TPTP Problem File: SLH0857^1.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 : SCC_Bloemen_Sequential/0000_SCC_Bloemen_Sequential/prob_00637_022320__5768642_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1448 ( 550 unt; 172 typ; 0 def)
% Number of atoms : 3608 (1171 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 11292 ( 431 ~; 54 |; 231 &;8840 @)
% ( 0 <=>;1736 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 19 ( 18 usr)
% Number of type conns : 572 ( 572 >; 0 *; 0 +; 0 <<)
% Number of symbols : 155 ( 154 usr; 13 con; 0-9 aty)
% Number of variables : 3608 ( 178 ^;3331 !; 99 ?;3608 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:50:06.361
%------------------------------------------------------------------------------
% Could-be-implicit typings (18)
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% Explicit typings (154)
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produc4031800376763917143od_v_v: product_prod_v_v > product_prod_v_v > produc206430290419586791od_v_v ).
thf(sy_c_Product__Type_OPair_001tf__v_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J,type,
produc3862955338007567901t_unit: v > sCC_Bl1394983891496994913t_unit > produc5741669702376414499t_unit ).
thf(sy_c_Product__Type_OPair_001tf__v_001tf__v,type,
product_Pair_v_v: v > v > product_prod_v_v ).
thf(sy_c_Relation_ODomain_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
domain6359000466948879308od_v_v: set_Pr2149350503807050951od_v_v > set_Product_prod_v_v ).
thf(sy_c_Relation_ODomain_001tf__v_001tf__v,type,
domain_v_v: set_Product_prod_v_v > set_v ).
thf(sy_c_Relation_OField_001t__Nat__Onat,type,
field_nat: set_Pr1261947904930325089at_nat > set_nat ).
thf(sy_c_Relation_OField_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
field_7153129647634986036od_v_v: set_Pr2149350503807050951od_v_v > set_Product_prod_v_v ).
thf(sy_c_Relation_OField_001tf__v,type,
field_v: set_Product_prod_v_v > set_v ).
thf(sy_c_Relation_OId_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
id_Product_prod_v_v: set_Pr2149350503807050951od_v_v ).
thf(sy_c_Relation_OId_001tf__v,type,
id_v: set_Product_prod_v_v ).
thf(sy_c_Relation_ORange_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
range_7878975032137371189od_v_v: set_Pr2149350503807050951od_v_v > set_Product_prod_v_v ).
thf(sy_c_Relation_ORange_001tf__v_001tf__v,type,
range_v_v: set_Product_prod_v_v > set_v ).
thf(sy_c_Relation_Orefl__on_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
refl_o4548774019903118566od_v_v: set_Product_prod_v_v > set_Pr2149350503807050951od_v_v > $o ).
thf(sy_c_Relation_Orefl__on_001tf__v,type,
refl_on_v: set_v > set_Product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_O_092_060S_062_001tf__v_001tf__a,type,
sCC_Bloemen_S_v_a: sCC_Bl1191828773336950226xt_v_a > v > set_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_O_092_060S_062__update_001tf__v_001tf__a,type,
sCC_Bl4466863261252462032te_v_a: ( ( v > set_v ) > v > set_v ) > sCC_Bl1191828773336950226xt_v_a > sCC_Bl1191828773336950226xt_v_a ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Oenv__ext_001tf__v_001tf__a,type,
sCC_Bl3252069758149425774xt_v_a: v > ( v > set_v ) > set_v > set_v > ( v > set_v ) > set_set_v > list_v > list_v > a > sCC_Bl1191828773336950226xt_v_a ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Oexplored_001tf__v_001tf__a,type,
sCC_Bl6885986953353844043ed_v_a: sCC_Bl1191828773336950226xt_v_a > set_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ostack_001tf__v_001tf__a,type,
sCC_Bl1791845272665611460ck_v_a: sCC_Bl1191828773336950226xt_v_a > list_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ostack__update_001tf__v_001tf__a,type,
sCC_Bl6443695954061435821te_v_a: ( list_v > list_v ) > sCC_Bl1191828773336950226xt_v_a > sCC_Bl1191828773336950226xt_v_a ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ovisited_001tf__v_001tf__a,type,
sCC_Bl1198488560823802982ed_v_a: sCC_Bl1191828773336950226xt_v_a > set_v ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_001t__Nat__Onat,type,
sCC_Bl8035451632035226289ph_nat: set_nat > ( nat > set_nat ) > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl8307124943676871238od_v_v: set_Product_prod_v_v > ( product_prod_v_v > set_Product_prod_v_v ) > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_001tf__v,type,
sCC_Bloemen_graph_v: set_v > ( v > set_v ) > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__scc_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl6242042402218619277od_v_v: ( product_prod_v_v > set_Product_prod_v_v ) > set_Product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__scc_001tf__v,type,
sCC_Bloemen_is_scc_v: ( v > set_v ) > set_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__subscc_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl2301996248249672505od_v_v: ( product_prod_v_v > set_Product_prod_v_v ) > set_Product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__subscc_001tf__v,type,
sCC_Bl5398416737448265317bscc_v: ( v > set_v ) > set_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl4981926079593201289od_v_v: ( product_prod_v_v > set_Product_prod_v_v ) > product_prod_v_v > product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable_001tf__v,type,
sCC_Bl649662514949026229able_v: ( v > set_v ) > v > v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable__avoiding_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl5370300055464682748od_v_v: ( product_prod_v_v > set_Product_prod_v_v ) > product_prod_v_v > product_prod_v_v > set_Pr2149350503807050951od_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable__avoiding_001tf__v,type,
sCC_Bl4291963740693775144ding_v: ( v > set_v ) > v > v > set_Product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable__end_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl4714988717384592488od_v_v: ( product_prod_v_v > set_Product_prod_v_v ) > product_prod_v_v > product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable__end_001tf__v,type,
sCC_Bl770211535891879572_end_v: ( v > set_v ) > v > v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Owf__env_001tf__v_001tf__a,type,
sCC_Bl4124178362578471481nv_v_a: ( v > set_v ) > sCC_Bl1191828773336950226xt_v_a > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Oprecedes_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl2026170059108282219od_v_v: product_prod_v_v > product_prod_v_v > list_P7986770385144383213od_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Oprecedes_001tf__v,type,
sCC_Bl4022239298816431255edes_v: v > v > list_v > $o ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
collec140062887454715474od_v_v: ( product_prod_v_v > $o ) > set_Product_prod_v_v ).
thf(sy_c_Set_OCollect_001tf__v,type,
collect_v: ( v > $o ) > set_v ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
insert5641704497130386615od_v_v: produc206430290419586791od_v_v > set_Pr2149350503807050951od_v_v > set_Pr2149350503807050951od_v_v ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
insert1338601472111419319od_v_v: product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
insert7504383016908236695od_v_v: set_Product_prod_v_v > set_se8455005133513928103od_v_v > set_se8455005133513928103od_v_v ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__v_J,type,
insert_set_v: set_v > set_set_v > set_set_v ).
thf(sy_c_Set_Oinsert_001tf__v,type,
insert_v2: v > set_v > set_v ).
thf(sy_c_Set_Ois__empty_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
is_emp8964507351669718201od_v_v: set_Product_prod_v_v > $o ).
thf(sy_c_Set_Ois__empty_001tf__v,type,
is_empty_v: set_v > $o ).
thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
is_sin9198872032823709915od_v_v: set_Product_prod_v_v > $o ).
thf(sy_c_Set_Ois__singleton_001tf__v,type,
is_singleton_v: set_v > $o ).
thf(sy_c_Set_Opairwise_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
pairwi5745945156428401490od_v_v: ( product_prod_v_v > product_prod_v_v > $o ) > set_Product_prod_v_v > $o ).
thf(sy_c_Set_Opairwise_001tf__v,type,
pairwise_v: ( v > v > $o ) > set_v > $o ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oremove_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
remove5001965847480235980od_v_v: product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Set_Oremove_001tf__v,type,
remove_v: v > set_v > set_v ).
thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
the_el5392834299063928540od_v_v: set_Product_prod_v_v > product_prod_v_v ).
thf(sy_c_Set_Othe__elem_001tf__v,type,
the_elem_v: set_v > v ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__v_Mtf__v_J_Mt__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
member3038538357316246288od_v_v: produc206430290419586791od_v_v > set_Pr2149350503807050951od_v_v > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
member7453568604450474000od_v_v: product_prod_v_v > set_Product_prod_v_v > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
member8406446414694345712od_v_v: set_Product_prod_v_v > set_se8455005133513928103od_v_v > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__v_J,type,
member_set_v: set_v > set_set_v > $o ).
thf(sy_c_member_001tf__v,type,
member_v2: v > set_v > $o ).
thf(sy_v_e,type,
e: sCC_Bl1191828773336950226xt_v_a ).
thf(sy_v_m,type,
m: v ).
thf(sy_v_successors,type,
successors: v > set_v ).
thf(sy_v_thesis,type,
thesis: $o ).
thf(sy_v_vertices,type,
vertices: set_v ).
% Relevant facts (1274)
thf(fact_0_assms_I3_J,axiom,
~ ( member_v2 @ m @ ( sCC_Bl6885986953353844043ed_v_a @ e ) ) ).
% assms(3)
thf(fact_1_assms_I2_J,axiom,
member_v2 @ m @ ( sCC_Bl1198488560823802982ed_v_a @ e ) ).
% assms(2)
thf(fact_2_that,axiom,
! [N: v] :
( ( member_v2 @ N @ ( set_v2 @ ( sCC_Bl1791845272665611460ck_v_a @ e ) ) )
=> ( ( member_v2 @ m @ ( sCC_Bloemen_S_v_a @ e @ N ) )
=> thesis ) ) ).
% that
thf(fact_3_fold__congs_I7_J,axiom,
! [R: sCC_Bl1191828773336950226xt_v_a,R2: sCC_Bl1191828773336950226xt_v_a,V: list_v,F: list_v > list_v,F2: list_v > list_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl1791845272665611460ck_v_a @ R2 )
= V )
=> ( ! [V2: list_v] :
( ( V = V2 )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl6443695954061435821te_v_a @ F @ R )
= ( sCC_Bl6443695954061435821te_v_a @ F2 @ R2 ) ) ) ) ) ).
% fold_congs(7)
thf(fact_4_unfold__congs_I7_J,axiom,
! [R: sCC_Bl1191828773336950226xt_v_a,R2: sCC_Bl1191828773336950226xt_v_a,V: list_v,F: list_v > list_v,F2: list_v > list_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl1791845272665611460ck_v_a @ R2 )
= V )
=> ( ! [V2: list_v] :
( ( V2 = V )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl6443695954061435821te_v_a @ F @ R )
= ( sCC_Bl6443695954061435821te_v_a @ F2 @ R2 ) ) ) ) ) ).
% unfold_congs(7)
thf(fact_5_fold__congs_I2_J,axiom,
! [R: sCC_Bl1191828773336950226xt_v_a,R2: sCC_Bl1191828773336950226xt_v_a,V: v > set_v,F: ( v > set_v ) > v > set_v,F2: ( v > set_v ) > v > set_v] :
( ( R = R2 )
=> ( ( ( sCC_Bloemen_S_v_a @ R2 )
= V )
=> ( ! [V2: v > set_v] :
( ( V = V2 )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl4466863261252462032te_v_a @ F @ R )
= ( sCC_Bl4466863261252462032te_v_a @ F2 @ R2 ) ) ) ) ) ).
% fold_congs(2)
thf(fact_6_unfold__congs_I2_J,axiom,
! [R: sCC_Bl1191828773336950226xt_v_a,R2: sCC_Bl1191828773336950226xt_v_a,V: v > set_v,F: ( v > set_v ) > v > set_v,F2: ( v > set_v ) > v > set_v] :
( ( R = R2 )
=> ( ( ( sCC_Bloemen_S_v_a @ R2 )
= V )
=> ( ! [V2: v > set_v] :
( ( V2 = V )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl4466863261252462032te_v_a @ F @ R )
= ( sCC_Bl4466863261252462032te_v_a @ F2 @ R2 ) ) ) ) ) ).
% unfold_congs(2)
thf(fact_7_assms_I1_J,axiom,
sCC_Bl4124178362578471481nv_v_a @ successors @ e ).
% assms(1)
thf(fact_8_precedes__refl,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X @ X @ Xs )
= ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_refl
thf(fact_9_precedes__refl,axiom,
! [X: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ X @ Xs )
= ( member_v2 @ X @ ( set_v2 @ Xs ) ) ) ).
% precedes_refl
thf(fact_10_precedes__mem_I2_J,axiom,
! [X: product_prod_v_v,Y: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ Xs )
=> ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_mem(2)
thf(fact_11_precedes__mem_I2_J,axiom,
! [X: v,Y: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( member_v2 @ Y @ ( set_v2 @ Xs ) ) ) ).
% precedes_mem(2)
thf(fact_12_precedes__mem_I1_J,axiom,
! [X: product_prod_v_v,Y: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ Xs )
=> ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_mem(1)
thf(fact_13_precedes__mem_I1_J,axiom,
! [X: v,Y: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( member_v2 @ X @ ( set_v2 @ Xs ) ) ) ).
% precedes_mem(1)
thf(fact_14_select__convs_I7_J,axiom,
! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: a] :
( ( sCC_Bl1791845272665611460ck_v_a @ ( sCC_Bl3252069758149425774xt_v_a @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Stack ) ).
% select_convs(7)
thf(fact_15_select__convs_I2_J,axiom,
! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: a] :
( ( sCC_Bloemen_S_v_a @ ( sCC_Bl3252069758149425774xt_v_a @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= S ) ).
% select_convs(2)
thf(fact_16_stack__visited,axiom,
! [E: sCC_Bl1191828773336950226xt_v_a,N: v] :
( ( sCC_Bl4124178362578471481nv_v_a @ successors @ E )
=> ( ( member_v2 @ N @ ( set_v2 @ ( sCC_Bl1791845272665611460ck_v_a @ E ) ) )
=> ( member_v2 @ N @ ( sCC_Bl1198488560823802982ed_v_a @ E ) ) ) ) ).
% stack_visited
thf(fact_17_in__set__member,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
= ( member6878703317195979394od_v_v @ Xs @ X ) ) ).
% in_set_member
thf(fact_18_in__set__member,axiom,
! [X: v,Xs: list_v] :
( ( member_v2 @ X @ ( set_v2 @ Xs ) )
= ( member_v @ Xs @ X ) ) ).
% in_set_member
thf(fact_19_select__convs_I4_J,axiom,
! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: a] :
( ( sCC_Bl1198488560823802982ed_v_a @ ( sCC_Bl3252069758149425774xt_v_a @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Visited ) ).
% select_convs(4)
thf(fact_20_select__convs_I3_J,axiom,
! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: a] :
( ( sCC_Bl6885986953353844043ed_v_a @ ( sCC_Bl3252069758149425774xt_v_a @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Explored ) ).
% select_convs(3)
thf(fact_21_update__convs_I7_J,axiom,
! [Stack2: list_v > list_v,Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: a] :
( ( sCC_Bl6443695954061435821te_v_a @ Stack2 @ ( sCC_Bl3252069758149425774xt_v_a @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= ( sCC_Bl3252069758149425774xt_v_a @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ ( Stack2 @ Stack ) @ Cstack @ More ) ) ).
% update_convs(7)
thf(fact_22_update__convs_I2_J,axiom,
! [S2: ( v > set_v ) > v > set_v,Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: a] :
( ( sCC_Bl4466863261252462032te_v_a @ S2 @ ( sCC_Bl3252069758149425774xt_v_a @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= ( sCC_Bl3252069758149425774xt_v_a @ Root @ ( S2 @ S ) @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) ) ).
% update_convs(2)
thf(fact_23_graph_Owf__env_Ocong,axiom,
sCC_Bl4124178362578471481nv_v_a = sCC_Bl4124178362578471481nv_v_a ).
% graph.wf_env.cong
thf(fact_24_stack__class,axiom,
! [E: sCC_Bl1191828773336950226xt_v_a,N: v,M: v] :
( ( sCC_Bl4124178362578471481nv_v_a @ successors @ E )
=> ( ( member_v2 @ N @ ( set_v2 @ ( sCC_Bl1791845272665611460ck_v_a @ E ) ) )
=> ( ( member_v2 @ M @ ( sCC_Bloemen_S_v_a @ E @ N ) )
=> ( member_v2 @ M @ ( minus_minus_set_v @ ( sCC_Bl1198488560823802982ed_v_a @ E ) @ ( sCC_Bl6885986953353844043ed_v_a @ E ) ) ) ) ) ) ).
% stack_class
thf(fact_25_reachable__end_Ocases,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ Y2 )
=> ~ ( member_v2 @ A2 @ ( successors @ Y2 ) ) ) ) ) ).
% reachable_end.cases
thf(fact_26_re__refl,axiom,
! [X: v] : ( sCC_Bl770211535891879572_end_v @ successors @ X @ X ) ).
% re_refl
thf(fact_27_re__succ,axiom,
! [X: v,Y: v,Z: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y )
=> ( ( member_v2 @ Z @ ( successors @ Y ) )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Z ) ) ) ).
% re_succ
thf(fact_28_reachable__end_Osimps,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A2 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: v,Y3: v,Z2: v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ Y3 )
& ( member_v2 @ Z2 @ ( successors @ Y3 ) ) ) ) ) ).
% reachable_end.simps
thf(fact_29_succ__re,axiom,
! [Y: v,X: v,Z: v] :
( ( member_v2 @ Y @ ( successors @ X ) )
=> ( ( sCC_Bl770211535891879572_end_v @ successors @ Y @ Z )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Z ) ) ) ).
% succ_re
thf(fact_30_graph_Ostack__visited,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1191828773336950226xt_v_a,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4124178362578471481nv_v_a @ Successors @ E )
=> ( ( member_v2 @ N @ ( set_v2 @ ( sCC_Bl1791845272665611460ck_v_a @ E ) ) )
=> ( member_v2 @ N @ ( sCC_Bl1198488560823802982ed_v_a @ E ) ) ) ) ) ).
% graph.stack_visited
thf(fact_31_ra__refl,axiom,
! [X: v,E2: set_Product_prod_v_v] : ( sCC_Bl4291963740693775144ding_v @ successors @ X @ X @ E2 ) ).
% ra_refl
thf(fact_32_ra__trans,axiom,
! [X: v,Y: v,E2: set_Product_prod_v_v,Z: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E2 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ Y @ Z @ E2 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Z @ E2 ) ) ) ).
% ra_trans
thf(fact_33_graph__axioms,axiom,
sCC_Bloemen_graph_v @ vertices @ successors ).
% graph_axioms
thf(fact_34_reachable_Ocases,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: v] :
( ( member_v2 @ Y2 @ ( successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ A2 ) ) ) ) ).
% reachable.cases
thf(fact_35_succ__reachable,axiom,
! [X: v,Y: v,Z: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( member_v2 @ Z @ ( successors @ Y ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% succ_reachable
thf(fact_36_reachable__trans,axiom,
! [X: v,Y: v,Z: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% reachable_trans
thf(fact_37_reachable__end__induct,axiom,
! [X: v,Y: v,P: v > v > $o] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ! [X3: v] : ( P @ X3 @ X3 )
=> ( ! [X3: v,Y2: v,Z3: v] :
( ( P @ X3 @ Y2 )
=> ( ( member_v2 @ Z3 @ ( successors @ Y2 ) )
=> ( P @ X3 @ Z3 ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% reachable_end_induct
thf(fact_38_reachable__edge,axiom,
! [Y: v,X: v] :
( ( member_v2 @ Y @ ( successors @ X ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% reachable_edge
thf(fact_39_reachable_Osimps,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A2 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: v,Y3: v,Z2: v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( member_v2 @ Y3 @ ( successors @ X2 ) )
& ( sCC_Bl649662514949026229able_v @ successors @ Y3 @ Z2 ) ) ) ) ).
% reachable.simps
thf(fact_40_reachable__succ,axiom,
! [Y: v,X: v,Z: v] :
( ( member_v2 @ Y @ ( successors @ X ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% reachable_succ
thf(fact_41_reachable__refl,axiom,
! [X: v] : ( sCC_Bl649662514949026229able_v @ successors @ X @ X ) ).
% reachable_refl
thf(fact_42_mem__Collect__eq,axiom,
! [A: v,P: v > $o] :
( ( member_v2 @ A @ ( collect_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_43_mem__Collect__eq,axiom,
! [A: product_prod_v_v,P: product_prod_v_v > $o] :
( ( member7453568604450474000od_v_v @ A @ ( collec140062887454715474od_v_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A3: set_v] :
( ( collect_v
@ ^ [X2: v] : ( member_v2 @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A3: set_Product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_46_ra__reachable,axiom,
! [X: v,Y: v,E2: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E2 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% ra_reachable
thf(fact_47_reachable__re,axiom,
! [X: v,Y: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y ) ) ).
% reachable_re
thf(fact_48_re__reachable,axiom,
! [X: v,Y: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% re_reachable
thf(fact_49_sccE,axiom,
! [S3: set_v,X: v,Y: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S3 )
=> ( ( member_v2 @ X @ S3 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ X )
=> ( member_v2 @ Y @ S3 ) ) ) ) ) ).
% sccE
thf(fact_50_graph_Ora__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,E2: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ X @ E2 ) ) ).
% graph.ra_refl
thf(fact_51_graph_Oreachable__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y: product_prod_v_v,X: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X ) )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ Z )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_succ
thf(fact_52_graph_Oreachable__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v2 @ Y @ ( Successors @ X ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_succ
thf(fact_53_graph_Oreachable__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ X ) ) ).
% graph.reachable_refl
thf(fact_54_graph_Oreachable__avoiding_Ocong,axiom,
sCC_Bl4291963740693775144ding_v = sCC_Bl4291963740693775144ding_v ).
% graph.reachable_avoiding.cong
thf(fact_55_graph_Ore__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Y )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.re_succ
thf(fact_56_graph_Ore__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y )
=> ( ( member_v2 @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Z ) ) ) ) ).
% graph.re_succ
thf(fact_57_graph_Ore__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ X ) ) ).
% graph.re_refl
thf(fact_58_graph_Oreachable__end__induct,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,P: product_prod_v_v > product_prod_v_v > $o] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ! [X3: product_prod_v_v] : ( P @ X3 @ X3 )
=> ( ! [X3: product_prod_v_v,Y2: product_prod_v_v,Z3: product_prod_v_v] :
( ( P @ X3 @ Y2 )
=> ( ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ Y2 ) )
=> ( P @ X3 @ Z3 ) ) )
=> ( P @ X @ Y ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_59_graph_Oreachable__end__induct,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,P: v > v > $o] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ! [X3: v] : ( P @ X3 @ X3 )
=> ( ! [X3: v,Y2: v,Z3: v] :
( ( P @ X3 @ Y2 )
=> ( ( member_v2 @ Z3 @ ( Successors @ Y2 ) )
=> ( P @ X3 @ Z3 ) ) )
=> ( P @ X @ Y ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_60_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A2 )
= ( ? [X2: product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: product_prod_v_v,Y3: product_prod_v_v,Z2: product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( sCC_Bl4714988717384592488od_v_v @ Successors @ X2 @ Y3 )
& ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_61_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A2 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: v,Y3: v,Z2: v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( sCC_Bl770211535891879572_end_v @ Successors @ X2 @ Y3 )
& ( member_v2 @ Z2 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_62_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: product_prod_v_v] :
( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ Y2 )
=> ~ ( member7453568604450474000od_v_v @ A2 @ ( Successors @ Y2 ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_63_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: v] :
( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ Y2 )
=> ~ ( member_v2 @ A2 @ ( Successors @ Y2 ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_64_graph_Oreachable__end_Ocong,axiom,
sCC_Bl770211535891879572_end_v = sCC_Bl770211535891879572_end_v ).
% graph.reachable_end.cong
thf(fact_65_graph_Oreachable__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_trans
thf(fact_66_graph_Oreachable_Osimps,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ A1 @ A2 )
= ( ? [X2: product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: product_prod_v_v,Y3: product_prod_v_v,Z2: product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( member7453568604450474000od_v_v @ Y3 @ ( Successors @ X2 ) )
& ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y3 @ Z2 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_67_graph_Oreachable_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ A1 @ A2 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A2 = X2 ) )
| ? [X2: v,Y3: v,Z2: v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( member_v2 @ Y3 @ ( Successors @ X2 ) )
& ( sCC_Bl649662514949026229able_v @ Successors @ Y3 @ Z2 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_68_graph_Oreachable_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y2 @ A2 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_69_graph_Oreachable_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: v] :
( ( member_v2 @ Y2 @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ A2 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_70_graph_Osucc__reachable,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_reachable
thf(fact_71_graph_Osucc__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( member_v2 @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_reachable
thf(fact_72_graph_Oreachable__edge,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y: product_prod_v_v,X: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y ) ) ) ).
% graph.reachable_edge
thf(fact_73_graph_Oreachable__edge,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v2 @ Y @ ( Successors @ X ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y ) ) ) ).
% graph.reachable_edge
thf(fact_74_graph_Oreachable_Ocong,axiom,
sCC_Bl649662514949026229able_v = sCC_Bl649662514949026229able_v ).
% graph.reachable.cong
thf(fact_75_graph_Oreachable__re,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y ) ) ) ).
% graph.reachable_re
thf(fact_76_graph_Ore__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y ) ) ) ).
% graph.re_reachable
thf(fact_77_graph_Ora__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E2: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E2 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y ) ) ) ).
% graph.ra_reachable
thf(fact_78_graph_Ora__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E2: set_Product_prod_v_v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E2 )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ Y @ Z @ E2 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Z @ E2 ) ) ) ) ).
% graph.ra_trans
thf(fact_79_graph_Osucc__re,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y: product_prod_v_v,X: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X ) )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ Y @ Z )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_re
thf(fact_80_graph_Osucc__re,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v2 @ Y @ ( Successors @ X ) )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ Y @ Z )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_re
thf(fact_81_graph_Ostack__class,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1191828773336950226xt_v_a,N: v,M: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4124178362578471481nv_v_a @ Successors @ E )
=> ( ( member_v2 @ N @ ( set_v2 @ ( sCC_Bl1791845272665611460ck_v_a @ E ) ) )
=> ( ( member_v2 @ M @ ( sCC_Bloemen_S_v_a @ E @ N ) )
=> ( member_v2 @ M @ ( minus_minus_set_v @ ( sCC_Bl1198488560823802982ed_v_a @ E ) @ ( sCC_Bl6885986953353844043ed_v_a @ E ) ) ) ) ) ) ) ).
% graph.stack_class
thf(fact_82_is__subscc__def,axiom,
! [S3: set_v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S3 )
= ( ! [X2: v] :
( ( member_v2 @ X2 @ S3 )
=> ! [Y3: v] :
( ( member_v2 @ Y3 @ S3 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y3 ) ) ) ) ) ).
% is_subscc_def
thf(fact_83_ra__empty,axiom,
! [X: v,Y: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ bot_bo723834152578015283od_v_v )
= ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% ra_empty
thf(fact_84_sclosed,axiom,
! [X4: v] :
( ( member_v2 @ X4 @ vertices )
=> ( ord_less_eq_set_v @ ( successors @ X4 ) @ vertices ) ) ).
% sclosed
thf(fact_85_Diff__iff,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v2 @ C @ ( minus_minus_set_v @ A3 @ B ) )
= ( ( member_v2 @ C @ A3 )
& ~ ( member_v2 @ C @ B ) ) ) ).
% Diff_iff
thf(fact_86_Diff__iff,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
= ( ( member7453568604450474000od_v_v @ C @ A3 )
& ~ ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% Diff_iff
thf(fact_87_DiffI,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v2 @ C @ A3 )
=> ( ~ ( member_v2 @ C @ B )
=> ( member_v2 @ C @ ( minus_minus_set_v @ A3 @ B ) ) ) ) ).
% DiffI
thf(fact_88_DiffI,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B ) ) ) ) ).
% DiffI
thf(fact_89_ra__mono,axiom,
! [X: v,Y: v,E2: set_Product_prod_v_v,E3: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E2 )
=> ( ( ord_le7336532860387713383od_v_v @ E3 @ E2 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E3 ) ) ) ).
% ra_mono
thf(fact_90_ra__cases,axiom,
! [X: v,Y: v,E2: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E2 )
=> ( ( X = Y )
| ? [Z3: v] :
( ( member_v2 @ Z3 @ ( successors @ X ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Z3 ) @ E2 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ Z3 @ Y @ E2 ) ) ) ) ).
% ra_cases
thf(fact_91_edge__ra,axiom,
! [Y: v,X: v,E2: set_Product_prod_v_v] :
( ( member_v2 @ Y @ ( successors @ X ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ E2 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E2 ) ) ) ).
% edge_ra
thf(fact_92_reachable__avoiding_Osimps,axiom,
! [A1: v,A2: v,A32: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A2 @ A32 )
= ( ? [X2: v,E4: set_Product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = X2 )
& ( A32 = E4 ) )
| ? [X2: v,Y3: v,E4: set_Product_prod_v_v,Z2: v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( A32 = E4 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y3 @ E4 )
& ( member_v2 @ Z2 @ ( successors @ Y3 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ Z2 ) @ E4 ) ) ) ) ).
% reachable_avoiding.simps
thf(fact_93_reachable__avoiding_Ocases,axiom,
! [A1: v,A2: v,A32: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A2 @ A32 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ Y2 @ A32 )
=> ( ( member_v2 @ A2 @ ( successors @ Y2 ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ A2 ) @ A32 ) ) ) ) ) ).
% reachable_avoiding.cases
thf(fact_94_ra__succ,axiom,
! [X: v,Y: v,E2: set_Product_prod_v_v,Z: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E2 )
=> ( ( member_v2 @ Z @ ( successors @ Y ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ Z ) @ E2 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Z @ E2 ) ) ) ) ).
% ra_succ
thf(fact_95_empty__iff,axiom,
! [C: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ C @ bot_bo723834152578015283od_v_v ) ).
% empty_iff
thf(fact_96_empty__iff,axiom,
! [C: v] :
~ ( member_v2 @ C @ bot_bot_set_v ) ).
% empty_iff
thf(fact_97_all__not__in__conv,axiom,
! [A3: set_Product_prod_v_v] :
( ( ! [X2: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ X2 @ A3 ) )
= ( A3 = bot_bo723834152578015283od_v_v ) ) ).
% all_not_in_conv
thf(fact_98_all__not__in__conv,axiom,
! [A3: set_v] :
( ( ! [X2: v] :
~ ( member_v2 @ X2 @ A3 ) )
= ( A3 = bot_bot_set_v ) ) ).
% all_not_in_conv
thf(fact_99_Collect__empty__eq,axiom,
! [P: product_prod_v_v > $o] :
( ( ( collec140062887454715474od_v_v @ P )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: product_prod_v_v] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_100_Collect__empty__eq,axiom,
! [P: v > $o] :
( ( ( collect_v @ P )
= bot_bot_set_v )
= ( ! [X2: v] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_101_empty__Collect__eq,axiom,
! [P: product_prod_v_v > $o] :
( ( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ P ) )
= ( ! [X2: product_prod_v_v] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_102_empty__Collect__eq,axiom,
! [P: v > $o] :
( ( bot_bot_set_v
= ( collect_v @ P ) )
= ( ! [X2: v] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_103_subsetI,axiom,
! [A3: set_v,B: set_v] :
( ! [X3: v] :
( ( member_v2 @ X3 @ A3 )
=> ( member_v2 @ X3 @ B ) )
=> ( ord_less_eq_set_v @ A3 @ B ) ) ).
% subsetI
thf(fact_104_subsetI,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ( member7453568604450474000od_v_v @ X3 @ B ) )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ).
% subsetI
thf(fact_105_subset__antisym,axiom,
! [A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( ord_less_eq_set_v @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_106_subset__antisym,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_107_subset__empty,axiom,
! [A3: set_v] :
( ( ord_less_eq_set_v @ A3 @ bot_bot_set_v )
= ( A3 = bot_bot_set_v ) ) ).
% subset_empty
thf(fact_108_subset__empty,axiom,
! [A3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
= ( A3 = bot_bo723834152578015283od_v_v ) ) ).
% subset_empty
thf(fact_109_empty__subsetI,axiom,
! [A3: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A3 ) ).
% empty_subsetI
thf(fact_110_empty__subsetI,axiom,
! [A3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A3 ) ).
% empty_subsetI
thf(fact_111_Diff__empty,axiom,
! [A3: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
= A3 ) ).
% Diff_empty
thf(fact_112_Diff__empty,axiom,
! [A3: set_v] :
( ( minus_minus_set_v @ A3 @ bot_bot_set_v )
= A3 ) ).
% Diff_empty
thf(fact_113_empty__Diff,axiom,
! [A3: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ bot_bo723834152578015283od_v_v @ A3 )
= bot_bo723834152578015283od_v_v ) ).
% empty_Diff
thf(fact_114_empty__Diff,axiom,
! [A3: set_v] :
( ( minus_minus_set_v @ bot_bot_set_v @ A3 )
= bot_bot_set_v ) ).
% empty_Diff
thf(fact_115_Diff__cancel,axiom,
! [A3: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ A3 )
= bot_bo723834152578015283od_v_v ) ).
% Diff_cancel
thf(fact_116_Diff__cancel,axiom,
! [A3: set_v] :
( ( minus_minus_set_v @ A3 @ A3 )
= bot_bot_set_v ) ).
% Diff_cancel
thf(fact_117_Diff__eq__empty__iff,axiom,
! [A3: set_v,B: set_v] :
( ( ( minus_minus_set_v @ A3 @ B )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ A3 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_118_Diff__eq__empty__iff,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_119_graph_Ois__scc__def,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S3: set_Product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S3 )
= ( ( S3 != bot_bo723834152578015283od_v_v )
& ( sCC_Bl2301996248249672505od_v_v @ Successors @ S3 )
& ! [S4: set_Product_prod_v_v] :
( ( ( ord_le7336532860387713383od_v_v @ S3 @ S4 )
& ( sCC_Bl2301996248249672505od_v_v @ Successors @ S4 ) )
=> ( S4 = S3 ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_120_graph_Ois__scc__def,axiom,
! [Vertices: set_v,Successors: v > set_v,S3: set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S3 )
= ( ( S3 != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ Successors @ S3 )
& ! [S4: set_v] :
( ( ( ord_less_eq_set_v @ S3 @ S4 )
& ( sCC_Bl5398416737448265317bscc_v @ Successors @ S4 ) )
=> ( S4 = S3 ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_121_emptyE,axiom,
! [A: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ A @ bot_bo723834152578015283od_v_v ) ).
% emptyE
thf(fact_122_emptyE,axiom,
! [A: v] :
~ ( member_v2 @ A @ bot_bot_set_v ) ).
% emptyE
thf(fact_123_in__mono,axiom,
! [A3: set_v,B: set_v,X: v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( member_v2 @ X @ A3 )
=> ( member_v2 @ X @ B ) ) ) ).
% in_mono
thf(fact_124_in__mono,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( member7453568604450474000od_v_v @ X @ B ) ) ) ).
% in_mono
thf(fact_125_subsetD,axiom,
! [A3: set_v,B: set_v,C: v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( member_v2 @ C @ A3 )
=> ( member_v2 @ C @ B ) ) ) ).
% subsetD
thf(fact_126_subsetD,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( member7453568604450474000od_v_v @ C @ A3 )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% subsetD
thf(fact_127_equals0D,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v] :
( ( A3 = bot_bo723834152578015283od_v_v )
=> ~ ( member7453568604450474000od_v_v @ A @ A3 ) ) ).
% equals0D
thf(fact_128_equals0D,axiom,
! [A3: set_v,A: v] :
( ( A3 = bot_bot_set_v )
=> ~ ( member_v2 @ A @ A3 ) ) ).
% equals0D
thf(fact_129_equals0I,axiom,
! [A3: set_Product_prod_v_v] :
( ! [Y2: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ Y2 @ A3 )
=> ( A3 = bot_bo723834152578015283od_v_v ) ) ).
% equals0I
thf(fact_130_equals0I,axiom,
! [A3: set_v] :
( ! [Y2: v] :
~ ( member_v2 @ Y2 @ A3 )
=> ( A3 = bot_bot_set_v ) ) ).
% equals0I
thf(fact_131_equalityE,axiom,
! [A3: set_v,B: set_v] :
( ( A3 = B )
=> ~ ( ( ord_less_eq_set_v @ A3 @ B )
=> ~ ( ord_less_eq_set_v @ B @ A3 ) ) ) ).
% equalityE
thf(fact_132_equalityE,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A3 = B )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ~ ( ord_le7336532860387713383od_v_v @ B @ A3 ) ) ) ).
% equalityE
thf(fact_133_subset__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B2: set_v] :
! [X2: v] :
( ( member_v2 @ X2 @ A4 )
=> ( member_v2 @ X2 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_134_subset__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A4 )
=> ( member7453568604450474000od_v_v @ X2 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_135_equalityD1,axiom,
! [A3: set_v,B: set_v] :
( ( A3 = B )
=> ( ord_less_eq_set_v @ A3 @ B ) ) ).
% equalityD1
thf(fact_136_equalityD1,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A3 = B )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ).
% equalityD1
thf(fact_137_equalityD2,axiom,
! [A3: set_v,B: set_v] :
( ( A3 = B )
=> ( ord_less_eq_set_v @ B @ A3 ) ) ).
% equalityD2
thf(fact_138_equalityD2,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A3 = B )
=> ( ord_le7336532860387713383od_v_v @ B @ A3 ) ) ).
% equalityD2
thf(fact_139_ex__in__conv,axiom,
! [A3: set_Product_prod_v_v] :
( ( ? [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A3 ) )
= ( A3 != bot_bo723834152578015283od_v_v ) ) ).
% ex_in_conv
thf(fact_140_ex__in__conv,axiom,
! [A3: set_v] :
( ( ? [X2: v] : ( member_v2 @ X2 @ A3 ) )
= ( A3 != bot_bot_set_v ) ) ).
% ex_in_conv
thf(fact_141_subset__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B2: set_v] :
! [T: v] :
( ( member_v2 @ T @ A4 )
=> ( member_v2 @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_142_subset__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
! [T: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ T @ A4 )
=> ( member7453568604450474000od_v_v @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_143_subset__refl,axiom,
! [A3: set_v] : ( ord_less_eq_set_v @ A3 @ A3 ) ).
% subset_refl
thf(fact_144_subset__refl,axiom,
! [A3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A3 @ A3 ) ).
% subset_refl
thf(fact_145_Collect__mono,axiom,
! [P: v > $o,Q: v > $o] :
( ! [X3: v] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) ) ) ).
% Collect_mono
thf(fact_146_Collect__mono,axiom,
! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ! [X3: product_prod_v_v] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) ) ) ).
% Collect_mono
thf(fact_147_subset__trans,axiom,
! [A3: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ord_less_eq_set_v @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_148_subset__trans,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ord_le7336532860387713383od_v_v @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_149_set__eq__subset,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A4: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A4 @ B2 )
& ( ord_less_eq_set_v @ B2 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_150_set__eq__subset,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
& ( ord_le7336532860387713383od_v_v @ B2 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_151_Collect__mono__iff,axiom,
! [P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) )
= ( ! [X2: v] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_152_Collect__mono__iff,axiom,
! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) )
= ( ! [X2: product_prod_v_v] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_153_graph_Ois__subscc_Ocong,axiom,
sCC_Bl5398416737448265317bscc_v = sCC_Bl5398416737448265317bscc_v ).
% graph.is_subscc.cong
thf(fact_154_graph_Ois__scc_Ocong,axiom,
sCC_Bloemen_is_scc_v = sCC_Bloemen_is_scc_v ).
% graph.is_scc.cong
thf(fact_155_Diff__mono,axiom,
! [A3: set_v,C2: set_v,D: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ C2 )
=> ( ( ord_less_eq_set_v @ D @ B )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ B ) @ ( minus_minus_set_v @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_156_Diff__mono,axiom,
! [A3: set_Product_prod_v_v,C2: set_Product_prod_v_v,D: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ D @ B )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ ( minus_4183494784930505774od_v_v @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_157_Diff__subset,axiom,
! [A3: set_v,B: set_v] : ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ B ) @ A3 ) ).
% Diff_subset
thf(fact_158_Diff__subset,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ A3 ) ).
% Diff_subset
thf(fact_159_double__diff,axiom,
! [A3: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ( minus_minus_set_v @ B @ ( minus_minus_set_v @ C2 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_160_double__diff,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ( minus_4183494784930505774od_v_v @ B @ ( minus_4183494784930505774od_v_v @ C2 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_161_subset__code_I1_J,axiom,
! [Xs: list_v,B: set_v] :
( ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ B )
= ( ! [X2: v] :
( ( member_v2 @ X2 @ ( set_v2 @ Xs ) )
=> ( member_v2 @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_162_subset__code_I1_J,axiom,
! [Xs: list_P7986770385144383213od_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ B )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( member7453568604450474000od_v_v @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_163_graph_Osclosed,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ Vertices )
=> ( ord_le7336532860387713383od_v_v @ ( Successors @ X4 ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_164_graph_Osclosed,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ! [X4: v] :
( ( member_v2 @ X4 @ Vertices )
=> ( ord_less_eq_set_v @ ( Successors @ X4 ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_165_graph_Oedge__ra,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y: product_prod_v_v,X: product_prod_v_v,E2: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X ) )
=> ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Y ) @ E2 )
=> ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y @ E2 ) ) ) ) ).
% graph.edge_ra
thf(fact_166_graph_Oedge__ra,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X: v,E2: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v2 @ Y @ ( Successors @ X ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ E2 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E2 ) ) ) ) ).
% graph.edge_ra
thf(fact_167_graph_Ora__cases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,E2: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y @ E2 )
=> ( ( X = Y )
| ? [Z3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ X ) )
& ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Z3 ) @ E2 )
& ( sCC_Bl5370300055464682748od_v_v @ Successors @ Z3 @ Y @ E2 ) ) ) ) ) ).
% graph.ra_cases
thf(fact_168_graph_Ora__cases,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E2: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E2 )
=> ( ( X = Y )
| ? [Z3: v] :
( ( member_v2 @ Z3 @ ( Successors @ X ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Z3 ) @ E2 )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ Z3 @ Y @ E2 ) ) ) ) ) ).
% graph.ra_cases
thf(fact_169_graph_Oreachable__avoiding_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v,A32: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ A2 @ A32 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: product_prod_v_v] :
( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ Y2 @ A32 )
=> ( ( member7453568604450474000od_v_v @ A2 @ ( Successors @ Y2 ) )
=> ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y2 @ A2 ) @ A32 ) ) ) ) ) ) ).
% graph.reachable_avoiding.cases
thf(fact_170_graph_Oreachable__avoiding_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v,A32: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ A2 @ A32 )
=> ( ( A2 != A1 )
=> ~ ! [Y2: v] :
( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ Y2 @ A32 )
=> ( ( member_v2 @ A2 @ ( Successors @ Y2 ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ A2 ) @ A32 ) ) ) ) ) ) ).
% graph.reachable_avoiding.cases
thf(fact_171_graph_Oreachable__avoiding_Osimps,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A2: product_prod_v_v,A32: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ A2 @ A32 )
= ( ? [X2: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v] :
( ( A1 = X2 )
& ( A2 = X2 )
& ( A32 = E4 ) )
| ? [X2: product_prod_v_v,Y3: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v,Z2: product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( A32 = E4 )
& ( sCC_Bl5370300055464682748od_v_v @ Successors @ X2 @ Y3 @ E4 )
& ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y3 ) )
& ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y3 @ Z2 ) @ E4 ) ) ) ) ) ).
% graph.reachable_avoiding.simps
thf(fact_172_graph_Oreachable__avoiding_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A2: v,A32: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ A2 @ A32 )
= ( ? [X2: v,E4: set_Product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = X2 )
& ( A32 = E4 ) )
| ? [X2: v,Y3: v,E4: set_Product_prod_v_v,Z2: v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( A32 = E4 )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y3 @ E4 )
& ( member_v2 @ Z2 @ ( Successors @ Y3 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ Z2 ) @ E4 ) ) ) ) ) ).
% graph.reachable_avoiding.simps
thf(fact_173_graph_Ora__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,E2: set_Pr2149350503807050951od_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y @ E2 )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y ) )
=> ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y @ Z ) @ E2 )
=> ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Z @ E2 ) ) ) ) ) ).
% graph.ra_succ
thf(fact_174_graph_Ora__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E2: set_Product_prod_v_v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E2 )
=> ( ( member_v2 @ Z @ ( Successors @ Y ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ Z ) @ E2 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Z @ E2 ) ) ) ) ) ).
% graph.ra_succ
thf(fact_175_graph_Ora__mono,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E2: set_Product_prod_v_v,E3: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E2 )
=> ( ( ord_le7336532860387713383od_v_v @ E3 @ E2 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E3 ) ) ) ) ).
% graph.ra_mono
thf(fact_176_DiffE,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v2 @ C @ ( minus_minus_set_v @ A3 @ B ) )
=> ~ ( ( member_v2 @ C @ A3 )
=> ( member_v2 @ C @ B ) ) ) ).
% DiffE
thf(fact_177_DiffE,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
=> ~ ( ( member7453568604450474000od_v_v @ C @ A3 )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% DiffE
thf(fact_178_DiffD1,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v2 @ C @ ( minus_minus_set_v @ A3 @ B ) )
=> ( member_v2 @ C @ A3 ) ) ).
% DiffD1
thf(fact_179_DiffD1,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
=> ( member7453568604450474000od_v_v @ C @ A3 ) ) ).
% DiffD1
thf(fact_180_DiffD2,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v2 @ C @ ( minus_minus_set_v @ A3 @ B ) )
=> ~ ( member_v2 @ C @ B ) ) ).
% DiffD2
thf(fact_181_DiffD2,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
=> ~ ( member7453568604450474000od_v_v @ C @ B ) ) ).
% DiffD2
thf(fact_182_graph_Ora__empty,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ bot_bo723834152578015283od_v_v )
= ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y ) ) ) ).
% graph.ra_empty
thf(fact_183_graph_Ois__subscc__def,axiom,
! [Vertices: set_v,Successors: v > set_v,S3: set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S3 )
= ( ! [X2: v] :
( ( member_v2 @ X2 @ S3 )
=> ! [Y3: v] :
( ( member_v2 @ Y3 @ S3 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y3 ) ) ) ) ) ) ).
% graph.is_subscc_def
thf(fact_184_graph_OsccE,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S3: set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S3 )
=> ( ( member7453568604450474000od_v_v @ X @ S3 )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ X )
=> ( member7453568604450474000od_v_v @ Y @ S3 ) ) ) ) ) ) ).
% graph.sccE
thf(fact_185_graph_OsccE,axiom,
! [Vertices: set_v,Successors: v > set_v,S3: set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S3 )
=> ( ( member_v2 @ X @ S3 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ X )
=> ( member_v2 @ Y @ S3 ) ) ) ) ) ) ).
% graph.sccE
thf(fact_186_is__scc__def,axiom,
! [S3: set_v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S3 )
= ( ( S3 != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S3 )
& ! [S4: set_v] :
( ( ( ord_less_eq_set_v @ S3 @ S4 )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S4 ) )
=> ( S4 = S3 ) ) ) ) ).
% is_scc_def
thf(fact_187_subscc__add,axiom,
! [S3: set_v,X: v,Y: v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S3 )
=> ( ( member_v2 @ X @ S3 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ X )
=> ( sCC_Bl5398416737448265317bscc_v @ successors @ ( insert_v2 @ Y @ S3 ) ) ) ) ) ) ).
% subscc_add
thf(fact_188_diff__shunt__var,axiom,
! [X: set_v,Y: set_v] :
( ( ( minus_minus_set_v @ X @ Y )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_189_diff__shunt__var,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ X @ Y )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_190_scc__partition,axiom,
! [S3: set_v,S5: set_v,X: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S3 )
=> ( ( sCC_Bloemen_is_scc_v @ successors @ S5 )
=> ( ( member_v2 @ X @ ( inf_inf_set_v @ S3 @ S5 ) )
=> ( S3 = S5 ) ) ) ) ).
% scc_partition
thf(fact_191_prod_Oinject,axiom,
! [X1: v,X22: v,Y1: v,Y22: v] :
( ( ( product_Pair_v_v @ X1 @ X22 )
= ( product_Pair_v_v @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_192_old_Oprod_Oinject,axiom,
! [A: v,B3: v,A5: v,B4: v] :
( ( ( product_Pair_v_v @ A @ B3 )
= ( product_Pair_v_v @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B3 = B4 ) ) ) ).
% old.prod.inject
thf(fact_193_order__refl,axiom,
! [X: set_v] : ( ord_less_eq_set_v @ X @ X ) ).
% order_refl
thf(fact_194_order__refl,axiom,
! [X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ X ) ).
% order_refl
thf(fact_195_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_196_dual__order_Orefl,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ A @ A ) ).
% dual_order.refl
thf(fact_197_dual__order_Orefl,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ A ) ).
% dual_order.refl
thf(fact_198_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_199_less__by__empty,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A3 = bot_bo723834152578015283od_v_v )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ).
% less_by_empty
thf(fact_200_insertCI,axiom,
! [A: v,B: set_v,B3: v] :
( ( ~ ( member_v2 @ A @ B )
=> ( A = B3 ) )
=> ( member_v2 @ A @ ( insert_v2 @ B3 @ B ) ) ) ).
% insertCI
thf(fact_201_insertCI,axiom,
! [A: product_prod_v_v,B: set_Product_prod_v_v,B3: product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ A @ B )
=> ( A = B3 ) )
=> ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B3 @ B ) ) ) ).
% insertCI
thf(fact_202_insert__iff,axiom,
! [A: v,B3: v,A3: set_v] :
( ( member_v2 @ A @ ( insert_v2 @ B3 @ A3 ) )
= ( ( A = B3 )
| ( member_v2 @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_203_insert__iff,axiom,
! [A: product_prod_v_v,B3: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B3 @ A3 ) )
= ( ( A = B3 )
| ( member7453568604450474000od_v_v @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_204_insert__absorb2,axiom,
! [X: v,A3: set_v] :
( ( insert_v2 @ X @ ( insert_v2 @ X @ A3 ) )
= ( insert_v2 @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_205_insert__absorb2,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X @ ( insert1338601472111419319od_v_v @ X @ A3 ) )
= ( insert1338601472111419319od_v_v @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_206_inf_Oidem,axiom,
! [A: set_v] :
( ( inf_inf_set_v @ A @ A )
= A ) ).
% inf.idem
thf(fact_207_inf__idem,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ X )
= X ) ).
% inf_idem
thf(fact_208_inf_Oleft__idem,axiom,
! [A: set_v,B3: set_v] :
( ( inf_inf_set_v @ A @ ( inf_inf_set_v @ A @ B3 ) )
= ( inf_inf_set_v @ A @ B3 ) ) ).
% inf.left_idem
thf(fact_209_inf__left__idem,axiom,
! [X: set_v,Y: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ X @ Y ) )
= ( inf_inf_set_v @ X @ Y ) ) ).
% inf_left_idem
thf(fact_210_inf_Oright__idem,axiom,
! [A: set_v,B3: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B3 ) @ B3 )
= ( inf_inf_set_v @ A @ B3 ) ) ).
% inf.right_idem
thf(fact_211_inf__right__idem,axiom,
! [X: set_v,Y: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X @ Y ) @ Y )
= ( inf_inf_set_v @ X @ Y ) ) ).
% inf_right_idem
thf(fact_212_IntI,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A3 )
=> ( ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).
% IntI
thf(fact_213_IntI,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v2 @ C @ A3 )
=> ( ( member_v2 @ C @ B )
=> ( member_v2 @ C @ ( inf_inf_set_v @ A3 @ B ) ) ) ) ).
% IntI
thf(fact_214_Int__iff,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
= ( ( member7453568604450474000od_v_v @ C @ A3 )
& ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% Int_iff
thf(fact_215_Int__iff,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v2 @ C @ ( inf_inf_set_v @ A3 @ B ) )
= ( ( member_v2 @ C @ A3 )
& ( member_v2 @ C @ B ) ) ) ).
% Int_iff
thf(fact_216_inf_Obounded__iff,axiom,
! [A: set_v,B3: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B3 @ C ) )
= ( ( ord_less_eq_set_v @ A @ B3 )
& ( ord_less_eq_set_v @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_217_inf_Obounded__iff,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B3 @ C ) )
= ( ( ord_le7336532860387713383od_v_v @ A @ B3 )
& ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_218_inf_Obounded__iff,axiom,
! [A: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B3 @ C ) )
= ( ( ord_less_eq_nat @ A @ B3 )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_219_le__inf__iff,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) )
= ( ( ord_less_eq_set_v @ X @ Y )
& ( ord_less_eq_set_v @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_220_le__inf__iff,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) )
= ( ( ord_le7336532860387713383od_v_v @ X @ Y )
& ( ord_le7336532860387713383od_v_v @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_221_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_222_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_223_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ bot_bot_set_v )
= bot_bot_set_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_224_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ X )
= bot_bo723834152578015283od_v_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_225_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ X )
= bot_bot_set_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_226_inf__bot__right,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% inf_bot_right
thf(fact_227_inf__bot__right,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ bot_bot_set_v )
= bot_bot_set_v ) ).
% inf_bot_right
thf(fact_228_inf__bot__left,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ X )
= bot_bo723834152578015283od_v_v ) ).
% inf_bot_left
thf(fact_229_inf__bot__left,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ X )
= bot_bot_set_v ) ).
% inf_bot_left
thf(fact_230_singletonI,axiom,
! [A: product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).
% singletonI
thf(fact_231_singletonI,axiom,
! [A: v] : ( member_v2 @ A @ ( insert_v2 @ A @ bot_bot_set_v ) ) ).
% singletonI
thf(fact_232_insert__subset,axiom,
! [X: v,A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ ( insert_v2 @ X @ A3 ) @ B )
= ( ( member_v2 @ X @ B )
& ( ord_less_eq_set_v @ A3 @ B ) ) ) ).
% insert_subset
thf(fact_233_insert__subset,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ X @ A3 ) @ B )
= ( ( member7453568604450474000od_v_v @ X @ B )
& ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ) ).
% insert_subset
thf(fact_234_Int__subset__iff,axiom,
! [C2: set_v,A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C2 @ ( inf_inf_set_v @ A3 @ B ) )
= ( ( ord_less_eq_set_v @ C2 @ A3 )
& ( ord_less_eq_set_v @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_235_Int__subset__iff,axiom,
! [C2: set_Product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
= ( ( ord_le7336532860387713383od_v_v @ C2 @ A3 )
& ( ord_le7336532860387713383od_v_v @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_236_Int__insert__left__if0,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_237_Int__insert__left__if0,axiom,
! [A: v,C2: set_v,B: set_v] :
( ~ ( member_v2 @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v2 @ A @ B ) @ C2 )
= ( inf_inf_set_v @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_238_Int__insert__left__if1,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C2 )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_239_Int__insert__left__if1,axiom,
! [A: v,C2: set_v,B: set_v] :
( ( member_v2 @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v2 @ A @ B ) @ C2 )
= ( insert_v2 @ A @ ( inf_inf_set_v @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_240_insert__inter__insert,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ).
% insert_inter_insert
thf(fact_241_insert__inter__insert,axiom,
! [A: v,A3: set_v,B: set_v] :
( ( inf_inf_set_v @ ( insert_v2 @ A @ A3 ) @ ( insert_v2 @ A @ B ) )
= ( insert_v2 @ A @ ( inf_inf_set_v @ A3 @ B ) ) ) ).
% insert_inter_insert
thf(fact_242_Int__insert__right__if0,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_243_Int__insert__right__if0,axiom,
! [A: v,A3: set_v,B: set_v] :
( ~ ( member_v2 @ A @ A3 )
=> ( ( inf_inf_set_v @ A3 @ ( insert_v2 @ A @ B ) )
= ( inf_inf_set_v @ A3 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_244_Int__insert__right__if1,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_245_Int__insert__right__if1,axiom,
! [A: v,A3: set_v,B: set_v] :
( ( member_v2 @ A @ A3 )
=> ( ( inf_inf_set_v @ A3 @ ( insert_v2 @ A @ B ) )
= ( insert_v2 @ A @ ( inf_inf_set_v @ A3 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_246_insert__Diff1,axiom,
! [X: v,B: set_v,A3: set_v] :
( ( member_v2 @ X @ B )
=> ( ( minus_minus_set_v @ ( insert_v2 @ X @ A3 ) @ B )
= ( minus_minus_set_v @ A3 @ B ) ) ) ).
% insert_Diff1
thf(fact_247_insert__Diff1,axiom,
! [X: product_prod_v_v,B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A3 ) @ B )
= ( minus_4183494784930505774od_v_v @ A3 @ B ) ) ) ).
% insert_Diff1
thf(fact_248_Diff__insert0,axiom,
! [X: v,A3: set_v,B: set_v] :
( ~ ( member_v2 @ X @ A3 )
=> ( ( minus_minus_set_v @ A3 @ ( insert_v2 @ X @ B ) )
= ( minus_minus_set_v @ A3 @ B ) ) ) ).
% Diff_insert0
thf(fact_249_Diff__insert0,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ B ) )
= ( minus_4183494784930505774od_v_v @ A3 @ B ) ) ) ).
% Diff_insert0
thf(fact_250_singleton__insert__inj__eq_H,axiom,
! [A: v,A3: set_v,B3: v] :
( ( ( insert_v2 @ A @ A3 )
= ( insert_v2 @ B3 @ bot_bot_set_v ) )
= ( ( A = B3 )
& ( ord_less_eq_set_v @ A3 @ ( insert_v2 @ B3 @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_251_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B3: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ A3 )
= ( insert1338601472111419319od_v_v @ B3 @ bot_bo723834152578015283od_v_v ) )
= ( ( A = B3 )
& ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B3 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_252_singleton__insert__inj__eq,axiom,
! [B3: v,A: v,A3: set_v] :
( ( ( insert_v2 @ B3 @ bot_bot_set_v )
= ( insert_v2 @ A @ A3 ) )
= ( ( A = B3 )
& ( ord_less_eq_set_v @ A3 @ ( insert_v2 @ B3 @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_253_singleton__insert__inj__eq,axiom,
! [B3: product_prod_v_v,A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ B3 @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ A @ A3 ) )
= ( ( A = B3 )
& ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B3 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_254_insert__disjoint_I1_J,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) @ B )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A @ B )
& ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v ) ) ) ).
% insert_disjoint(1)
thf(fact_255_insert__disjoint_I1_J,axiom,
! [A: v,A3: set_v,B: set_v] :
( ( ( inf_inf_set_v @ ( insert_v2 @ A @ A3 ) @ B )
= bot_bot_set_v )
= ( ~ ( member_v2 @ A @ B )
& ( ( inf_inf_set_v @ A3 @ B )
= bot_bot_set_v ) ) ) ).
% insert_disjoint(1)
thf(fact_256_insert__disjoint_I2_J,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) @ B ) )
= ( ~ ( member7453568604450474000od_v_v @ A @ B )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_257_insert__disjoint_I2_J,axiom,
! [A: v,A3: set_v,B: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ ( insert_v2 @ A @ A3 ) @ B ) )
= ( ~ ( member_v2 @ A @ B )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A3 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_258_disjoint__insert_I1_J,axiom,
! [B: set_Product_prod_v_v,A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ A3 ) )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A @ B )
& ( ( inf_in6271465464967711157od_v_v @ B @ A3 )
= bot_bo723834152578015283od_v_v ) ) ) ).
% disjoint_insert(1)
thf(fact_259_disjoint__insert_I1_J,axiom,
! [B: set_v,A: v,A3: set_v] :
( ( ( inf_inf_set_v @ B @ ( insert_v2 @ A @ A3 ) )
= bot_bot_set_v )
= ( ~ ( member_v2 @ A @ B )
& ( ( inf_inf_set_v @ B @ A3 )
= bot_bot_set_v ) ) ) ).
% disjoint_insert(1)
thf(fact_260_disjoint__insert_I2_J,axiom,
! [A3: set_Product_prod_v_v,B3: product_prod_v_v,B: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B3 @ B ) ) )
= ( ~ ( member7453568604450474000od_v_v @ B3 @ A3 )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_261_disjoint__insert_I2_J,axiom,
! [A3: set_v,B3: v,B: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ A3 @ ( insert_v2 @ B3 @ B ) ) )
= ( ~ ( member_v2 @ B3 @ A3 )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A3 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_262_insert__Diff__single,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
= ( insert1338601472111419319od_v_v @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_263_insert__Diff__single,axiom,
! [A: v,A3: set_v] :
( ( insert_v2 @ A @ ( minus_minus_set_v @ A3 @ ( insert_v2 @ A @ bot_bot_set_v ) ) )
= ( insert_v2 @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_264_Diff__disjoint,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B @ A3 ) )
= bot_bo723834152578015283od_v_v ) ).
% Diff_disjoint
thf(fact_265_Diff__disjoint,axiom,
! [A3: set_v,B: set_v] :
( ( inf_inf_set_v @ A3 @ ( minus_minus_set_v @ B @ A3 ) )
= bot_bot_set_v ) ).
% Diff_disjoint
thf(fact_266_inf__sup__aci_I4_J,axiom,
! [X: set_v,Y: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ X @ Y ) )
= ( inf_inf_set_v @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_267_inf__sup__aci_I3_J,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ Y @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_268_inf__sup__aci_I2_J,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X @ Y ) @ Z )
= ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_269_inf__sup__aci_I1_J,axiom,
( inf_inf_set_v
= ( ^ [X2: set_v,Y3: set_v] : ( inf_inf_set_v @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_270_inf_Oassoc,axiom,
! [A: set_v,B3: set_v,C: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B3 ) @ C )
= ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B3 @ C ) ) ) ).
% inf.assoc
thf(fact_271_inf__assoc,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X @ Y ) @ Z )
= ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_272_inf_Ocommute,axiom,
( inf_inf_set_v
= ( ^ [A6: set_v,B5: set_v] : ( inf_inf_set_v @ B5 @ A6 ) ) ) ).
% inf.commute
thf(fact_273_inf__commute,axiom,
( inf_inf_set_v
= ( ^ [X2: set_v,Y3: set_v] : ( inf_inf_set_v @ Y3 @ X2 ) ) ) ).
% inf_commute
thf(fact_274_inf_Oleft__commute,axiom,
! [B3: set_v,A: set_v,C: set_v] :
( ( inf_inf_set_v @ B3 @ ( inf_inf_set_v @ A @ C ) )
= ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B3 @ C ) ) ) ).
% inf.left_commute
thf(fact_275_inf__left__commute,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ Y @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_276_IntE,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
=> ~ ( ( member7453568604450474000od_v_v @ C @ A3 )
=> ~ ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% IntE
thf(fact_277_IntE,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v2 @ C @ ( inf_inf_set_v @ A3 @ B ) )
=> ~ ( ( member_v2 @ C @ A3 )
=> ~ ( member_v2 @ C @ B ) ) ) ).
% IntE
thf(fact_278_IntD1,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
=> ( member7453568604450474000od_v_v @ C @ A3 ) ) ).
% IntD1
thf(fact_279_IntD1,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v2 @ C @ ( inf_inf_set_v @ A3 @ B ) )
=> ( member_v2 @ C @ A3 ) ) ).
% IntD1
thf(fact_280_IntD2,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ).
% IntD2
thf(fact_281_IntD2,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v2 @ C @ ( inf_inf_set_v @ A3 @ B ) )
=> ( member_v2 @ C @ B ) ) ).
% IntD2
thf(fact_282_insertE,axiom,
! [A: v,B3: v,A3: set_v] :
( ( member_v2 @ A @ ( insert_v2 @ B3 @ A3 ) )
=> ( ( A != B3 )
=> ( member_v2 @ A @ A3 ) ) ) ).
% insertE
thf(fact_283_insertE,axiom,
! [A: product_prod_v_v,B3: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B3 @ A3 ) )
=> ( ( A != B3 )
=> ( member7453568604450474000od_v_v @ A @ A3 ) ) ) ).
% insertE
thf(fact_284_insertI1,axiom,
! [A: v,B: set_v] : ( member_v2 @ A @ ( insert_v2 @ A @ B ) ) ).
% insertI1
thf(fact_285_insertI1,axiom,
! [A: product_prod_v_v,B: set_Product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ B ) ) ).
% insertI1
thf(fact_286_insertI2,axiom,
! [A: v,B: set_v,B3: v] :
( ( member_v2 @ A @ B )
=> ( member_v2 @ A @ ( insert_v2 @ B3 @ B ) ) ) ).
% insertI2
thf(fact_287_insertI2,axiom,
! [A: product_prod_v_v,B: set_Product_prod_v_v,B3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ B )
=> ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B3 @ B ) ) ) ).
% insertI2
thf(fact_288_Int__assoc,axiom,
! [A3: set_v,B: set_v,C2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A3 @ B ) @ C2 )
= ( inf_inf_set_v @ A3 @ ( inf_inf_set_v @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_289_Int__absorb,axiom,
! [A3: set_v] :
( ( inf_inf_set_v @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_290_Set_Oset__insert,axiom,
! [X: v,A3: set_v] :
( ( member_v2 @ X @ A3 )
=> ~ ! [B6: set_v] :
( ( A3
= ( insert_v2 @ X @ B6 ) )
=> ( member_v2 @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_291_Set_Oset__insert,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A3 )
=> ~ ! [B6: set_Product_prod_v_v] :
( ( A3
= ( insert1338601472111419319od_v_v @ X @ B6 ) )
=> ( member7453568604450474000od_v_v @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_292_Int__commute,axiom,
( inf_inf_set_v
= ( ^ [A4: set_v,B2: set_v] : ( inf_inf_set_v @ B2 @ A4 ) ) ) ).
% Int_commute
thf(fact_293_insert__ident,axiom,
! [X: v,A3: set_v,B: set_v] :
( ~ ( member_v2 @ X @ A3 )
=> ( ~ ( member_v2 @ X @ B )
=> ( ( ( insert_v2 @ X @ A3 )
= ( insert_v2 @ X @ B ) )
= ( A3 = B ) ) ) ) ).
% insert_ident
thf(fact_294_insert__ident,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ X @ B )
=> ( ( ( insert1338601472111419319od_v_v @ X @ A3 )
= ( insert1338601472111419319od_v_v @ X @ B ) )
= ( A3 = B ) ) ) ) ).
% insert_ident
thf(fact_295_insert__absorb,axiom,
! [A: v,A3: set_v] :
( ( member_v2 @ A @ A3 )
=> ( ( insert_v2 @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_296_insert__absorb,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( insert1338601472111419319od_v_v @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_297_insert__eq__iff,axiom,
! [A: v,A3: set_v,B3: v,B: set_v] :
( ~ ( member_v2 @ A @ A3 )
=> ( ~ ( member_v2 @ B3 @ B )
=> ( ( ( insert_v2 @ A @ A3 )
= ( insert_v2 @ B3 @ B ) )
= ( ( ( A = B3 )
=> ( A3 = B ) )
& ( ( A != B3 )
=> ? [C3: set_v] :
( ( A3
= ( insert_v2 @ B3 @ C3 ) )
& ~ ( member_v2 @ B3 @ C3 )
& ( B
= ( insert_v2 @ A @ C3 ) )
& ~ ( member_v2 @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_298_insert__eq__iff,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B3: product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ B3 @ B )
=> ( ( ( insert1338601472111419319od_v_v @ A @ A3 )
= ( insert1338601472111419319od_v_v @ B3 @ B ) )
= ( ( ( A = B3 )
=> ( A3 = B ) )
& ( ( A != B3 )
=> ? [C3: set_Product_prod_v_v] :
( ( A3
= ( insert1338601472111419319od_v_v @ B3 @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ B3 @ C3 )
& ( B
= ( insert1338601472111419319od_v_v @ A @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_299_insert__commute,axiom,
! [X: v,Y: v,A3: set_v] :
( ( insert_v2 @ X @ ( insert_v2 @ Y @ A3 ) )
= ( insert_v2 @ Y @ ( insert_v2 @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_300_insert__commute,axiom,
! [X: product_prod_v_v,Y: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X @ ( insert1338601472111419319od_v_v @ Y @ A3 ) )
= ( insert1338601472111419319od_v_v @ Y @ ( insert1338601472111419319od_v_v @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_301_Int__insert__left,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C2 )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_302_Int__insert__left,axiom,
! [A: v,C2: set_v,B: set_v] :
( ( ( member_v2 @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v2 @ A @ B ) @ C2 )
= ( insert_v2 @ A @ ( inf_inf_set_v @ B @ C2 ) ) ) )
& ( ~ ( member_v2 @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v2 @ A @ B ) @ C2 )
= ( inf_inf_set_v @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_303_Int__left__absorb,axiom,
! [A3: set_v,B: set_v] :
( ( inf_inf_set_v @ A3 @ ( inf_inf_set_v @ A3 @ B ) )
= ( inf_inf_set_v @ A3 @ B ) ) ).
% Int_left_absorb
thf(fact_304_Int__insert__right,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_305_Int__insert__right,axiom,
! [A: v,A3: set_v,B: set_v] :
( ( ( member_v2 @ A @ A3 )
=> ( ( inf_inf_set_v @ A3 @ ( insert_v2 @ A @ B ) )
= ( insert_v2 @ A @ ( inf_inf_set_v @ A3 @ B ) ) ) )
& ( ~ ( member_v2 @ A @ A3 )
=> ( ( inf_inf_set_v @ A3 @ ( insert_v2 @ A @ B ) )
= ( inf_inf_set_v @ A3 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_306_Int__left__commute,axiom,
! [A3: set_v,B: set_v,C2: set_v] :
( ( inf_inf_set_v @ A3 @ ( inf_inf_set_v @ B @ C2 ) )
= ( inf_inf_set_v @ B @ ( inf_inf_set_v @ A3 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_307_mk__disjoint__insert,axiom,
! [A: v,A3: set_v] :
( ( member_v2 @ A @ A3 )
=> ? [B6: set_v] :
( ( A3
= ( insert_v2 @ A @ B6 ) )
& ~ ( member_v2 @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_308_mk__disjoint__insert,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A3 )
=> ? [B6: set_Product_prod_v_v] :
( ( A3
= ( insert1338601472111419319od_v_v @ A @ B6 ) )
& ~ ( member7453568604450474000od_v_v @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_309_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_v,K: set_v,B3: set_v,A: set_v] :
( ( B
= ( inf_inf_set_v @ K @ B3 ) )
=> ( ( inf_inf_set_v @ A @ B )
= ( inf_inf_set_v @ K @ ( inf_inf_set_v @ A @ B3 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_310_boolean__algebra__cancel_Oinf1,axiom,
! [A3: set_v,K: set_v,A: set_v,B3: set_v] :
( ( A3
= ( inf_inf_set_v @ K @ A ) )
=> ( ( inf_inf_set_v @ A3 @ B3 )
= ( inf_inf_set_v @ K @ ( inf_inf_set_v @ A @ B3 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_311_bot__set__def,axiom,
( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ bot_bo8461541820394803818_v_v_o ) ) ).
% bot_set_def
thf(fact_312_bot__set__def,axiom,
( bot_bot_set_v
= ( collect_v @ bot_bot_v_o ) ) ).
% bot_set_def
thf(fact_313_inf_OcoboundedI2,axiom,
! [B3: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B3 @ C )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B3 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_314_inf_OcoboundedI2,axiom,
! [B3: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B3 @ C )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B3 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_315_inf_OcoboundedI2,axiom,
! [B3: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B3 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_316_inf_OcoboundedI1,axiom,
! [A: set_v,C: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B3 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_317_inf_OcoboundedI1,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B3 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_318_inf_OcoboundedI1,axiom,
! [A: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B3 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_319_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [B5: set_v,A6: set_v] :
( ( inf_inf_set_v @ A6 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_320_inf_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B5: set_Product_prod_v_v,A6: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A6 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_321_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A6: nat] :
( ( inf_inf_nat @ A6 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_322_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [A6: set_v,B5: set_v] :
( ( inf_inf_set_v @ A6 @ B5 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_323_inf_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A6: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A6 @ B5 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_324_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B5: nat] :
( ( inf_inf_nat @ A6 @ B5 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_325_inf_Ocobounded2,axiom,
! [A: set_v,B3: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_326_inf_Ocobounded2,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_327_inf_Ocobounded2,axiom,
! [A: nat,B3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_328_inf_Ocobounded1,axiom,
! [A: set_v,B3: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B3 ) @ A ) ).
% inf.cobounded1
thf(fact_329_inf_Ocobounded1,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B3 ) @ A ) ).
% inf.cobounded1
thf(fact_330_inf_Ocobounded1,axiom,
! [A: nat,B3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B3 ) @ A ) ).
% inf.cobounded1
thf(fact_331_inf_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A6: set_v,B5: set_v] :
( A6
= ( inf_inf_set_v @ A6 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_332_inf_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A6: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( A6
= ( inf_in6271465464967711157od_v_v @ A6 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_333_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B5: nat] :
( A6
= ( inf_inf_nat @ A6 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_334_inf__greatest,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( ord_less_eq_set_v @ X @ Z )
=> ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_335_inf__greatest,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( ord_le7336532860387713383od_v_v @ X @ Z )
=> ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_336_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_337_inf_OboundedI,axiom,
! [A: set_v,B3: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B3 )
=> ( ( ord_less_eq_set_v @ A @ C )
=> ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B3 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_338_inf_OboundedI,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B3 )
=> ( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B3 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_339_inf_OboundedI,axiom,
! [A: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B3 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_340_inf_OboundedE,axiom,
! [A: set_v,B3: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B3 @ C ) )
=> ~ ( ( ord_less_eq_set_v @ A @ B3 )
=> ~ ( ord_less_eq_set_v @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_341_inf_OboundedE,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B3 @ C ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A @ B3 )
=> ~ ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_342_inf_OboundedE,axiom,
! [A: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B3 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B3 )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_343_inf__absorb2,axiom,
! [Y: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y @ X )
=> ( ( inf_inf_set_v @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_344_inf__absorb2,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( ( inf_in6271465464967711157od_v_v @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_345_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_346_inf__absorb1,axiom,
! [X: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( inf_inf_set_v @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_347_inf__absorb1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( inf_in6271465464967711157od_v_v @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_348_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_349_inf_Oabsorb2,axiom,
! [B3: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B3 @ A )
=> ( ( inf_inf_set_v @ A @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_350_inf_Oabsorb2,axiom,
! [B3: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B3 @ A )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_351_inf_Oabsorb2,axiom,
! [B3: nat,A: nat] :
( ( ord_less_eq_nat @ B3 @ A )
=> ( ( inf_inf_nat @ A @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_352_inf_Oabsorb1,axiom,
! [A: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ A @ B3 )
=> ( ( inf_inf_set_v @ A @ B3 )
= A ) ) ).
% inf.absorb1
thf(fact_353_inf_Oabsorb1,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B3 )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B3 )
= A ) ) ).
% inf.absorb1
thf(fact_354_inf_Oabsorb1,axiom,
! [A: nat,B3: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( inf_inf_nat @ A @ B3 )
= A ) ) ).
% inf.absorb1
thf(fact_355_le__iff__inf,axiom,
( ord_less_eq_set_v
= ( ^ [X2: set_v,Y3: set_v] :
( ( inf_inf_set_v @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_356_le__iff__inf,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_357_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( inf_inf_nat @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_358_inf__unique,axiom,
! [F: set_v > set_v > set_v,X: set_v,Y: set_v] :
( ! [X3: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: set_v,Y2: set_v,Z3: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ( ord_less_eq_set_v @ X3 @ Z3 )
=> ( ord_less_eq_set_v @ X3 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_set_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_359_inf__unique,axiom,
! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ( ord_le7336532860387713383od_v_v @ X3 @ Z3 )
=> ( ord_le7336532860387713383od_v_v @ X3 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_360_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ X3 @ Z3 )
=> ( ord_less_eq_nat @ X3 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_361_inf_OorderI,axiom,
! [A: set_v,B3: set_v] :
( ( A
= ( inf_inf_set_v @ A @ B3 ) )
=> ( ord_less_eq_set_v @ A @ B3 ) ) ).
% inf.orderI
thf(fact_362_inf_OorderI,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( A
= ( inf_in6271465464967711157od_v_v @ A @ B3 ) )
=> ( ord_le7336532860387713383od_v_v @ A @ B3 ) ) ).
% inf.orderI
thf(fact_363_inf_OorderI,axiom,
! [A: nat,B3: nat] :
( ( A
= ( inf_inf_nat @ A @ B3 ) )
=> ( ord_less_eq_nat @ A @ B3 ) ) ).
% inf.orderI
thf(fact_364_inf_OorderE,axiom,
! [A: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ A @ B3 )
=> ( A
= ( inf_inf_set_v @ A @ B3 ) ) ) ).
% inf.orderE
thf(fact_365_inf_OorderE,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B3 )
=> ( A
= ( inf_in6271465464967711157od_v_v @ A @ B3 ) ) ) ).
% inf.orderE
thf(fact_366_inf_OorderE,axiom,
! [A: nat,B3: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( A
= ( inf_inf_nat @ A @ B3 ) ) ) ).
% inf.orderE
thf(fact_367_le__infI2,axiom,
! [B3: set_v,X: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B3 @ X )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B3 ) @ X ) ) ).
% le_infI2
thf(fact_368_le__infI2,axiom,
! [B3: set_Product_prod_v_v,X: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B3 @ X )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B3 ) @ X ) ) ).
% le_infI2
thf(fact_369_le__infI2,axiom,
! [B3: nat,X: nat,A: nat] :
( ( ord_less_eq_nat @ B3 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B3 ) @ X ) ) ).
% le_infI2
thf(fact_370_le__infI1,axiom,
! [A: set_v,X: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ A @ X )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B3 ) @ X ) ) ).
% le_infI1
thf(fact_371_le__infI1,axiom,
! [A: set_Product_prod_v_v,X: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ X )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B3 ) @ X ) ) ).
% le_infI1
thf(fact_372_le__infI1,axiom,
! [A: nat,X: nat,B3: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B3 ) @ X ) ) ).
% le_infI1
thf(fact_373_inf__mono,axiom,
! [A: set_v,C: set_v,B3: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ( ord_less_eq_set_v @ B3 @ D2 )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B3 ) @ ( inf_inf_set_v @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_374_inf__mono,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B3: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B3 ) @ ( inf_in6271465464967711157od_v_v @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_375_inf__mono,axiom,
! [A: nat,C: nat,B3: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B3 @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B3 ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_376_le__infI,axiom,
! [X: set_v,A: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ X @ A )
=> ( ( ord_less_eq_set_v @ X @ B3 )
=> ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ A @ B3 ) ) ) ) ).
% le_infI
thf(fact_377_le__infI,axiom,
! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ A )
=> ( ( ord_le7336532860387713383od_v_v @ X @ B3 )
=> ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ A @ B3 ) ) ) ) ).
% le_infI
thf(fact_378_le__infI,axiom,
! [X: nat,A: nat,B3: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ( ord_less_eq_nat @ X @ B3 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B3 ) ) ) ) ).
% le_infI
thf(fact_379_le__infE,axiom,
! [X: set_v,A: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ A @ B3 ) )
=> ~ ( ( ord_less_eq_set_v @ X @ A )
=> ~ ( ord_less_eq_set_v @ X @ B3 ) ) ) ).
% le_infE
thf(fact_380_le__infE,axiom,
! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ A @ B3 ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ X @ A )
=> ~ ( ord_le7336532860387713383od_v_v @ X @ B3 ) ) ) ).
% le_infE
thf(fact_381_le__infE,axiom,
! [X: nat,A: nat,B3: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B3 ) )
=> ~ ( ( ord_less_eq_nat @ X @ A )
=> ~ ( ord_less_eq_nat @ X @ B3 ) ) ) ).
% le_infE
thf(fact_382_inf__le2,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_383_inf__le2,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_384_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_385_inf__le1,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_386_inf__le1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_387_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_388_inf__sup__ord_I1_J,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_389_inf__sup__ord_I1_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_390_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_391_inf__sup__ord_I2_J,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_392_inf__sup__ord_I2_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_393_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_394_singleton__inject,axiom,
! [A: product_prod_v_v,B3: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ B3 @ bot_bo723834152578015283od_v_v ) )
=> ( A = B3 ) ) ).
% singleton_inject
thf(fact_395_singleton__inject,axiom,
! [A: v,B3: v] :
( ( ( insert_v2 @ A @ bot_bot_set_v )
= ( insert_v2 @ B3 @ bot_bot_set_v ) )
=> ( A = B3 ) ) ).
% singleton_inject
thf(fact_396_insert__not__empty,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A @ A3 )
!= bot_bo723834152578015283od_v_v ) ).
% insert_not_empty
thf(fact_397_insert__not__empty,axiom,
! [A: v,A3: set_v] :
( ( insert_v2 @ A @ A3 )
!= bot_bot_set_v ) ).
% insert_not_empty
thf(fact_398_doubleton__eq__iff,axiom,
! [A: product_prod_v_v,B3: product_prod_v_v,C: product_prod_v_v,D2: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ ( insert1338601472111419319od_v_v @ B3 @ bot_bo723834152578015283od_v_v ) )
= ( insert1338601472111419319od_v_v @ C @ ( insert1338601472111419319od_v_v @ D2 @ bot_bo723834152578015283od_v_v ) ) )
= ( ( ( A = C )
& ( B3 = D2 ) )
| ( ( A = D2 )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_399_doubleton__eq__iff,axiom,
! [A: v,B3: v,C: v,D2: v] :
( ( ( insert_v2 @ A @ ( insert_v2 @ B3 @ bot_bot_set_v ) )
= ( insert_v2 @ C @ ( insert_v2 @ D2 @ bot_bot_set_v ) ) )
= ( ( ( A = C )
& ( B3 = D2 ) )
| ( ( A = D2 )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_400_singleton__iff,axiom,
! [B3: product_prod_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B3 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
= ( B3 = A ) ) ).
% singleton_iff
thf(fact_401_singleton__iff,axiom,
! [B3: v,A: v] :
( ( member_v2 @ B3 @ ( insert_v2 @ A @ bot_bot_set_v ) )
= ( B3 = A ) ) ).
% singleton_iff
thf(fact_402_singletonD,axiom,
! [B3: product_prod_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B3 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
=> ( B3 = A ) ) ).
% singletonD
thf(fact_403_singletonD,axiom,
! [B3: v,A: v] :
( ( member_v2 @ B3 @ ( insert_v2 @ A @ bot_bot_set_v ) )
=> ( B3 = A ) ) ).
% singletonD
thf(fact_404_disjoint__iff__not__equal,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ! [Y3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ B )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_405_disjoint__iff__not__equal,axiom,
! [A3: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A3 @ B )
= bot_bot_set_v )
= ( ! [X2: v] :
( ( member_v2 @ X2 @ A3 )
=> ! [Y3: v] :
( ( member_v2 @ Y3 @ B )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_406_Int__empty__right,axiom,
! [A3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_right
thf(fact_407_Int__empty__right,axiom,
! [A3: set_v] :
( ( inf_inf_set_v @ A3 @ bot_bot_set_v )
= bot_bot_set_v ) ).
% Int_empty_right
thf(fact_408_Int__empty__left,axiom,
! [B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ B )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_left
thf(fact_409_Int__empty__left,axiom,
! [B: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ B )
= bot_bot_set_v ) ).
% Int_empty_left
thf(fact_410_disjoint__iff,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ~ ( member7453568604450474000od_v_v @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_411_disjoint__iff,axiom,
! [A3: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A3 @ B )
= bot_bot_set_v )
= ( ! [X2: v] :
( ( member_v2 @ X2 @ A3 )
=> ~ ( member_v2 @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_412_Int__emptyI,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ~ ( member7453568604450474000od_v_v @ X3 @ B ) )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v ) ) ).
% Int_emptyI
thf(fact_413_Int__emptyI,axiom,
! [A3: set_v,B: set_v] :
( ! [X3: v] :
( ( member_v2 @ X3 @ A3 )
=> ~ ( member_v2 @ X3 @ B ) )
=> ( ( inf_inf_set_v @ A3 @ B )
= bot_bot_set_v ) ) ).
% Int_emptyI
thf(fact_414_subset__insertI2,axiom,
! [A3: set_v,B: set_v,B3: v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ord_less_eq_set_v @ A3 @ ( insert_v2 @ B3 @ B ) ) ) ).
% subset_insertI2
thf(fact_415_subset__insertI2,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,B3: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B3 @ B ) ) ) ).
% subset_insertI2
thf(fact_416_subset__insertI,axiom,
! [B: set_v,A: v] : ( ord_less_eq_set_v @ B @ ( insert_v2 @ A @ B ) ) ).
% subset_insertI
thf(fact_417_subset__insertI,axiom,
! [B: set_Product_prod_v_v,A: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ B ) ) ).
% subset_insertI
thf(fact_418_subset__insert,axiom,
! [X: v,A3: set_v,B: set_v] :
( ~ ( member_v2 @ X @ A3 )
=> ( ( ord_less_eq_set_v @ A3 @ ( insert_v2 @ X @ B ) )
= ( ord_less_eq_set_v @ A3 @ B ) ) ) ).
% subset_insert
thf(fact_419_subset__insert,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ B ) )
= ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ) ).
% subset_insert
thf(fact_420_insert__mono,axiom,
! [C2: set_v,D: set_v,A: v] :
( ( ord_less_eq_set_v @ C2 @ D )
=> ( ord_less_eq_set_v @ ( insert_v2 @ A @ C2 ) @ ( insert_v2 @ A @ D ) ) ) ).
% insert_mono
thf(fact_421_insert__mono,axiom,
! [C2: set_Product_prod_v_v,D: set_Product_prod_v_v,A: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ D )
=> ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ A @ C2 ) @ ( insert1338601472111419319od_v_v @ A @ D ) ) ) ).
% insert_mono
thf(fact_422_Int__Collect__mono,axiom,
! [A3: set_v,B: set_v,P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ! [X3: v] :
( ( member_v2 @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ ( collect_v @ P ) ) @ ( inf_inf_set_v @ B @ ( collect_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_423_Int__Collect__mono,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ ( collec140062887454715474od_v_v @ P ) ) @ ( inf_in6271465464967711157od_v_v @ B @ ( collec140062887454715474od_v_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_424_Int__greatest,axiom,
! [C2: set_v,A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C2 @ A3 )
=> ( ( ord_less_eq_set_v @ C2 @ B )
=> ( ord_less_eq_set_v @ C2 @ ( inf_inf_set_v @ A3 @ B ) ) ) ) ).
% Int_greatest
thf(fact_425_Int__greatest,axiom,
! [C2: set_Product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ C2 @ B )
=> ( ord_le7336532860387713383od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) ) ) ) ).
% Int_greatest
thf(fact_426_Int__absorb2,axiom,
! [A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( inf_inf_set_v @ A3 @ B )
= A3 ) ) ).
% Int_absorb2
thf(fact_427_Int__absorb2,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= A3 ) ) ).
% Int_absorb2
thf(fact_428_Int__absorb1,axiom,
! [B: set_v,A3: set_v] :
( ( ord_less_eq_set_v @ B @ A3 )
=> ( ( inf_inf_set_v @ A3 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_429_Int__absorb1,axiom,
! [B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_430_Int__lower2,axiom,
! [A3: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B ) @ B ) ).
% Int_lower2
thf(fact_431_Int__lower2,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ B ) ).
% Int_lower2
thf(fact_432_Int__lower1,axiom,
! [A3: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B ) @ A3 ) ).
% Int_lower1
thf(fact_433_Int__lower1,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ A3 ) ).
% Int_lower1
thf(fact_434_Int__mono,axiom,
! [A3: set_v,C2: set_v,B: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A3 @ C2 )
=> ( ( ord_less_eq_set_v @ B @ D )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B ) @ ( inf_inf_set_v @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_435_Int__mono,axiom,
! [A3: set_Product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ ( inf_in6271465464967711157od_v_v @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_436_insert__Diff__if,axiom,
! [X: v,B: set_v,A3: set_v] :
( ( ( member_v2 @ X @ B )
=> ( ( minus_minus_set_v @ ( insert_v2 @ X @ A3 ) @ B )
= ( minus_minus_set_v @ A3 @ B ) ) )
& ( ~ ( member_v2 @ X @ B )
=> ( ( minus_minus_set_v @ ( insert_v2 @ X @ A3 ) @ B )
= ( insert_v2 @ X @ ( minus_minus_set_v @ A3 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_437_insert__Diff__if,axiom,
! [X: product_prod_v_v,B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ X @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A3 ) @ B )
= ( minus_4183494784930505774od_v_v @ A3 @ B ) ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A3 ) @ B )
= ( insert1338601472111419319od_v_v @ X @ ( minus_4183494784930505774od_v_v @ A3 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_438_Diff__Int__distrib2,axiom,
! [A3: set_v,B: set_v,C2: set_v] :
( ( inf_inf_set_v @ ( minus_minus_set_v @ A3 @ B ) @ C2 )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ C2 ) @ ( inf_inf_set_v @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_439_Diff__Int__distrib,axiom,
! [C2: set_v,A3: set_v,B: set_v] :
( ( inf_inf_set_v @ C2 @ ( minus_minus_set_v @ A3 @ B ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ C2 @ A3 ) @ ( inf_inf_set_v @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_440_Diff__Diff__Int,axiom,
! [A3: set_v,B: set_v] :
( ( minus_minus_set_v @ A3 @ ( minus_minus_set_v @ A3 @ B ) )
= ( inf_inf_set_v @ A3 @ B ) ) ).
% Diff_Diff_Int
thf(fact_441_Diff__Int2,axiom,
! [A3: set_v,C2: set_v,B: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ C2 ) @ ( inf_inf_set_v @ B @ C2 ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_442_Int__Diff,axiom,
! [A3: set_v,B: set_v,C2: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ B ) @ C2 )
= ( inf_inf_set_v @ A3 @ ( minus_minus_set_v @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_443_graph_Odfss_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,X: produc5741669702376414499t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ~ ! [V2: v,E5: sCC_Bl1394983891496994913t_unit] :
( X
!= ( produc3862955338007567901t_unit @ V2 @ E5 ) ) ) ).
% graph.dfss.cases
thf(fact_444_subset__singleton__iff,axiom,
! [X5: set_v,A: v] :
( ( ord_less_eq_set_v @ X5 @ ( insert_v2 @ A @ bot_bot_set_v ) )
= ( ( X5 = bot_bot_set_v )
| ( X5
= ( insert_v2 @ A @ bot_bot_set_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_445_subset__singleton__iff,axiom,
! [X5: set_Product_prod_v_v,A: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X5 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
= ( ( X5 = bot_bo723834152578015283od_v_v )
| ( X5
= ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_446_subset__singletonD,axiom,
! [A3: set_v,X: v] :
( ( ord_less_eq_set_v @ A3 @ ( insert_v2 @ X @ bot_bot_set_v ) )
=> ( ( A3 = bot_bot_set_v )
| ( A3
= ( insert_v2 @ X @ bot_bot_set_v ) ) ) ) ).
% subset_singletonD
thf(fact_447_subset__singletonD,axiom,
! [A3: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
=> ( ( A3 = bot_bo723834152578015283od_v_v )
| ( A3
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% subset_singletonD
thf(fact_448_Diff__insert,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ).
% Diff_insert
thf(fact_449_Diff__insert,axiom,
! [A3: set_v,A: v,B: set_v] :
( ( minus_minus_set_v @ A3 @ ( insert_v2 @ A @ B ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A3 @ B ) @ ( insert_v2 @ A @ bot_bot_set_v ) ) ) ).
% Diff_insert
thf(fact_450_insert__Diff,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( insert1338601472111419319od_v_v @ A @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_451_insert__Diff,axiom,
! [A: v,A3: set_v] :
( ( member_v2 @ A @ A3 )
=> ( ( insert_v2 @ A @ ( minus_minus_set_v @ A3 @ ( insert_v2 @ A @ bot_bot_set_v ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_452_Diff__insert2,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) @ B ) ) ).
% Diff_insert2
thf(fact_453_Diff__insert2,axiom,
! [A3: set_v,A: v,B: set_v] :
( ( minus_minus_set_v @ A3 @ ( insert_v2 @ A @ B ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A3 @ ( insert_v2 @ A @ bot_bot_set_v ) ) @ B ) ) ).
% Diff_insert2
thf(fact_454_Diff__insert__absorb,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A3 ) @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_455_Diff__insert__absorb,axiom,
! [X: v,A3: set_v] :
( ~ ( member_v2 @ X @ A3 )
=> ( ( minus_minus_set_v @ ( insert_v2 @ X @ A3 ) @ ( insert_v2 @ X @ bot_bot_set_v ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_456_Diff__triv,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v )
=> ( ( minus_4183494784930505774od_v_v @ A3 @ B )
= A3 ) ) ).
% Diff_triv
thf(fact_457_Diff__triv,axiom,
! [A3: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A3 @ B )
= bot_bot_set_v )
=> ( ( minus_minus_set_v @ A3 @ B )
= A3 ) ) ).
% Diff_triv
thf(fact_458_Int__Diff__disjoint,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
= bot_bo723834152578015283od_v_v ) ).
% Int_Diff_disjoint
thf(fact_459_Int__Diff__disjoint,axiom,
! [A3: set_v,B: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A3 @ B ) @ ( minus_minus_set_v @ A3 @ B ) )
= bot_bot_set_v ) ).
% Int_Diff_disjoint
thf(fact_460_subset__Diff__insert,axiom,
! [A3: set_v,B: set_v,X: v,C2: set_v] :
( ( ord_less_eq_set_v @ A3 @ ( minus_minus_set_v @ B @ ( insert_v2 @ X @ C2 ) ) )
= ( ( ord_less_eq_set_v @ A3 @ ( minus_minus_set_v @ B @ C2 ) )
& ~ ( member_v2 @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_461_subset__Diff__insert,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,X: product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B @ ( insert1338601472111419319od_v_v @ X @ C2 ) ) )
= ( ( ord_le7336532860387713383od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B @ C2 ) )
& ~ ( member7453568604450474000od_v_v @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_462_graph_Oscc__partition,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S3: set_Product_prod_v_v,S5: set_Product_prod_v_v,X: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S3 )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S5 )
=> ( ( member7453568604450474000od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ S3 @ S5 ) )
=> ( S3 = S5 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_463_graph_Oscc__partition,axiom,
! [Vertices: set_v,Successors: v > set_v,S3: set_v,S5: set_v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S3 )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S5 )
=> ( ( member_v2 @ X @ ( inf_inf_set_v @ S3 @ S5 ) )
=> ( S3 = S5 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_464_subset__insert__iff,axiom,
! [A3: set_v,X: v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ ( insert_v2 @ X @ B ) )
= ( ( ( member_v2 @ X @ A3 )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ ( insert_v2 @ X @ bot_bot_set_v ) ) @ B ) )
& ( ~ ( member_v2 @ X @ A3 )
=> ( ord_less_eq_set_v @ A3 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_465_subset__insert__iff,axiom,
! [A3: set_Product_prod_v_v,X: product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ B ) )
= ( ( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_466_Diff__single__insert,axiom,
! [A3: set_v,X: v,B: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ ( insert_v2 @ X @ bot_bot_set_v ) ) @ B )
=> ( ord_less_eq_set_v @ A3 @ ( insert_v2 @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_467_Diff__single__insert,axiom,
! [A3: set_Product_prod_v_v,X: product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B )
=> ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_468_order__antisym__conv,axiom,
! [Y: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y @ X )
=> ( ( ord_less_eq_set_v @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_469_order__antisym__conv,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( ( ord_le7336532860387713383od_v_v @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_470_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_471_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_472_ord__le__eq__subst,axiom,
! [A: set_v,B3: set_v,F: set_v > set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_473_ord__le__eq__subst,axiom,
! [A: set_v,B3: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_474_ord__le__eq__subst,axiom,
! [A: set_v,B3: set_v,F: set_v > nat,C: nat] :
( ( ord_less_eq_set_v @ A @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_475_ord__le__eq__subst,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_476_ord__le__eq__subst,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_477_ord__le__eq__subst,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v,F: set_Product_prod_v_v > nat,C: nat] :
( ( ord_le7336532860387713383od_v_v @ A @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_478_ord__le__eq__subst,axiom,
! [A: nat,B3: nat,F: nat > set_v,C: set_v] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_479_ord__le__eq__subst,axiom,
! [A: nat,B3: nat,F: nat > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_480_ord__le__eq__subst,axiom,
! [A: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_481_ord__eq__le__subst,axiom,
! [A: set_v,F: set_v > set_v,B3: set_v,C: set_v] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_v @ B3 @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_482_ord__eq__le__subst,axiom,
! [A: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B3: set_v,C: set_v] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_v @ B3 @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_483_ord__eq__le__subst,axiom,
! [A: nat,F: set_v > nat,B3: set_v,C: set_v] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_v @ B3 @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_484_ord__eq__le__subst,axiom,
! [A: set_v,F: set_Product_prod_v_v > set_v,B3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_485_ord__eq__le__subst,axiom,
! [A: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_486_ord__eq__le__subst,axiom,
! [A: nat,F: set_Product_prod_v_v > nat,B3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_487_ord__eq__le__subst,axiom,
! [A: set_v,F: nat > set_v,B3: nat,C: nat] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_488_ord__eq__le__subst,axiom,
! [A: set_Product_prod_v_v,F: nat > set_Product_prod_v_v,B3: nat,C: nat] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_489_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B3: nat,C: nat] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_490_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_491_order__eq__refl,axiom,
! [X: set_v,Y: set_v] :
( ( X = Y )
=> ( ord_less_eq_set_v @ X @ Y ) ) ).
% order_eq_refl
thf(fact_492_order__eq__refl,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( X = Y )
=> ( ord_le7336532860387713383od_v_v @ X @ Y ) ) ).
% order_eq_refl
thf(fact_493_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_494_order__subst2,axiom,
! [A: set_v,B3: set_v,F: set_v > set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B3 )
=> ( ( ord_less_eq_set_v @ ( F @ B3 ) @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_495_order__subst2,axiom,
! [A: set_v,B3: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ B3 )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B3 ) @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_496_order__subst2,axiom,
! [A: set_v,B3: set_v,F: set_v > nat,C: nat] :
( ( ord_less_eq_set_v @ A @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_497_order__subst2,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B3 )
=> ( ( ord_less_eq_set_v @ ( F @ B3 ) @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_498_order__subst2,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B3 )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B3 ) @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_499_order__subst2,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v,F: set_Product_prod_v_v > nat,C: nat] :
( ( ord_le7336532860387713383od_v_v @ A @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_500_order__subst2,axiom,
! [A: nat,B3: nat,F: nat > set_v,C: set_v] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_set_v @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_501_order__subst2,axiom,
! [A: nat,B3: nat,F: nat > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_502_order__subst2,axiom,
! [A: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_503_order__subst1,axiom,
! [A: set_v,F: set_v > set_v,B3: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_v @ B3 @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_504_order__subst1,axiom,
! [A: set_v,F: set_Product_prod_v_v > set_v,B3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ ( F @ B3 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_505_order__subst1,axiom,
! [A: set_v,F: nat > set_v,B3: nat,C: nat] :
( ( ord_less_eq_set_v @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_506_order__subst1,axiom,
! [A: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B3: set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_v @ B3 @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_507_order__subst1,axiom,
! [A: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( F @ B3 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_508_order__subst1,axiom,
! [A: set_Product_prod_v_v,F: nat > set_Product_prod_v_v,B3: nat,C: nat] :
( ( ord_le7336532860387713383od_v_v @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_509_order__subst1,axiom,
! [A: nat,F: set_v > nat,B3: set_v,C: set_v] :
( ( ord_less_eq_nat @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_v @ B3 @ C )
=> ( ! [X3: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_510_order__subst1,axiom,
! [A: nat,F: set_Product_prod_v_v > nat,B3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_nat @ A @ ( F @ B3 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ C )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_511_order__subst1,axiom,
! [A: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_512_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A6: set_v,B5: set_v] :
( ( ord_less_eq_set_v @ A6 @ B5 )
& ( ord_less_eq_set_v @ B5 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_513_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A6: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A6 @ B5 )
& ( ord_le7336532860387713383od_v_v @ B5 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_514_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
= ( ^ [A6: nat,B5: nat] :
( ( ord_less_eq_nat @ A6 @ B5 )
& ( ord_less_eq_nat @ B5 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_515_antisym,axiom,
! [A: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ A @ B3 )
=> ( ( ord_less_eq_set_v @ B3 @ A )
=> ( A = B3 ) ) ) ).
% antisym
thf(fact_516_antisym,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B3 )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ A )
=> ( A = B3 ) ) ) ).
% antisym
thf(fact_517_antisym,axiom,
! [A: nat,B3: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A )
=> ( A = B3 ) ) ) ).
% antisym
thf(fact_518_dual__order_Otrans,axiom,
! [B3: set_v,A: set_v,C: set_v] :
( ( ord_less_eq_set_v @ B3 @ A )
=> ( ( ord_less_eq_set_v @ C @ B3 )
=> ( ord_less_eq_set_v @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_519_dual__order_Otrans,axiom,
! [B3: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B3 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ C @ B3 )
=> ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_520_dual__order_Otrans,axiom,
! [B3: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_521_dual__order_Oantisym,axiom,
! [B3: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B3 @ A )
=> ( ( ord_less_eq_set_v @ A @ B3 )
=> ( A = B3 ) ) ) ).
% dual_order.antisym
thf(fact_522_dual__order_Oantisym,axiom,
! [B3: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B3 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ A @ B3 )
=> ( A = B3 ) ) ) ).
% dual_order.antisym
thf(fact_523_dual__order_Oantisym,axiom,
! [B3: nat,A: nat] :
( ( ord_less_eq_nat @ B3 @ A )
=> ( ( ord_less_eq_nat @ A @ B3 )
=> ( A = B3 ) ) ) ).
% dual_order.antisym
thf(fact_524_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A6: set_v,B5: set_v] :
( ( ord_less_eq_set_v @ B5 @ A6 )
& ( ord_less_eq_set_v @ A6 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_525_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A6: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B5 @ A6 )
& ( ord_le7336532860387713383od_v_v @ A6 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_526_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
= ( ^ [A6: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A6 )
& ( ord_less_eq_nat @ A6 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_527_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B3: nat] :
( ! [A7: nat,B7: nat] :
( ( ord_less_eq_nat @ A7 @ B7 )
=> ( P @ A7 @ B7 ) )
=> ( ! [A7: nat,B7: nat] :
( ( P @ B7 @ A7 )
=> ( P @ A7 @ B7 ) )
=> ( P @ A @ B3 ) ) ) ).
% linorder_wlog
thf(fact_528_order__trans,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( ord_less_eq_set_v @ Y @ Z )
=> ( ord_less_eq_set_v @ X @ Z ) ) ) ).
% order_trans
thf(fact_529_order__trans,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( ord_le7336532860387713383od_v_v @ Y @ Z )
=> ( ord_le7336532860387713383od_v_v @ X @ Z ) ) ) ).
% order_trans
thf(fact_530_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_531_order_Otrans,axiom,
! [A: set_v,B3: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B3 )
=> ( ( ord_less_eq_set_v @ B3 @ C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% order.trans
thf(fact_532_order_Otrans,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B3 )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% order.trans
thf(fact_533_order_Otrans,axiom,
! [A: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_534_order__antisym,axiom,
! [X: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( ord_less_eq_set_v @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_535_order__antisym,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_536_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_537_ord__le__eq__trans,axiom,
! [A: set_v,B3: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_538_ord__le__eq__trans,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B3 )
=> ( ( B3 = C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_539_ord__le__eq__trans,axiom,
! [A: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_540_ord__eq__le__trans,axiom,
! [A: set_v,B3: set_v,C: set_v] :
( ( A = B3 )
=> ( ( ord_less_eq_set_v @ B3 @ C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_541_ord__eq__le__trans,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A = B3 )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_542_ord__eq__le__trans,axiom,
! [A: nat,B3: nat,C: nat] :
( ( A = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_543_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [X2: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X2 @ Y3 )
& ( ord_less_eq_set_v @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_544_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ Y3 )
& ( ord_le7336532860387713383od_v_v @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_545_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_546_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_547_nle__le,axiom,
! [A: nat,B3: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B3 ) )
= ( ( ord_less_eq_nat @ B3 @ A )
& ( B3 != A ) ) ) ).
% nle_le
thf(fact_548_Pair__inject,axiom,
! [A: v,B3: v,A5: v,B4: v] :
( ( ( product_Pair_v_v @ A @ B3 )
= ( product_Pair_v_v @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B3 != B4 ) ) ) ).
% Pair_inject
thf(fact_549_prod__cases,axiom,
! [P: product_prod_v_v > $o,P2: product_prod_v_v] :
( ! [A7: v,B7: v] : ( P @ ( product_Pair_v_v @ A7 @ B7 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_550_surj__pair,axiom,
! [P2: product_prod_v_v] :
? [X3: v,Y2: v] :
( P2
= ( product_Pair_v_v @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_551_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_v_v] :
~ ! [A7: v,B7: v] :
( Y
!= ( product_Pair_v_v @ A7 @ B7 ) ) ).
% old.prod.exhaust
thf(fact_552_graph_Osubscc__add,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S3: set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl2301996248249672505od_v_v @ Successors @ S3 )
=> ( ( member7453568604450474000od_v_v @ X @ S3 )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ X )
=> ( sCC_Bl2301996248249672505od_v_v @ Successors @ ( insert1338601472111419319od_v_v @ Y @ S3 ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_553_graph_Osubscc__add,axiom,
! [Vertices: set_v,Successors: v > set_v,S3: set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S3 )
=> ( ( member_v2 @ X @ S3 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ X )
=> ( sCC_Bl5398416737448265317bscc_v @ Successors @ ( insert_v2 @ Y @ S3 ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_554_bot_Oextremum__uniqueI,axiom,
! [A: set_v] :
( ( ord_less_eq_set_v @ A @ bot_bot_set_v )
=> ( A = bot_bot_set_v ) ) ).
% bot.extremum_uniqueI
thf(fact_555_bot_Oextremum__uniqueI,axiom,
! [A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ bot_bo723834152578015283od_v_v )
=> ( A = bot_bo723834152578015283od_v_v ) ) ).
% bot.extremum_uniqueI
thf(fact_556_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_557_bot_Oextremum__unique,axiom,
! [A: set_v] :
( ( ord_less_eq_set_v @ A @ bot_bot_set_v )
= ( A = bot_bot_set_v ) ) ).
% bot.extremum_unique
thf(fact_558_bot_Oextremum__unique,axiom,
! [A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ bot_bo723834152578015283od_v_v )
= ( A = bot_bo723834152578015283od_v_v ) ) ).
% bot.extremum_unique
thf(fact_559_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_560_bot_Oextremum,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A ) ).
% bot.extremum
thf(fact_561_bot_Oextremum,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A ) ).
% bot.extremum
thf(fact_562_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_563_the__elem__eq,axiom,
! [X: product_prod_v_v] :
( ( the_el5392834299063928540od_v_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= X ) ).
% the_elem_eq
thf(fact_564_the__elem__eq,axiom,
! [X: v] :
( ( the_elem_v @ ( insert_v2 @ X @ bot_bot_set_v ) )
= X ) ).
% the_elem_eq
thf(fact_565_is__singletonI,axiom,
! [X: product_prod_v_v] : ( is_sin9198872032823709915od_v_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ).
% is_singletonI
thf(fact_566_is__singletonI,axiom,
! [X: v] : ( is_singleton_v @ ( insert_v2 @ X @ bot_bot_set_v ) ) ).
% is_singletonI
thf(fact_567_vfin,axiom,
finite_finite_v @ vertices ).
% vfin
thf(fact_568_subrelI,axiom,
! [R: set_Product_prod_v_v,S6: set_Product_prod_v_v] :
( ! [X3: v,Y2: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y2 ) @ R )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y2 ) @ S6 ) )
=> ( ord_le7336532860387713383od_v_v @ R @ S6 ) ) ).
% subrelI
thf(fact_569_set__removeAll,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( remove481895986417801203od_v_v @ X @ Xs ) )
= ( minus_4183494784930505774od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ).
% set_removeAll
thf(fact_570_set__removeAll,axiom,
! [X: v,Xs: list_v] :
( ( set_v2 @ ( removeAll_v @ X @ Xs ) )
= ( minus_minus_set_v @ ( set_v2 @ Xs ) @ ( insert_v2 @ X @ bot_bot_set_v ) ) ) ).
% set_removeAll
thf(fact_571_insert__subsetI,axiom,
! [X: v,A3: set_v,X5: set_v] :
( ( member_v2 @ X @ A3 )
=> ( ( ord_less_eq_set_v @ X5 @ A3 )
=> ( ord_less_eq_set_v @ ( insert_v2 @ X @ X5 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_572_insert__subsetI,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,X5: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ X5 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ X @ X5 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_573_subset__emptyI,axiom,
! [A3: set_v] :
( ! [X3: v] :
~ ( member_v2 @ X3 @ A3 )
=> ( ord_less_eq_set_v @ A3 @ bot_bot_set_v ) ) ).
% subset_emptyI
thf(fact_574_subset__emptyI,axiom,
! [A3: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ A3 @ bot_bo723834152578015283od_v_v ) ) ).
% subset_emptyI
thf(fact_575_List_Ofinite__set,axiom,
! [Xs: list_v] : ( finite_finite_v @ ( set_v2 @ Xs ) ) ).
% List.finite_set
thf(fact_576_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_577_removeAll__id,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( remove481895986417801203od_v_v @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_578_removeAll__id,axiom,
! [X: v,Xs: list_v] :
( ~ ( member_v2 @ X @ ( set_v2 @ Xs ) )
=> ( ( removeAll_v @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_579_finite__list,axiom,
! [A3: set_v] :
( ( finite_finite_v @ A3 )
=> ? [Xs2: list_v] :
( ( set_v2 @ Xs2 )
= A3 ) ) ).
% finite_list
thf(fact_580_finite__list,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ? [Xs2: list_nat] :
( ( set_nat2 @ Xs2 )
= A3 ) ) ).
% finite_list
thf(fact_581_graph_Ovfin,axiom,
! [Vertices: set_nat,Successors: nat > set_nat] :
( ( sCC_Bl8035451632035226289ph_nat @ Vertices @ Successors )
=> ( finite_finite_nat @ Vertices ) ) ).
% graph.vfin
thf(fact_582_graph_Ovfin,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( finite_finite_v @ Vertices ) ) ).
% graph.vfin
thf(fact_583_graph_Ointro,axiom,
! [Vertices: set_nat,Successors: nat > set_nat] :
( ( finite_finite_nat @ Vertices )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ Vertices )
=> ( ord_less_eq_set_nat @ ( Successors @ X3 ) @ Vertices ) )
=> ( sCC_Bl8035451632035226289ph_nat @ Vertices @ Successors ) ) ) ).
% graph.intro
thf(fact_584_graph_Ointro,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ Vertices )
=> ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ Vertices )
=> ( ord_le7336532860387713383od_v_v @ ( Successors @ X3 ) @ Vertices ) )
=> ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors ) ) ) ).
% graph.intro
thf(fact_585_graph_Ointro,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( finite_finite_v @ Vertices )
=> ( ! [X3: v] :
( ( member_v2 @ X3 @ Vertices )
=> ( ord_less_eq_set_v @ ( Successors @ X3 ) @ Vertices ) )
=> ( sCC_Bloemen_graph_v @ Vertices @ Successors ) ) ) ).
% graph.intro
thf(fact_586_SCC__Bloemen__Sequential_Ograph__def,axiom,
( sCC_Bl8035451632035226289ph_nat
= ( ^ [Vertices2: set_nat,Successors2: nat > set_nat] :
( ( finite_finite_nat @ Vertices2 )
& ! [X2: nat] :
( ( member_nat @ X2 @ Vertices2 )
=> ( ord_less_eq_set_nat @ ( Successors2 @ X2 ) @ Vertices2 ) ) ) ) ) ).
% SCC_Bloemen_Sequential.graph_def
thf(fact_587_SCC__Bloemen__Sequential_Ograph__def,axiom,
( sCC_Bl8307124943676871238od_v_v
= ( ^ [Vertices2: set_Product_prod_v_v,Successors2: product_prod_v_v > set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ Vertices2 )
& ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ Vertices2 )
=> ( ord_le7336532860387713383od_v_v @ ( Successors2 @ X2 ) @ Vertices2 ) ) ) ) ) ).
% SCC_Bloemen_Sequential.graph_def
thf(fact_588_SCC__Bloemen__Sequential_Ograph__def,axiom,
( sCC_Bloemen_graph_v
= ( ^ [Vertices2: set_v,Successors2: v > set_v] :
( ( finite_finite_v @ Vertices2 )
& ! [X2: v] :
( ( member_v2 @ X2 @ Vertices2 )
=> ( ord_less_eq_set_v @ ( Successors2 @ X2 ) @ Vertices2 ) ) ) ) ) ).
% SCC_Bloemen_Sequential.graph_def
thf(fact_589_is__singleton__the__elem,axiom,
( is_sin9198872032823709915od_v_v
= ( ^ [A4: set_Product_prod_v_v] :
( A4
= ( insert1338601472111419319od_v_v @ ( the_el5392834299063928540od_v_v @ A4 ) @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% is_singleton_the_elem
thf(fact_590_is__singleton__the__elem,axiom,
( is_singleton_v
= ( ^ [A4: set_v] :
( A4
= ( insert_v2 @ ( the_elem_v @ A4 ) @ bot_bot_set_v ) ) ) ) ).
% is_singleton_the_elem
thf(fact_591_is__singletonI_H,axiom,
! [A3: set_Product_prod_v_v] :
( ( A3 != bot_bo723834152578015283od_v_v )
=> ( ! [X3: product_prod_v_v,Y2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ( ( member7453568604450474000od_v_v @ Y2 @ A3 )
=> ( X3 = Y2 ) ) )
=> ( is_sin9198872032823709915od_v_v @ A3 ) ) ) ).
% is_singletonI'
thf(fact_592_is__singletonI_H,axiom,
! [A3: set_v] :
( ( A3 != bot_bot_set_v )
=> ( ! [X3: v,Y2: v] :
( ( member_v2 @ X3 @ A3 )
=> ( ( member_v2 @ Y2 @ A3 )
=> ( X3 = Y2 ) ) )
=> ( is_singleton_v @ A3 ) ) ) ).
% is_singletonI'
thf(fact_593_ssubst__Pair__rhs,axiom,
! [R: v,S6: v,R3: set_Product_prod_v_v,S7: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ R @ S6 ) @ R3 )
=> ( ( S7 = S6 )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ R @ S7 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_594_diff__right__commute,axiom,
! [A: nat,C: nat,B3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B3 )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B3 ) @ C ) ) ).
% diff_right_commute
thf(fact_595_bot__empty__eq,axiom,
( bot_bo8461541820394803818_v_v_o
= ( ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ).
% bot_empty_eq
thf(fact_596_bot__empty__eq,axiom,
( bot_bot_v_o
= ( ^ [X2: v] : ( member_v2 @ X2 @ bot_bot_set_v ) ) ) ).
% bot_empty_eq
thf(fact_597_is__singletonE,axiom,
! [A3: set_Product_prod_v_v] :
( ( is_sin9198872032823709915od_v_v @ A3 )
=> ~ ! [X3: product_prod_v_v] :
( A3
!= ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) ) ) ).
% is_singletonE
thf(fact_598_is__singletonE,axiom,
! [A3: set_v] :
( ( is_singleton_v @ A3 )
=> ~ ! [X3: v] :
( A3
!= ( insert_v2 @ X3 @ bot_bot_set_v ) ) ) ).
% is_singletonE
thf(fact_599_is__singleton__def,axiom,
( is_sin9198872032823709915od_v_v
= ( ^ [A4: set_Product_prod_v_v] :
? [X2: product_prod_v_v] :
( A4
= ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% is_singleton_def
thf(fact_600_is__singleton__def,axiom,
( is_singleton_v
= ( ^ [A4: set_v] :
? [X2: v] :
( A4
= ( insert_v2 @ X2 @ bot_bot_set_v ) ) ) ) ).
% is_singleton_def
thf(fact_601_finite__Diff__insert,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) ) )
= ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_602_finite__Diff__insert,axiom,
! [A3: set_v,A: v,B: set_v] :
( ( finite_finite_v @ ( minus_minus_set_v @ A3 @ ( insert_v2 @ A @ B ) ) )
= ( finite_finite_v @ ( minus_minus_set_v @ A3 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_603_finite__Diff__insert,axiom,
! [A3: set_nat,A: nat,B: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ B ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_604_finite__Diff,axiom,
! [A3: set_v,B: set_v] :
( ( finite_finite_v @ A3 )
=> ( finite_finite_v @ ( minus_minus_set_v @ A3 @ B ) ) ) ).
% finite_Diff
thf(fact_605_finite__Diff,axiom,
! [A3: set_nat,B: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B ) ) ) ).
% finite_Diff
thf(fact_606_finite__Diff2,axiom,
! [B: set_v,A3: set_v] :
( ( finite_finite_v @ B )
=> ( ( finite_finite_v @ ( minus_minus_set_v @ A3 @ B ) )
= ( finite_finite_v @ A3 ) ) ) ).
% finite_Diff2
thf(fact_607_finite__Diff2,axiom,
! [B: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B ) )
= ( finite_finite_nat @ A3 ) ) ) ).
% finite_Diff2
thf(fact_608_finite__Int,axiom,
! [F3: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F3 )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F3 @ G ) ) ) ).
% finite_Int
thf(fact_609_finite__Int,axiom,
! [F3: set_v,G: set_v] :
( ( ( finite_finite_v @ F3 )
| ( finite_finite_v @ G ) )
=> ( finite_finite_v @ ( inf_inf_set_v @ F3 @ G ) ) ) ).
% finite_Int
thf(fact_610_finite__remove__induct,axiom,
! [B: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( A8 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A8 @ B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_611_finite__remove__induct,axiom,
! [B: set_v,P: set_v > $o] :
( ( finite_finite_v @ B )
=> ( ( P @ bot_bot_set_v )
=> ( ! [A8: set_v] :
( ( finite_finite_v @ A8 )
=> ( ( A8 != bot_bot_set_v )
=> ( ( ord_less_eq_set_v @ A8 @ B )
=> ( ! [X4: v] :
( ( member_v2 @ X4 @ A8 )
=> ( P @ ( minus_minus_set_v @ A8 @ ( insert_v2 @ X4 @ bot_bot_set_v ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_612_finite__remove__induct,axiom,
! [B: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ B )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [A8: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A8 )
=> ( ( A8 != bot_bo723834152578015283od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ A8 @ B )
=> ( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A8 )
=> ( P @ ( minus_4183494784930505774od_v_v @ A8 @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_613_remove__induct,axiom,
! [P: set_nat > $o,B: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B )
=> ( P @ B ) )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( A8 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A8 @ B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_614_remove__induct,axiom,
! [P: set_v > $o,B: set_v] :
( ( P @ bot_bot_set_v )
=> ( ( ~ ( finite_finite_v @ B )
=> ( P @ B ) )
=> ( ! [A8: set_v] :
( ( finite_finite_v @ A8 )
=> ( ( A8 != bot_bot_set_v )
=> ( ( ord_less_eq_set_v @ A8 @ B )
=> ( ! [X4: v] :
( ( member_v2 @ X4 @ A8 )
=> ( P @ ( minus_minus_set_v @ A8 @ ( insert_v2 @ X4 @ bot_bot_set_v ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_615_remove__induct,axiom,
! [P: set_Product_prod_v_v > $o,B: set_Product_prod_v_v] :
( ( P @ bot_bo723834152578015283od_v_v )
=> ( ( ~ ( finite3348123685078250256od_v_v @ B )
=> ( P @ B ) )
=> ( ! [A8: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A8 )
=> ( ( A8 != bot_bo723834152578015283od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ A8 @ B )
=> ( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A8 )
=> ( P @ ( minus_4183494784930505774od_v_v @ A8 @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_616_finite__insert,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) )
= ( finite3348123685078250256od_v_v @ A3 ) ) ).
% finite_insert
thf(fact_617_finite__insert,axiom,
! [A: v,A3: set_v] :
( ( finite_finite_v @ ( insert_v2 @ A @ A3 ) )
= ( finite_finite_v @ A3 ) ) ).
% finite_insert
thf(fact_618_finite__insert,axiom,
! [A: nat,A3: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A3 ) )
= ( finite_finite_nat @ A3 ) ) ).
% finite_insert
thf(fact_619_finite__has__maximal2,axiom,
! [A3: set_set_v,A: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( member_set_v @ A @ A3 )
=> ? [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
& ( ord_less_eq_set_v @ A @ X3 )
& ! [Xa: set_v] :
( ( member_set_v @ Xa @ A3 )
=> ( ( ord_less_eq_set_v @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_620_finite__has__maximal2,axiom,
! [A3: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( member8406446414694345712od_v_v @ A @ A3 )
=> ? [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A3 )
& ( ord_le7336532860387713383od_v_v @ A @ X3 )
& ! [Xa: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_621_finite__has__maximal2,axiom,
! [A3: set_nat,A: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A @ A3 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_622_finite__has__minimal2,axiom,
! [A3: set_set_v,A: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( member_set_v @ A @ A3 )
=> ? [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
& ( ord_less_eq_set_v @ X3 @ A )
& ! [Xa: set_v] :
( ( member_set_v @ Xa @ A3 )
=> ( ( ord_less_eq_set_v @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_623_finite__has__minimal2,axiom,
! [A3: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( member8406446414694345712od_v_v @ A @ A3 )
=> ? [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A3 )
& ( ord_le7336532860387713383od_v_v @ X3 @ A )
& ! [Xa: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_624_finite__has__minimal2,axiom,
! [A3: set_nat,A: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A @ A3 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_625_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_626_finite_OemptyI,axiom,
finite3348123685078250256od_v_v @ bot_bo723834152578015283od_v_v ).
% finite.emptyI
thf(fact_627_finite_OemptyI,axiom,
finite_finite_v @ bot_bot_set_v ).
% finite.emptyI
thf(fact_628_infinite__imp__nonempty,axiom,
! [S3: set_nat] :
( ~ ( finite_finite_nat @ S3 )
=> ( S3 != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_629_infinite__imp__nonempty,axiom,
! [S3: set_Product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ S3 )
=> ( S3 != bot_bo723834152578015283od_v_v ) ) ).
% infinite_imp_nonempty
thf(fact_630_infinite__imp__nonempty,axiom,
! [S3: set_v] :
( ~ ( finite_finite_v @ S3 )
=> ( S3 != bot_bot_set_v ) ) ).
% infinite_imp_nonempty
thf(fact_631_finite__subset,axiom,
! [A3: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A3 ) ) ) ).
% finite_subset
thf(fact_632_finite__subset,axiom,
! [A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( finite_finite_v @ B )
=> ( finite_finite_v @ A3 ) ) ) ).
% finite_subset
thf(fact_633_finite__subset,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( finite3348123685078250256od_v_v @ B )
=> ( finite3348123685078250256od_v_v @ A3 ) ) ) ).
% finite_subset
thf(fact_634_infinite__super,axiom,
! [S3: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S3 @ T2 )
=> ( ~ ( finite_finite_nat @ S3 )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_635_infinite__super,axiom,
! [S3: set_v,T2: set_v] :
( ( ord_less_eq_set_v @ S3 @ T2 )
=> ( ~ ( finite_finite_v @ S3 )
=> ~ ( finite_finite_v @ T2 ) ) ) ).
% infinite_super
thf(fact_636_infinite__super,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ S3 @ T2 )
=> ( ~ ( finite3348123685078250256od_v_v @ S3 )
=> ~ ( finite3348123685078250256od_v_v @ T2 ) ) ) ).
% infinite_super
thf(fact_637_rev__finite__subset,axiom,
! [B: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A3 @ B )
=> ( finite_finite_nat @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_638_rev__finite__subset,axiom,
! [B: set_v,A3: set_v] :
( ( finite_finite_v @ B )
=> ( ( ord_less_eq_set_v @ A3 @ B )
=> ( finite_finite_v @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_639_rev__finite__subset,axiom,
! [B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ B )
=> ( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( finite3348123685078250256od_v_v @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_640_finite_OinsertI,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A3 )
=> ( finite3348123685078250256od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) ) ) ).
% finite.insertI
thf(fact_641_finite_OinsertI,axiom,
! [A3: set_v,A: v] :
( ( finite_finite_v @ A3 )
=> ( finite_finite_v @ ( insert_v2 @ A @ A3 ) ) ) ).
% finite.insertI
thf(fact_642_finite_OinsertI,axiom,
! [A3: set_nat,A: nat] :
( ( finite_finite_nat @ A3 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A3 ) ) ) ).
% finite.insertI
thf(fact_643_Diff__infinite__finite,axiom,
! [T2: set_v,S3: set_v] :
( ( finite_finite_v @ T2 )
=> ( ~ ( finite_finite_v @ S3 )
=> ~ ( finite_finite_v @ ( minus_minus_set_v @ S3 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_644_Diff__infinite__finite,axiom,
! [T2: set_nat,S3: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S3 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S3 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_645_finite__has__minimal,axiom,
! [A3: set_set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ? [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
& ! [Xa: set_v] :
( ( member_set_v @ Xa @ A3 )
=> ( ( ord_less_eq_set_v @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_646_finite__has__minimal,axiom,
! [A3: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ? [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A3 )
& ! [Xa: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_647_finite__has__minimal,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_648_finite__has__maximal,axiom,
! [A3: set_set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ? [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
& ! [Xa: set_v] :
( ( member_set_v @ Xa @ A3 )
=> ( ( ord_less_eq_set_v @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_649_finite__has__maximal,axiom,
! [A3: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ? [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A3 )
& ! [Xa: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_650_finite__has__maximal,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_651_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A8: set_nat] :
( ? [A7: nat] :
( A
= ( insert_nat @ A7 @ A8 ) )
=> ~ ( finite_finite_nat @ A8 ) ) ) ) ).
% finite.cases
thf(fact_652_finite_Ocases,axiom,
! [A: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A )
=> ( ( A != bot_bo723834152578015283od_v_v )
=> ~ ! [A8: set_Product_prod_v_v] :
( ? [A7: product_prod_v_v] :
( A
= ( insert1338601472111419319od_v_v @ A7 @ A8 ) )
=> ~ ( finite3348123685078250256od_v_v @ A8 ) ) ) ) ).
% finite.cases
thf(fact_653_finite_Ocases,axiom,
! [A: set_v] :
( ( finite_finite_v @ A )
=> ( ( A != bot_bot_set_v )
=> ~ ! [A8: set_v] :
( ? [A7: v] :
( A
= ( insert_v2 @ A7 @ A8 ) )
=> ~ ( finite_finite_v @ A8 ) ) ) ) ).
% finite.cases
thf(fact_654_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A6: set_nat] :
( ( A6 = bot_bot_set_nat )
| ? [A4: set_nat,B5: nat] :
( ( A6
= ( insert_nat @ B5 @ A4 ) )
& ( finite_finite_nat @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_655_finite_Osimps,axiom,
( finite3348123685078250256od_v_v
= ( ^ [A6: set_Product_prod_v_v] :
( ( A6 = bot_bo723834152578015283od_v_v )
| ? [A4: set_Product_prod_v_v,B5: product_prod_v_v] :
( ( A6
= ( insert1338601472111419319od_v_v @ B5 @ A4 ) )
& ( finite3348123685078250256od_v_v @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_656_finite_Osimps,axiom,
( finite_finite_v
= ( ^ [A6: set_v] :
( ( A6 = bot_bot_set_v )
| ? [A4: set_v,B5: v] :
( ( A6
= ( insert_v2 @ B5 @ A4 ) )
& ( finite_finite_v @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_657_finite__induct,axiom,
! [F3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ~ ( member_nat @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X3 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_658_finite__induct,axiom,
! [F3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [X3: product_prod_v_v,F4: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F4 )
=> ( ~ ( member7453568604450474000od_v_v @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert1338601472111419319od_v_v @ X3 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_659_finite__induct,axiom,
! [F3: set_v,P: set_v > $o] :
( ( finite_finite_v @ F3 )
=> ( ( P @ bot_bot_set_v )
=> ( ! [X3: v,F4: set_v] :
( ( finite_finite_v @ F4 )
=> ( ~ ( member_v2 @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_v2 @ X3 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_660_finite__ne__induct,axiom,
! [F3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ! [X3: nat] : ( P @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ! [X3: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( F4 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X3 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_661_finite__ne__induct,axiom,
! [F3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( F3 != bot_bo723834152578015283od_v_v )
=> ( ! [X3: product_prod_v_v] : ( P @ ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) )
=> ( ! [X3: product_prod_v_v,F4: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F4 )
=> ( ( F4 != bot_bo723834152578015283od_v_v )
=> ( ~ ( member7453568604450474000od_v_v @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert1338601472111419319od_v_v @ X3 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_662_finite__ne__induct,axiom,
! [F3: set_v,P: set_v > $o] :
( ( finite_finite_v @ F3 )
=> ( ( F3 != bot_bot_set_v )
=> ( ! [X3: v] : ( P @ ( insert_v2 @ X3 @ bot_bot_set_v ) )
=> ( ! [X3: v,F4: set_v] :
( ( finite_finite_v @ F4 )
=> ( ( F4 != bot_bot_set_v )
=> ( ~ ( member_v2 @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_v2 @ X3 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_663_infinite__finite__induct,axiom,
! [P: set_nat > $o,A3: set_nat] :
( ! [A8: set_nat] :
( ~ ( finite_finite_nat @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ~ ( member_nat @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X3 @ F4 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_664_infinite__finite__induct,axiom,
! [P: set_Product_prod_v_v > $o,A3: set_Product_prod_v_v] :
( ! [A8: set_Product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [X3: product_prod_v_v,F4: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F4 )
=> ( ~ ( member7453568604450474000od_v_v @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert1338601472111419319od_v_v @ X3 @ F4 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_665_infinite__finite__induct,axiom,
! [P: set_v > $o,A3: set_v] :
( ! [A8: set_v] :
( ~ ( finite_finite_v @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_v )
=> ( ! [X3: v,F4: set_v] :
( ( finite_finite_v @ F4 )
=> ( ~ ( member_v2 @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_v2 @ X3 @ F4 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_666_finite__subset__induct_H,axiom,
! [F3: set_nat,A3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A7: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat @ A7 @ A3 )
=> ( ( ord_less_eq_set_nat @ F4 @ A3 )
=> ( ~ ( member_nat @ A7 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ A7 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_667_finite__subset__induct_H,axiom,
! [F3: set_v,A3: set_v,P: set_v > $o] :
( ( finite_finite_v @ F3 )
=> ( ( ord_less_eq_set_v @ F3 @ A3 )
=> ( ( P @ bot_bot_set_v )
=> ( ! [A7: v,F4: set_v] :
( ( finite_finite_v @ F4 )
=> ( ( member_v2 @ A7 @ A3 )
=> ( ( ord_less_eq_set_v @ F4 @ A3 )
=> ( ~ ( member_v2 @ A7 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_v2 @ A7 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_668_finite__subset__induct_H,axiom,
! [F3: set_Product_prod_v_v,A3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( ord_le7336532860387713383od_v_v @ F3 @ A3 )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [A7: product_prod_v_v,F4: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F4 )
=> ( ( member7453568604450474000od_v_v @ A7 @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ F4 @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ A7 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert1338601472111419319od_v_v @ A7 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_669_finite__subset__induct,axiom,
! [F3: set_nat,A3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A7: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat @ A7 @ A3 )
=> ( ~ ( member_nat @ A7 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ A7 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_670_finite__subset__induct,axiom,
! [F3: set_v,A3: set_v,P: set_v > $o] :
( ( finite_finite_v @ F3 )
=> ( ( ord_less_eq_set_v @ F3 @ A3 )
=> ( ( P @ bot_bot_set_v )
=> ( ! [A7: v,F4: set_v] :
( ( finite_finite_v @ F4 )
=> ( ( member_v2 @ A7 @ A3 )
=> ( ~ ( member_v2 @ A7 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_v2 @ A7 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_671_finite__subset__induct,axiom,
! [F3: set_Product_prod_v_v,A3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( ord_le7336532860387713383od_v_v @ F3 @ A3 )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [A7: product_prod_v_v,F4: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F4 )
=> ( ( member7453568604450474000od_v_v @ A7 @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ A7 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert1338601472111419319od_v_v @ A7 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_672_infinite__remove,axiom,
! [S3: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S3 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_673_infinite__remove,axiom,
! [S3: set_Product_prod_v_v,A: product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ S3 )
=> ~ ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ S3 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% infinite_remove
thf(fact_674_infinite__remove,axiom,
! [S3: set_v,A: v] :
( ~ ( finite_finite_v @ S3 )
=> ~ ( finite_finite_v @ ( minus_minus_set_v @ S3 @ ( insert_v2 @ A @ bot_bot_set_v ) ) ) ) ).
% infinite_remove
thf(fact_675_infinite__coinduct,axiom,
! [X5: set_nat > $o,A3: set_nat] :
( ( X5 @ A3 )
=> ( ! [A8: set_nat] :
( ( X5 @ A8 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A8 )
& ( ( X5 @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_676_infinite__coinduct,axiom,
! [X5: set_Product_prod_v_v > $o,A3: set_Product_prod_v_v] :
( ( X5 @ A3 )
=> ( ! [A8: set_Product_prod_v_v] :
( ( X5 @ A8 )
=> ? [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A8 )
& ( ( X5 @ ( minus_4183494784930505774od_v_v @ A8 @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) )
| ~ ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A8 @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) ) ) ) )
=> ~ ( finite3348123685078250256od_v_v @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_677_infinite__coinduct,axiom,
! [X5: set_v > $o,A3: set_v] :
( ( X5 @ A3 )
=> ( ! [A8: set_v] :
( ( X5 @ A8 )
=> ? [X4: v] :
( ( member_v2 @ X4 @ A8 )
& ( ( X5 @ ( minus_minus_set_v @ A8 @ ( insert_v2 @ X4 @ bot_bot_set_v ) ) )
| ~ ( finite_finite_v @ ( minus_minus_set_v @ A8 @ ( insert_v2 @ X4 @ bot_bot_set_v ) ) ) ) ) )
=> ~ ( finite_finite_v @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_678_finite__empty__induct,axiom,
! [A3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A3 )
=> ( ( P @ A3 )
=> ( ! [A7: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( member_nat @ A7 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ A7 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_679_finite__empty__induct,axiom,
! [A3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ A3 )
=> ( ( P @ A3 )
=> ( ! [A7: product_prod_v_v,A8: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A8 )
=> ( ( member7453568604450474000od_v_v @ A7 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_4183494784930505774od_v_v @ A8 @ ( insert1338601472111419319od_v_v @ A7 @ bot_bo723834152578015283od_v_v ) ) ) ) ) )
=> ( P @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% finite_empty_induct
thf(fact_680_finite__empty__induct,axiom,
! [A3: set_v,P: set_v > $o] :
( ( finite_finite_v @ A3 )
=> ( ( P @ A3 )
=> ( ! [A7: v,A8: set_v] :
( ( finite_finite_v @ A8 )
=> ( ( member_v2 @ A7 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_minus_set_v @ A8 @ ( insert_v2 @ A7 @ bot_bot_set_v ) ) ) ) ) )
=> ( P @ bot_bot_set_v ) ) ) ) ).
% finite_empty_induct
thf(fact_681_ra__add__edge,axiom,
! [X: v,Y: v,E2: set_Product_prod_v_v,V3: v,W: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E2 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ ( sup_su414716646722978715od_v_v @ E2 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ V3 @ ( sup_su414716646722978715od_v_v @ E2 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
& ( sCC_Bl4291963740693775144ding_v @ successors @ W @ Y @ ( sup_su414716646722978715od_v_v @ E2 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ).
% ra_add_edge
thf(fact_682_finite__ranking__induct,axiom,
! [S3: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,S8: set_nat] :
( ( finite_finite_nat @ S8 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S8 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S8 )
=> ( P @ ( insert_nat @ X3 @ S8 ) ) ) ) )
=> ( P @ S3 ) ) ) ) ).
% finite_ranking_induct
thf(fact_683_finite__ranking__induct,axiom,
! [S3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o,F: product_prod_v_v > nat] :
( ( finite3348123685078250256od_v_v @ S3 )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [X3: product_prod_v_v,S8: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ S8 )
=> ( ! [Y5: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y5 @ S8 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S8 )
=> ( P @ ( insert1338601472111419319od_v_v @ X3 @ S8 ) ) ) ) )
=> ( P @ S3 ) ) ) ) ).
% finite_ranking_induct
thf(fact_684_finite__ranking__induct,axiom,
! [S3: set_v,P: set_v > $o,F: v > nat] :
( ( finite_finite_v @ S3 )
=> ( ( P @ bot_bot_set_v )
=> ( ! [X3: v,S8: set_v] :
( ( finite_finite_v @ S8 )
=> ( ! [Y5: v] :
( ( member_v2 @ Y5 @ S8 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S8 )
=> ( P @ ( insert_v2 @ X3 @ S8 ) ) ) ) )
=> ( P @ S3 ) ) ) ) ).
% finite_ranking_induct
thf(fact_685_Collect__empty__eq__bot,axiom,
! [P: product_prod_v_v > $o] :
( ( ( collec140062887454715474od_v_v @ P )
= bot_bo723834152578015283od_v_v )
= ( P = bot_bo8461541820394803818_v_v_o ) ) ).
% Collect_empty_eq_bot
thf(fact_686_Collect__empty__eq__bot,axiom,
! [P: v > $o] :
( ( ( collect_v @ P )
= bot_bot_set_v )
= ( P = bot_bot_v_o ) ) ).
% Collect_empty_eq_bot
thf(fact_687_remove__def,axiom,
( remove5001965847480235980od_v_v
= ( ^ [X2: product_prod_v_v,A4: set_Product_prod_v_v] : ( minus_4183494784930505774od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% remove_def
thf(fact_688_remove__def,axiom,
( remove_v
= ( ^ [X2: v,A4: set_v] : ( minus_minus_set_v @ A4 @ ( insert_v2 @ X2 @ bot_bot_set_v ) ) ) ) ).
% remove_def
thf(fact_689_List_Oset__insert,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( insert4539780211034306307od_v_v @ X @ Xs ) )
= ( insert1338601472111419319od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_690_List_Oset__insert,axiom,
! [X: v,Xs: list_v] :
( ( set_v2 @ ( insert_v @ X @ Xs ) )
= ( insert_v2 @ X @ ( set_v2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_691_sup_Oright__idem,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B3 ) @ B3 )
= ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ).
% sup.right_idem
thf(fact_692_sup__left__idem,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y ) )
= ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% sup_left_idem
thf(fact_693_sup_Oleft__idem,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B3 ) )
= ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ).
% sup.left_idem
thf(fact_694_sup__idem,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ X )
= X ) ).
% sup_idem
thf(fact_695_sup_Oidem,axiom,
! [A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ A )
= A ) ).
% sup.idem
thf(fact_696_Un__iff,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v2 @ C @ ( sup_sup_set_v @ A3 @ B ) )
= ( ( member_v2 @ C @ A3 )
| ( member_v2 @ C @ B ) ) ) ).
% Un_iff
thf(fact_697_Un__iff,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
= ( ( member7453568604450474000od_v_v @ C @ A3 )
| ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% Un_iff
thf(fact_698_UnCI,axiom,
! [C: v,B: set_v,A3: set_v] :
( ( ~ ( member_v2 @ C @ B )
=> ( member_v2 @ C @ A3 ) )
=> ( member_v2 @ C @ ( sup_sup_set_v @ A3 @ B ) ) ) ).
% UnCI
thf(fact_699_UnCI,axiom,
! [C: product_prod_v_v,B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ A3 ) )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ) ).
% UnCI
thf(fact_700_member__remove,axiom,
! [X: v,Y: v,A3: set_v] :
( ( member_v2 @ X @ ( remove_v @ Y @ A3 ) )
= ( ( member_v2 @ X @ A3 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_701_member__remove,axiom,
! [X: product_prod_v_v,Y: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( remove5001965847480235980od_v_v @ Y @ A3 ) )
= ( ( member7453568604450474000od_v_v @ X @ A3 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_702_le__sup__iff,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ X @ Y ) @ Z )
= ( ( ord_less_eq_set_v @ X @ Z )
& ( ord_less_eq_set_v @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_703_le__sup__iff,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ Z )
= ( ( ord_le7336532860387713383od_v_v @ X @ Z )
& ( ord_le7336532860387713383od_v_v @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_704_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_705_sup_Obounded__iff,axiom,
! [B3: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B3 @ C ) @ A )
= ( ( ord_less_eq_set_v @ B3 @ A )
& ( ord_less_eq_set_v @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_706_sup_Obounded__iff,axiom,
! [B3: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B3 @ C ) @ A )
= ( ( ord_le7336532860387713383od_v_v @ B3 @ A )
& ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_707_sup_Obounded__iff,axiom,
! [B3: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A )
= ( ( ord_less_eq_nat @ B3 @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_708_sup__bot__left,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ X )
= X ) ).
% sup_bot_left
thf(fact_709_sup__bot__left,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ X )
= X ) ).
% sup_bot_left
thf(fact_710_sup__bot__right,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ bot_bo723834152578015283od_v_v )
= X ) ).
% sup_bot_right
thf(fact_711_sup__bot__right,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ X @ bot_bot_set_v )
= X ) ).
% sup_bot_right
thf(fact_712_bot__eq__sup__iff,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( sup_su414716646722978715od_v_v @ X @ Y ) )
= ( ( X = bot_bo723834152578015283od_v_v )
& ( Y = bot_bo723834152578015283od_v_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_713_bot__eq__sup__iff,axiom,
! [X: set_v,Y: set_v] :
( ( bot_bot_set_v
= ( sup_sup_set_v @ X @ Y ) )
= ( ( X = bot_bot_set_v )
& ( Y = bot_bot_set_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_714_sup__eq__bot__iff,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ X @ Y )
= bot_bo723834152578015283od_v_v )
= ( ( X = bot_bo723834152578015283od_v_v )
& ( Y = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_715_sup__eq__bot__iff,axiom,
! [X: set_v,Y: set_v] :
( ( ( sup_sup_set_v @ X @ Y )
= bot_bot_set_v )
= ( ( X = bot_bot_set_v )
& ( Y = bot_bot_set_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_716_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A @ B3 )
= bot_bo723834152578015283od_v_v )
= ( ( A = bot_bo723834152578015283od_v_v )
& ( B3 = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_717_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_v,B3: set_v] :
( ( ( sup_sup_set_v @ A @ B3 )
= bot_bot_set_v )
= ( ( A = bot_bot_set_v )
& ( B3 = bot_bot_set_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_718_sup__bot_Oleft__neutral,axiom,
! [A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_719_sup__bot_Oleft__neutral,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_720_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( sup_su414716646722978715od_v_v @ A @ B3 ) )
= ( ( A = bot_bo723834152578015283od_v_v )
& ( B3 = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_721_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_v,B3: set_v] :
( ( bot_bot_set_v
= ( sup_sup_set_v @ A @ B3 ) )
= ( ( A = bot_bot_set_v )
& ( B3 = bot_bot_set_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_722_sup__bot_Oright__neutral,axiom,
! [A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ bot_bo723834152578015283od_v_v )
= A ) ).
% sup_bot.right_neutral
thf(fact_723_sup__bot_Oright__neutral,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ A @ bot_bot_set_v )
= A ) ).
% sup_bot.right_neutral
thf(fact_724_sup__inf__absorb,axiom,
! [X: set_v,Y: set_v] :
( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_725_sup__inf__absorb,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_726_inf__sup__absorb,axiom,
! [X: set_v,Y: set_v] :
( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_727_inf__sup__absorb,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_728_Un__empty,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A3 @ B )
= bot_bo723834152578015283od_v_v )
= ( ( A3 = bot_bo723834152578015283od_v_v )
& ( B = bot_bo723834152578015283od_v_v ) ) ) ).
% Un_empty
thf(fact_729_Un__empty,axiom,
! [A3: set_v,B: set_v] :
( ( ( sup_sup_set_v @ A3 @ B )
= bot_bot_set_v )
= ( ( A3 = bot_bot_set_v )
& ( B = bot_bot_set_v ) ) ) ).
% Un_empty
thf(fact_730_finite__Un,axiom,
! [F3: set_v,G: set_v] :
( ( finite_finite_v @ ( sup_sup_set_v @ F3 @ G ) )
= ( ( finite_finite_v @ F3 )
& ( finite_finite_v @ G ) ) ) ).
% finite_Un
thf(fact_731_finite__Un,axiom,
! [F3: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F3 @ G ) )
= ( ( finite_finite_nat @ F3 )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_732_finite__Un,axiom,
! [F3: set_Product_prod_v_v,G: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ ( sup_su414716646722978715od_v_v @ F3 @ G ) )
= ( ( finite3348123685078250256od_v_v @ F3 )
& ( finite3348123685078250256od_v_v @ G ) ) ) ).
% finite_Un
thf(fact_733_Un__subset__iff,axiom,
! [A3: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A3 @ B ) @ C2 )
= ( ( ord_less_eq_set_v @ A3 @ C2 )
& ( ord_less_eq_set_v @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_734_Un__subset__iff,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ C2 )
= ( ( ord_le7336532860387713383od_v_v @ A3 @ C2 )
& ( ord_le7336532860387713383od_v_v @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_735_Un__insert__right,axiom,
! [A3: set_v,A: v,B: set_v] :
( ( sup_sup_set_v @ A3 @ ( insert_v2 @ A @ B ) )
= ( insert_v2 @ A @ ( sup_sup_set_v @ A3 @ B ) ) ) ).
% Un_insert_right
thf(fact_736_Un__insert__right,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ) ).
% Un_insert_right
thf(fact_737_Un__insert__left,axiom,
! [A: v,B: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( insert_v2 @ A @ B ) @ C2 )
= ( insert_v2 @ A @ ( sup_sup_set_v @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_738_Un__insert__left,axiom,
! [A: product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C2 )
= ( insert1338601472111419319od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_739_Un__Int__eq_I1_J,axiom,
! [S3: set_v,T2: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ S3 @ T2 ) @ S3 )
= S3 ) ).
% Un_Int_eq(1)
thf(fact_740_Un__Int__eq_I1_J,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S3 @ T2 ) @ S3 )
= S3 ) ).
% Un_Int_eq(1)
thf(fact_741_Un__Int__eq_I2_J,axiom,
! [S3: set_v,T2: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ S3 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_742_Un__Int__eq_I2_J,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S3 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_743_Un__Int__eq_I3_J,axiom,
! [S3: set_v,T2: set_v] :
( ( inf_inf_set_v @ S3 @ ( sup_sup_set_v @ S3 @ T2 ) )
= S3 ) ).
% Un_Int_eq(3)
thf(fact_744_Un__Int__eq_I3_J,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ S3 @ ( sup_su414716646722978715od_v_v @ S3 @ T2 ) )
= S3 ) ).
% Un_Int_eq(3)
thf(fact_745_Un__Int__eq_I4_J,axiom,
! [T2: set_v,S3: set_v] :
( ( inf_inf_set_v @ T2 @ ( sup_sup_set_v @ S3 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_746_Un__Int__eq_I4_J,axiom,
! [T2: set_Product_prod_v_v,S3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ T2 @ ( sup_su414716646722978715od_v_v @ S3 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_747_Int__Un__eq_I1_J,axiom,
! [S3: set_v,T2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ S3 @ T2 ) @ S3 )
= S3 ) ).
% Int_Un_eq(1)
thf(fact_748_Int__Un__eq_I1_J,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S3 @ T2 ) @ S3 )
= S3 ) ).
% Int_Un_eq(1)
thf(fact_749_Int__Un__eq_I2_J,axiom,
! [S3: set_v,T2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ S3 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_750_Int__Un__eq_I2_J,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S3 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_751_Int__Un__eq_I3_J,axiom,
! [S3: set_v,T2: set_v] :
( ( sup_sup_set_v @ S3 @ ( inf_inf_set_v @ S3 @ T2 ) )
= S3 ) ).
% Int_Un_eq(3)
thf(fact_752_Int__Un__eq_I3_J,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ S3 @ ( inf_in6271465464967711157od_v_v @ S3 @ T2 ) )
= S3 ) ).
% Int_Un_eq(3)
thf(fact_753_Int__Un__eq_I4_J,axiom,
! [T2: set_v,S3: set_v] :
( ( sup_sup_set_v @ T2 @ ( inf_inf_set_v @ S3 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_754_Int__Un__eq_I4_J,axiom,
! [T2: set_Product_prod_v_v,S3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ T2 @ ( inf_in6271465464967711157od_v_v @ S3 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_755_Un__Diff__cancel,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B @ A3 ) )
= ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ).
% Un_Diff_cancel
thf(fact_756_Un__Diff__cancel2,axiom,
! [B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ B @ A3 ) @ A3 )
= ( sup_su414716646722978715od_v_v @ B @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_757_in__set__insert,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ( insert4539780211034306307od_v_v @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_758_in__set__insert,axiom,
! [X: v,Xs: list_v] :
( ( member_v2 @ X @ ( set_v2 @ Xs ) )
=> ( ( insert_v @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_759_sup_OcoboundedI2,axiom,
! [C: set_v,B3: set_v,A: set_v] :
( ( ord_less_eq_set_v @ C @ B3 )
=> ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_760_sup_OcoboundedI2,axiom,
! [C: set_Product_prod_v_v,B3: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ B3 )
=> ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_761_sup_OcoboundedI2,axiom,
! [C: nat,B3: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_762_sup_OcoboundedI1,axiom,
! [C: set_v,A: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ C @ A )
=> ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_763_sup_OcoboundedI1,axiom,
! [C: set_Product_prod_v_v,A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_764_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B3: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_765_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [A6: set_v,B5: set_v] :
( ( sup_sup_set_v @ A6 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_766_sup_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A6: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A6 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_767_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B5: nat] :
( ( sup_sup_nat @ A6 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_768_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [B5: set_v,A6: set_v] :
( ( sup_sup_set_v @ A6 @ B5 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_769_sup_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B5: set_Product_prod_v_v,A6: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A6 @ B5 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_770_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A6: nat] :
( ( sup_sup_nat @ A6 @ B5 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_771_sup_Ocobounded2,axiom,
! [B3: set_v,A: set_v] : ( ord_less_eq_set_v @ B3 @ ( sup_sup_set_v @ A @ B3 ) ) ).
% sup.cobounded2
thf(fact_772_sup_Ocobounded2,axiom,
! [B3: set_Product_prod_v_v,A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B3 @ ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ).
% sup.cobounded2
thf(fact_773_sup_Ocobounded2,axiom,
! [B3: nat,A: nat] : ( ord_less_eq_nat @ B3 @ ( sup_sup_nat @ A @ B3 ) ) ).
% sup.cobounded2
thf(fact_774_sup_Ocobounded1,axiom,
! [A: set_v,B3: set_v] : ( ord_less_eq_set_v @ A @ ( sup_sup_set_v @ A @ B3 ) ) ).
% sup.cobounded1
thf(fact_775_sup_Ocobounded1,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ).
% sup.cobounded1
thf(fact_776_sup_Ocobounded1,axiom,
! [A: nat,B3: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B3 ) ) ).
% sup.cobounded1
thf(fact_777_sup_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [B5: set_v,A6: set_v] :
( A6
= ( sup_sup_set_v @ A6 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_778_sup_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B5: set_Product_prod_v_v,A6: set_Product_prod_v_v] :
( A6
= ( sup_su414716646722978715od_v_v @ A6 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_779_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A6: nat] :
( A6
= ( sup_sup_nat @ A6 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_780_sup_OboundedI,axiom,
! [B3: set_v,A: set_v,C: set_v] :
( ( ord_less_eq_set_v @ B3 @ A )
=> ( ( ord_less_eq_set_v @ C @ A )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ B3 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_781_sup_OboundedI,axiom,
! [B3: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B3 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B3 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_782_sup_OboundedI,axiom,
! [B3: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_783_sup_OboundedE,axiom,
! [B3: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B3 @ C ) @ A )
=> ~ ( ( ord_less_eq_set_v @ B3 @ A )
=> ~ ( ord_less_eq_set_v @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_784_sup_OboundedE,axiom,
! [B3: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B3 @ C ) @ A )
=> ~ ( ( ord_le7336532860387713383od_v_v @ B3 @ A )
=> ~ ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_785_sup_OboundedE,axiom,
! [B3: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B3 @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_786_sup__absorb2,axiom,
! [X: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( sup_sup_set_v @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_787_sup__absorb2,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( sup_su414716646722978715od_v_v @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_788_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_789_sup__absorb1,axiom,
! [Y: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y @ X )
=> ( ( sup_sup_set_v @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_790_sup__absorb1,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( ( sup_su414716646722978715od_v_v @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_791_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_792_sup_Oabsorb2,axiom,
! [A: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ A @ B3 )
=> ( ( sup_sup_set_v @ A @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_793_sup_Oabsorb2,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B3 )
=> ( ( sup_su414716646722978715od_v_v @ A @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_794_sup_Oabsorb2,axiom,
! [A: nat,B3: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( sup_sup_nat @ A @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_795_sup_Oabsorb1,axiom,
! [B3: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B3 @ A )
=> ( ( sup_sup_set_v @ A @ B3 )
= A ) ) ).
% sup.absorb1
thf(fact_796_sup_Oabsorb1,axiom,
! [B3: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B3 @ A )
=> ( ( sup_su414716646722978715od_v_v @ A @ B3 )
= A ) ) ).
% sup.absorb1
thf(fact_797_sup_Oabsorb1,axiom,
! [B3: nat,A: nat] :
( ( ord_less_eq_nat @ B3 @ A )
=> ( ( sup_sup_nat @ A @ B3 )
= A ) ) ).
% sup.absorb1
thf(fact_798_sup__unique,axiom,
! [F: set_v > set_v > set_v,X: set_v,Y: set_v] :
( ! [X3: set_v,Y2: set_v] : ( ord_less_eq_set_v @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_v,Y2: set_v] : ( ord_less_eq_set_v @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_v,Y2: set_v,Z3: set_v] :
( ( ord_less_eq_set_v @ Y2 @ X3 )
=> ( ( ord_less_eq_set_v @ Z3 @ X3 )
=> ( ord_less_eq_set_v @ ( F @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_set_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_799_sup__unique,axiom,
! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y2 @ X3 )
=> ( ( ord_le7336532860387713383od_v_v @ Z3 @ X3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_su414716646722978715od_v_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_800_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ( ord_less_eq_nat @ ( F @ Y2 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_801_sup_OorderI,axiom,
! [A: set_v,B3: set_v] :
( ( A
= ( sup_sup_set_v @ A @ B3 ) )
=> ( ord_less_eq_set_v @ B3 @ A ) ) ).
% sup.orderI
thf(fact_802_sup_OorderI,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( A
= ( sup_su414716646722978715od_v_v @ A @ B3 ) )
=> ( ord_le7336532860387713383od_v_v @ B3 @ A ) ) ).
% sup.orderI
thf(fact_803_sup_OorderI,axiom,
! [A: nat,B3: nat] :
( ( A
= ( sup_sup_nat @ A @ B3 ) )
=> ( ord_less_eq_nat @ B3 @ A ) ) ).
% sup.orderI
thf(fact_804_sup_OorderE,axiom,
! [B3: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B3 @ A )
=> ( A
= ( sup_sup_set_v @ A @ B3 ) ) ) ).
% sup.orderE
thf(fact_805_sup_OorderE,axiom,
! [B3: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B3 @ A )
=> ( A
= ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ) ).
% sup.orderE
thf(fact_806_sup_OorderE,axiom,
! [B3: nat,A: nat] :
( ( ord_less_eq_nat @ B3 @ A )
=> ( A
= ( sup_sup_nat @ A @ B3 ) ) ) ).
% sup.orderE
thf(fact_807_le__iff__sup,axiom,
( ord_less_eq_set_v
= ( ^ [X2: set_v,Y3: set_v] :
( ( sup_sup_set_v @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_808_le__iff__sup,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_809_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( sup_sup_nat @ X2 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_810_sup__least,axiom,
! [Y: set_v,X: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ Y @ X )
=> ( ( ord_less_eq_set_v @ Z @ X )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_811_sup__least,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( ( ord_le7336532860387713383od_v_v @ Z @ X )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_812_sup__least,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_813_sup__mono,axiom,
! [A: set_v,C: set_v,B3: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ( ord_less_eq_set_v @ B3 @ D2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B3 ) @ ( sup_sup_set_v @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_814_sup__mono,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B3: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B3 ) @ ( sup_su414716646722978715od_v_v @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_815_sup__mono,axiom,
! [A: nat,C: nat,B3: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B3 @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B3 ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_816_sup_Omono,axiom,
! [C: set_v,A: set_v,D2: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ C @ A )
=> ( ( ord_less_eq_set_v @ D2 @ B3 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ C @ D2 ) @ ( sup_sup_set_v @ A @ B3 ) ) ) ) ).
% sup.mono
thf(fact_817_sup_Omono,axiom,
! [C: set_Product_prod_v_v,A: set_Product_prod_v_v,D2: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ( ord_le7336532860387713383od_v_v @ D2 @ B3 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ C @ D2 ) @ ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ) ) ).
% sup.mono
thf(fact_818_sup_Omono,axiom,
! [C: nat,A: nat,D2: nat,B3: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D2 @ B3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A @ B3 ) ) ) ) ).
% sup.mono
thf(fact_819_le__supI2,axiom,
! [X: set_v,B3: set_v,A: set_v] :
( ( ord_less_eq_set_v @ X @ B3 )
=> ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ A @ B3 ) ) ) ).
% le_supI2
thf(fact_820_le__supI2,axiom,
! [X: set_Product_prod_v_v,B3: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ B3 )
=> ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ) ).
% le_supI2
thf(fact_821_le__supI2,axiom,
! [X: nat,B3: nat,A: nat] :
( ( ord_less_eq_nat @ X @ B3 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B3 ) ) ) ).
% le_supI2
thf(fact_822_le__supI1,axiom,
! [X: set_v,A: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ X @ A )
=> ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ A @ B3 ) ) ) ).
% le_supI1
thf(fact_823_le__supI1,axiom,
! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ A )
=> ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ) ).
% le_supI1
thf(fact_824_le__supI1,axiom,
! [X: nat,A: nat,B3: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B3 ) ) ) ).
% le_supI1
thf(fact_825_sup__ge2,axiom,
! [Y: set_v,X: set_v] : ( ord_less_eq_set_v @ Y @ ( sup_sup_set_v @ X @ Y ) ) ).
% sup_ge2
thf(fact_826_sup__ge2,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y @ ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% sup_ge2
thf(fact_827_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_828_sup__ge1,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ X @ Y ) ) ).
% sup_ge1
thf(fact_829_sup__ge1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% sup_ge1
thf(fact_830_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_831_le__supI,axiom,
! [A: set_v,X: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ A @ X )
=> ( ( ord_less_eq_set_v @ B3 @ X )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B3 ) @ X ) ) ) ).
% le_supI
thf(fact_832_le__supI,axiom,
! [A: set_Product_prod_v_v,X: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ X )
=> ( ( ord_le7336532860387713383od_v_v @ B3 @ X )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B3 ) @ X ) ) ) ).
% le_supI
thf(fact_833_le__supI,axiom,
! [A: nat,X: nat,B3: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ( ord_less_eq_nat @ B3 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B3 ) @ X ) ) ) ).
% le_supI
thf(fact_834_le__supE,axiom,
! [A: set_v,B3: set_v,X: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B3 ) @ X )
=> ~ ( ( ord_less_eq_set_v @ A @ X )
=> ~ ( ord_less_eq_set_v @ B3 @ X ) ) ) ).
% le_supE
thf(fact_835_le__supE,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B3 ) @ X )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A @ X )
=> ~ ( ord_le7336532860387713383od_v_v @ B3 @ X ) ) ) ).
% le_supE
thf(fact_836_le__supE,axiom,
! [A: nat,B3: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B3 ) @ X )
=> ~ ( ( ord_less_eq_nat @ A @ X )
=> ~ ( ord_less_eq_nat @ B3 @ X ) ) ) ).
% le_supE
thf(fact_837_inf__sup__ord_I3_J,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_838_inf__sup__ord_I3_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_839_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_840_inf__sup__ord_I4_J,axiom,
! [Y: set_v,X: set_v] : ( ord_less_eq_set_v @ Y @ ( sup_sup_set_v @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_841_inf__sup__ord_I4_J,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y @ ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_842_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_843_boolean__algebra__cancel_Osup2,axiom,
! [B: set_Product_prod_v_v,K: set_Product_prod_v_v,B3: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( B
= ( sup_su414716646722978715od_v_v @ K @ B3 ) )
=> ( ( sup_su414716646722978715od_v_v @ A @ B )
= ( sup_su414716646722978715od_v_v @ K @ ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_844_boolean__algebra__cancel_Osup1,axiom,
! [A3: set_Product_prod_v_v,K: set_Product_prod_v_v,A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( A3
= ( sup_su414716646722978715od_v_v @ K @ A ) )
=> ( ( sup_su414716646722978715od_v_v @ A3 @ B3 )
= ( sup_su414716646722978715od_v_v @ K @ ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_845_Un__left__commute,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B @ C2 ) )
= ( sup_su414716646722978715od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A3 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_846_Un__left__absorb,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
= ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ).
% Un_left_absorb
thf(fact_847_Un__commute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B2 @ A4 ) ) ) ).
% Un_commute
thf(fact_848_Un__absorb,axiom,
! [A3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_849_Un__assoc,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ C2 )
= ( sup_su414716646722978715od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_850_ball__Un,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ( P @ X2 ) )
& ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_851_bex__Un,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A3 )
& ( P @ X2 ) )
| ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_852_UnI2,axiom,
! [C: v,B: set_v,A3: set_v] :
( ( member_v2 @ C @ B )
=> ( member_v2 @ C @ ( sup_sup_set_v @ A3 @ B ) ) ) ).
% UnI2
thf(fact_853_UnI2,axiom,
! [C: product_prod_v_v,B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ B )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ) ).
% UnI2
thf(fact_854_UnI1,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v2 @ C @ A3 )
=> ( member_v2 @ C @ ( sup_sup_set_v @ A3 @ B ) ) ) ).
% UnI1
thf(fact_855_UnI1,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A3 )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ) ).
% UnI1
thf(fact_856_UnE,axiom,
! [C: v,A3: set_v,B: set_v] :
( ( member_v2 @ C @ ( sup_sup_set_v @ A3 @ B ) )
=> ( ~ ( member_v2 @ C @ A3 )
=> ( member_v2 @ C @ B ) ) ) ).
% UnE
thf(fact_857_UnE,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
=> ( ~ ( member7453568604450474000od_v_v @ C @ A3 )
=> ( member7453568604450474000od_v_v @ C @ B ) ) ) ).
% UnE
thf(fact_858_sup__left__commute,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) )
= ( sup_su414716646722978715od_v_v @ Y @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_859_sup_Oleft__commute,axiom,
! [B3: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ B3 @ ( sup_su414716646722978715od_v_v @ A @ C ) )
= ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B3 @ C ) ) ) ).
% sup.left_commute
thf(fact_860_sup__commute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_861_sup_Ocommute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A6: set_Product_prod_v_v,B5: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B5 @ A6 ) ) ) ).
% sup.commute
thf(fact_862_sup__assoc,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ Z )
= ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_863_sup_Oassoc,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B3 ) @ C )
= ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B3 @ C ) ) ) ).
% sup.assoc
thf(fact_864_inf__sup__aci_I5_J,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_865_inf__sup__aci_I6_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ Z )
= ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_866_inf__sup__aci_I7_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) )
= ( sup_su414716646722978715od_v_v @ Y @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_867_inf__sup__aci_I8_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y ) )
= ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_868_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ bot_bo723834152578015283od_v_v )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_869_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ X @ bot_bot_set_v )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_870_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y ) @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_871_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_872_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y ) @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_873_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_874_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_v,Z: set_v,X: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ Y @ Z ) @ X )
= ( sup_sup_set_v @ ( inf_inf_set_v @ Y @ X ) @ ( inf_inf_set_v @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_875_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ Z ) @ X )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y @ X ) @ ( inf_in6271465464967711157od_v_v @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_876_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_v,Z: set_v,X: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ Y @ Z ) @ X )
= ( inf_inf_set_v @ ( sup_sup_set_v @ Y @ X ) @ ( sup_sup_set_v @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_877_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) @ X )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ X ) @ ( sup_su414716646722978715od_v_v @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_878_sup__inf__distrib2,axiom,
! [Y: set_v,Z: set_v,X: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ Y @ Z ) @ X )
= ( inf_inf_set_v @ ( sup_sup_set_v @ Y @ X ) @ ( sup_sup_set_v @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_879_sup__inf__distrib2,axiom,
! [Y: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) @ X )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ X ) @ ( sup_su414716646722978715od_v_v @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_880_sup__inf__distrib1,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y ) @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_881_sup__inf__distrib1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_882_inf__sup__distrib2,axiom,
! [Y: set_v,Z: set_v,X: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ Y @ Z ) @ X )
= ( sup_sup_set_v @ ( inf_inf_set_v @ Y @ X ) @ ( inf_inf_set_v @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_883_inf__sup__distrib2,axiom,
! [Y: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ Z ) @ X )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y @ X ) @ ( inf_in6271465464967711157od_v_v @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_884_inf__sup__distrib1,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y ) @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_885_inf__sup__distrib1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_886_distrib__imp2,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ! [X3: set_v,Y2: set_v,Z3: set_v] :
( ( sup_sup_set_v @ X3 @ ( inf_inf_set_v @ Y2 @ Z3 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X3 @ Y2 ) @ ( sup_sup_set_v @ X3 @ Z3 ) ) )
=> ( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y ) @ ( inf_inf_set_v @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_887_distrib__imp2,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X3 @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z3 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X3 @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X3 @ Z3 ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_888_distrib__imp1,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ! [X3: set_v,Y2: set_v,Z3: set_v] :
( ( inf_inf_set_v @ X3 @ ( sup_sup_set_v @ Y2 @ Z3 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X3 @ Y2 ) @ ( inf_inf_set_v @ X3 @ Z3 ) ) )
=> ( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y ) @ ( sup_sup_set_v @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_889_distrib__imp1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ Y2 @ Z3 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X3 @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X3 @ Z3 ) ) )
=> ( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_890_Un__empty__left,axiom,
! [B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ B )
= B ) ).
% Un_empty_left
thf(fact_891_Un__empty__left,axiom,
! [B: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ B )
= B ) ).
% Un_empty_left
thf(fact_892_Un__empty__right,axiom,
! [A3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
= A3 ) ).
% Un_empty_right
thf(fact_893_Un__empty__right,axiom,
! [A3: set_v] :
( ( sup_sup_set_v @ A3 @ bot_bot_set_v )
= A3 ) ).
% Un_empty_right
thf(fact_894_infinite__Un,axiom,
! [S3: set_v,T2: set_v] :
( ( ~ ( finite_finite_v @ ( sup_sup_set_v @ S3 @ T2 ) ) )
= ( ~ ( finite_finite_v @ S3 )
| ~ ( finite_finite_v @ T2 ) ) ) ).
% infinite_Un
thf(fact_895_infinite__Un,axiom,
! [S3: set_nat,T2: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S3 @ T2 ) ) )
= ( ~ ( finite_finite_nat @ S3 )
| ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_896_infinite__Un,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( ~ ( finite3348123685078250256od_v_v @ ( sup_su414716646722978715od_v_v @ S3 @ T2 ) ) )
= ( ~ ( finite3348123685078250256od_v_v @ S3 )
| ~ ( finite3348123685078250256od_v_v @ T2 ) ) ) ).
% infinite_Un
thf(fact_897_Un__infinite,axiom,
! [S3: set_v,T2: set_v] :
( ~ ( finite_finite_v @ S3 )
=> ~ ( finite_finite_v @ ( sup_sup_set_v @ S3 @ T2 ) ) ) ).
% Un_infinite
thf(fact_898_Un__infinite,axiom,
! [S3: set_nat,T2: set_nat] :
( ~ ( finite_finite_nat @ S3 )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S3 @ T2 ) ) ) ).
% Un_infinite
thf(fact_899_Un__infinite,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ S3 )
=> ~ ( finite3348123685078250256od_v_v @ ( sup_su414716646722978715od_v_v @ S3 @ T2 ) ) ) ).
% Un_infinite
thf(fact_900_finite__UnI,axiom,
! [F3: set_v,G: set_v] :
( ( finite_finite_v @ F3 )
=> ( ( finite_finite_v @ G )
=> ( finite_finite_v @ ( sup_sup_set_v @ F3 @ G ) ) ) ) ).
% finite_UnI
thf(fact_901_finite__UnI,axiom,
! [F3: set_nat,G: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( finite_finite_nat @ G )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F3 @ G ) ) ) ) ).
% finite_UnI
thf(fact_902_finite__UnI,axiom,
! [F3: set_Product_prod_v_v,G: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( finite3348123685078250256od_v_v @ G )
=> ( finite3348123685078250256od_v_v @ ( sup_su414716646722978715od_v_v @ F3 @ G ) ) ) ) ).
% finite_UnI
thf(fact_903_Un__mono,axiom,
! [A3: set_v,C2: set_v,B: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A3 @ C2 )
=> ( ( ord_less_eq_set_v @ B @ D )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A3 @ B ) @ ( sup_sup_set_v @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_904_Un__mono,axiom,
! [A3: set_Product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ ( sup_su414716646722978715od_v_v @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_905_Un__least,axiom,
! [A3: set_v,C2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ C2 )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A3 @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_906_Un__least,axiom,
! [A3: set_Product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_907_Un__upper1,axiom,
! [A3: set_v,B: set_v] : ( ord_less_eq_set_v @ A3 @ ( sup_sup_set_v @ A3 @ B ) ) ).
% Un_upper1
thf(fact_908_Un__upper1,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ).
% Un_upper1
thf(fact_909_Un__upper2,axiom,
! [B: set_v,A3: set_v] : ( ord_less_eq_set_v @ B @ ( sup_sup_set_v @ A3 @ B ) ) ).
% Un_upper2
thf(fact_910_Un__upper2,axiom,
! [B: set_Product_prod_v_v,A3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A3 @ B ) ) ).
% Un_upper2
thf(fact_911_Un__absorb1,axiom,
! [A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( sup_sup_set_v @ A3 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_912_Un__absorb1,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( sup_su414716646722978715od_v_v @ A3 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_913_Un__absorb2,axiom,
! [B: set_v,A3: set_v] :
( ( ord_less_eq_set_v @ B @ A3 )
=> ( ( sup_sup_set_v @ A3 @ B )
= A3 ) ) ).
% Un_absorb2
thf(fact_914_Un__absorb2,axiom,
! [B: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A3 )
=> ( ( sup_su414716646722978715od_v_v @ A3 @ B )
= A3 ) ) ).
% Un_absorb2
thf(fact_915_subset__UnE,axiom,
! [C2: set_v,A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C2 @ ( sup_sup_set_v @ A3 @ B ) )
=> ~ ! [A9: set_v] :
( ( ord_less_eq_set_v @ A9 @ A3 )
=> ! [B8: set_v] :
( ( ord_less_eq_set_v @ B8 @ B )
=> ( C2
!= ( sup_sup_set_v @ A9 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_916_subset__UnE,axiom,
! [C2: set_Product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
=> ~ ! [A9: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A9 @ A3 )
=> ! [B8: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B8 @ B )
=> ( C2
!= ( sup_su414716646722978715od_v_v @ A9 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_917_subset__Un__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B2: set_v] :
( ( sup_sup_set_v @ A4 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_918_subset__Un__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_919_Un__Int__distrib2,axiom,
! [B: set_v,C2: set_v,A3: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ B @ C2 ) @ A3 )
= ( inf_inf_set_v @ ( sup_sup_set_v @ B @ A3 ) @ ( sup_sup_set_v @ C2 @ A3 ) ) ) ).
% Un_Int_distrib2
thf(fact_920_Un__Int__distrib2,axiom,
! [B: set_Product_prod_v_v,C2: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) @ A3 )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B @ A3 ) @ ( sup_su414716646722978715od_v_v @ C2 @ A3 ) ) ) ).
% Un_Int_distrib2
thf(fact_921_Int__Un__distrib2,axiom,
! [B: set_v,C2: set_v,A3: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ B @ C2 ) @ A3 )
= ( sup_sup_set_v @ ( inf_inf_set_v @ B @ A3 ) @ ( inf_inf_set_v @ C2 @ A3 ) ) ) ).
% Int_Un_distrib2
thf(fact_922_Int__Un__distrib2,axiom,
! [B: set_Product_prod_v_v,C2: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B @ C2 ) @ A3 )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B @ A3 ) @ ( inf_in6271465464967711157od_v_v @ C2 @ A3 ) ) ) ).
% Int_Un_distrib2
thf(fact_923_Un__Int__distrib,axiom,
! [A3: set_v,B: set_v,C2: set_v] :
( ( sup_sup_set_v @ A3 @ ( inf_inf_set_v @ B @ C2 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ A3 @ B ) @ ( sup_sup_set_v @ A3 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_924_Un__Int__distrib,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ ( sup_su414716646722978715od_v_v @ A3 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_925_Int__Un__distrib,axiom,
! [A3: set_v,B: set_v,C2: set_v] :
( ( inf_inf_set_v @ A3 @ ( sup_sup_set_v @ B @ C2 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ A3 @ B ) @ ( inf_inf_set_v @ A3 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_926_Int__Un__distrib,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B @ C2 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ ( inf_in6271465464967711157od_v_v @ A3 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_927_Un__Int__crazy,axiom,
! [A3: set_v,B: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ A3 @ B ) @ ( inf_inf_set_v @ B @ C2 ) ) @ ( inf_inf_set_v @ C2 @ A3 ) )
= ( inf_inf_set_v @ ( inf_inf_set_v @ ( sup_sup_set_v @ A3 @ B ) @ ( sup_sup_set_v @ B @ C2 ) ) @ ( sup_sup_set_v @ C2 @ A3 ) ) ) ).
% Un_Int_crazy
thf(fact_928_Un__Int__crazy,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) @ ( inf_in6271465464967711157od_v_v @ C2 @ A3 ) )
= ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) @ ( sup_su414716646722978715od_v_v @ C2 @ A3 ) ) ) ).
% Un_Int_crazy
thf(fact_929_Un__Diff,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ C2 )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ C2 ) @ ( minus_4183494784930505774od_v_v @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_930_distrib__sup__le,axiom,
! [X: set_v,Y: set_v,Z: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) ) @ ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y ) @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_931_distrib__sup__le,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) ) @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_932_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_933_distrib__inf__le,axiom,
! [X: set_v,Y: set_v,Z: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y ) @ ( inf_inf_set_v @ X @ Z ) ) @ ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_934_distrib__inf__le,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) @ ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_935_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_936_singleton__Un__iff,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v )
= ( sup_su414716646722978715od_v_v @ A3 @ B ) )
= ( ( ( A3 = bot_bo723834152578015283od_v_v )
& ( B
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A3
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B = bot_bo723834152578015283od_v_v ) )
| ( ( A3
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_937_singleton__Un__iff,axiom,
! [X: v,A3: set_v,B: set_v] :
( ( ( insert_v2 @ X @ bot_bot_set_v )
= ( sup_sup_set_v @ A3 @ B ) )
= ( ( ( A3 = bot_bot_set_v )
& ( B
= ( insert_v2 @ X @ bot_bot_set_v ) ) )
| ( ( A3
= ( insert_v2 @ X @ bot_bot_set_v ) )
& ( B = bot_bot_set_v ) )
| ( ( A3
= ( insert_v2 @ X @ bot_bot_set_v ) )
& ( B
= ( insert_v2 @ X @ bot_bot_set_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_938_Un__singleton__iff,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A3 @ B )
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= ( ( ( A3 = bot_bo723834152578015283od_v_v )
& ( B
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A3
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B = bot_bo723834152578015283od_v_v ) )
| ( ( A3
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_939_Un__singleton__iff,axiom,
! [A3: set_v,B: set_v,X: v] :
( ( ( sup_sup_set_v @ A3 @ B )
= ( insert_v2 @ X @ bot_bot_set_v ) )
= ( ( ( A3 = bot_bot_set_v )
& ( B
= ( insert_v2 @ X @ bot_bot_set_v ) ) )
| ( ( A3
= ( insert_v2 @ X @ bot_bot_set_v ) )
& ( B = bot_bot_set_v ) )
| ( ( A3
= ( insert_v2 @ X @ bot_bot_set_v ) )
& ( B
= ( insert_v2 @ X @ bot_bot_set_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_940_insert__is__Un,axiom,
( insert1338601472111419319od_v_v
= ( ^ [A6: product_prod_v_v] : ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A6 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% insert_is_Un
thf(fact_941_insert__is__Un,axiom,
( insert_v2
= ( ^ [A6: v] : ( sup_sup_set_v @ ( insert_v2 @ A6 @ bot_bot_set_v ) ) ) ) ).
% insert_is_Un
thf(fact_942_Un__Int__assoc__eq,axiom,
! [A3: set_v,B: set_v,C2: set_v] :
( ( ( sup_sup_set_v @ ( inf_inf_set_v @ A3 @ B ) @ C2 )
= ( inf_inf_set_v @ A3 @ ( sup_sup_set_v @ B @ C2 ) ) )
= ( ord_less_eq_set_v @ C2 @ A3 ) ) ).
% Un_Int_assoc_eq
thf(fact_943_Un__Int__assoc__eq,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) )
= ( ord_le7336532860387713383od_v_v @ C2 @ A3 ) ) ).
% Un_Int_assoc_eq
thf(fact_944_Diff__subset__conv,axiom,
! [A3: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ B ) @ C2 )
= ( ord_less_eq_set_v @ A3 @ ( sup_sup_set_v @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_945_Diff__subset__conv,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ C2 )
= ( ord_le7336532860387713383od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_946_Diff__partition,axiom,
! [A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( sup_sup_set_v @ A3 @ ( minus_minus_set_v @ B @ A3 ) )
= B ) ) ).
% Diff_partition
thf(fact_947_Diff__partition,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( sup_su414716646722978715od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B @ A3 ) )
= B ) ) ).
% Diff_partition
thf(fact_948_Diff__Un,axiom,
! [A3: set_v,B: set_v,C2: set_v] :
( ( minus_minus_set_v @ A3 @ ( sup_sup_set_v @ B @ C2 ) )
= ( inf_inf_set_v @ ( minus_minus_set_v @ A3 @ B ) @ ( minus_minus_set_v @ A3 @ C2 ) ) ) ).
% Diff_Un
thf(fact_949_Diff__Un,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B @ C2 ) )
= ( inf_in6271465464967711157od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ ( minus_4183494784930505774od_v_v @ A3 @ C2 ) ) ) ).
% Diff_Un
thf(fact_950_Diff__Int,axiom,
! [A3: set_v,B: set_v,C2: set_v] :
( ( minus_minus_set_v @ A3 @ ( inf_inf_set_v @ B @ C2 ) )
= ( sup_sup_set_v @ ( minus_minus_set_v @ A3 @ B ) @ ( minus_minus_set_v @ A3 @ C2 ) ) ) ).
% Diff_Int
thf(fact_951_Diff__Int,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ ( minus_4183494784930505774od_v_v @ A3 @ C2 ) ) ) ).
% Diff_Int
thf(fact_952_Int__Diff__Un,axiom,
! [A3: set_v,B: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ A3 @ B ) @ ( minus_minus_set_v @ A3 @ B ) )
= A3 ) ).
% Int_Diff_Un
thf(fact_953_Int__Diff__Un,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) @ ( minus_4183494784930505774od_v_v @ A3 @ B ) )
= A3 ) ).
% Int_Diff_Un
thf(fact_954_Un__Diff__Int,axiom,
! [A3: set_v,B: set_v] :
( ( sup_sup_set_v @ ( minus_minus_set_v @ A3 @ B ) @ ( inf_inf_set_v @ A3 @ B ) )
= A3 ) ).
% Un_Diff_Int
thf(fact_955_Un__Diff__Int,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B ) @ ( inf_in6271465464967711157od_v_v @ A3 @ B ) )
= A3 ) ).
% Un_Diff_Int
thf(fact_956_graph_Ora__add__edge,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E2: set_Product_prod_v_v,V3: v,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E2 )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ ( sup_su414716646722978715od_v_v @ E2 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ V3 @ ( sup_su414716646722978715od_v_v @ E2 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ W @ Y @ ( sup_su414716646722978715od_v_v @ E2 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ).
% graph.ra_add_edge
thf(fact_957_remove__code_I1_J,axiom,
! [X: v,Xs: list_v] :
( ( remove_v @ X @ ( set_v2 @ Xs ) )
= ( set_v2 @ ( removeAll_v @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_958_set__union,axiom,
! [Xs: list_v,Ys: list_v] :
( ( set_v2 @ ( union_v @ Xs @ Ys ) )
= ( sup_sup_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ Ys ) ) ) ).
% set_union
thf(fact_959_set__union,axiom,
! [Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( union_4602324378607836129od_v_v @ Xs @ Ys ) )
= ( sup_su414716646722978715od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ Ys ) ) ) ).
% set_union
thf(fact_960_arg__min__least,axiom,
! [S3: set_nat,Y: nat,F: nat > nat] :
( ( finite_finite_nat @ S3 )
=> ( ( S3 != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S3 )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_961_arg__min__least,axiom,
! [S3: set_Product_prod_v_v,Y: product_prod_v_v,F: product_prod_v_v > nat] :
( ( finite3348123685078250256od_v_v @ S3 )
=> ( ( S3 != bot_bo723834152578015283od_v_v )
=> ( ( member7453568604450474000od_v_v @ Y @ S3 )
=> ( ord_less_eq_nat @ ( F @ ( lattic8210663337261790558_v_nat @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_962_arg__min__least,axiom,
! [S3: set_v,Y: v,F: v > nat] :
( ( finite_finite_v @ S3 )
=> ( ( S3 != bot_bot_set_v )
=> ( ( member_v2 @ Y @ S3 )
=> ( ord_less_eq_nat @ ( F @ ( lattic4614502217191357578_v_nat @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_963_Field__insert,axiom,
! [A: product_prod_v_v,B3: product_prod_v_v,R: set_Pr2149350503807050951od_v_v] :
( ( field_7153129647634986036od_v_v @ ( insert5641704497130386615od_v_v @ ( produc4031800376763917143od_v_v @ A @ B3 ) @ R ) )
= ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A @ ( insert1338601472111419319od_v_v @ B3 @ bot_bo723834152578015283od_v_v ) ) @ ( field_7153129647634986036od_v_v @ R ) ) ) ).
% Field_insert
thf(fact_964_Field__insert,axiom,
! [A: v,B3: v,R: set_Product_prod_v_v] :
( ( field_v @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ A @ B3 ) @ R ) )
= ( sup_sup_set_v @ ( insert_v2 @ A @ ( insert_v2 @ B3 @ bot_bot_set_v ) ) @ ( field_v @ R ) ) ) ).
% Field_insert
thf(fact_965_Sup__fin_Oremove,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ X @ A3 )
=> ( ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A3 )
= X ) )
& ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A3 )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_966_Sup__fin_Oremove,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( member8406446414694345712od_v_v @ X @ A3 )
=> ( ( ( ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) )
= bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ A3 )
= X ) )
& ( ( ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) )
!= bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ A3 )
= ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_967_Sup__fin_Oinsert__remove,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A3 ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A3 ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_968_Sup__fin_Oinsert__remove,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( ( ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) )
= bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X @ A3 ) )
= X ) )
& ( ( ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) )
!= bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X @ A3 ) )
= ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_969_Field__empty,axiom,
( ( field_7153129647634986036od_v_v @ bot_bo3282589961317712691od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Field_empty
thf(fact_970_Field__empty,axiom,
( ( field_v @ bot_bo723834152578015283od_v_v )
= bot_bot_set_v ) ).
% Field_empty
thf(fact_971_Field__Un,axiom,
! [R: set_Pr2149350503807050951od_v_v,S6: set_Pr2149350503807050951od_v_v] :
( ( field_7153129647634986036od_v_v @ ( sup_su1742609618068805275od_v_v @ R @ S6 ) )
= ( sup_su414716646722978715od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ ( field_7153129647634986036od_v_v @ S6 ) ) ) ).
% Field_Un
thf(fact_972_Field__Un,axiom,
! [R: set_Product_prod_v_v,S6: set_Product_prod_v_v] :
( ( field_v @ ( sup_su414716646722978715od_v_v @ R @ S6 ) )
= ( sup_sup_set_v @ ( field_v @ R ) @ ( field_v @ S6 ) ) ) ).
% Field_Un
thf(fact_973_inf__Sup__absorb,axiom,
! [A3: set_nat,A: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A @ A3 )
=> ( ( inf_inf_nat @ A @ ( lattic1093996805478795353in_nat @ A3 ) )
= A ) ) ) ).
% inf_Sup_absorb
thf(fact_974_inf__Sup__absorb,axiom,
! [A3: set_set_v,A: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( member_set_v @ A @ A3 )
=> ( ( inf_inf_set_v @ A @ ( lattic2918178447194608042_set_v @ A3 ) )
= A ) ) ) ).
% inf_Sup_absorb
thf(fact_975_Sup__fin_Oinsert,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A3 ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A3 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_976_Sup__fin_Oinsert,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X @ A3 ) )
= ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ A3 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_977_FieldI2,axiom,
! [I: product_prod_v_v,J: product_prod_v_v,R3: set_Pr2149350503807050951od_v_v] :
( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ I @ J ) @ R3 )
=> ( member7453568604450474000od_v_v @ J @ ( field_7153129647634986036od_v_v @ R3 ) ) ) ).
% FieldI2
thf(fact_978_FieldI2,axiom,
! [I: v,J: v,R3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ I @ J ) @ R3 )
=> ( member_v2 @ J @ ( field_v @ R3 ) ) ) ).
% FieldI2
thf(fact_979_FieldI1,axiom,
! [I: product_prod_v_v,J: product_prod_v_v,R3: set_Pr2149350503807050951od_v_v] :
( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ I @ J ) @ R3 )
=> ( member7453568604450474000od_v_v @ I @ ( field_7153129647634986036od_v_v @ R3 ) ) ) ).
% FieldI1
thf(fact_980_FieldI1,axiom,
! [I: v,J: v,R3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ I @ J ) @ R3 )
=> ( member_v2 @ I @ ( field_v @ R3 ) ) ) ).
% FieldI1
thf(fact_981_finite__Field,axiom,
! [R: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ R )
=> ( finite_finite_v @ ( field_v @ R ) ) ) ).
% finite_Field
thf(fact_982_finite__Field,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ R )
=> ( finite_finite_nat @ ( field_nat @ R ) ) ) ).
% finite_Field
thf(fact_983_mono__Field,axiom,
! [R: set_Pr2149350503807050951od_v_v,S6: set_Pr2149350503807050951od_v_v] :
( ( ord_le6241436655786843239od_v_v @ R @ S6 )
=> ( ord_le7336532860387713383od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ ( field_7153129647634986036od_v_v @ S6 ) ) ) ).
% mono_Field
thf(fact_984_mono__Field,axiom,
! [R: set_Product_prod_v_v,S6: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ R @ S6 )
=> ( ord_less_eq_set_v @ ( field_v @ R ) @ ( field_v @ S6 ) ) ) ).
% mono_Field
thf(fact_985_Sup__fin_OcoboundedI,axiom,
! [A3: set_set_v,A: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( member_set_v @ A @ A3 )
=> ( ord_less_eq_set_v @ A @ ( lattic2918178447194608042_set_v @ A3 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_986_Sup__fin_OcoboundedI,axiom,
! [A3: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( member8406446414694345712od_v_v @ A @ A3 )
=> ( ord_le7336532860387713383od_v_v @ A @ ( lattic5151207300795964030od_v_v @ A3 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_987_Sup__fin_OcoboundedI,axiom,
! [A3: set_nat,A: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A @ A3 )
=> ( ord_less_eq_nat @ A @ ( lattic1093996805478795353in_nat @ A3 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_988_Sup__fin_Oin__idem,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ X @ A3 )
=> ( ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A3 ) )
= ( lattic1093996805478795353in_nat @ A3 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_989_Sup__fin_Oin__idem,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( member8406446414694345712od_v_v @ X @ A3 )
=> ( ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ A3 ) )
= ( lattic5151207300795964030od_v_v @ A3 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_990_Sup__fin_Obounded__iff,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A3 ) @ X )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A3 )
=> ( ord_less_eq_set_v @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_991_Sup__fin_Obounded__iff,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ X )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_992_Sup__fin_Obounded__iff,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A3 ) @ X )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_993_Sup__fin_OboundedI,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ! [A7: set_v] :
( ( member_set_v @ A7 @ A3 )
=> ( ord_less_eq_set_v @ A7 @ X ) )
=> ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A3 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_994_Sup__fin_OboundedI,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ! [A7: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A7 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ A7 @ X ) )
=> ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_995_Sup__fin_OboundedI,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ! [A7: nat] :
( ( member_nat @ A7 @ A3 )
=> ( ord_less_eq_nat @ A7 @ X ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A3 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_996_Sup__fin_OboundedE,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A3 ) @ X )
=> ! [A10: set_v] :
( ( member_set_v @ A10 @ A3 )
=> ( ord_less_eq_set_v @ A10 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_997_Sup__fin_OboundedE,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ X )
=> ! [A10: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A10 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ A10 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_998_Sup__fin_OboundedE,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A3 ) @ X )
=> ! [A10: nat] :
( ( member_nat @ A10 @ A3 )
=> ( ord_less_eq_nat @ A10 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_999_Sup__fin_Osubset__imp,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ B )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( finite_finite_set_v @ B )
=> ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A3 ) @ ( lattic2918178447194608042_set_v @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1000_Sup__fin_Osubset__imp,axiom,
! [A3: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A3 @ B )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( finite6084192165098772208od_v_v @ B )
=> ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ ( lattic5151207300795964030od_v_v @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1001_Sup__fin_Osubset__imp,axiom,
! [A3: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A3 ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1002_Sup__fin_Osubset,axiom,
! [A3: set_nat,B: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A3 )
=> ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B ) @ ( lattic1093996805478795353in_nat @ A3 ) )
= ( lattic1093996805478795353in_nat @ A3 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1003_Sup__fin_Osubset,axiom,
! [A3: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( B != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le4714265922333009223od_v_v @ B @ A3 )
=> ( ( sup_su414716646722978715od_v_v @ ( lattic5151207300795964030od_v_v @ B ) @ ( lattic5151207300795964030od_v_v @ A3 ) )
= ( lattic5151207300795964030od_v_v @ A3 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1004_Sup__fin_Oinsert__not__elem,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ~ ( member_nat @ X @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A3 ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A3 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1005_Sup__fin_Oinsert__not__elem,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ~ ( member8406446414694345712od_v_v @ X @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X @ A3 ) )
= ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ A3 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1006_Sup__fin_Oclosed,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ! [X3: nat,Y2: nat] : ( member_nat @ ( sup_sup_nat @ X3 @ Y2 ) @ ( insert_nat @ X3 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic1093996805478795353in_nat @ A3 ) @ A3 ) ) ) ) ).
% Sup_fin.closed
thf(fact_1007_Sup__fin_Oclosed,axiom,
! [A3: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ! [X3: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( member8406446414694345712od_v_v @ ( sup_su414716646722978715od_v_v @ X3 @ Y2 ) @ ( insert7504383016908236695od_v_v @ X3 @ ( insert7504383016908236695od_v_v @ Y2 @ bot_bo3497076220358800403od_v_v ) ) )
=> ( member8406446414694345712od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ A3 ) ) ) ) ).
% Sup_fin.closed
thf(fact_1008_Sup__fin_Ounion,axiom,
! [A3: set_nat,B: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A3 @ B ) )
= ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A3 ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_1009_Sup__fin_Ounion,axiom,
! [A3: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( finite6084192165098772208od_v_v @ B )
=> ( ( B != bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( sup_su335656005089752955od_v_v @ A3 @ B ) )
= ( sup_su414716646722978715od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ ( lattic5151207300795964030od_v_v @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_1010_Inf__fin_Oinsert__remove,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A3 ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A3 ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1011_Inf__fin_Oinsert__remove,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X @ A3 ) )
= X ) )
& ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
!= bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X @ A3 ) )
= ( inf_inf_set_v @ X @ ( lattic8209813555532694032_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1012_Inf__fin_Oremove,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ X @ A3 )
=> ( ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A3 )
= X ) )
& ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A3 )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1013_Inf__fin_Oremove,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( member_set_v @ X @ A3 )
=> ( ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ A3 )
= X ) )
& ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
!= bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ A3 )
= ( inf_inf_set_v @ X @ ( lattic8209813555532694032_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1014_refl__on__singleton,axiom,
! [X: product_prod_v_v] : ( refl_o4548774019903118566od_v_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) @ ( insert5641704497130386615od_v_v @ ( produc4031800376763917143od_v_v @ X @ X ) @ bot_bo3282589961317712691od_v_v ) ) ).
% refl_on_singleton
thf(fact_1015_refl__on__singleton,axiom,
! [X: v] : ( refl_on_v @ ( insert_v2 @ X @ bot_bot_set_v ) @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ X @ X ) @ bot_bo723834152578015283od_v_v ) ) ).
% refl_on_singleton
thf(fact_1016_Inf__fin_Oinsert,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A3 ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1017_Inf__fin_Oinsert,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X @ A3 ) )
= ( inf_inf_set_v @ X @ ( lattic8209813555532694032_set_v @ A3 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1018_sup__Inf__absorb,axiom,
! [A3: set_nat,A: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A @ A3 )
=> ( ( sup_sup_nat @ ( lattic5238388535129920115in_nat @ A3 ) @ A )
= A ) ) ) ).
% sup_Inf_absorb
thf(fact_1019_sup__Inf__absorb,axiom,
! [A3: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( member8406446414694345712od_v_v @ A @ A3 )
=> ( ( sup_su414716646722978715od_v_v @ ( lattic4767070952889939172od_v_v @ A3 ) @ A )
= A ) ) ) ).
% sup_Inf_absorb
thf(fact_1020_refl__onD,axiom,
! [A3: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v] :
( ( refl_o4548774019903118566od_v_v @ A3 @ R )
=> ( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_1021_refl__onD,axiom,
! [A3: set_v,R: set_Product_prod_v_v,A: v] :
( ( refl_on_v @ A3 @ R )
=> ( ( member_v2 @ A @ A3 )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_1022_refl__onD1,axiom,
! [A3: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,X: product_prod_v_v,Y: product_prod_v_v] :
( ( refl_o4548774019903118566od_v_v @ A3 @ R )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Y ) @ R )
=> ( member7453568604450474000od_v_v @ X @ A3 ) ) ) ).
% refl_onD1
thf(fact_1023_refl__onD1,axiom,
! [A3: set_v,R: set_Product_prod_v_v,X: v,Y: v] :
( ( refl_on_v @ A3 @ R )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ R )
=> ( member_v2 @ X @ A3 ) ) ) ).
% refl_onD1
thf(fact_1024_refl__onD2,axiom,
! [A3: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,X: product_prod_v_v,Y: product_prod_v_v] :
( ( refl_o4548774019903118566od_v_v @ A3 @ R )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Y ) @ R )
=> ( member7453568604450474000od_v_v @ Y @ A3 ) ) ) ).
% refl_onD2
thf(fact_1025_refl__onD2,axiom,
! [A3: set_v,R: set_Product_prod_v_v,X: v,Y: v] :
( ( refl_on_v @ A3 @ R )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ R )
=> ( member_v2 @ Y @ A3 ) ) ) ).
% refl_onD2
thf(fact_1026_refl__on__Int,axiom,
! [A3: set_v,R: set_Product_prod_v_v,B: set_v,S6: set_Product_prod_v_v] :
( ( refl_on_v @ A3 @ R )
=> ( ( refl_on_v @ B @ S6 )
=> ( refl_on_v @ ( inf_inf_set_v @ A3 @ B ) @ ( inf_in6271465464967711157od_v_v @ R @ S6 ) ) ) ) ).
% refl_on_Int
thf(fact_1027_Inf__fin_OcoboundedI,axiom,
! [A3: set_set_v,A: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( member_set_v @ A @ A3 )
=> ( ord_less_eq_set_v @ ( lattic8209813555532694032_set_v @ A3 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1028_Inf__fin_OcoboundedI,axiom,
! [A3: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( member8406446414694345712od_v_v @ A @ A3 )
=> ( ord_le7336532860387713383od_v_v @ ( lattic4767070952889939172od_v_v @ A3 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1029_Inf__fin_OcoboundedI,axiom,
! [A3: set_nat,A: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A @ A3 )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A3 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1030_refl__on__empty,axiom,
refl_o4548774019903118566od_v_v @ bot_bo723834152578015283od_v_v @ bot_bo3282589961317712691od_v_v ).
% refl_on_empty
thf(fact_1031_refl__on__empty,axiom,
refl_on_v @ bot_bot_set_v @ bot_bo723834152578015283od_v_v ).
% refl_on_empty
thf(fact_1032_Inf__fin_Oin__idem,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ X @ A3 )
=> ( ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A3 ) )
= ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_1033_Inf__fin_Oin__idem,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( member_set_v @ X @ A3 )
=> ( ( inf_inf_set_v @ X @ ( lattic8209813555532694032_set_v @ A3 ) )
= ( lattic8209813555532694032_set_v @ A3 ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_1034_refl__on__Un,axiom,
! [A3: set_v,R: set_Product_prod_v_v,B: set_v,S6: set_Product_prod_v_v] :
( ( refl_on_v @ A3 @ R )
=> ( ( refl_on_v @ B @ S6 )
=> ( refl_on_v @ ( sup_sup_set_v @ A3 @ B ) @ ( sup_su414716646722978715od_v_v @ R @ S6 ) ) ) ) ).
% refl_on_Un
thf(fact_1035_refl__on__Un,axiom,
! [A3: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,B: set_Product_prod_v_v,S6: set_Pr2149350503807050951od_v_v] :
( ( refl_o4548774019903118566od_v_v @ A3 @ R )
=> ( ( refl_o4548774019903118566od_v_v @ B @ S6 )
=> ( refl_o4548774019903118566od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) @ ( sup_su1742609618068805275od_v_v @ R @ S6 ) ) ) ) ).
% refl_on_Un
thf(fact_1036_Inf__fin_OboundedE,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( ord_less_eq_set_v @ X @ ( lattic8209813555532694032_set_v @ A3 ) )
=> ! [A10: set_v] :
( ( member_set_v @ A10 @ A3 )
=> ( ord_less_eq_set_v @ X @ A10 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1037_Inf__fin_OboundedE,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ X @ ( lattic4767070952889939172od_v_v @ A3 ) )
=> ! [A10: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A10 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ X @ A10 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1038_Inf__fin_OboundedE,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A3 ) )
=> ! [A10: nat] :
( ( member_nat @ A10 @ A3 )
=> ( ord_less_eq_nat @ X @ A10 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1039_Inf__fin_OboundedI,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ! [A7: set_v] :
( ( member_set_v @ A7 @ A3 )
=> ( ord_less_eq_set_v @ X @ A7 ) )
=> ( ord_less_eq_set_v @ X @ ( lattic8209813555532694032_set_v @ A3 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1040_Inf__fin_OboundedI,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ! [A7: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A7 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ X @ A7 ) )
=> ( ord_le7336532860387713383od_v_v @ X @ ( lattic4767070952889939172od_v_v @ A3 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1041_Inf__fin_OboundedI,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ! [A7: nat] :
( ( member_nat @ A7 @ A3 )
=> ( ord_less_eq_nat @ X @ A7 ) )
=> ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1042_Inf__fin_Obounded__iff,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( ord_less_eq_set_v @ X @ ( lattic8209813555532694032_set_v @ A3 ) )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A3 )
=> ( ord_less_eq_set_v @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1043_Inf__fin_Obounded__iff,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ X @ ( lattic4767070952889939172od_v_v @ A3 ) )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1044_Inf__fin_Obounded__iff,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A3 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1045_Inf__fin_Osubset__imp,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ B )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( finite_finite_set_v @ B )
=> ( ord_less_eq_set_v @ ( lattic8209813555532694032_set_v @ B ) @ ( lattic8209813555532694032_set_v @ A3 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1046_Inf__fin_Osubset__imp,axiom,
! [A3: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A3 @ B )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( finite6084192165098772208od_v_v @ B )
=> ( ord_le7336532860387713383od_v_v @ ( lattic4767070952889939172od_v_v @ B ) @ ( lattic4767070952889939172od_v_v @ A3 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1047_Inf__fin_Osubset__imp,axiom,
! [A3: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1048_Inf__fin_Osubset,axiom,
! [A3: set_nat,B: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A3 )
=> ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A3 ) )
= ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_1049_Inf__fin_Osubset,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( B != bot_bot_set_set_v )
=> ( ( ord_le5216385588623774835_set_v @ B @ A3 )
=> ( ( inf_inf_set_v @ ( lattic8209813555532694032_set_v @ B ) @ ( lattic8209813555532694032_set_v @ A3 ) )
= ( lattic8209813555532694032_set_v @ A3 ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_1050_Inf__fin_Oinsert__not__elem,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ~ ( member_nat @ X @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A3 ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_1051_Inf__fin_Oinsert__not__elem,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ~ ( member_set_v @ X @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X @ A3 ) )
= ( inf_inf_set_v @ X @ ( lattic8209813555532694032_set_v @ A3 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_1052_Inf__fin_Oclosed,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ! [X3: nat,Y2: nat] : ( member_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ ( insert_nat @ X3 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic5238388535129920115in_nat @ A3 ) @ A3 ) ) ) ) ).
% Inf_fin.closed
thf(fact_1053_Inf__fin_Oclosed,axiom,
! [A3: set_set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ! [X3: set_v,Y2: set_v] : ( member_set_v @ ( inf_inf_set_v @ X3 @ Y2 ) @ ( insert_set_v @ X3 @ ( insert_set_v @ Y2 @ bot_bot_set_set_v ) ) )
=> ( member_set_v @ ( lattic8209813555532694032_set_v @ A3 ) @ A3 ) ) ) ) ).
% Inf_fin.closed
thf(fact_1054_Inf__fin_Ounion,axiom,
! [A3: set_nat,B: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( sup_sup_set_nat @ A3 @ B ) )
= ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ A3 ) @ ( lattic5238388535129920115in_nat @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_1055_Inf__fin_Ounion,axiom,
! [A3: set_set_v,B: set_set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( finite_finite_set_v @ B )
=> ( ( B != bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ ( sup_sup_set_set_v @ A3 @ B ) )
= ( inf_inf_set_v @ ( lattic8209813555532694032_set_v @ A3 ) @ ( lattic8209813555532694032_set_v @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_1056_Inf__fin__le__Sup__fin,axiom,
! [A3: set_set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ord_less_eq_set_v @ ( lattic8209813555532694032_set_v @ A3 ) @ ( lattic2918178447194608042_set_v @ A3 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_1057_Inf__fin__le__Sup__fin,axiom,
! [A3: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ord_le7336532860387713383od_v_v @ ( lattic4767070952889939172od_v_v @ A3 ) @ ( lattic5151207300795964030od_v_v @ A3 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_1058_Inf__fin__le__Sup__fin,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A3 ) @ ( lattic1093996805478795353in_nat @ A3 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_1059_refl__on__domain,axiom,
! [A3: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v,B3: product_prod_v_v] :
( ( refl_o4548774019903118566od_v_v @ A3 @ R )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A @ B3 ) @ R )
=> ( ( member7453568604450474000od_v_v @ A @ A3 )
& ( member7453568604450474000od_v_v @ B3 @ A3 ) ) ) ) ).
% refl_on_domain
thf(fact_1060_refl__on__domain,axiom,
! [A3: set_v,R: set_Product_prod_v_v,A: v,B3: v] :
( ( refl_on_v @ A3 @ R )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B3 ) @ R )
=> ( ( member_v2 @ A @ A3 )
& ( member_v2 @ B3 @ A3 ) ) ) ) ).
% refl_on_domain
thf(fact_1061_Set_Ois__empty__def,axiom,
( is_emp8964507351669718201od_v_v
= ( ^ [A4: set_Product_prod_v_v] : ( A4 = bot_bo723834152578015283od_v_v ) ) ) ).
% Set.is_empty_def
thf(fact_1062_Set_Ois__empty__def,axiom,
( is_empty_v
= ( ^ [A4: set_v] : ( A4 = bot_bot_set_v ) ) ) ).
% Set.is_empty_def
thf(fact_1063_linear__order__on__singleton,axiom,
! [X: product_prod_v_v] : ( order_6462556390437124636od_v_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) @ ( insert5641704497130386615od_v_v @ ( produc4031800376763917143od_v_v @ X @ X ) @ bot_bo3282589961317712691od_v_v ) ) ).
% linear_order_on_singleton
thf(fact_1064_linear__order__on__singleton,axiom,
! [X: v] : ( order_8768733634509060168r_on_v @ ( insert_v2 @ X @ bot_bot_set_v ) @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ X @ X ) @ bot_bo723834152578015283od_v_v ) ) ).
% linear_order_on_singleton
thf(fact_1065_lnear__order__on__empty,axiom,
order_6462556390437124636od_v_v @ bot_bo723834152578015283od_v_v @ bot_bo3282589961317712691od_v_v ).
% lnear_order_on_empty
thf(fact_1066_lnear__order__on__empty,axiom,
order_8768733634509060168r_on_v @ bot_bot_set_v @ bot_bo723834152578015283od_v_v ).
% lnear_order_on_empty
thf(fact_1067_subset__code_I2_J,axiom,
! [A3: set_v,Ys: list_v] :
( ( ord_less_eq_set_v @ A3 @ ( coset_v @ Ys ) )
= ( ! [X2: v] :
( ( member_v2 @ X2 @ ( set_v2 @ Ys ) )
=> ~ ( member_v2 @ X2 @ A3 ) ) ) ) ).
% subset_code(2)
thf(fact_1068_subset__code_I2_J,axiom,
! [A3: set_Product_prod_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ ( coset_766761627116920666od_v_v @ Ys ) )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Ys ) )
=> ~ ( member7453568604450474000od_v_v @ X2 @ A3 ) ) ) ) ).
% subset_code(2)
thf(fact_1069_insert__code_I2_J,axiom,
! [X: v,Xs: list_v] :
( ( insert_v2 @ X @ ( coset_v @ Xs ) )
= ( coset_v @ ( removeAll_v @ X @ Xs ) ) ) ).
% insert_code(2)
thf(fact_1070_insert__code_I2_J,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( insert1338601472111419319od_v_v @ X @ ( coset_766761627116920666od_v_v @ Xs ) )
= ( coset_766761627116920666od_v_v @ ( remove481895986417801203od_v_v @ X @ Xs ) ) ) ).
% insert_code(2)
thf(fact_1071_finite__Linear__order__induct,axiom,
! [R: set_Product_prod_v_v,X: v,P: v > $o] :
( ( order_8768733634509060168r_on_v @ ( field_v @ R ) @ R )
=> ( ( member_v2 @ X @ ( field_v @ R ) )
=> ( ( finite3348123685078250256od_v_v @ R )
=> ( ! [X3: v] :
( ( member_v2 @ X3 @ ( field_v @ R ) )
=> ( ! [Y5: v] :
( ( member_v2 @ Y5 @ ( order_aboveS_v @ R @ X3 ) )
=> ( P @ Y5 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% finite_Linear_order_induct
thf(fact_1072_finite__Linear__order__induct,axiom,
! [R: set_Pr2149350503807050951od_v_v,X: product_prod_v_v,P: product_prod_v_v > $o] :
( ( order_6462556390437124636od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
=> ( ( member7453568604450474000od_v_v @ X @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( finite5952053201251911184od_v_v @ R )
=> ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ! [Y5: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y5 @ ( order_1156346741491923410od_v_v @ R @ X3 ) )
=> ( P @ Y5 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% finite_Linear_order_induct
thf(fact_1073_is__empty__set,axiom,
! [Xs: list_v] :
( ( is_empty_v @ ( set_v2 @ Xs ) )
= ( null_v @ Xs ) ) ).
% is_empty_set
thf(fact_1074_Linear__order__Well__order__iff,axiom,
! [R: set_Product_prod_v_v] :
( ( order_8768733634509060168r_on_v @ ( field_v @ R ) @ R )
=> ( ( order_6972113574731384241r_on_v @ ( field_v @ R ) @ R )
= ( ! [A4: set_v] :
( ( ord_less_eq_set_v @ A4 @ ( field_v @ R ) )
=> ( ( A4 != bot_bot_set_v )
=> ? [X2: v] :
( ( member_v2 @ X2 @ A4 )
& ! [Y3: v] :
( ( member_v2 @ Y3 @ A4 )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_1075_Linear__order__Well__order__iff,axiom,
! [R: set_Pr2149350503807050951od_v_v] :
( ( order_6462556390437124636od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
=> ( ( order_7541072052284126853od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
= ( ! [A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( A4 != bot_bo723834152578015283od_v_v )
=> ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A4 )
& ! [Y3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ A4 )
=> ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X2 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_1076_well__order__on__domain,axiom,
! [A3: set_Product_prod_v_v,R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v,B3: product_prod_v_v] :
( ( order_7541072052284126853od_v_v @ A3 @ R )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A @ B3 ) @ R )
=> ( ( member7453568604450474000od_v_v @ A @ A3 )
& ( member7453568604450474000od_v_v @ B3 @ A3 ) ) ) ) ).
% well_order_on_domain
thf(fact_1077_well__order__on__domain,axiom,
! [A3: set_v,R: set_Product_prod_v_v,A: v,B3: v] :
( ( order_6972113574731384241r_on_v @ A3 @ R )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B3 ) @ R )
=> ( ( member_v2 @ A @ A3 )
& ( member_v2 @ B3 @ A3 ) ) ) ) ).
% well_order_on_domain
thf(fact_1078_well__order__on__empty,axiom,
order_7541072052284126853od_v_v @ bot_bo723834152578015283od_v_v @ bot_bo3282589961317712691od_v_v ).
% well_order_on_empty
thf(fact_1079_well__order__on__empty,axiom,
order_6972113574731384241r_on_v @ bot_bot_set_v @ bot_bo723834152578015283od_v_v ).
% well_order_on_empty
thf(fact_1080_finite__Partial__order__induct,axiom,
! [R: set_Product_prod_v_v,X: v,P: v > $o] :
( ( order_5272072345360262664r_on_v @ ( field_v @ R ) @ R )
=> ( ( member_v2 @ X @ ( field_v @ R ) )
=> ( ( finite3348123685078250256od_v_v @ R )
=> ( ! [X3: v] :
( ( member_v2 @ X3 @ ( field_v @ R ) )
=> ( ! [Y5: v] :
( ( member_v2 @ Y5 @ ( order_aboveS_v @ R @ X3 ) )
=> ( P @ Y5 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% finite_Partial_order_induct
thf(fact_1081_finite__Partial__order__induct,axiom,
! [R: set_Pr2149350503807050951od_v_v,X: product_prod_v_v,P: product_prod_v_v > $o] :
( ( order_4212533993404950492od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
=> ( ( member7453568604450474000od_v_v @ X @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( finite5952053201251911184od_v_v @ R )
=> ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ! [Y5: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y5 @ ( order_1156346741491923410od_v_v @ R @ X3 ) )
=> ( P @ Y5 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% finite_Partial_order_induct
thf(fact_1082_underS__incl__iff,axiom,
! [R: set_Product_prod_v_v,A: v,B3: v] :
( ( order_8768733634509060168r_on_v @ ( field_v @ R ) @ R )
=> ( ( member_v2 @ A @ ( field_v @ R ) )
=> ( ( member_v2 @ B3 @ ( field_v @ R ) )
=> ( ( ord_less_eq_set_v @ ( order_underS_v @ R @ A ) @ ( order_underS_v @ R @ B3 ) )
= ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B3 ) @ R ) ) ) ) ) ).
% underS_incl_iff
thf(fact_1083_underS__incl__iff,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v,B3: product_prod_v_v] :
( ( order_6462556390437124636od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
=> ( ( member7453568604450474000od_v_v @ A @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( member7453568604450474000od_v_v @ B3 @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( ord_le7336532860387713383od_v_v @ ( order_5211820470575790509od_v_v @ R @ A ) @ ( order_5211820470575790509od_v_v @ R @ B3 ) )
= ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A @ B3 ) @ R ) ) ) ) ) ).
% underS_incl_iff
thf(fact_1084_underS__I,axiom,
! [I: product_prod_v_v,J: product_prod_v_v,R3: set_Pr2149350503807050951od_v_v] :
( ( I != J )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ I @ J ) @ R3 )
=> ( member7453568604450474000od_v_v @ I @ ( order_5211820470575790509od_v_v @ R3 @ J ) ) ) ) ).
% underS_I
thf(fact_1085_underS__I,axiom,
! [I: v,J: v,R3: set_Product_prod_v_v] :
( ( I != J )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ I @ J ) @ R3 )
=> ( member_v2 @ I @ ( order_underS_v @ R3 @ J ) ) ) ) ).
% underS_I
thf(fact_1086_underS__E,axiom,
! [I: product_prod_v_v,R3: set_Pr2149350503807050951od_v_v,J: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ I @ ( order_5211820470575790509od_v_v @ R3 @ J ) )
=> ( ( I != J )
& ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ I @ J ) @ R3 ) ) ) ).
% underS_E
thf(fact_1087_underS__E,axiom,
! [I: v,R3: set_Product_prod_v_v,J: v] :
( ( member_v2 @ I @ ( order_underS_v @ R3 @ J ) )
=> ( ( I != J )
& ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ I @ J ) @ R3 ) ) ) ).
% underS_E
thf(fact_1088_BNF__Least__Fixpoint_OunderS__Field,axiom,
! [I: v,R3: set_Product_prod_v_v,J: v] :
( ( member_v2 @ I @ ( order_underS_v @ R3 @ J ) )
=> ( member_v2 @ I @ ( field_v @ R3 ) ) ) ).
% BNF_Least_Fixpoint.underS_Field
thf(fact_1089_BNF__Least__Fixpoint_OunderS__Field,axiom,
! [I: product_prod_v_v,R3: set_Pr2149350503807050951od_v_v,J: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ I @ ( order_5211820470575790509od_v_v @ R3 @ J ) )
=> ( member7453568604450474000od_v_v @ I @ ( field_7153129647634986036od_v_v @ R3 ) ) ) ).
% BNF_Least_Fixpoint.underS_Field
thf(fact_1090_underS__empty,axiom,
! [A: product_prod_v_v,R: set_Pr2149350503807050951od_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( order_5211820470575790509od_v_v @ R @ A )
= bot_bo723834152578015283od_v_v ) ) ).
% underS_empty
thf(fact_1091_underS__empty,axiom,
! [A: v,R: set_Product_prod_v_v] :
( ~ ( member_v2 @ A @ ( field_v @ R ) )
=> ( ( order_underS_v @ R @ A )
= bot_bot_set_v ) ) ).
% underS_empty
thf(fact_1092_Order__Relation_OunderS__Field,axiom,
! [R: set_Product_prod_v_v,A: v] : ( ord_less_eq_set_v @ ( order_underS_v @ R @ A ) @ ( field_v @ R ) ) ).
% Order_Relation.underS_Field
thf(fact_1093_Order__Relation_OunderS__Field,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( order_5211820470575790509od_v_v @ R @ A ) @ ( field_7153129647634986036od_v_v @ R ) ) ).
% Order_Relation.underS_Field
thf(fact_1094_partial__order__on__empty,axiom,
order_4212533993404950492od_v_v @ bot_bo723834152578015283od_v_v @ bot_bo3282589961317712691od_v_v ).
% partial_order_on_empty
thf(fact_1095_partial__order__on__empty,axiom,
order_5272072345360262664r_on_v @ bot_bot_set_v @ bot_bo723834152578015283od_v_v ).
% partial_order_on_empty
thf(fact_1096_Refl__under__underS,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v] :
( ( refl_o4548774019903118566od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
=> ( ( member7453568604450474000od_v_v @ A @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( order_6892855479609198156od_v_v @ R @ A )
= ( sup_su414716646722978715od_v_v @ ( order_5211820470575790509od_v_v @ R @ A ) @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% Refl_under_underS
thf(fact_1097_Refl__under__underS,axiom,
! [R: set_Product_prod_v_v,A: v] :
( ( refl_on_v @ ( field_v @ R ) @ R )
=> ( ( member_v2 @ A @ ( field_v @ R ) )
=> ( ( order_under_v @ R @ A )
= ( sup_sup_set_v @ ( order_underS_v @ R @ A ) @ ( insert_v2 @ A @ bot_bot_set_v ) ) ) ) ) ).
% Refl_under_underS
thf(fact_1098_Range__insert,axiom,
! [A: v,B3: v,R: set_Product_prod_v_v] :
( ( range_v_v @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ A @ B3 ) @ R ) )
= ( insert_v2 @ B3 @ ( range_v_v @ R ) ) ) ).
% Range_insert
thf(fact_1099_Domain__insert,axiom,
! [A: v,B3: v,R: set_Product_prod_v_v] :
( ( domain_v_v @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ A @ B3 ) @ R ) )
= ( insert_v2 @ A @ ( domain_v_v @ R ) ) ) ).
% Domain_insert
thf(fact_1100_Domain__empty,axiom,
( ( domain_v_v @ bot_bo723834152578015283od_v_v )
= bot_bot_set_v ) ).
% Domain_empty
thf(fact_1101_Range__empty,axiom,
( ( range_v_v @ bot_bo723834152578015283od_v_v )
= bot_bot_set_v ) ).
% Range_empty
thf(fact_1102_Field__def,axiom,
( field_7153129647634986036od_v_v
= ( ^ [R4: set_Pr2149350503807050951od_v_v] : ( sup_su414716646722978715od_v_v @ ( domain6359000466948879308od_v_v @ R4 ) @ ( range_7878975032137371189od_v_v @ R4 ) ) ) ) ).
% Field_def
thf(fact_1103_Domain_Ocases,axiom,
! [A: v,R: set_Product_prod_v_v] :
( ( member_v2 @ A @ ( domain_v_v @ R ) )
=> ~ ! [B7: v] :
~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B7 ) @ R ) ) ).
% Domain.cases
thf(fact_1104_Domain_Osimps,axiom,
! [A: v,R: set_Product_prod_v_v] :
( ( member_v2 @ A @ ( domain_v_v @ R ) )
= ( ? [A6: v,B5: v] :
( ( A = A6 )
& ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A6 @ B5 ) @ R ) ) ) ) ).
% Domain.simps
thf(fact_1105_Domain_ODomainI,axiom,
! [A: v,B3: v,R: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B3 ) @ R )
=> ( member_v2 @ A @ ( domain_v_v @ R ) ) ) ).
% Domain.DomainI
thf(fact_1106_DomainE,axiom,
! [A: v,R: set_Product_prod_v_v] :
( ( member_v2 @ A @ ( domain_v_v @ R ) )
=> ~ ! [B7: v] :
~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B7 ) @ R ) ) ).
% DomainE
thf(fact_1107_Domain__iff,axiom,
! [A: v,R: set_Product_prod_v_v] :
( ( member_v2 @ A @ ( domain_v_v @ R ) )
= ( ? [Y3: v] : ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ Y3 ) @ R ) ) ) ).
% Domain_iff
thf(fact_1108_Range__iff,axiom,
! [A: v,R: set_Product_prod_v_v] :
( ( member_v2 @ A @ ( range_v_v @ R ) )
= ( ? [Y3: v] : ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ A ) @ R ) ) ) ).
% Range_iff
thf(fact_1109_RangeE,axiom,
! [B3: v,R: set_Product_prod_v_v] :
( ( member_v2 @ B3 @ ( range_v_v @ R ) )
=> ~ ! [A7: v] :
~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A7 @ B3 ) @ R ) ) ).
% RangeE
thf(fact_1110_Range_Ointros,axiom,
! [A: v,B3: v,R: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B3 ) @ R )
=> ( member_v2 @ B3 @ ( range_v_v @ R ) ) ) ).
% Range.intros
thf(fact_1111_Range_Osimps,axiom,
! [A: v,R: set_Product_prod_v_v] :
( ( member_v2 @ A @ ( range_v_v @ R ) )
= ( ? [A6: v,B5: v] :
( ( A = B5 )
& ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A6 @ B5 ) @ R ) ) ) ) ).
% Range.simps
thf(fact_1112_Range_Ocases,axiom,
! [A: v,R: set_Product_prod_v_v] :
( ( member_v2 @ A @ ( range_v_v @ R ) )
=> ~ ! [A7: v] :
~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A7 @ A ) @ R ) ) ).
% Range.cases
thf(fact_1113_Domain__mono,axiom,
! [R: set_Product_prod_v_v,S6: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ R @ S6 )
=> ( ord_less_eq_set_v @ ( domain_v_v @ R ) @ ( domain_v_v @ S6 ) ) ) ).
% Domain_mono
thf(fact_1114_Domain__empty__iff,axiom,
! [R: set_Product_prod_v_v] :
( ( ( domain_v_v @ R )
= bot_bot_set_v )
= ( R = bot_bo723834152578015283od_v_v ) ) ).
% Domain_empty_iff
thf(fact_1115_Range__mono,axiom,
! [R: set_Product_prod_v_v,S6: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ R @ S6 )
=> ( ord_less_eq_set_v @ ( range_v_v @ R ) @ ( range_v_v @ S6 ) ) ) ).
% Range_mono
thf(fact_1116_Range__empty__iff,axiom,
! [R: set_Product_prod_v_v] :
( ( ( range_v_v @ R )
= bot_bot_set_v )
= ( R = bot_bo723834152578015283od_v_v ) ) ).
% Range_empty_iff
thf(fact_1117_Domain__Un__eq,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( domain_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
= ( sup_sup_set_v @ ( domain_v_v @ A3 ) @ ( domain_v_v @ B ) ) ) ).
% Domain_Un_eq
thf(fact_1118_under__Field,axiom,
! [R: set_Product_prod_v_v,A: v] : ( ord_less_eq_set_v @ ( order_under_v @ R @ A ) @ ( field_v @ R ) ) ).
% under_Field
thf(fact_1119_under__Field,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( order_6892855479609198156od_v_v @ R @ A ) @ ( field_7153129647634986036od_v_v @ R ) ) ).
% under_Field
thf(fact_1120_Range__Un__eq,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( range_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B ) )
= ( sup_sup_set_v @ ( range_v_v @ A3 ) @ ( range_v_v @ B ) ) ) ).
% Range_Un_eq
thf(fact_1121_underS__subset__under,axiom,
! [R: set_Product_prod_v_v,A: v] : ( ord_less_eq_set_v @ ( order_underS_v @ R @ A ) @ ( order_under_v @ R @ A ) ) ).
% underS_subset_under
thf(fact_1122_underS__subset__under,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( order_5211820470575790509od_v_v @ R @ A ) @ ( order_6892855479609198156od_v_v @ R @ A ) ) ).
% underS_subset_under
thf(fact_1123_Sup__fin_Oeq__fold,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A3 ) )
= ( finite_fold_nat_nat @ sup_sup_nat @ X @ A3 ) ) ) ).
% Sup_fin.eq_fold
thf(fact_1124_Sup__fin_Oeq__fold,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X @ A3 ) )
= ( finite8952066981541671560od_v_v @ sup_su414716646722978715od_v_v @ X @ A3 ) ) ) ).
% Sup_fin.eq_fold
thf(fact_1125_Inf__fin_Oeq__fold,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A3 ) )
= ( finite_fold_nat_nat @ inf_inf_nat @ X @ A3 ) ) ) ).
% Inf_fin.eq_fold
thf(fact_1126_Inf__fin_Oeq__fold,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X @ A3 ) )
= ( finite338946655151718280_set_v @ inf_inf_set_v @ X @ A3 ) ) ) ).
% Inf_fin.eq_fold
thf(fact_1127_pairwise__alt,axiom,
( pairwi5745945156428401490od_v_v
= ( ^ [R5: product_prod_v_v > product_prod_v_v > $o,S9: set_Product_prod_v_v] :
! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ S9 )
=> ! [Y3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ ( minus_4183494784930505774od_v_v @ S9 @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) )
=> ( R5 @ X2 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_1128_pairwise__alt,axiom,
( pairwise_v
= ( ^ [R5: v > v > $o,S9: set_v] :
! [X2: v] :
( ( member_v2 @ X2 @ S9 )
=> ! [Y3: v] :
( ( member_v2 @ Y3 @ ( minus_minus_set_v @ S9 @ ( insert_v2 @ X2 @ bot_bot_set_v ) ) )
=> ( R5 @ X2 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_1129_pairwise__empty,axiom,
! [P: product_prod_v_v > product_prod_v_v > $o] : ( pairwi5745945156428401490od_v_v @ P @ bot_bo723834152578015283od_v_v ) ).
% pairwise_empty
thf(fact_1130_pairwise__empty,axiom,
! [P: v > v > $o] : ( pairwise_v @ P @ bot_bot_set_v ) ).
% pairwise_empty
thf(fact_1131_pairwiseD,axiom,
! [R3: v > v > $o,S3: set_v,X: v,Y: v] :
( ( pairwise_v @ R3 @ S3 )
=> ( ( member_v2 @ X @ S3 )
=> ( ( member_v2 @ Y @ S3 )
=> ( ( X != Y )
=> ( R3 @ X @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_1132_pairwiseD,axiom,
! [R3: product_prod_v_v > product_prod_v_v > $o,S3: set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v] :
( ( pairwi5745945156428401490od_v_v @ R3 @ S3 )
=> ( ( member7453568604450474000od_v_v @ X @ S3 )
=> ( ( member7453568604450474000od_v_v @ Y @ S3 )
=> ( ( X != Y )
=> ( R3 @ X @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_1133_pairwiseI,axiom,
! [S3: set_v,R3: v > v > $o] :
( ! [X3: v,Y2: v] :
( ( member_v2 @ X3 @ S3 )
=> ( ( member_v2 @ Y2 @ S3 )
=> ( ( X3 != Y2 )
=> ( R3 @ X3 @ Y2 ) ) ) )
=> ( pairwise_v @ R3 @ S3 ) ) ).
% pairwiseI
thf(fact_1134_pairwiseI,axiom,
! [S3: set_Product_prod_v_v,R3: product_prod_v_v > product_prod_v_v > $o] :
( ! [X3: product_prod_v_v,Y2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ S3 )
=> ( ( member7453568604450474000od_v_v @ Y2 @ S3 )
=> ( ( X3 != Y2 )
=> ( R3 @ X3 @ Y2 ) ) ) )
=> ( pairwi5745945156428401490od_v_v @ R3 @ S3 ) ) ).
% pairwiseI
thf(fact_1135_pairwise__insert,axiom,
! [R: v > v > $o,X: v,S6: set_v] :
( ( pairwise_v @ R @ ( insert_v2 @ X @ S6 ) )
= ( ! [Y3: v] :
( ( ( member_v2 @ Y3 @ S6 )
& ( Y3 != X ) )
=> ( ( R @ X @ Y3 )
& ( R @ Y3 @ X ) ) )
& ( pairwise_v @ R @ S6 ) ) ) ).
% pairwise_insert
thf(fact_1136_pairwise__insert,axiom,
! [R: product_prod_v_v > product_prod_v_v > $o,X: product_prod_v_v,S6: set_Product_prod_v_v] :
( ( pairwi5745945156428401490od_v_v @ R @ ( insert1338601472111419319od_v_v @ X @ S6 ) )
= ( ! [Y3: product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ Y3 @ S6 )
& ( Y3 != X ) )
=> ( ( R @ X @ Y3 )
& ( R @ Y3 @ X ) ) )
& ( pairwi5745945156428401490od_v_v @ R @ S6 ) ) ) ).
% pairwise_insert
thf(fact_1137_pairwise__mono,axiom,
! [P: v > v > $o,A3: set_v,Q: v > v > $o,B: set_v] :
( ( pairwise_v @ P @ A3 )
=> ( ! [X3: v,Y2: v] :
( ( P @ X3 @ Y2 )
=> ( Q @ X3 @ Y2 ) )
=> ( ( ord_less_eq_set_v @ B @ A3 )
=> ( pairwise_v @ Q @ B ) ) ) ) ).
% pairwise_mono
thf(fact_1138_pairwise__mono,axiom,
! [P: product_prod_v_v > product_prod_v_v > $o,A3: set_Product_prod_v_v,Q: product_prod_v_v > product_prod_v_v > $o,B: set_Product_prod_v_v] :
( ( pairwi5745945156428401490od_v_v @ P @ A3 )
=> ( ! [X3: product_prod_v_v,Y2: product_prod_v_v] :
( ( P @ X3 @ Y2 )
=> ( Q @ X3 @ Y2 ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ A3 )
=> ( pairwi5745945156428401490od_v_v @ Q @ B ) ) ) ) ).
% pairwise_mono
thf(fact_1139_pairwise__subset,axiom,
! [P: v > v > $o,S3: set_v,T2: set_v] :
( ( pairwise_v @ P @ S3 )
=> ( ( ord_less_eq_set_v @ T2 @ S3 )
=> ( pairwise_v @ P @ T2 ) ) ) ).
% pairwise_subset
thf(fact_1140_pairwise__subset,axiom,
! [P: product_prod_v_v > product_prod_v_v > $o,S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( pairwi5745945156428401490od_v_v @ P @ S3 )
=> ( ( ord_le7336532860387713383od_v_v @ T2 @ S3 )
=> ( pairwi5745945156428401490od_v_v @ P @ T2 ) ) ) ).
% pairwise_subset
thf(fact_1141_pairwise__singleton,axiom,
! [P: product_prod_v_v > product_prod_v_v > $o,A3: product_prod_v_v] : ( pairwi5745945156428401490od_v_v @ P @ ( insert1338601472111419319od_v_v @ A3 @ bot_bo723834152578015283od_v_v ) ) ).
% pairwise_singleton
thf(fact_1142_pairwise__singleton,axiom,
! [P: v > v > $o,A3: v] : ( pairwise_v @ P @ ( insert_v2 @ A3 @ bot_bot_set_v ) ) ).
% pairwise_singleton
thf(fact_1143_union__fold__insert,axiom,
! [A3: set_v,B: set_v] :
( ( finite_finite_v @ A3 )
=> ( ( sup_sup_set_v @ A3 @ B )
= ( finite_fold_v_set_v @ insert_v2 @ B @ A3 ) ) ) ).
% union_fold_insert
thf(fact_1144_union__fold__insert,axiom,
! [A3: set_nat,B: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( sup_sup_set_nat @ A3 @ B )
= ( finite5529483035118572448et_nat @ insert_nat @ B @ A3 ) ) ) ).
% union_fold_insert
thf(fact_1145_union__fold__insert,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A3 )
=> ( ( sup_su414716646722978715od_v_v @ A3 @ B )
= ( finite6851115414092367464od_v_v @ insert1338601472111419319od_v_v @ B @ A3 ) ) ) ).
% union_fold_insert
thf(fact_1146_minus__fold__remove,axiom,
! [A3: set_v,B: set_v] :
( ( finite_finite_v @ A3 )
=> ( ( minus_minus_set_v @ B @ A3 )
= ( finite_fold_v_set_v @ remove_v @ B @ A3 ) ) ) ).
% minus_fold_remove
thf(fact_1147_minus__fold__remove,axiom,
! [A3: set_nat,B: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( minus_minus_set_nat @ B @ A3 )
= ( finite5529483035118572448et_nat @ remove_nat @ B @ A3 ) ) ) ).
% minus_fold_remove
thf(fact_1148_Linear__order__in__diff__Id,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v,B3: product_prod_v_v] :
( ( order_6462556390437124636od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ R )
=> ( ( member7453568604450474000od_v_v @ A @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( member7453568604450474000od_v_v @ B3 @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A @ B3 ) @ R )
= ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ B3 @ A ) @ ( minus_5255927943254941998od_v_v @ R @ id_Product_prod_v_v ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1149_Linear__order__in__diff__Id,axiom,
! [R: set_Product_prod_v_v,A: v,B3: v] :
( ( order_8768733634509060168r_on_v @ ( field_v @ R ) @ R )
=> ( ( member_v2 @ A @ ( field_v @ R ) )
=> ( ( member_v2 @ B3 @ ( field_v @ R ) )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B3 ) @ R )
= ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ B3 @ A ) @ ( minus_4183494784930505774od_v_v @ R @ id_v ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1150_psubset__insert__iff,axiom,
! [A3: set_v,X: v,B: set_v] :
( ( ord_less_set_v @ A3 @ ( insert_v2 @ X @ B ) )
= ( ( ( member_v2 @ X @ B )
=> ( ord_less_set_v @ A3 @ B ) )
& ( ~ ( member_v2 @ X @ B )
=> ( ( ( member_v2 @ X @ A3 )
=> ( ord_less_set_v @ ( minus_minus_set_v @ A3 @ ( insert_v2 @ X @ bot_bot_set_v ) ) @ B ) )
& ( ~ ( member_v2 @ X @ A3 )
=> ( ord_less_eq_set_v @ A3 @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1151_psubset__insert__iff,axiom,
! [A3: set_Product_prod_v_v,X: product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ B ) )
= ( ( ( member7453568604450474000od_v_v @ X @ B )
=> ( ord_le4186455585809229939od_v_v @ A3 @ B ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ B )
=> ( ( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ord_le4186455585809229939od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1152_psubsetI,axiom,
! [A3: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( A3 != B )
=> ( ord_less_set_v @ A3 @ B ) ) ) ).
% psubsetI
thf(fact_1153_psubsetI,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( A3 != B )
=> ( ord_le4186455585809229939od_v_v @ A3 @ B ) ) ) ).
% psubsetI
thf(fact_1154_pair__in__Id__conv,axiom,
! [A: v,B3: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B3 ) @ id_v )
= ( A = B3 ) ) ).
% pair_in_Id_conv
thf(fact_1155_IdI,axiom,
! [A: v] : ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ A ) @ id_v ) ).
% IdI
thf(fact_1156_psubset__imp__ex__mem,axiom,
! [A3: set_v,B: set_v] :
( ( ord_less_set_v @ A3 @ B )
=> ? [B7: v] : ( member_v2 @ B7 @ ( minus_minus_set_v @ B @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1157_psubset__imp__ex__mem,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ A3 @ B )
=> ? [B7: product_prod_v_v] : ( member7453568604450474000od_v_v @ B7 @ ( minus_4183494784930505774od_v_v @ B @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1158_inf_Ostrict__coboundedI2,axiom,
! [B3: set_v,C: set_v,A: set_v] :
( ( ord_less_set_v @ B3 @ C )
=> ( ord_less_set_v @ ( inf_inf_set_v @ A @ B3 ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_1159_inf_Ostrict__coboundedI2,axiom,
! [B3: nat,C: nat,A: nat] :
( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B3 ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_1160_inf_Ostrict__coboundedI1,axiom,
! [A: set_v,C: set_v,B3: set_v] :
( ( ord_less_set_v @ A @ C )
=> ( ord_less_set_v @ ( inf_inf_set_v @ A @ B3 ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_1161_inf_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B3: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B3 ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_1162_inf_Ostrict__order__iff,axiom,
( ord_less_set_v
= ( ^ [A6: set_v,B5: set_v] :
( ( A6
= ( inf_inf_set_v @ A6 @ B5 ) )
& ( A6 != B5 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_1163_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A6: nat,B5: nat] :
( ( A6
= ( inf_inf_nat @ A6 @ B5 ) )
& ( A6 != B5 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_1164_inf_Ostrict__boundedE,axiom,
! [A: set_v,B3: set_v,C: set_v] :
( ( ord_less_set_v @ A @ ( inf_inf_set_v @ B3 @ C ) )
=> ~ ( ( ord_less_set_v @ A @ B3 )
=> ~ ( ord_less_set_v @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_1165_inf_Ostrict__boundedE,axiom,
! [A: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A @ ( inf_inf_nat @ B3 @ C ) )
=> ~ ( ( ord_less_nat @ A @ B3 )
=> ~ ( ord_less_nat @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_1166_inf_Oabsorb4,axiom,
! [B3: set_v,A: set_v] :
( ( ord_less_set_v @ B3 @ A )
=> ( ( inf_inf_set_v @ A @ B3 )
= B3 ) ) ).
% inf.absorb4
thf(fact_1167_inf_Oabsorb4,axiom,
! [B3: nat,A: nat] :
( ( ord_less_nat @ B3 @ A )
=> ( ( inf_inf_nat @ A @ B3 )
= B3 ) ) ).
% inf.absorb4
thf(fact_1168_inf_Oabsorb3,axiom,
! [A: set_v,B3: set_v] :
( ( ord_less_set_v @ A @ B3 )
=> ( ( inf_inf_set_v @ A @ B3 )
= A ) ) ).
% inf.absorb3
thf(fact_1169_inf_Oabsorb3,axiom,
! [A: nat,B3: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( inf_inf_nat @ A @ B3 )
= A ) ) ).
% inf.absorb3
thf(fact_1170_less__infI2,axiom,
! [B3: set_v,X: set_v,A: set_v] :
( ( ord_less_set_v @ B3 @ X )
=> ( ord_less_set_v @ ( inf_inf_set_v @ A @ B3 ) @ X ) ) ).
% less_infI2
thf(fact_1171_less__infI2,axiom,
! [B3: nat,X: nat,A: nat] :
( ( ord_less_nat @ B3 @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B3 ) @ X ) ) ).
% less_infI2
thf(fact_1172_less__infI1,axiom,
! [A: set_v,X: set_v,B3: set_v] :
( ( ord_less_set_v @ A @ X )
=> ( ord_less_set_v @ ( inf_inf_set_v @ A @ B3 ) @ X ) ) ).
% less_infI1
thf(fact_1173_less__infI1,axiom,
! [A: nat,X: nat,B3: nat] :
( ( ord_less_nat @ A @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B3 ) @ X ) ) ).
% less_infI1
thf(fact_1174_bot_Onot__eq__extremum,axiom,
! [A: set_Product_prod_v_v] :
( ( A != bot_bo723834152578015283od_v_v )
= ( ord_le4186455585809229939od_v_v @ bot_bo723834152578015283od_v_v @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1175_bot_Onot__eq__extremum,axiom,
! [A: set_v] :
( ( A != bot_bot_set_v )
= ( ord_less_set_v @ bot_bot_set_v @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1176_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1177_bot_Oextremum__strict,axiom,
! [A: set_Product_prod_v_v] :
~ ( ord_le4186455585809229939od_v_v @ A @ bot_bo723834152578015283od_v_v ) ).
% bot.extremum_strict
thf(fact_1178_bot_Oextremum__strict,axiom,
! [A: set_v] :
~ ( ord_less_set_v @ A @ bot_bot_set_v ) ).
% bot.extremum_strict
thf(fact_1179_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_1180_not__psubset__empty,axiom,
! [A3: set_Product_prod_v_v] :
~ ( ord_le4186455585809229939od_v_v @ A3 @ bot_bo723834152578015283od_v_v ) ).
% not_psubset_empty
thf(fact_1181_not__psubset__empty,axiom,
! [A3: set_v] :
~ ( ord_less_set_v @ A3 @ bot_bot_set_v ) ).
% not_psubset_empty
thf(fact_1182_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B2: set_v] :
( ( ord_less_set_v @ A4 @ B2 )
| ( A4 = B2 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1183_subset__iff__psubset__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ A4 @ B2 )
| ( A4 = B2 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1184_subset__psubset__trans,axiom,
! [A3: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A3 @ B )
=> ( ( ord_less_set_v @ B @ C2 )
=> ( ord_less_set_v @ A3 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_1185_subset__psubset__trans,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ( ord_le4186455585809229939od_v_v @ B @ C2 )
=> ( ord_le4186455585809229939od_v_v @ A3 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_1186_subset__not__subset__eq,axiom,
( ord_less_set_v
= ( ^ [A4: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A4 @ B2 )
& ~ ( ord_less_eq_set_v @ B2 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1187_subset__not__subset__eq,axiom,
( ord_le4186455585809229939od_v_v
= ( ^ [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
& ~ ( ord_le7336532860387713383od_v_v @ B2 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1188_psubset__subset__trans,axiom,
! [A3: set_v,B: set_v,C2: set_v] :
( ( ord_less_set_v @ A3 @ B )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ord_less_set_v @ A3 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_1189_psubset__subset__trans,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ A3 @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ord_le4186455585809229939od_v_v @ A3 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_1190_psubset__imp__subset,axiom,
! [A3: set_v,B: set_v] :
( ( ord_less_set_v @ A3 @ B )
=> ( ord_less_eq_set_v @ A3 @ B ) ) ).
% psubset_imp_subset
thf(fact_1191_psubset__imp__subset,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ A3 @ B )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B ) ) ).
% psubset_imp_subset
thf(fact_1192_psubset__eq,axiom,
( ord_less_set_v
= ( ^ [A4: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A4 @ B2 )
& ( A4 != B2 ) ) ) ) ).
% psubset_eq
thf(fact_1193_psubset__eq,axiom,
( ord_le4186455585809229939od_v_v
= ( ^ [A4: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B2 )
& ( A4 != B2 ) ) ) ) ).
% psubset_eq
thf(fact_1194_psubsetE,axiom,
! [A3: set_v,B: set_v] :
( ( ord_less_set_v @ A3 @ B )
=> ~ ( ( ord_less_eq_set_v @ A3 @ B )
=> ( ord_less_eq_set_v @ B @ A3 ) ) ) ).
% psubsetE
thf(fact_1195_psubsetE,axiom,
! [A3: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ A3 @ B )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A3 @ B )
=> ( ord_le7336532860387713383od_v_v @ B @ A3 ) ) ) ).
% psubsetE
thf(fact_1196_less__supI1,axiom,
! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ X @ A )
=> ( ord_le4186455585809229939od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ) ).
% less_supI1
thf(fact_1197_less__supI1,axiom,
! [X: nat,A: nat,B3: nat] :
( ( ord_less_nat @ X @ A )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B3 ) ) ) ).
% less_supI1
thf(fact_1198_less__supI2,axiom,
! [X: set_Product_prod_v_v,B3: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ X @ B3 )
=> ( ord_le4186455585809229939od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ) ).
% less_supI2
thf(fact_1199_less__supI2,axiom,
! [X: nat,B3: nat,A: nat] :
( ( ord_less_nat @ X @ B3 )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B3 ) ) ) ).
% less_supI2
thf(fact_1200_sup_Oabsorb3,axiom,
! [B3: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ B3 @ A )
=> ( ( sup_su414716646722978715od_v_v @ A @ B3 )
= A ) ) ).
% sup.absorb3
thf(fact_1201_sup_Oabsorb3,axiom,
! [B3: nat,A: nat] :
( ( ord_less_nat @ B3 @ A )
=> ( ( sup_sup_nat @ A @ B3 )
= A ) ) ).
% sup.absorb3
thf(fact_1202_sup_Oabsorb4,axiom,
! [A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ A @ B3 )
=> ( ( sup_su414716646722978715od_v_v @ A @ B3 )
= B3 ) ) ).
% sup.absorb4
thf(fact_1203_sup_Oabsorb4,axiom,
! [A: nat,B3: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( sup_sup_nat @ A @ B3 )
= B3 ) ) ).
% sup.absorb4
thf(fact_1204_sup_Ostrict__boundedE,axiom,
! [B3: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ ( sup_su414716646722978715od_v_v @ B3 @ C ) @ A )
=> ~ ( ( ord_le4186455585809229939od_v_v @ B3 @ A )
=> ~ ( ord_le4186455585809229939od_v_v @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_1205_sup_Ostrict__boundedE,axiom,
! [B3: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B3 @ C ) @ A )
=> ~ ( ( ord_less_nat @ B3 @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_1206_sup_Ostrict__order__iff,axiom,
( ord_le4186455585809229939od_v_v
= ( ^ [B5: set_Product_prod_v_v,A6: set_Product_prod_v_v] :
( ( A6
= ( sup_su414716646722978715od_v_v @ A6 @ B5 ) )
& ( A6 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1207_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B5: nat,A6: nat] :
( ( A6
= ( sup_sup_nat @ A6 @ B5 ) )
& ( A6 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1208_sup_Ostrict__coboundedI1,axiom,
! [C: set_Product_prod_v_v,A: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ C @ A )
=> ( ord_le4186455585809229939od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_1209_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B3: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B3 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_1210_sup_Ostrict__coboundedI2,axiom,
! [C: set_Product_prod_v_v,B3: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ C @ B3 )
=> ( ord_le4186455585809229939od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B3 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_1211_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B3: nat,A: nat] :
( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B3 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_1212_IdE,axiom,
! [P2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ P2 @ id_v )
=> ~ ! [X3: v] :
( P2
!= ( product_Pair_v_v @ X3 @ X3 ) ) ) ).
% IdE
thf(fact_1213_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_1214_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_1215_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_1216_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_1217_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_1218_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_1219_order__less__subst2,axiom,
! [A: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_1220_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_1221_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_1222_ord__less__eq__subst,axiom,
! [A: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_1223_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B3: nat,C: nat] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1224_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_1225_order__less__asym_H,axiom,
! [A: nat,B3: nat] :
( ( ord_less_nat @ A @ B3 )
=> ~ ( ord_less_nat @ B3 @ A ) ) ).
% order_less_asym'
thf(fact_1226_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_1227_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_1228_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_1229_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: nat,A: nat] :
( ( ord_less_nat @ B3 @ A )
=> ( A != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_1230_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B3: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( A != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_1231_dual__order_Ostrict__trans,axiom,
! [B3: nat,A: nat,C: nat] :
( ( ord_less_nat @ B3 @ A )
=> ( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_1232_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_1233_order_Ostrict__trans,axiom,
! [A: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_1234_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B3: nat] :
( ! [A7: nat,B7: nat] :
( ( ord_less_nat @ A7 @ B7 )
=> ( P @ A7 @ B7 ) )
=> ( ! [A7: nat] : ( P @ A7 @ A7 )
=> ( ! [A7: nat,B7: nat] :
( ( P @ B7 @ A7 )
=> ( P @ A7 @ B7 ) )
=> ( P @ A @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_1235_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
? [N2: nat] :
( ( P4 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P4 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_1236_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_1237_dual__order_Oasym,axiom,
! [B3: nat,A: nat] :
( ( ord_less_nat @ B3 @ A )
=> ~ ( ord_less_nat @ A @ B3 ) ) ).
% dual_order.asym
thf(fact_1238_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_1239_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_1240_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_1241_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I2: nat] :
( ( ord_less_nat @ K2 @ I2 )
=> ( P @ I2 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1242_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1243_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1244_le__diff__iff_H,axiom,
! [A: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B3 ) )
= ( ord_less_eq_nat @ B3 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1245_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1246_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1247_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1248_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1249_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1250_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B3 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1251_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_1252_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_1253_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1254_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_1255_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1256_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1257_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1258_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1259_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1260_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1261_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1262_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1263_diff__less__mono,axiom,
! [A: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B3 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1264_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1265_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1266_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_1267_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_1268_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_1269_bounded__nat__set__is__finite,axiom,
! [N3: set_nat,N: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N3 )
=> ( ord_less_nat @ X3 @ N ) )
=> ( finite_finite_nat @ N3 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1270_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M2: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N4 )
=> ( ord_less_nat @ X2 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1271_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M3: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M3 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1272_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M2: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N4 )
=> ( ord_less_eq_nat @ X2 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1273_bdd__above__nat,axiom,
condit2214826472909112428ve_nat = finite_finite_nat ).
% bdd_above_nat
% Conjectures (2)
thf(conj_0,hypothesis,
! [N5: v] :
( ( member_v2 @ N5 @ ( set_v2 @ ( sCC_Bl1791845272665611460ck_v_a @ e ) ) )
=> ( ( member_v2 @ m @ ( sCC_Bloemen_S_v_a @ e @ N5 ) )
=> thesis ) ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------