire7103218114511261257raph_a @ VH1 @ EH1 @ S @ ( undire7777452895879145676dges_a @ edges @ S ) ) ) ) ) ) ) ).
% induced_edges_union_subgraph_single
thf(fact_1248_ulgraph__axioms,axiom,
undire7251896706689453996raph_a @ vertices @ edges ).
% ulgraph_axioms
thf(fact_1249_gnew_Oulgraph__axioms,axiom,
undire7251896706689453996raph_a @ vertices @ e ).
% gnew.ulgraph_axioms
thf(fact_1250_is__complement__edges,axiom,
! [V4: set_a,E4: set_set_a] :
( ( undire8013100667316154652ment_a @ vertices @ edges @ ( produc2116933609460601975_set_a @ V4 @ E4 ) )
= ( ( vertices = V4 )
& ( ( undire4625228487420481630dges_a @ vertices @ edges )
= E4 ) ) ) ).
% is_complement_edges
thf(fact_1251_gnew_Ois__complement__edges,axiom,
! [V4: set_a,E4: set_set_a] :
( ( undire8013100667316154652ment_a @ vertices @ e @ ( produc2116933609460601975_set_a @ V4 @ E4 ) )
= ( ( vertices = V4 )
& ( ( undire4625228487420481630dges_a @ vertices @ e )
= E4 ) ) ) ).
% gnew.is_complement_edges
thf(fact_1252_complement__edges__def,axiom,
( ( undire4625228487420481630dges_a @ vertices @ edges )
= ( minus_5736297505244876581_set_a @ ( undire2918257014606996450dges_a @ vertices ) @ edges ) ) ).
% complement_edges_def
thf(fact_1253_gnew_Ocomplement__edges__def,axiom,
( ( undire4625228487420481630dges_a @ vertices @ e )
= ( minus_5736297505244876581_set_a @ ( undire2918257014606996450dges_a @ vertices ) @ e ) ) ).
% gnew.complement_edges_def
thf(fact_1254_edge__density__def,axiom,
! [X5: set_a,Y7: set_a] :
( ( undire297304480579013331sity_a @ edges @ X5 @ Y7 )
= ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ edges @ X5 @ Y7 ) ) ) @ ( semiri5074537144036343181t_real @ ( times_times_nat @ ( finite_card_a @ X5 ) @ ( finite_card_a @ Y7 ) ) ) ) ) ).
% edge_density_def
thf(fact_1255_diff__diff__cancel,axiom,
! [I: nat,N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( minus_minus_nat @ N3 @ ( minus_minus_nat @ N3 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1256_diff__is__0__eq,axiom,
! [M4: nat,N3: nat] :
( ( ( minus_minus_nat @ M4 @ N3 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M4 @ N3 ) ) ).
% diff_is_0_eq
thf(fact_1257_diff__is__0__eq_H,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ( minus_minus_nat @ M4 @ N3 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1258_gnew_Oedge__density__def,axiom,
! [X5: set_a,Y7: set_a] :
( ( undire297304480579013331sity_a @ e @ X5 @ Y7 )
= ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( finite4795055649997197647od_a_a @ ( undire8383842906760478443ween_a @ e @ X5 @ Y7 ) ) ) @ ( semiri5074537144036343181t_real @ ( times_times_nat @ ( finite_card_a @ X5 ) @ ( finite_card_a @ Y7 ) ) ) ) ) ).
% gnew.edge_density_def
thf(fact_1259_diff__le__mono2,axiom,
! [M4: nat,N3: nat,L: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N3 ) @ ( minus_minus_nat @ L @ M4 ) ) ) ).
% diff_le_mono2
thf(fact_1260_le__diff__iff_H,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1261_diff__le__self,axiom,
! [M4: nat,N3: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ N3 ) @ M4 ) ).
% diff_le_self
thf(fact_1262_diff__le__mono,axiom,
! [M4: nat,N3: nat,L: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ L ) @ ( minus_minus_nat @ N3 @ L ) ) ) ).
% diff_le_mono
thf(fact_1263_Nat_Odiff__diff__eq,axiom,
! [K: nat,M4: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N3 @ K ) )
= ( minus_minus_nat @ M4 @ N3 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1264_le__diff__iff,axiom,
! [K: nat,M4: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N3 @ K ) )
= ( ord_less_eq_nat @ M4 @ N3 ) ) ) ) ).
% le_diff_iff
thf(fact_1265_eq__diff__iff,axiom,
! [K: nat,M4: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( ( minus_minus_nat @ M4 @ K )
= ( minus_minus_nat @ N3 @ K ) )
= ( M4 = N3 ) ) ) ) ).
% eq_diff_iff
thf(fact_1266_real__of__nat__div3,axiom,
! [N3: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N3 @ X ) ) ) @ one_one_real ) ).
% real_of_nat_div3
thf(fact_1267_complete__real,axiom,
! [S: set_real] :
( ? [X6: real] : ( member_real @ X6 @ S )
=> ( ? [Z6: real] :
! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ord_less_eq_real @ X4 @ Z6 ) )
=> ? [Y5: real] :
( ! [X6: real] :
( ( member_real @ X6 @ S )
=> ( ord_less_eq_real @ X6 @ Y5 ) )
& ! [Z6: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ord_less_eq_real @ X4 @ Z6 ) )
=> ( ord_less_eq_real @ Y5 @ Z6 ) ) ) ) ) ).
% complete_real
thf(fact_1268_real__of__nat__div4,axiom,
! [N3: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N3 @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_1269_real__of__nat__div2,axiom,
! [N3: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N3 @ X ) ) ) ) ).
% real_of_nat_div2
thf(fact_1270_times__div__less__eq__dividend,axiom,
! [N3: nat,M4: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N3 @ ( divide_divide_nat @ M4 @ N3 ) ) @ M4 ) ).
% times_div_less_eq_dividend
thf(fact_1271_div__times__less__eq__dividend,axiom,
! [M4: nat,N3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M4 @ N3 ) @ N3 ) @ M4 ) ).
% div_times_less_eq_dividend
thf(fact_1272_div__le__mono,axiom,
! [M4: nat,N3: nat,K: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M4 @ K ) @ ( divide_divide_nat @ N3 @ K ) ) ) ).
% div_le_mono
thf(fact_1273_div__le__dividend,axiom,
! [M4: nat,N3: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M4 @ N3 ) @ M4 ) ).
% div_le_dividend
thf(fact_1274_mult__le__cancel2,axiom,
! [M4: nat,K: nat,N3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M4 @ K ) @ ( times_times_nat @ N3 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M4 @ N3 ) ) ) ).
% mult_le_cancel2
thf(fact_1275_div__greater__zero__iff,axiom,
! [M4: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M4 @ N3 ) )
= ( ( ord_less_eq_nat @ N3 @ M4 )
& ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% div_greater_zero_iff
thf(fact_1276_div__le__mono2,axiom,
! [M4: nat,N3: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N3 ) @ ( divide_divide_nat @ K @ M4 ) ) ) ) ).
% div_le_mono2
% Conjectures (1)
thf(conj_0,conjecture,
member_set_a @ ( insert_a @ x @ ( insert_a @ y @ bot_bot_set_a ) ) @ edges ).
%------------------------------------------------------------------------------