TPTP Problem File: SLH0863^1.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_02522_086887__6373340_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1537 ( 615 unt; 252 typ; 0 def)
% Number of atoms : 3743 (1284 equ; 0 cnn)
% Maximal formula atoms : 16 ( 2 avg)
% Number of connectives : 12072 ( 448 ~; 52 |; 310 &;9591 @)
% ( 0 <=>;1671 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 7 avg)
% Number of types : 26 ( 25 usr)
% Number of type conns : 898 ( 898 >; 0 *; 0 +; 0 <<)
% Number of symbols : 230 ( 227 usr; 19 con; 0-9 aty)
% Number of variables : 3600 ( 180 ^;3318 !; 102 ?;3600 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:54:08.192
%------------------------------------------------------------------------------
% Could-be-implicit typings (25)
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thf(sy_c_Set_Oimage_001t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
image_2941599411333419807od_v_v: ( list_P7986770385144383213od_v_v > set_Product_prod_v_v ) > set_li2323639185124838733od_v_v > set_se8455005133513928103od_v_v ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__v_J_001t__List__Olist_Itf__v_J,type,
image_list_v_list_v: ( list_v > list_v ) > set_list_v > set_list_v ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__v_J_001t__Set__Oset_Itf__v_J,type,
image_list_v_set_v: ( list_v > set_v ) > set_list_v > set_set_v ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
image_781944334261467077od_v_v: ( product_prod_v_v > product_prod_v_v ) > set_Product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001tf__v,type,
image_6152814753742948081_v_v_v: ( product_prod_v_v > v ) > set_Product_prod_v_v > set_v ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
image_5212826947168092101od_v_v: ( set_Product_prod_v_v > set_Product_prod_v_v ) > set_se8455005133513928103od_v_v > set_se8455005133513928103od_v_v ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__v_J_001t__Set__Oset_Itf__v_J,type,
image_set_v_set_v: ( set_v > set_v ) > set_set_v > set_set_v ).
thf(sy_c_Set_Oimage_001tf__v_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
image_9222788639401671577od_v_v: ( v > product_prod_v_v ) > set_v > set_Product_prod_v_v ).
thf(sy_c_Set_Oimage_001tf__v_001tf__v,type,
image_v_v: ( v > v ) > set_v > set_v ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
insert4087971119735676093od_v_v: list_P7986770385144383213od_v_v > set_li2323639185124838733od_v_v > set_li2323639185124838733od_v_v ).
thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__v_J,type,
insert_list_v: list_v > set_list_v > set_list_v ).
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_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J,type,
insert6563505994279474003t_unit: produc5741669702376414499t_unit > set_Pr6425124735969554649t_unit > set_Pr6425124735969554649t_unit ).
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__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J,type,
insert3615455318616551057t_unit: sCC_Bl1394983891496994913t_unit > set_SC5159739854165745687t_unit > set_SC5159739854165745687t_unit ).
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__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_Set_Ovimage_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
vimage6257782490871955835od_v_v: ( product_prod_v_v > product_prod_v_v ) > set_Product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Set_Ovimage_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001tf__v,type,
vimage3896114166191421095_v_v_v: ( product_prod_v_v > v ) > set_v > set_Product_prod_v_v ).
thf(sy_c_Set_Ovimage_001tf__v_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
vimage6966088051850144591od_v_v: ( v > product_prod_v_v ) > set_Product_prod_v_v > set_v ).
thf(sy_c_Set_Ovimage_001tf__v_001tf__v,type,
vimage_v_v: ( v > v ) > set_v > set_v ).
thf(sy_c_Sum__Type_OInl_001t__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J_001t__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J,type,
sum_In526841707622398774t_unit: produc5741669702376414499t_unit > sum_su8181647976486975269t_unit ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__List__Olist_Itf__v_J_Mt__List__Olist_Itf__v_J_J,type,
accp_P8827133495749382256list_v: ( produc1391462591744249447list_v > produc1391462591744249447list_v > $o ) > produc1391462591744249447list_v > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J_Mt__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J_J,type,
accp_S2303753412255344476t_unit: ( sum_su8181647976486975269t_unit > sum_su8181647976486975269t_unit > $o ) > sum_su8181647976486975269t_unit > $o ).
thf(sy_c_Zorn_Ochains_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
chains2766520108962750135od_v_v: set_se8455005133513928103od_v_v > set_se2157405561750842759od_v_v ).
thf(sy_c_Zorn_Ochains_001tf__v,type,
chains_v: set_set_v > set_set_set_v ).
thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
member4190458934886417558od_v_v: list_P7986770385144383213od_v_v > set_li2323639185124838733od_v_v > $o ).
thf(sy_c_member_001t__List__Olist_Itf__v_J,type,
member_list_v: list_v > set_list_v > $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_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J,type,
member7924940910754673978t_unit: produc5741669702376414499t_unit > set_Pr6425124735969554649t_unit > $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_It__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_J,type,
member5511408251247217616od_v_v: set_se8455005133513928103od_v_v > set_se2157405561750842759od_v_v > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__v_J_J,type,
member_set_set_v: set_set_v > set_set_set_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_v: v > set_v > $o ).
thf(sy_v_S_H____,type,
s: set_v ).
thf(sy_v_S____,type,
s2: set_v ).
thf(sy_v_e,type,
e: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_e1,type,
e1: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_e_H,type,
e2: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_successors,type,
successors: v > set_v ).
thf(sy_v_v,type,
v2: v ).
thf(sy_v_vertices,type,
vertices: set_v ).
thf(sy_v_x____,type,
x: v ).
% Relevant facts (1279)
thf(fact_0_wf_H,axiom,
sCC_Bl9196236973127232072t_unit @ successors @ e2 ).
% wf'
thf(fact_1__092_060open_062x_A_092_060in_062_Aexplored_Ae_H_092_060close_062,axiom,
member_v @ x @ ( sCC_Bl157864678168468314t_unit @ e2 ) ).
% \<open>x \<in> explored e'\<close>
thf(fact_2_xv,axiom,
( ( sCC_Bl649662514949026229able_v @ successors @ v2 @ x )
& ( sCC_Bl649662514949026229able_v @ successors @ x @ v2 ) ) ).
% xv
thf(fact_3_dfs__dfss__rel_Ocong,axiom,
sCC_Bl907557413677168252_rel_v = sCC_Bl907557413677168252_rel_v ).
% dfs_dfss_rel.cong
thf(fact_4_sub__env__trans,axiom,
! [E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
=> ( ( sCC_Bl5768913643336123637t_unit @ E2 @ E3 )
=> ( sCC_Bl5768913643336123637t_unit @ E @ E3 ) ) ) ).
% sub_env_trans
thf(fact_5__C3_C,axiom,
sCC_Bl6082031138996704384t_unit @ successors @ v2 @ e1 @ e2 ).
% "3"
thf(fact_6_e_H__def,axiom,
( e2
= ( sCC_Bloemen_dfss_v @ successors @ v2 @ e1 ) ) ).
% e'_def
thf(fact_7_reachable_Ocases,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y: v] :
( ( member_v @ Y @ ( successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Y @ A2 ) ) ) ) ).
% reachable.cases
thf(fact_8_reachable__refl,axiom,
! [X: v] : ( sCC_Bl649662514949026229able_v @ successors @ X @ X ) ).
% reachable_refl
thf(fact_9_reachable__succ,axiom,
! [Y2: v,X: v,Z: v] :
( ( member_v @ Y2 @ ( successors @ X ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ Z )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% reachable_succ
thf(fact_10_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_v @ Y3 @ ( successors @ X2 ) )
& ( sCC_Bl649662514949026229able_v @ successors @ Y3 @ Z2 ) ) ) ) ).
% reachable.simps
thf(fact_11_reachable__edge,axiom,
! [Y2: v,X: v] :
( ( member_v @ Y2 @ ( successors @ X ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 ) ) ).
% reachable_edge
thf(fact_12_reachable__end__induct,axiom,
! [X: v,Y2: v,P: v > v > $o] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 )
=> ( ! [X3: v] : ( P @ X3 @ X3 )
=> ( ! [X3: v,Y: v,Z3: v] :
( ( P @ X3 @ Y )
=> ( ( member_v @ Z3 @ ( successors @ Y ) )
=> ( P @ X3 @ Z3 ) ) )
=> ( P @ X @ Y2 ) ) ) ) ).
% reachable_end_induct
thf(fact_13_reachable__trans,axiom,
! [X: v,Y2: v,Z: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ Z )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% reachable_trans
thf(fact_14_succ__reachable,axiom,
! [X: v,Y2: v,Z: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 )
=> ( ( member_v @ Z @ ( successors @ Y2 ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% succ_reachable
thf(fact_15_True,axiom,
( s2
= ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) ) ).
% True
thf(fact_16_S_H__def_I1_J,axiom,
s != s2 ).
% S'_def(1)
thf(fact_17__092_060open_062_092_060And_062n_O_An_A_092_060in_062_A_092_060S_062_Ae_H_An_092_060close_062,axiom,
! [N: v] : ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ e2 @ N ) ) ).
% \<open>\<And>n. n \<in> \<S> e' n\<close>
thf(fact_18_S__reflexive,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ).
% S_reflexive
thf(fact_19__092_060open_062_092_060not_062_Ais__scc_AS_092_060close_062,axiom,
~ ( sCC_Bloemen_is_scc_v @ successors @ s2 ) ).
% \<open>\<not> is_scc S\<close>
thf(fact_20__092_060open_062x_A_092_060in_062_AS_H_A_092_060and_062_Ax_A_092_060notin_062_AS_092_060close_062,axiom,
( ( member_v @ x @ s )
& ~ ( member_v @ x @ s2 ) ) ).
% \<open>x \<in> S' \<and> x \<notin> S\<close>
thf(fact_21_graph_Odfss_Ocong,axiom,
sCC_Bloemen_dfss_v = sCC_Bloemen_dfss_v ).
% graph.dfss.cong
thf(fact_22_graph_Opost__dfss_Ocong,axiom,
sCC_Bl6082031138996704384t_unit = sCC_Bl6082031138996704384t_unit ).
% graph.post_dfss.cong
thf(fact_23_graph_Oreachable_Ocong,axiom,
sCC_Bl649662514949026229able_v = sCC_Bl649662514949026229able_v ).
% graph.reachable.cong
thf(fact_24_graph_Owf__env_Ocong,axiom,
sCC_Bl9196236973127232072t_unit = sCC_Bl9196236973127232072t_unit ).
% graph.wf_env.cong
thf(fact_25__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_O_Ax_A_092_060in_062_AS_H_A_092_060and_062_Ax_A_092_060notin_062_AS_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X3: v] :
~ ( ( member_v @ X3 @ s )
& ~ ( member_v @ X3 @ s2 ) ) ).
% \<open>\<And>thesis. (\<And>x. x \<in> S' \<and> x \<notin> S \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_26_sccE,axiom,
! [S: set_v,X: v,Y2: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
=> ( ( member_v @ X @ S )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ X )
=> ( member_v @ Y2 @ S ) ) ) ) ) ).
% sccE
thf(fact_27_subscc,axiom,
sCC_Bl5398416737448265317bscc_v @ successors @ s2 ).
% subscc
thf(fact_28_is__subscc__def,axiom,
! [S: set_v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
= ( ! [X2: v] :
( ( member_v @ X2 @ S )
=> ! [Y3: v] :
( ( member_v @ Y3 @ S )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y3 ) ) ) ) ) ).
% is_subscc_def
thf(fact_29_re__reachable,axiom,
! [X: v,Y2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y2 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 ) ) ).
% re_reachable
thf(fact_30_reachable__re,axiom,
! [X: v,Y2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y2 ) ) ).
% reachable_re
thf(fact_31_local_Owf,axiom,
sCC_Bl9196236973127232072t_unit @ successors @ e ).
% local.wf
thf(fact_32_S_H__def_I3_J,axiom,
sCC_Bl5398416737448265317bscc_v @ successors @ s ).
% S'_def(3)
thf(fact_33__C1_C,axiom,
sCC_Bl36166008131615352t_unit @ successors @ v2 @ e ).
% "1"
thf(fact_34_sub,axiom,
sCC_Bl5768913643336123637t_unit @ e @ e1 ).
% sub
thf(fact_35_succ__re,axiom,
! [Y2: v,X: v,Z: v] :
( ( member_v @ Y2 @ ( successors @ X ) )
=> ( ( sCC_Bl770211535891879572_end_v @ successors @ Y2 @ Z )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Z ) ) ) ).
% succ_re
thf(fact_36_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_v @ Z2 @ ( successors @ Y3 ) ) ) ) ) ).
% reachable_end.simps
thf(fact_37_reachable__end_Ocases,axiom,
! [A1: v,A2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A2 )
=> ( ( A2 != A1 )
=> ~ ! [Y: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ Y )
=> ~ ( member_v @ A2 @ ( successors @ Y ) ) ) ) ) ).
% reachable_end.cases
thf(fact_38_re__refl,axiom,
! [X: v] : ( sCC_Bl770211535891879572_end_v @ successors @ X @ X ) ).
% re_refl
thf(fact_39_re__succ,axiom,
! [X: v,Y2: v,Z: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y2 )
=> ( ( member_v @ Z @ ( successors @ Y2 ) )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Z ) ) ) ).
% re_succ
thf(fact_40__092_060open_062sub__env_Ae_Ae_H_092_060close_062,axiom,
sCC_Bl5768913643336123637t_unit @ e @ e2 ).
% \<open>sub_env e e'\<close>
thf(fact_41_init__env__pre__dfs,axiom,
! [V: v] : ( sCC_Bl36166008131615352t_unit @ successors @ V @ ( sCC_Bl7693227186847904995_env_v @ V ) ) ).
% init_env_pre_dfs
thf(fact_42_mem__Collect__eq,axiom,
! [A: v,P: v > $o] :
( ( member_v @ 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_v @ 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_S_H__def_I2_J,axiom,
ord_less_eq_set_v @ s2 @ s ).
% S'_def(2)
thf(fact_47_graph_Oreachable__end_Ocong,axiom,
sCC_Bl770211535891879572_end_v = sCC_Bl770211535891879572_end_v ).
% graph.reachable_end.cong
thf(fact_48_graph_Ois__subscc_Ocong,axiom,
sCC_Bl5398416737448265317bscc_v = sCC_Bl5398416737448265317bscc_v ).
% graph.is_subscc.cong
thf(fact_49_graph_Ois__scc_Ocong,axiom,
sCC_Bloemen_is_scc_v = sCC_Bloemen_is_scc_v ).
% graph.is_scc.cong
thf(fact_50_graph_Opre__dfs_Ocong,axiom,
sCC_Bl36166008131615352t_unit = sCC_Bl36166008131615352t_unit ).
% graph.pre_dfs.cong
thf(fact_51__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062S_H_O_A_092_060lbrakk_062S_H_A_092_060noteq_062_AS_059_AS_A_092_060subseteq_062_AS_H_059_Ais__subscc_AS_H_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [S2: set_v] :
( ( S2 != s2 )
=> ( ( ord_less_eq_set_v @ s2 @ S2 )
=> ~ ( sCC_Bl5398416737448265317bscc_v @ successors @ S2 ) ) ) ).
% \<open>\<And>thesis. (\<And>S'. \<lbrakk>S' \<noteq> S; S \<subseteq> S'; is_subscc S'\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_52_subscc__add,axiom,
! [S: set_v,X: v,Y2: v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
=> ( ( member_v @ X @ S )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y2 @ X )
=> ( sCC_Bl5398416737448265317bscc_v @ successors @ ( insert_v2 @ Y2 @ S ) ) ) ) ) ) ).
% subscc_add
thf(fact_53_v,axiom,
~ ( member_v @ v2 @ ( sCC_Bl4645233313691564917t_unit @ e ) ) ).
% v
thf(fact_54_scc__partition,axiom,
! [S: set_v,S3: set_v,X: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
=> ( ( sCC_Bloemen_is_scc_v @ successors @ S3 )
=> ( ( member_v @ X @ ( inf_inf_set_v @ S @ S3 ) )
=> ( S = S3 ) ) ) ) ).
% scc_partition
thf(fact_55_ra__reachable,axiom,
! [X: v,Y2: v,E4: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E4 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 ) ) ).
% ra_reachable
thf(fact_56__092_060open_062S_A_092_060noteq_062_A_123_125_092_060close_062,axiom,
s2 != bot_bot_set_v ).
% \<open>S \<noteq> {}\<close>
thf(fact_57_ra__trans,axiom,
! [X: v,Y2: v,E4: set_Product_prod_v_v,Z: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ Y2 @ Z @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Z @ E4 ) ) ) ).
% ra_trans
thf(fact_58_ra__refl,axiom,
! [X: v,E4: set_Product_prod_v_v] : ( sCC_Bl4291963740693775144ding_v @ successors @ X @ X @ E4 ) ).
% ra_refl
thf(fact_59_graph__axioms,axiom,
sCC_Bloemen_graph_v @ vertices @ successors ).
% graph_axioms
thf(fact_60_pre__dfss__pre__dfs,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( sCC_Bl36166008131615352t_unit @ successors @ W @ E ) ) ) ) ).
% pre_dfss_pre_dfs
thf(fact_61_is__scc__def,axiom,
! [S: set_v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
= ( ( S != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
& ! [S4: set_v] :
( ( ( ord_less_eq_set_v @ S @ S4 )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S4 ) )
=> ( S4 = S ) ) ) ) ).
% is_scc_def
thf(fact_62_sclosed,axiom,
! [X4: v] :
( ( member_v @ X4 @ vertices )
=> ( ord_less_eq_set_v @ ( successors @ X4 ) @ vertices ) ) ).
% sclosed
thf(fact_63_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_64_graph_Osclosed,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ! [X4: v] :
( ( member_v @ X4 @ Vertices )
=> ( ord_less_eq_set_v @ ( Successors @ X4 ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_65_graph_Ora__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,E4: set_Product_prod_v_v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ Y2 @ Z @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Z @ E4 ) ) ) ) ).
% graph.ra_trans
thf(fact_66_graph_Oreachable__avoiding_Ocong,axiom,
sCC_Bl4291963740693775144ding_v = sCC_Bl4291963740693775144ding_v ).
% graph.reachable_avoiding.cong
thf(fact_67_graph_Ora__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,E4: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ X @ E4 ) ) ).
% graph.ra_refl
thf(fact_68_graph_Opre__dfss_Ocong,axiom,
sCC_Bl1748261141445803503t_unit = sCC_Bl1748261141445803503t_unit ).
% graph.pre_dfss.cong
thf(fact_69_graph_Opre__dfss__pre__dfs,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( sCC_Bl36166008131615352t_unit @ Successors @ W @ E ) ) ) ) ) ).
% graph.pre_dfss_pre_dfs
thf(fact_70_graph_Ora__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,E4: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E4 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 ) ) ) ).
% graph.ra_reachable
thf(fact_71_graph_Oscc__partition,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v,S3: set_Product_prod_v_v,X: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S3 )
=> ( ( member7453568604450474000od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ S @ S3 ) )
=> ( S = S3 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_72_graph_Oscc__partition,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,S3: set_v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S3 )
=> ( ( member_v @ X @ ( inf_inf_set_v @ S @ S3 ) )
=> ( S = S3 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_73_graph_Ois__scc__def,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
= ( ( S != bot_bo723834152578015283od_v_v )
& ( sCC_Bl2301996248249672505od_v_v @ Successors @ S )
& ! [S4: set_Product_prod_v_v] :
( ( ( ord_le7336532860387713383od_v_v @ S @ S4 )
& ( sCC_Bl2301996248249672505od_v_v @ Successors @ S4 ) )
=> ( S4 = S ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_74_graph_Ois__scc__def,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
= ( ( S != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
& ! [S4: set_v] :
( ( ( ord_less_eq_set_v @ S @ S4 )
& ( sCC_Bl5398416737448265317bscc_v @ Successors @ S4 ) )
=> ( S4 = S ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_75_graph_Osubscc__add,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v,X: product_prod_v_v,Y2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl2301996248249672505od_v_v @ Successors @ S )
=> ( ( member7453568604450474000od_v_v @ X @ S )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y2 )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y2 @ X )
=> ( sCC_Bl2301996248249672505od_v_v @ Successors @ ( insert1338601472111419319od_v_v @ Y2 @ S ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_76_graph_Osubscc__add,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,X: v,Y2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
=> ( ( member_v @ X @ S )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ X )
=> ( sCC_Bl5398416737448265317bscc_v @ Successors @ ( insert_v2 @ Y2 @ S ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_77_graph_Oreachable__edge,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y2: product_prod_v_v,X: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y2 ) ) ) ).
% graph.reachable_edge
thf(fact_78_graph_Oreachable__edge,axiom,
! [Vertices: set_v,Successors: v > set_v,Y2: v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y2 @ ( Successors @ X ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 ) ) ) ).
% graph.reachable_edge
thf(fact_79_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,Y2: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y2 )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y2 ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_reachable
thf(fact_80_graph_Osucc__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 )
=> ( ( member_v @ Z @ ( Successors @ Y2 ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_reachable
thf(fact_81_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 )
=> ~ ! [Y: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ A2 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_82_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 )
=> ~ ! [Y: v] :
( ( member_v @ Y @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Y @ A2 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_83_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_84_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_v @ Y3 @ ( Successors @ X2 ) )
& ( sCC_Bl649662514949026229able_v @ Successors @ Y3 @ Z2 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_85_graph_Oreachable__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ Z )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_trans
thf(fact_86_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,Y2: 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 @ Y2 )
=> ( ! [X3: product_prod_v_v] : ( P @ X3 @ X3 )
=> ( ! [X3: product_prod_v_v,Y: product_prod_v_v,Z3: product_prod_v_v] :
( ( P @ X3 @ Y )
=> ( ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ Y ) )
=> ( P @ X3 @ Z3 ) ) )
=> ( P @ X @ Y2 ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_87_graph_Oreachable__end__induct,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,P: v > v > $o] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 )
=> ( ! [X3: v] : ( P @ X3 @ X3 )
=> ( ! [X3: v,Y: v,Z3: v] :
( ( P @ X3 @ Y )
=> ( ( member_v @ Z3 @ ( Successors @ Y ) )
=> ( P @ X3 @ Z3 ) ) )
=> ( P @ X @ Y2 ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_88_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_89_graph_Oreachable__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y2: product_prod_v_v,X: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X ) )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y2 @ Z )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_succ
thf(fact_90_graph_Oreachable__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,Y2: v,X: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y2 @ ( Successors @ X ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ Z )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_succ
thf(fact_91_graph_Osucc__re,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y2: product_prod_v_v,X: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X ) )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ Y2 @ Z )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_re
thf(fact_92_graph_Osucc__re,axiom,
! [Vertices: set_v,Successors: v > set_v,Y2: v,X: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y2 @ ( Successors @ X ) )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ Y2 @ Z )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_re
thf(fact_93_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 )
=> ~ ! [Y: product_prod_v_v] :
( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ Y )
=> ~ ( member7453568604450474000od_v_v @ A2 @ ( Successors @ Y ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_94_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 )
=> ~ ! [Y: v] :
( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ Y )
=> ~ ( member_v @ A2 @ ( Successors @ Y ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_95_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_96_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_v @ Z2 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_97_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_98_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,Y2: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Y2 )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y2 ) )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.re_succ
thf(fact_99_graph_Ore__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y2 )
=> ( ( member_v @ Z @ ( Successors @ Y2 ) )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Z ) ) ) ) ).
% graph.re_succ
thf(fact_100_graph_Osub__env__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
=> ( ( sCC_Bl5768913643336123637t_unit @ E2 @ E3 )
=> ( sCC_Bl5768913643336123637t_unit @ E @ E3 ) ) ) ) ).
% graph.sub_env_trans
thf(fact_101_graph_Oinit__env__pre__dfs,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl36166008131615352t_unit @ Successors @ V @ ( sCC_Bl7693227186847904995_env_v @ V ) ) ) ).
% graph.init_env_pre_dfs
thf(fact_102_graph_OS__reflexive,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ).
% graph.S_reflexive
thf(fact_103_graph_Ore__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y2 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 ) ) ) ).
% graph.re_reachable
thf(fact_104_graph_Oreachable__re,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y2 ) ) ) ).
% graph.reachable_re
thf(fact_105_graph_Ois__subscc__def,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
= ( ! [X2: v] :
( ( member_v @ X2 @ S )
=> ! [Y3: v] :
( ( member_v @ Y3 @ S )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y3 ) ) ) ) ) ) ).
% graph.is_subscc_def
thf(fact_106_graph_OsccE,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v,X: product_prod_v_v,Y2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
=> ( ( member7453568604450474000od_v_v @ X @ S )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y2 )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y2 @ X )
=> ( member7453568604450474000od_v_v @ Y2 @ S ) ) ) ) ) ) ).
% graph.sccE
thf(fact_107_graph_OsccE,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,X: v,Y2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
=> ( ( member_v @ X @ S )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y2 @ X )
=> ( member_v @ Y2 @ S ) ) ) ) ) ) ).
% graph.sccE
thf(fact_108_disjoint__insert_I2_J,axiom,
! [A3: set_v,B: v,B2: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ A3 @ ( insert_v2 @ B @ B2 ) ) )
= ( ~ ( member_v @ B @ A3 )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A3 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_109_disjoint__insert_I2_J,axiom,
! [A3: set_Product_prod_v_v,B: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B @ B2 ) ) )
= ( ~ ( member7453568604450474000od_v_v @ B @ A3 )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_110_disjoint__insert_I1_J,axiom,
! [B2: set_v,A: v,A3: set_v] :
( ( ( inf_inf_set_v @ B2 @ ( insert_v2 @ A @ A3 ) )
= bot_bot_set_v )
= ( ~ ( member_v @ A @ B2 )
& ( ( inf_inf_set_v @ B2 @ A3 )
= bot_bot_set_v ) ) ) ).
% disjoint_insert(1)
thf(fact_111_disjoint__insert_I1_J,axiom,
! [B2: set_Product_prod_v_v,A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ A @ A3 ) )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A @ B2 )
& ( ( inf_in6271465464967711157od_v_v @ B2 @ A3 )
= bot_bo723834152578015283od_v_v ) ) ) ).
% disjoint_insert(1)
thf(fact_112_insert__disjoint_I2_J,axiom,
! [A: v,A3: set_v,B2: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ ( insert_v2 @ A @ A3 ) @ B2 ) )
= ( ~ ( member_v @ A @ B2 )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A3 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_113_insert__disjoint_I2_J,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) @ B2 ) )
= ( ~ ( member7453568604450474000od_v_v @ A @ B2 )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_114_insert__disjoint_I1_J,axiom,
! [A: v,A3: set_v,B2: set_v] :
( ( ( inf_inf_set_v @ ( insert_v2 @ A @ A3 ) @ B2 )
= bot_bot_set_v )
= ( ~ ( member_v @ A @ B2 )
& ( ( inf_inf_set_v @ A3 @ B2 )
= bot_bot_set_v ) ) ) ).
% insert_disjoint(1)
thf(fact_115_insert__disjoint_I1_J,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) @ B2 )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A @ B2 )
& ( ( inf_in6271465464967711157od_v_v @ A3 @ B2 )
= bot_bo723834152578015283od_v_v ) ) ) ).
% insert_disjoint(1)
thf(fact_116_singleton__insert__inj__eq,axiom,
! [B: v,A: v,A3: set_v] :
( ( ( insert_v2 @ B @ bot_bot_set_v )
= ( insert_v2 @ A @ A3 ) )
= ( ( A = B )
& ( ord_less_eq_set_v @ A3 @ ( insert_v2 @ B @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_117_singleton__insert__inj__eq,axiom,
! [B: product_prod_v_v,A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ A @ A3 ) )
= ( ( A = B )
& ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_118_singleton__insert__inj__eq_H,axiom,
! [A: v,A3: set_v,B: v] :
( ( ( insert_v2 @ A @ A3 )
= ( insert_v2 @ B @ bot_bot_set_v ) )
= ( ( A = B )
& ( ord_less_eq_set_v @ A3 @ ( insert_v2 @ B @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_119_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ A3 )
= ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) )
= ( ( A = B )
& ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_120_Int__insert__left__if0,axiom,
! [A: product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ C )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B2 ) @ C )
= ( inf_in6271465464967711157od_v_v @ B2 @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_121_Int__insert__left__if0,axiom,
! [A: v,C: set_v,B2: set_v] :
( ~ ( member_v @ A @ C )
=> ( ( inf_inf_set_v @ ( insert_v2 @ A @ B2 ) @ C )
= ( inf_inf_set_v @ B2 @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_122_Int__insert__left__if1,axiom,
! [A: product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ C )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B2 ) @ C )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_123_Int__insert__left__if1,axiom,
! [A: v,C: set_v,B2: set_v] :
( ( member_v @ A @ C )
=> ( ( inf_inf_set_v @ ( insert_v2 @ A @ B2 ) @ C )
= ( insert_v2 @ A @ ( inf_inf_set_v @ B2 @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_124_insert__inter__insert,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_125_insert__inter__insert,axiom,
! [A: v,A3: set_v,B2: set_v] :
( ( inf_inf_set_v @ ( insert_v2 @ A @ A3 ) @ ( insert_v2 @ A @ B2 ) )
= ( insert_v2 @ A @ ( inf_inf_set_v @ A3 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_126_Int__insert__right__if0,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_127_Int__insert__right__if0,axiom,
! [A: v,A3: set_v,B2: set_v] :
( ~ ( member_v @ A @ A3 )
=> ( ( inf_inf_set_v @ A3 @ ( insert_v2 @ A @ B2 ) )
= ( inf_inf_set_v @ A3 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_128_ra__mono,axiom,
! [X: v,Y2: v,E4: set_Product_prod_v_v,E5: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E4 )
=> ( ( ord_le7336532860387713383od_v_v @ E5 @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E5 ) ) ) ).
% ra_mono
thf(fact_129_ra__empty,axiom,
! [X: v,Y2: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ bot_bo723834152578015283od_v_v )
= ( sCC_Bl649662514949026229able_v @ successors @ X @ Y2 ) ) ).
% ra_empty
thf(fact_130_empty__Collect__eq,axiom,
! [P: v > $o] :
( ( bot_bot_set_v
= ( collect_v @ P ) )
= ( ! [X2: v] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_131_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_132_Collect__empty__eq,axiom,
! [P: v > $o] :
( ( ( collect_v @ P )
= bot_bot_set_v )
= ( ! [X2: v] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_133_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_134_all__not__in__conv,axiom,
! [A3: set_v] :
( ( ! [X2: v] :
~ ( member_v @ X2 @ A3 ) )
= ( A3 = bot_bot_set_v ) ) ).
% all_not_in_conv
thf(fact_135_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_136_empty__iff,axiom,
! [C2: v] :
~ ( member_v @ C2 @ bot_bot_set_v ) ).
% empty_iff
thf(fact_137_empty__iff,axiom,
! [C2: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ C2 @ bot_bo723834152578015283od_v_v ) ).
% empty_iff
thf(fact_138_subset__antisym,axiom,
! [A3: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A3 @ B2 )
=> ( ( ord_less_eq_set_v @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% subset_antisym
thf(fact_139_subset__antisym,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% subset_antisym
thf(fact_140_subsetI,axiom,
! [A3: set_v,B2: set_v] :
( ! [X3: v] :
( ( member_v @ X3 @ A3 )
=> ( member_v @ X3 @ B2 ) )
=> ( ord_less_eq_set_v @ A3 @ B2 ) ) ).
% subsetI
thf(fact_141_subsetI,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ( member7453568604450474000od_v_v @ X3 @ B2 ) )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B2 ) ) ).
% subsetI
thf(fact_142_insert__absorb2,axiom,
! [X: v,A3: set_v] :
( ( insert_v2 @ X @ ( insert_v2 @ X @ A3 ) )
= ( insert_v2 @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_143_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_144_insert__iff,axiom,
! [A: v,B: v,A3: set_v] :
( ( member_v @ A @ ( insert_v2 @ B @ A3 ) )
= ( ( A = B )
| ( member_v @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_145_insert__iff,axiom,
! [A: product_prod_v_v,B: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ A3 ) )
= ( ( A = B )
| ( member7453568604450474000od_v_v @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_146_insertCI,axiom,
! [A: v,B2: set_v,B: v] :
( ( ~ ( member_v @ A @ B2 )
=> ( A = B ) )
=> ( member_v @ A @ ( insert_v2 @ B @ B2 ) ) ) ).
% insertCI
thf(fact_147_insertCI,axiom,
! [A: product_prod_v_v,B2: set_Product_prod_v_v,B: product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ A @ B2 )
=> ( A = B ) )
=> ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ B2 ) ) ) ).
% insertCI
thf(fact_148_Int__iff,axiom,
! [C2: product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) )
= ( ( member7453568604450474000od_v_v @ C2 @ A3 )
& ( member7453568604450474000od_v_v @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_149_Int__iff,axiom,
! [C2: v,A3: set_v,B2: set_v] :
( ( member_v @ C2 @ ( inf_inf_set_v @ A3 @ B2 ) )
= ( ( member_v @ C2 @ A3 )
& ( member_v @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_150_IntI,axiom,
! [C2: product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ A3 )
=> ( ( member7453568604450474000od_v_v @ C2 @ B2 )
=> ( member7453568604450474000od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) ) ) ) ).
% IntI
thf(fact_151_IntI,axiom,
! [C2: v,A3: set_v,B2: set_v] :
( ( member_v @ C2 @ A3 )
=> ( ( member_v @ C2 @ B2 )
=> ( member_v @ C2 @ ( inf_inf_set_v @ A3 @ B2 ) ) ) ) ).
% IntI
thf(fact_152_empty__subsetI,axiom,
! [A3: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A3 ) ).
% empty_subsetI
thf(fact_153_empty__subsetI,axiom,
! [A3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A3 ) ).
% empty_subsetI
thf(fact_154_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_155_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_156_singletonI,axiom,
! [A: v] : ( member_v @ A @ ( insert_v2 @ A @ bot_bot_set_v ) ) ).
% singletonI
thf(fact_157_singletonI,axiom,
! [A: product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).
% singletonI
thf(fact_158_insert__subset,axiom,
! [X: v,A3: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ ( insert_v2 @ X @ A3 ) @ B2 )
= ( ( member_v @ X @ B2 )
& ( ord_less_eq_set_v @ A3 @ B2 ) ) ) ).
% insert_subset
thf(fact_159_insert__subset,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ X @ A3 ) @ B2 )
= ( ( member7453568604450474000od_v_v @ X @ B2 )
& ( ord_le7336532860387713383od_v_v @ A3 @ B2 ) ) ) ).
% insert_subset
thf(fact_160_Int__subset__iff,axiom,
! [C: set_v,A3: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ C @ ( inf_inf_set_v @ A3 @ B2 ) )
= ( ( ord_less_eq_set_v @ C @ A3 )
& ( ord_less_eq_set_v @ C @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_161_Int__subset__iff,axiom,
! [C: set_Product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) )
= ( ( ord_le7336532860387713383od_v_v @ C @ A3 )
& ( ord_le7336532860387713383od_v_v @ C @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_162_Int__insert__right__if1,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_163_Int__insert__right__if1,axiom,
! [A: v,A3: set_v,B2: set_v] :
( ( member_v @ A @ A3 )
=> ( ( inf_inf_set_v @ A3 @ ( insert_v2 @ A @ B2 ) )
= ( insert_v2 @ A @ ( inf_inf_set_v @ A3 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_164_graph_Ora__mono,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,E4: set_Product_prod_v_v,E5: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E4 )
=> ( ( ord_le7336532860387713383od_v_v @ E5 @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E5 ) ) ) ) ).
% graph.ra_mono
thf(fact_165_ex__in__conv,axiom,
! [A3: set_v] :
( ( ? [X2: v] : ( member_v @ X2 @ A3 ) )
= ( A3 != bot_bot_set_v ) ) ).
% ex_in_conv
thf(fact_166_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_167_equals0I,axiom,
! [A3: set_v] :
( ! [Y: v] :
~ ( member_v @ Y @ A3 )
=> ( A3 = bot_bot_set_v ) ) ).
% equals0I
thf(fact_168_equals0I,axiom,
! [A3: set_Product_prod_v_v] :
( ! [Y: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ Y @ A3 )
=> ( A3 = bot_bo723834152578015283od_v_v ) ) ).
% equals0I
thf(fact_169_equals0D,axiom,
! [A3: set_v,A: v] :
( ( A3 = bot_bot_set_v )
=> ~ ( member_v @ A @ A3 ) ) ).
% equals0D
thf(fact_170_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_171_emptyE,axiom,
! [A: v] :
~ ( member_v @ A @ bot_bot_set_v ) ).
% emptyE
thf(fact_172_emptyE,axiom,
! [A: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ A @ bot_bo723834152578015283od_v_v ) ).
% emptyE
thf(fact_173_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_174_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_175_set__eq__subset,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A4: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ A4 @ B3 )
& ( ord_less_eq_set_v @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_176_set__eq__subset,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B3 )
& ( ord_le7336532860387713383od_v_v @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_177_subset__trans,axiom,
! [A3: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A3 @ B2 )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ord_less_eq_set_v @ A3 @ C ) ) ) ).
% subset_trans
thf(fact_178_subset__trans,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ord_le7336532860387713383od_v_v @ A3 @ C ) ) ) ).
% subset_trans
thf(fact_179_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_180_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_181_subset__refl,axiom,
! [A3: set_v] : ( ord_less_eq_set_v @ A3 @ A3 ) ).
% subset_refl
thf(fact_182_subset__refl,axiom,
! [A3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A3 @ A3 ) ).
% subset_refl
thf(fact_183_subset__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B3: set_v] :
! [T: v] :
( ( member_v @ T @ A4 )
=> ( member_v @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_184_subset__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
! [T: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ T @ A4 )
=> ( member7453568604450474000od_v_v @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_185_equalityD2,axiom,
! [A3: set_v,B2: set_v] :
( ( A3 = B2 )
=> ( ord_less_eq_set_v @ B2 @ A3 ) ) ).
% equalityD2
thf(fact_186_equalityD2,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A3 = B2 )
=> ( ord_le7336532860387713383od_v_v @ B2 @ A3 ) ) ).
% equalityD2
thf(fact_187_equalityD1,axiom,
! [A3: set_v,B2: set_v] :
( ( A3 = B2 )
=> ( ord_less_eq_set_v @ A3 @ B2 ) ) ).
% equalityD1
thf(fact_188_equalityD1,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A3 = B2 )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B2 ) ) ).
% equalityD1
thf(fact_189_subset__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B3: set_v] :
! [X2: v] :
( ( member_v @ X2 @ A4 )
=> ( member_v @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_190_subset__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A4 )
=> ( member7453568604450474000od_v_v @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_191_equalityE,axiom,
! [A3: set_v,B2: set_v] :
( ( A3 = B2 )
=> ~ ( ( ord_less_eq_set_v @ A3 @ B2 )
=> ~ ( ord_less_eq_set_v @ B2 @ A3 ) ) ) ).
% equalityE
thf(fact_192_equalityE,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A3 = B2 )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
=> ~ ( ord_le7336532860387713383od_v_v @ B2 @ A3 ) ) ) ).
% equalityE
thf(fact_193_subsetD,axiom,
! [A3: set_v,B2: set_v,C2: v] :
( ( ord_less_eq_set_v @ A3 @ B2 )
=> ( ( member_v @ C2 @ A3 )
=> ( member_v @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_194_subsetD,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
=> ( ( member7453568604450474000od_v_v @ C2 @ A3 )
=> ( member7453568604450474000od_v_v @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_195_in__mono,axiom,
! [A3: set_v,B2: set_v,X: v] :
( ( ord_less_eq_set_v @ A3 @ B2 )
=> ( ( member_v @ X @ A3 )
=> ( member_v @ X @ B2 ) ) ) ).
% in_mono
thf(fact_196_in__mono,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
=> ( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( member7453568604450474000od_v_v @ X @ B2 ) ) ) ).
% in_mono
thf(fact_197_mk__disjoint__insert,axiom,
! [A: v,A3: set_v] :
( ( member_v @ A @ A3 )
=> ? [B4: set_v] :
( ( A3
= ( insert_v2 @ A @ B4 ) )
& ~ ( member_v @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_198_mk__disjoint__insert,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A3 )
=> ? [B4: set_Product_prod_v_v] :
( ( A3
= ( insert1338601472111419319od_v_v @ A @ B4 ) )
& ~ ( member7453568604450474000od_v_v @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_199_insert__commute,axiom,
! [X: v,Y2: v,A3: set_v] :
( ( insert_v2 @ X @ ( insert_v2 @ Y2 @ A3 ) )
= ( insert_v2 @ Y2 @ ( insert_v2 @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_200_insert__commute,axiom,
! [X: product_prod_v_v,Y2: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X @ ( insert1338601472111419319od_v_v @ Y2 @ A3 ) )
= ( insert1338601472111419319od_v_v @ Y2 @ ( insert1338601472111419319od_v_v @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_201_insert__eq__iff,axiom,
! [A: v,A3: set_v,B: v,B2: set_v] :
( ~ ( member_v @ A @ A3 )
=> ( ~ ( member_v @ B @ B2 )
=> ( ( ( insert_v2 @ A @ A3 )
= ( insert_v2 @ B @ B2 ) )
= ( ( ( A = B )
=> ( A3 = B2 ) )
& ( ( A != B )
=> ? [C3: set_v] :
( ( A3
= ( insert_v2 @ B @ C3 ) )
& ~ ( member_v @ B @ C3 )
& ( B2
= ( insert_v2 @ A @ C3 ) )
& ~ ( member_v @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_202_insert__eq__iff,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: product_prod_v_v,B2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ B @ B2 )
=> ( ( ( insert1338601472111419319od_v_v @ A @ A3 )
= ( insert1338601472111419319od_v_v @ B @ B2 ) )
= ( ( ( A = B )
=> ( A3 = B2 ) )
& ( ( A != B )
=> ? [C3: set_Product_prod_v_v] :
( ( A3
= ( insert1338601472111419319od_v_v @ B @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ B @ C3 )
& ( B2
= ( insert1338601472111419319od_v_v @ A @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_203_insert__absorb,axiom,
! [A: v,A3: set_v] :
( ( member_v @ A @ A3 )
=> ( ( insert_v2 @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_204_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_205_insert__ident,axiom,
! [X: v,A3: set_v,B2: set_v] :
( ~ ( member_v @ X @ A3 )
=> ( ~ ( member_v @ X @ B2 )
=> ( ( ( insert_v2 @ X @ A3 )
= ( insert_v2 @ X @ B2 ) )
= ( A3 = B2 ) ) ) ) ).
% insert_ident
thf(fact_206_insert__ident,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ X @ B2 )
=> ( ( ( insert1338601472111419319od_v_v @ X @ A3 )
= ( insert1338601472111419319od_v_v @ X @ B2 ) )
= ( A3 = B2 ) ) ) ) ).
% insert_ident
thf(fact_207_Set_Oset__insert,axiom,
! [X: v,A3: set_v] :
( ( member_v @ X @ A3 )
=> ~ ! [B4: set_v] :
( ( A3
= ( insert_v2 @ X @ B4 ) )
=> ( member_v @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_208_Set_Oset__insert,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A3 )
=> ~ ! [B4: set_Product_prod_v_v] :
( ( A3
= ( insert1338601472111419319od_v_v @ X @ B4 ) )
=> ( member7453568604450474000od_v_v @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_209_insertI2,axiom,
! [A: v,B2: set_v,B: v] :
( ( member_v @ A @ B2 )
=> ( member_v @ A @ ( insert_v2 @ B @ B2 ) ) ) ).
% insertI2
thf(fact_210_insertI2,axiom,
! [A: product_prod_v_v,B2: set_Product_prod_v_v,B: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ B2 )
=> ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ B2 ) ) ) ).
% insertI2
thf(fact_211_insertI1,axiom,
! [A: v,B2: set_v] : ( member_v @ A @ ( insert_v2 @ A @ B2 ) ) ).
% insertI1
thf(fact_212_insertI1,axiom,
! [A: product_prod_v_v,B2: set_Product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ B2 ) ) ).
% insertI1
thf(fact_213_insertE,axiom,
! [A: v,B: v,A3: set_v] :
( ( member_v @ A @ ( insert_v2 @ B @ A3 ) )
=> ( ( A != B )
=> ( member_v @ A @ A3 ) ) ) ).
% insertE
thf(fact_214_insertE,axiom,
! [A: product_prod_v_v,B: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ A3 ) )
=> ( ( A != B )
=> ( member7453568604450474000od_v_v @ A @ A3 ) ) ) ).
% insertE
thf(fact_215_Int__left__commute,axiom,
! [A3: set_v,B2: set_v,C: set_v] :
( ( inf_inf_set_v @ A3 @ ( inf_inf_set_v @ B2 @ C ) )
= ( inf_inf_set_v @ B2 @ ( inf_inf_set_v @ A3 @ C ) ) ) ).
% Int_left_commute
thf(fact_216_Int__left__absorb,axiom,
! [A3: set_v,B2: set_v] :
( ( inf_inf_set_v @ A3 @ ( inf_inf_set_v @ A3 @ B2 ) )
= ( inf_inf_set_v @ A3 @ B2 ) ) ).
% Int_left_absorb
thf(fact_217_Int__commute,axiom,
( inf_inf_set_v
= ( ^ [A4: set_v,B3: set_v] : ( inf_inf_set_v @ B3 @ A4 ) ) ) ).
% Int_commute
thf(fact_218_Int__absorb,axiom,
! [A3: set_v] :
( ( inf_inf_set_v @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_219_Int__assoc,axiom,
! [A3: set_v,B2: set_v,C: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A3 @ B2 ) @ C )
= ( inf_inf_set_v @ A3 @ ( inf_inf_set_v @ B2 @ C ) ) ) ).
% Int_assoc
thf(fact_220_IntD2,axiom,
! [C2: product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) )
=> ( member7453568604450474000od_v_v @ C2 @ B2 ) ) ).
% IntD2
thf(fact_221_IntD2,axiom,
! [C2: v,A3: set_v,B2: set_v] :
( ( member_v @ C2 @ ( inf_inf_set_v @ A3 @ B2 ) )
=> ( member_v @ C2 @ B2 ) ) ).
% IntD2
thf(fact_222_IntD1,axiom,
! [C2: product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) )
=> ( member7453568604450474000od_v_v @ C2 @ A3 ) ) ).
% IntD1
thf(fact_223_IntD1,axiom,
! [C2: v,A3: set_v,B2: set_v] :
( ( member_v @ C2 @ ( inf_inf_set_v @ A3 @ B2 ) )
=> ( member_v @ C2 @ A3 ) ) ).
% IntD1
thf(fact_224_IntE,axiom,
! [C2: product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) )
=> ~ ( ( member7453568604450474000od_v_v @ C2 @ A3 )
=> ~ ( member7453568604450474000od_v_v @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_225_IntE,axiom,
! [C2: v,A3: set_v,B2: set_v] :
( ( member_v @ C2 @ ( inf_inf_set_v @ A3 @ B2 ) )
=> ~ ( ( member_v @ C2 @ A3 )
=> ~ ( member_v @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_226_graph_Ora__empty,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ bot_bo723834152578015283od_v_v )
= ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y2 ) ) ) ).
% graph.ra_empty
thf(fact_227_singleton__inject,axiom,
! [A: v,B: v] :
( ( ( insert_v2 @ A @ bot_bot_set_v )
= ( insert_v2 @ B @ bot_bot_set_v ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_228_singleton__inject,axiom,
! [A: product_prod_v_v,B: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_229_insert__not__empty,axiom,
! [A: v,A3: set_v] :
( ( insert_v2 @ A @ A3 )
!= bot_bot_set_v ) ).
% insert_not_empty
thf(fact_230_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_231_doubleton__eq__iff,axiom,
! [A: v,B: v,C2: v,D: v] :
( ( ( insert_v2 @ A @ ( insert_v2 @ B @ bot_bot_set_v ) )
= ( insert_v2 @ C2 @ ( insert_v2 @ D @ bot_bot_set_v ) ) )
= ( ( ( A = C2 )
& ( B = D ) )
| ( ( A = D )
& ( B = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_232_doubleton__eq__iff,axiom,
! [A: product_prod_v_v,B: product_prod_v_v,C2: product_prod_v_v,D: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) )
= ( insert1338601472111419319od_v_v @ C2 @ ( insert1338601472111419319od_v_v @ D @ bot_bo723834152578015283od_v_v ) ) )
= ( ( ( A = C2 )
& ( B = D ) )
| ( ( A = D )
& ( B = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_233_singleton__iff,axiom,
! [B: v,A: v] :
( ( member_v @ B @ ( insert_v2 @ A @ bot_bot_set_v ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_234_singleton__iff,axiom,
! [B: product_prod_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_235_singletonD,axiom,
! [B: v,A: v] :
( ( member_v @ B @ ( insert_v2 @ A @ bot_bot_set_v ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_236_singletonD,axiom,
! [B: product_prod_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_237_subset__insertI2,axiom,
! [A3: set_v,B2: set_v,B: v] :
( ( ord_less_eq_set_v @ A3 @ B2 )
=> ( ord_less_eq_set_v @ A3 @ ( insert_v2 @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_238_subset__insertI2,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,B: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
=> ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_239_subset__insertI,axiom,
! [B2: set_v,A: v] : ( ord_less_eq_set_v @ B2 @ ( insert_v2 @ A @ B2 ) ) ).
% subset_insertI
thf(fact_240_subset__insertI,axiom,
! [B2: set_Product_prod_v_v,A: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ A @ B2 ) ) ).
% subset_insertI
thf(fact_241_subset__insert,axiom,
! [X: v,A3: set_v,B2: set_v] :
( ~ ( member_v @ X @ A3 )
=> ( ( ord_less_eq_set_v @ A3 @ ( insert_v2 @ X @ B2 ) )
= ( ord_less_eq_set_v @ A3 @ B2 ) ) ) ).
% subset_insert
thf(fact_242_subset__insert,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ B2 ) )
= ( ord_le7336532860387713383od_v_v @ A3 @ B2 ) ) ) ).
% subset_insert
thf(fact_243_insert__mono,axiom,
! [C: set_v,D2: set_v,A: v] :
( ( ord_less_eq_set_v @ C @ D2 )
=> ( ord_less_eq_set_v @ ( insert_v2 @ A @ C ) @ ( insert_v2 @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_244_insert__mono,axiom,
! [C: set_Product_prod_v_v,D2: set_Product_prod_v_v,A: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ A @ C ) @ ( insert1338601472111419319od_v_v @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_245_disjoint__iff__not__equal,axiom,
! [A3: set_v,B2: set_v] :
( ( ( inf_inf_set_v @ A3 @ B2 )
= bot_bot_set_v )
= ( ! [X2: v] :
( ( member_v @ X2 @ A3 )
=> ! [Y3: v] :
( ( member_v @ Y3 @ B2 )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_246_disjoint__iff__not__equal,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A3 @ B2 )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ! [Y3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ B2 )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_247_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_248_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_249_Int__empty__left,axiom,
! [B2: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ B2 )
= bot_bot_set_v ) ).
% Int_empty_left
thf(fact_250_Int__empty__left,axiom,
! [B2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ B2 )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_left
thf(fact_251_disjoint__iff,axiom,
! [A3: set_v,B2: set_v] :
( ( ( inf_inf_set_v @ A3 @ B2 )
= bot_bot_set_v )
= ( ! [X2: v] :
( ( member_v @ X2 @ A3 )
=> ~ ( member_v @ X2 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_252_disjoint__iff,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A3 @ B2 )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ~ ( member7453568604450474000od_v_v @ X2 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_253_Int__emptyI,axiom,
! [A3: set_v,B2: set_v] :
( ! [X3: v] :
( ( member_v @ X3 @ A3 )
=> ~ ( member_v @ X3 @ B2 ) )
=> ( ( inf_inf_set_v @ A3 @ B2 )
= bot_bot_set_v ) ) ).
% Int_emptyI
thf(fact_254_Int__emptyI,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ~ ( member7453568604450474000od_v_v @ X3 @ B2 ) )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ B2 )
= bot_bo723834152578015283od_v_v ) ) ).
% Int_emptyI
thf(fact_255_Int__Collect__mono,axiom,
! [A3: set_v,B2: set_v,P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ A3 @ B2 )
=> ( ! [X3: v] :
( ( member_v @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ ( collect_v @ P ) ) @ ( inf_inf_set_v @ B2 @ ( collect_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_256_Int__Collect__mono,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
=> ( ! [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 @ B2 @ ( collec140062887454715474od_v_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_257_Int__greatest,axiom,
! [C: set_v,A3: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ C @ A3 )
=> ( ( ord_less_eq_set_v @ C @ B2 )
=> ( ord_less_eq_set_v @ C @ ( inf_inf_set_v @ A3 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_258_Int__greatest,axiom,
! [C: set_Product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ C @ B2 )
=> ( ord_le7336532860387713383od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_259_Int__absorb2,axiom,
! [A3: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A3 @ B2 )
=> ( ( inf_inf_set_v @ A3 @ B2 )
= A3 ) ) ).
% Int_absorb2
thf(fact_260_Int__absorb2,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ B2 )
= A3 ) ) ).
% Int_absorb2
thf(fact_261_Int__absorb1,axiom,
! [B2: set_v,A3: set_v] :
( ( ord_less_eq_set_v @ B2 @ A3 )
=> ( ( inf_inf_set_v @ A3 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_262_Int__absorb1,axiom,
! [B2: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_263_Int__lower2,axiom,
! [A3: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_264_Int__lower2,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_265_Int__lower1,axiom,
! [A3: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B2 ) @ A3 ) ).
% Int_lower1
thf(fact_266_Int__lower1,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) @ A3 ) ).
% Int_lower1
thf(fact_267_Int__mono,axiom,
! [A3: set_v,C: set_v,B2: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A3 @ C )
=> ( ( ord_less_eq_set_v @ B2 @ D2 )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B2 ) @ ( inf_inf_set_v @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_268_Int__mono,axiom,
! [A3: set_Product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) @ ( inf_in6271465464967711157od_v_v @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_269_Int__insert__right,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_270_Int__insert__right,axiom,
! [A: v,A3: set_v,B2: set_v] :
( ( ( member_v @ A @ A3 )
=> ( ( inf_inf_set_v @ A3 @ ( insert_v2 @ A @ B2 ) )
= ( insert_v2 @ A @ ( inf_inf_set_v @ A3 @ B2 ) ) ) )
& ( ~ ( member_v @ A @ A3 )
=> ( ( inf_inf_set_v @ A3 @ ( insert_v2 @ A @ B2 ) )
= ( inf_inf_set_v @ A3 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_271_Int__insert__left,axiom,
! [A: product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A @ C )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B2 ) @ C )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A @ C )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B2 ) @ C )
= ( inf_in6271465464967711157od_v_v @ B2 @ C ) ) ) ) ).
% Int_insert_left
thf(fact_272_Int__insert__left,axiom,
! [A: v,C: set_v,B2: set_v] :
( ( ( member_v @ A @ C )
=> ( ( inf_inf_set_v @ ( insert_v2 @ A @ B2 ) @ C )
= ( insert_v2 @ A @ ( inf_inf_set_v @ B2 @ C ) ) ) )
& ( ~ ( member_v @ A @ C )
=> ( ( inf_inf_set_v @ ( insert_v2 @ A @ B2 ) @ C )
= ( inf_inf_set_v @ B2 @ C ) ) ) ) ).
% Int_insert_left
thf(fact_273_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_274_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_275_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_276_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_277_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_278_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_279_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_280_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_281_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_282_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_283_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_284_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_285_inf_Obounded__iff,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B @ C2 ) )
= ( ( ord_less_eq_set_v @ A @ B )
& ( ord_less_eq_set_v @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_286_inf_Obounded__iff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) )
= ( ( ord_le7336532860387713383od_v_v @ A @ B )
& ( ord_le7336532860387713383od_v_v @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_287_le__inf__iff,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) )
= ( ( ord_less_eq_set_v @ X @ Y2 )
& ( ord_less_eq_set_v @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_288_le__inf__iff,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) )
= ( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
& ( ord_le7336532860387713383od_v_v @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_289_inf__right__idem,axiom,
! [X: set_v,Y2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ Y2 )
= ( inf_inf_set_v @ X @ Y2 ) ) ).
% inf_right_idem
thf(fact_290_inf_Oright__idem,axiom,
! [A: set_v,B: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B ) @ B )
= ( inf_inf_set_v @ A @ B ) ) ).
% inf.right_idem
thf(fact_291_inf__left__idem,axiom,
! [X: set_v,Y2: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ X @ Y2 ) )
= ( inf_inf_set_v @ X @ Y2 ) ) ).
% inf_left_idem
thf(fact_292_inf_Oidem,axiom,
! [A: set_v] :
( ( inf_inf_set_v @ A @ A )
= A ) ).
% inf.idem
thf(fact_293_inf__idem,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ X )
= X ) ).
% inf_idem
thf(fact_294_inf_Oleft__idem,axiom,
! [A: set_v,B: set_v] :
( ( inf_inf_set_v @ A @ ( inf_inf_set_v @ A @ B ) )
= ( inf_inf_set_v @ A @ B ) ) ).
% inf.left_idem
thf(fact_295_bot__set__def,axiom,
( bot_bot_set_v
= ( collect_v @ bot_bot_v_o ) ) ).
% bot_set_def
thf(fact_296_bot__set__def,axiom,
( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ bot_bo8461541820394803818_v_v_o ) ) ).
% bot_set_def
thf(fact_297_inf__sup__aci_I4_J,axiom,
! [X: set_v,Y2: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ X @ Y2 ) )
= ( inf_inf_set_v @ X @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_298_inf__sup__aci_I3_J,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) )
= ( inf_inf_set_v @ Y2 @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_299_inf__sup__aci_I2_J,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ Z )
= ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_300_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_301_inf_Oassoc,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B ) @ C2 )
= ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B @ C2 ) ) ) ).
% inf.assoc
thf(fact_302_inf__assoc,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ Z )
= ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) ) ) ).
% inf_assoc
thf(fact_303_inf_Ocommute,axiom,
( inf_inf_set_v
= ( ^ [A5: set_v,B5: set_v] : ( inf_inf_set_v @ B5 @ A5 ) ) ) ).
% inf.commute
thf(fact_304_inf__commute,axiom,
( inf_inf_set_v
= ( ^ [X2: set_v,Y3: set_v] : ( inf_inf_set_v @ Y3 @ X2 ) ) ) ).
% inf_commute
thf(fact_305_boolean__algebra__cancel_Oinf1,axiom,
! [A3: set_v,K: set_v,A: set_v,B: set_v] :
( ( A3
= ( inf_inf_set_v @ K @ A ) )
=> ( ( inf_inf_set_v @ A3 @ B )
= ( inf_inf_set_v @ K @ ( inf_inf_set_v @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_306_boolean__algebra__cancel_Oinf2,axiom,
! [B2: set_v,K: set_v,B: set_v,A: set_v] :
( ( B2
= ( inf_inf_set_v @ K @ B ) )
=> ( ( inf_inf_set_v @ A @ B2 )
= ( inf_inf_set_v @ K @ ( inf_inf_set_v @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_307_inf_Oleft__commute,axiom,
! [B: set_v,A: set_v,C2: set_v] :
( ( inf_inf_set_v @ B @ ( inf_inf_set_v @ A @ C2 ) )
= ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B @ C2 ) ) ) ).
% inf.left_commute
thf(fact_308_inf__left__commute,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) )
= ( inf_inf_set_v @ Y2 @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_309_inf__sup__ord_I2_J,axiom,
! [X: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_310_inf__sup__ord_I2_J,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_311_inf__sup__ord_I1_J,axiom,
! [X: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ X ) ).
% inf_sup_ord(1)
thf(fact_312_inf__sup__ord_I1_J,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ X ) ).
% inf_sup_ord(1)
thf(fact_313_inf__le1,axiom,
! [X: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ X ) ).
% inf_le1
thf(fact_314_inf__le1,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ X ) ).
% inf_le1
thf(fact_315_inf__le2,axiom,
! [X: set_v,Y2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_316_inf__le2,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_317_le__infE,axiom,
! [X: set_v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ A @ B ) )
=> ~ ( ( ord_less_eq_set_v @ X @ A )
=> ~ ( ord_less_eq_set_v @ X @ B ) ) ) ).
% le_infE
thf(fact_318_le__infE,axiom,
! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ A @ B ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ X @ A )
=> ~ ( ord_le7336532860387713383od_v_v @ X @ B ) ) ) ).
% le_infE
thf(fact_319_le__infI,axiom,
! [X: set_v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ X @ A )
=> ( ( ord_less_eq_set_v @ X @ B )
=> ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ A @ B ) ) ) ) ).
% le_infI
thf(fact_320_le__infI,axiom,
! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ A )
=> ( ( ord_le7336532860387713383od_v_v @ X @ B )
=> ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ) ).
% le_infI
thf(fact_321_inf__mono,axiom,
! [A: set_v,C2: set_v,B: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A @ C2 )
=> ( ( ord_less_eq_set_v @ B @ D )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ ( inf_inf_set_v @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_322_inf__mono,axiom,
! [A: 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 @ A @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ ( inf_in6271465464967711157od_v_v @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_323_le__infI1,axiom,
! [A: set_v,X: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ X )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_324_le__infI1,axiom,
! [A: set_Product_prod_v_v,X: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ X )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_325_le__infI2,axiom,
! [B: set_v,X: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ X )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_326_le__infI2,axiom,
! [B: set_Product_prod_v_v,X: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ X )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_327_inf_OorderE,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( A
= ( inf_inf_set_v @ A @ B ) ) ) ).
% inf.orderE
thf(fact_328_inf_OorderE,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( A
= ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ).
% inf.orderE
thf(fact_329_inf_OorderI,axiom,
! [A: set_v,B: set_v] :
( ( A
= ( inf_inf_set_v @ A @ B ) )
=> ( ord_less_eq_set_v @ A @ B ) ) ).
% inf.orderI
thf(fact_330_inf_OorderI,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A
= ( inf_in6271465464967711157od_v_v @ A @ B ) )
=> ( ord_le7336532860387713383od_v_v @ A @ B ) ) ).
% inf.orderI
thf(fact_331_inf__unique,axiom,
! [F: set_v > set_v > set_v,X: set_v,Y2: set_v] :
( ! [X3: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( F @ X3 @ Y ) @ X3 )
=> ( ! [X3: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( F @ X3 @ Y ) @ Y )
=> ( ! [X3: set_v,Y: set_v,Z3: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ( ord_less_eq_set_v @ X3 @ Z3 )
=> ( ord_less_eq_set_v @ X3 @ ( F @ Y @ Z3 ) ) ) )
=> ( ( inf_inf_set_v @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_332_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,Y2: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X3 @ Y ) @ X3 )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X3 @ Y ) @ Y )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ( ord_le7336532860387713383od_v_v @ X3 @ Z3 )
=> ( ord_le7336532860387713383od_v_v @ X3 @ ( F @ Y @ Z3 ) ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_333_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_334_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_335_inf_Oabsorb1,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( inf_inf_set_v @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_336_inf_Oabsorb1,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_337_inf_Oabsorb2,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( inf_inf_set_v @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_338_inf_Oabsorb2,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_339_inf__absorb1,axiom,
! [X: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X @ Y2 )
=> ( ( inf_inf_set_v @ X @ Y2 )
= X ) ) ).
% inf_absorb1
thf(fact_340_inf__absorb1,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
=> ( ( inf_in6271465464967711157od_v_v @ X @ Y2 )
= X ) ) ).
% inf_absorb1
thf(fact_341_inf__absorb2,axiom,
! [Y2: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y2 @ X )
=> ( ( inf_inf_set_v @ X @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_342_inf__absorb2,axiom,
! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y2 @ X )
=> ( ( inf_in6271465464967711157od_v_v @ X @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_343_inf_OboundedE,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B @ C2 ) )
=> ~ ( ( ord_less_eq_set_v @ A @ B )
=> ~ ( ord_less_eq_set_v @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_344_inf_OboundedE,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ~ ( ord_le7336532860387713383od_v_v @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_345_inf_OboundedI,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ A @ C2 )
=> ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_346_inf_OboundedI,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ A @ C2 )
=> ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_347_inf__greatest,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X @ Y2 )
=> ( ( ord_less_eq_set_v @ X @ Z )
=> ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_348_inf__greatest,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
=> ( ( ord_le7336532860387713383od_v_v @ X @ Z )
=> ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_349_inf_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A5: set_v,B5: set_v] :
( A5
= ( inf_inf_set_v @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_350_inf_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( A5
= ( inf_in6271465464967711157od_v_v @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_351_inf_Ocobounded1,axiom,
! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_352_inf_Ocobounded1,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_353_inf_Ocobounded2,axiom,
! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_354_inf_Ocobounded2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_355_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [A5: set_v,B5: set_v] :
( ( inf_inf_set_v @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_356_inf_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_357_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [B5: set_v,A5: set_v] :
( ( inf_inf_set_v @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_358_inf_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B5: set_Product_prod_v_v,A5: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_359_inf_OcoboundedI1,axiom,
! [A: set_v,C2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ C2 )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_360_inf_OcoboundedI1,axiom,
! [A: set_Product_prod_v_v,C2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C2 )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_361_inf_OcoboundedI2,axiom,
! [B: set_v,C2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ C2 )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_362_inf_OcoboundedI2,axiom,
! [B: set_Product_prod_v_v,C2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_363_the__elem__eq,axiom,
! [X: v] :
( ( the_elem_v @ ( insert_v2 @ X @ bot_bot_set_v ) )
= X ) ).
% the_elem_eq
thf(fact_364_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_365_order__refl,axiom,
! [X: set_v] : ( ord_less_eq_set_v @ X @ X ) ).
% order_refl
thf(fact_366_order__refl,axiom,
! [X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ X ) ).
% order_refl
thf(fact_367_dual__order_Orefl,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ A @ A ) ).
% dual_order.refl
thf(fact_368_dual__order_Orefl,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ A ) ).
% dual_order.refl
thf(fact_369_reachable__avoiding_Ocases,axiom,
! [A1: v,A2: v,A32: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A2 @ A32 )
=> ( ( A2 != A1 )
=> ~ ! [Y: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ Y @ A32 )
=> ( ( member_v @ A2 @ ( successors @ Y ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ A2 ) @ A32 ) ) ) ) ) ).
% reachable_avoiding.cases
thf(fact_370_ra__succ,axiom,
! [X: v,Y2: v,E4: set_Product_prod_v_v,Z: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E4 )
=> ( ( member_v @ Z @ ( successors @ Y2 ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ Z ) @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Z @ E4 ) ) ) ) ).
% ra_succ
thf(fact_371_reachable__avoiding_Osimps,axiom,
! [A1: v,A2: v,A32: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A2 @ A32 )
= ( ? [X2: v,E6: set_Product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = X2 )
& ( A32 = E6 ) )
| ? [X2: v,Y3: v,E6: set_Product_prod_v_v,Z2: v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( A32 = E6 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y3 @ E6 )
& ( member_v @ Z2 @ ( successors @ Y3 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ Z2 ) @ E6 ) ) ) ) ).
% reachable_avoiding.simps
thf(fact_372_ra__cases,axiom,
! [X: v,Y2: v,E4: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E4 )
=> ( ( X = Y2 )
| ? [Z3: v] :
( ( member_v @ Z3 @ ( successors @ X ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Z3 ) @ E4 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ Z3 @ Y2 @ E4 ) ) ) ) ).
% ra_cases
thf(fact_373_edge__ra,axiom,
! [Y2: v,X: v,E4: set_Product_prod_v_v] :
( ( member_v @ Y2 @ ( successors @ X ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y2 ) @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E4 ) ) ) ).
% edge_ra
thf(fact_374_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,Y2: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y2 @ E4 )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y2 ) )
=> ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y2 @ Z ) @ E4 )
=> ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Z @ E4 ) ) ) ) ) ).
% graph.ra_succ
thf(fact_375_graph_Ora__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,E4: set_Product_prod_v_v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E4 )
=> ( ( member_v @ Z @ ( Successors @ Y2 ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y2 @ Z ) @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Z @ E4 ) ) ) ) ) ).
% graph.ra_succ
thf(fact_376_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,E6: set_Pr2149350503807050951od_v_v] :
( ( A1 = X2 )
& ( A2 = X2 )
& ( A32 = E6 ) )
| ? [X2: product_prod_v_v,Y3: product_prod_v_v,E6: set_Pr2149350503807050951od_v_v,Z2: product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( A32 = E6 )
& ( sCC_Bl5370300055464682748od_v_v @ Successors @ X2 @ Y3 @ E6 )
& ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y3 ) )
& ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y3 @ Z2 ) @ E6 ) ) ) ) ) ).
% graph.reachable_avoiding.simps
thf(fact_377_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,E6: set_Product_prod_v_v] :
( ( A1 = X2 )
& ( A2 = X2 )
& ( A32 = E6 ) )
| ? [X2: v,Y3: v,E6: set_Product_prod_v_v,Z2: v] :
( ( A1 = X2 )
& ( A2 = Z2 )
& ( A32 = E6 )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ X2 @ Y3 @ E6 )
& ( member_v @ Z2 @ ( Successors @ Y3 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ Z2 ) @ E6 ) ) ) ) ) ).
% graph.reachable_avoiding.simps
thf(fact_378_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 )
=> ~ ! [Y: product_prod_v_v] :
( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ Y @ A32 )
=> ( ( member7453568604450474000od_v_v @ A2 @ ( Successors @ Y ) )
=> ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y @ A2 ) @ A32 ) ) ) ) ) ) ).
% graph.reachable_avoiding.cases
thf(fact_379_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 )
=> ~ ! [Y: v] :
( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ Y @ A32 )
=> ( ( member_v @ A2 @ ( Successors @ Y ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ A2 ) @ A32 ) ) ) ) ) ) ).
% graph.reachable_avoiding.cases
thf(fact_380_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,Y2: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y2 @ E4 )
=> ( ( X = Y2 )
| ? [Z3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ X ) )
& ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Z3 ) @ E4 )
& ( sCC_Bl5370300055464682748od_v_v @ Successors @ Z3 @ Y2 @ E4 ) ) ) ) ) ).
% graph.ra_cases
thf(fact_381_graph_Ora__cases,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,E4: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E4 )
=> ( ( X = Y2 )
| ? [Z3: v] :
( ( member_v @ Z3 @ ( Successors @ X ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Z3 ) @ E4 )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ Z3 @ Y2 @ E4 ) ) ) ) ) ).
% graph.ra_cases
thf(fact_382_graph_Oedge__ra,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y2: product_prod_v_v,X: product_prod_v_v,E4: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y2 @ ( Successors @ X ) )
=> ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Y2 ) @ E4 )
=> ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y2 @ E4 ) ) ) ) ).
% graph.edge_ra
thf(fact_383_graph_Oedge__ra,axiom,
! [Vertices: set_v,Successors: v > set_v,Y2: v,X: v,E4: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y2 @ ( Successors @ X ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y2 ) @ E4 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E4 ) ) ) ) ).
% graph.edge_ra
thf(fact_384_order__antisym__conv,axiom,
! [Y2: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y2 @ X )
=> ( ( ord_less_eq_set_v @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_385_order__antisym__conv,axiom,
! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y2 @ X )
=> ( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_386_ord__le__eq__subst,axiom,
! [A: set_v,B: set_v,F: set_v > set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_387_ord__le__eq__subst,axiom,
! [A: set_v,B: set_v,F: set_v > set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_388_ord__le__eq__subst,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C2: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_389_ord__le__eq__subst,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_390_ord__eq__le__subst,axiom,
! [A: set_v,F: set_v > set_v,B: set_v,C2: set_v] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_391_ord__eq__le__subst,axiom,
! [A: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B: set_v,C2: set_v] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_392_ord__eq__le__subst,axiom,
! [A: set_v,F: set_Product_prod_v_v > set_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( A
= ( F @ B ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_393_ord__eq__le__subst,axiom,
! [A: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( A
= ( F @ B ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_394_order__eq__refl,axiom,
! [X: set_v,Y2: set_v] :
( ( X = Y2 )
=> ( ord_less_eq_set_v @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_395_order__eq__refl,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( X = Y2 )
=> ( ord_le7336532860387713383od_v_v @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_396_order__subst2,axiom,
! [A: set_v,B: set_v,F: set_v > set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ ( F @ B ) @ C2 )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_397_order__subst2,axiom,
! [A: set_v,B: set_v,F: set_v > set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B ) @ C2 )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_398_order__subst2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C2: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_less_eq_set_v @ ( F @ B ) @ C2 )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_399_order__subst2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B ) @ C2 )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_400_order__subst1,axiom,
! [A: set_v,F: set_v > set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_401_order__subst1,axiom,
! [A: set_v,F: set_Product_prod_v_v > set_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ ( F @ B ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_402_order__subst1,axiom,
! [A: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B: set_v,C2: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_403_order__subst1,axiom,
! [A: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( F @ B ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_404_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A5: set_v,B5: set_v] :
( ( ord_less_eq_set_v @ A5 @ B5 )
& ( ord_less_eq_set_v @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_405_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A5 @ B5 )
& ( ord_le7336532860387713383od_v_v @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_406_antisym,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_407_antisym,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_408_dual__order_Otrans,axiom,
! [B: set_v,A: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( ord_less_eq_set_v @ C2 @ B )
=> ( ord_less_eq_set_v @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_409_dual__order_Otrans,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( ord_le7336532860387713383od_v_v @ C2 @ B )
=> ( ord_le7336532860387713383od_v_v @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_410_dual__order_Oantisym,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( ord_less_eq_set_v @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_411_dual__order_Oantisym,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_412_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_v,Z4: set_v] : ( Y4 = Z4 ) )
= ( ^ [A5: set_v,B5: set_v] :
( ( ord_less_eq_set_v @ B5 @ A5 )
& ( ord_less_eq_set_v @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_413_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y4 = Z4 ) )
= ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B5 @ A5 )
& ( ord_le7336532860387713383od_v_v @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_414_order__trans,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X @ Y2 )
=> ( ( ord_less_eq_set_v @ Y2 @ Z )
=> ( ord_less_eq_set_v @ X @ Z ) ) ) ).
% order_trans
thf(fact_415_order__trans,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
=> ( ( ord_le7336532860387713383od_v_v @ Y2 @ Z )
=> ( ord_le7336532860387713383od_v_v @ X @ Z ) ) ) ).
% order_trans
thf(fact_416_order_Otrans,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ord_less_eq_set_v @ A @ C2 ) ) ) ).
% order.trans
thf(fact_417_order_Otrans,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ord_le7336532860387713383od_v_v @ A @ C2 ) ) ) ).
% order.trans
thf(fact_418_order__antisym,axiom,
! [X: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X @ Y2 )
=> ( ( ord_less_eq_set_v @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_419_order__antisym,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
=> ( ( ord_le7336532860387713383od_v_v @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_420_ord__le__eq__trans,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_set_v @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_421_ord__le__eq__trans,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( B = C2 )
=> ( ord_le7336532860387713383od_v_v @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_422_ord__eq__le__trans,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( A = B )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ord_less_eq_set_v @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_423_ord__eq__le__trans,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( A = B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ord_le7336532860387713383od_v_v @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_424_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_425_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_426_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_427_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_428_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_429_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_430_bot_Oextremum,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A ) ).
% bot.extremum
thf(fact_431_bot_Oextremum,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A ) ).
% bot.extremum
thf(fact_432_avoiding__explored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,X: v,Y2: v,E4: set_Product_prod_v_v,W: v,V: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E4 )
=> ( ~ ( member_v @ Y2 @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ).
% avoiding_explored
thf(fact_433_ra__add__edge,axiom,
! [X: v,Y2: v,E4: set_Product_prod_v_v,V: v,W: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y2 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ V @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
& ( sCC_Bl4291963740693775144ding_v @ successors @ W @ Y2 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ).
% ra_add_edge
thf(fact_434_bot__empty__eq,axiom,
( bot_bot_v_o
= ( ^ [X2: v] : ( member_v @ X2 @ bot_bot_set_v ) ) ) ).
% bot_empty_eq
thf(fact_435_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_436_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_437_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_438_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_439_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_440_less__by__empty,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A3 = bot_bo723834152578015283od_v_v )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B2 ) ) ).
% less_by_empty
thf(fact_441_is__singletonI,axiom,
! [X: v] : ( is_singleton_v @ ( insert_v2 @ X @ bot_bot_set_v ) ) ).
% is_singletonI
thf(fact_442_is__singletonI,axiom,
! [X: product_prod_v_v] : ( is_sin9198872032823709915od_v_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ).
% is_singletonI
thf(fact_443_sup_Oright__idem,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ B )
= ( sup_su414716646722978715od_v_v @ A @ B ) ) ).
% sup.right_idem
thf(fact_444_sup__left__idem,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) )
= ( sup_su414716646722978715od_v_v @ X @ Y2 ) ) ).
% sup_left_idem
thf(fact_445_sup_Oleft__idem,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B ) )
= ( sup_su414716646722978715od_v_v @ A @ B ) ) ).
% sup.left_idem
thf(fact_446_sup__idem,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ X )
= X ) ).
% sup_idem
thf(fact_447_sup_Oidem,axiom,
! [A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ A )
= A ) ).
% sup.idem
thf(fact_448_Un__iff,axiom,
! [C2: v,A3: set_v,B2: set_v] :
( ( member_v @ C2 @ ( sup_sup_set_v @ A3 @ B2 ) )
= ( ( member_v @ C2 @ A3 )
| ( member_v @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_449_Un__iff,axiom,
! [C2: product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) )
= ( ( member7453568604450474000od_v_v @ C2 @ A3 )
| ( member7453568604450474000od_v_v @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_450_UnCI,axiom,
! [C2: v,B2: set_v,A3: set_v] :
( ( ~ ( member_v @ C2 @ B2 )
=> ( member_v @ C2 @ A3 ) )
=> ( member_v @ C2 @ ( sup_sup_set_v @ A3 @ B2 ) ) ) ).
% UnCI
thf(fact_451_UnCI,axiom,
! [C2: product_prod_v_v,B2: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ C2 @ B2 )
=> ( member7453568604450474000od_v_v @ C2 @ A3 ) )
=> ( member7453568604450474000od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) ) ) ).
% UnCI
thf(fact_452_le__sup__iff,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ X @ Y2 ) @ Z )
= ( ( ord_less_eq_set_v @ X @ Z )
& ( ord_less_eq_set_v @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_453_le__sup__iff,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) @ Z )
= ( ( ord_le7336532860387713383od_v_v @ X @ Z )
& ( ord_le7336532860387713383od_v_v @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_454_sup_Obounded__iff,axiom,
! [B: set_v,C2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B @ C2 ) @ A )
= ( ( ord_less_eq_set_v @ B @ A )
& ( ord_less_eq_set_v @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_455_sup_Obounded__iff,axiom,
! [B: set_Product_prod_v_v,C2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B @ C2 ) @ A )
= ( ( ord_le7336532860387713383od_v_v @ B @ A )
& ( ord_le7336532860387713383od_v_v @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_456_sup__bot__left,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ X )
= X ) ).
% sup_bot_left
thf(fact_457_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_458_sup__bot__right,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ X @ bot_bot_set_v )
= X ) ).
% sup_bot_right
thf(fact_459_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_460_bot__eq__sup__iff,axiom,
! [X: set_v,Y2: set_v] :
( ( bot_bot_set_v
= ( sup_sup_set_v @ X @ Y2 ) )
= ( ( X = bot_bot_set_v )
& ( Y2 = bot_bot_set_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_461_bot__eq__sup__iff,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( sup_su414716646722978715od_v_v @ X @ Y2 ) )
= ( ( X = bot_bo723834152578015283od_v_v )
& ( Y2 = bot_bo723834152578015283od_v_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_462_sup__eq__bot__iff,axiom,
! [X: set_v,Y2: set_v] :
( ( ( sup_sup_set_v @ X @ Y2 )
= bot_bot_set_v )
= ( ( X = bot_bot_set_v )
& ( Y2 = bot_bot_set_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_463_sup__eq__bot__iff,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ X @ Y2 )
= bot_bo723834152578015283od_v_v )
= ( ( X = bot_bo723834152578015283od_v_v )
& ( Y2 = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_464_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_v,B: set_v] :
( ( ( sup_sup_set_v @ A @ B )
= bot_bot_set_v )
= ( ( A = bot_bot_set_v )
& ( B = bot_bot_set_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_465_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A @ B )
= bot_bo723834152578015283od_v_v )
= ( ( A = bot_bo723834152578015283od_v_v )
& ( B = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_466_sup__bot_Oleft__neutral,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_467_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_468_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_v,B: set_v] :
( ( bot_bot_set_v
= ( sup_sup_set_v @ A @ B ) )
= ( ( A = bot_bot_set_v )
& ( B = bot_bot_set_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_469_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( sup_su414716646722978715od_v_v @ A @ B ) )
= ( ( A = bot_bo723834152578015283od_v_v )
& ( B = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_470_sup__bot_Oright__neutral,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ A @ bot_bot_set_v )
= A ) ).
% sup_bot.right_neutral
thf(fact_471_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_472_inf__sup__absorb,axiom,
! [X: set_v,Y2: set_v] :
( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ X @ Y2 ) )
= X ) ).
% inf_sup_absorb
thf(fact_473_inf__sup__absorb,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) )
= X ) ).
% inf_sup_absorb
thf(fact_474_sup__inf__absorb,axiom,
! [X: set_v,Y2: set_v] :
( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ X @ Y2 ) )
= X ) ).
% sup_inf_absorb
thf(fact_475_sup__inf__absorb,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) )
= X ) ).
% sup_inf_absorb
thf(fact_476_Un__empty,axiom,
! [A3: set_v,B2: set_v] :
( ( ( sup_sup_set_v @ A3 @ B2 )
= bot_bot_set_v )
= ( ( A3 = bot_bot_set_v )
& ( B2 = bot_bot_set_v ) ) ) ).
% Un_empty
thf(fact_477_Un__empty,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A3 @ B2 )
= bot_bo723834152578015283od_v_v )
= ( ( A3 = bot_bo723834152578015283od_v_v )
& ( B2 = bot_bo723834152578015283od_v_v ) ) ) ).
% Un_empty
thf(fact_478_Un__subset__iff,axiom,
! [A3: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A3 @ B2 ) @ C )
= ( ( ord_less_eq_set_v @ A3 @ C )
& ( ord_less_eq_set_v @ B2 @ C ) ) ) ).
% Un_subset_iff
thf(fact_479_Un__subset__iff,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) @ C )
= ( ( ord_le7336532860387713383od_v_v @ A3 @ C )
& ( ord_le7336532860387713383od_v_v @ B2 @ C ) ) ) ).
% Un_subset_iff
thf(fact_480_Un__insert__left,axiom,
! [A: v,B2: set_v,C: set_v] :
( ( sup_sup_set_v @ ( insert_v2 @ A @ B2 ) @ C )
= ( insert_v2 @ A @ ( sup_sup_set_v @ B2 @ C ) ) ) ).
% Un_insert_left
thf(fact_481_Un__insert__left,axiom,
! [A: product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A @ B2 ) @ C )
= ( insert1338601472111419319od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B2 @ C ) ) ) ).
% Un_insert_left
thf(fact_482_Un__insert__right,axiom,
! [A3: set_v,A: v,B2: set_v] :
( ( sup_sup_set_v @ A3 @ ( insert_v2 @ A @ B2 ) )
= ( insert_v2 @ A @ ( sup_sup_set_v @ A3 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_483_Un__insert__right,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( insert1338601472111419319od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_484_Int__Un__eq_I4_J,axiom,
! [T2: set_v,S: set_v] :
( ( sup_sup_set_v @ T2 @ ( inf_inf_set_v @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_485_Int__Un__eq_I4_J,axiom,
! [T2: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ T2 @ ( inf_in6271465464967711157od_v_v @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_486_Int__Un__eq_I3_J,axiom,
! [S: set_v,T2: set_v] :
( ( sup_sup_set_v @ S @ ( inf_inf_set_v @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_487_Int__Un__eq_I3_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ S @ ( inf_in6271465464967711157od_v_v @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_488_Int__Un__eq_I2_J,axiom,
! [S: set_v,T2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_489_Int__Un__eq_I2_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_490_Int__Un__eq_I1_J,axiom,
! [S: set_v,T2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_491_Int__Un__eq_I1_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_492_Un__Int__eq_I4_J,axiom,
! [T2: set_v,S: set_v] :
( ( inf_inf_set_v @ T2 @ ( sup_sup_set_v @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_493_Un__Int__eq_I4_J,axiom,
! [T2: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ T2 @ ( sup_su414716646722978715od_v_v @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_494_Un__Int__eq_I3_J,axiom,
! [S: set_v,T2: set_v] :
( ( inf_inf_set_v @ S @ ( sup_sup_set_v @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_495_Un__Int__eq_I3_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ S @ ( sup_su414716646722978715od_v_v @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_496_Un__Int__eq_I2_J,axiom,
! [S: set_v,T2: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_497_Un__Int__eq_I2_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_498_Un__Int__eq_I1_J,axiom,
! [S: set_v,T2: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_499_Un__Int__eq_I1_J,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_500_Un__left__commute,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B2 @ C ) )
= ( sup_su414716646722978715od_v_v @ B2 @ ( sup_su414716646722978715od_v_v @ A3 @ C ) ) ) ).
% Un_left_commute
thf(fact_501_Un__left__absorb,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) )
= ( sup_su414716646722978715od_v_v @ A3 @ B2 ) ) ).
% Un_left_absorb
thf(fact_502_Un__commute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B3 @ A4 ) ) ) ).
% Un_commute
thf(fact_503_Un__absorb,axiom,
! [A3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_504_Un__assoc,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) @ C )
= ( sup_su414716646722978715od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B2 @ C ) ) ) ).
% Un_assoc
thf(fact_505_ball__Un,axiom,
! [A3: set_Product_prod_v_v,B2: 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 @ B2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ( P @ X2 ) )
& ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ B2 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_506_bex__Un,axiom,
! [A3: set_Product_prod_v_v,B2: 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 @ B2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A3 )
& ( P @ X2 ) )
| ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ B2 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_507_UnI2,axiom,
! [C2: v,B2: set_v,A3: set_v] :
( ( member_v @ C2 @ B2 )
=> ( member_v @ C2 @ ( sup_sup_set_v @ A3 @ B2 ) ) ) ).
% UnI2
thf(fact_508_UnI2,axiom,
! [C2: product_prod_v_v,B2: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ B2 )
=> ( member7453568604450474000od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) ) ) ).
% UnI2
thf(fact_509_UnI1,axiom,
! [C2: v,A3: set_v,B2: set_v] :
( ( member_v @ C2 @ A3 )
=> ( member_v @ C2 @ ( sup_sup_set_v @ A3 @ B2 ) ) ) ).
% UnI1
thf(fact_510_UnI1,axiom,
! [C2: product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ A3 )
=> ( member7453568604450474000od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) ) ) ).
% UnI1
thf(fact_511_UnE,axiom,
! [C2: v,A3: set_v,B2: set_v] :
( ( member_v @ C2 @ ( sup_sup_set_v @ A3 @ B2 ) )
=> ( ~ ( member_v @ C2 @ A3 )
=> ( member_v @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_512_UnE,axiom,
! [C2: product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) )
=> ( ~ ( member7453568604450474000od_v_v @ C2 @ A3 )
=> ( member7453568604450474000od_v_v @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_513_sup__left__commute,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
= ( sup_su414716646722978715od_v_v @ Y2 @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_514_sup_Oleft__commute,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A @ C2 ) )
= ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) ) ).
% sup.left_commute
thf(fact_515_boolean__algebra__cancel_Osup2,axiom,
! [B2: set_Product_prod_v_v,K: set_Product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( B2
= ( sup_su414716646722978715od_v_v @ K @ B ) )
=> ( ( sup_su414716646722978715od_v_v @ A @ B2 )
= ( sup_su414716646722978715od_v_v @ K @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_516_boolean__algebra__cancel_Osup1,axiom,
! [A3: set_Product_prod_v_v,K: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A3
= ( sup_su414716646722978715od_v_v @ K @ A ) )
=> ( ( sup_su414716646722978715od_v_v @ A3 @ B )
= ( sup_su414716646722978715od_v_v @ K @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_517_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_518_sup_Ocommute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B5 @ A5 ) ) ) ).
% sup.commute
thf(fact_519_sup__assoc,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) @ Z )
= ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_520_sup_Oassoc,axiom,
! [A: 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 @ A @ B ) @ C2 )
= ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C2 ) ) ) ).
% sup.assoc
thf(fact_521_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_522_inf__sup__aci_I6_J,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) @ Z )
= ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_523_inf__sup__aci_I7_J,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
= ( sup_su414716646722978715od_v_v @ Y2 @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_524_inf__sup__aci_I8_J,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) )
= ( sup_su414716646722978715od_v_v @ X @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_525_sup_OcoboundedI2,axiom,
! [C2: set_v,B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ C2 @ B )
=> ( ord_less_eq_set_v @ C2 @ ( sup_sup_set_v @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_526_sup_OcoboundedI2,axiom,
! [C2: set_Product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ B )
=> ( ord_le7336532860387713383od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_527_sup_OcoboundedI1,axiom,
! [C2: set_v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C2 @ A )
=> ( ord_less_eq_set_v @ C2 @ ( sup_sup_set_v @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_528_sup_OcoboundedI1,axiom,
! [C2: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ A )
=> ( ord_le7336532860387713383od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_529_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [A5: set_v,B5: set_v] :
( ( sup_sup_set_v @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_530_sup_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_531_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [B5: set_v,A5: set_v] :
( ( sup_sup_set_v @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_532_sup_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B5: set_Product_prod_v_v,A5: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_533_sup_Ocobounded2,axiom,
! [B: set_v,A: set_v] : ( ord_less_eq_set_v @ B @ ( sup_sup_set_v @ A @ B ) ) ).
% sup.cobounded2
thf(fact_534_sup_Ocobounded2,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ).
% sup.cobounded2
thf(fact_535_sup_Ocobounded1,axiom,
! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ A @ ( sup_sup_set_v @ A @ B ) ) ).
% sup.cobounded1
thf(fact_536_sup_Ocobounded1,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ).
% sup.cobounded1
thf(fact_537_sup_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [B5: set_v,A5: set_v] :
( A5
= ( sup_sup_set_v @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_538_sup_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B5: set_Product_prod_v_v,A5: set_Product_prod_v_v] :
( A5
= ( sup_su414716646722978715od_v_v @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_539_sup_OboundedI,axiom,
! [B: set_v,A: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( ord_less_eq_set_v @ C2 @ A )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ B @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_540_sup_OboundedI,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( ord_le7336532860387713383od_v_v @ C2 @ A )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_541_sup_OboundedE,axiom,
! [B: set_v,C2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B @ C2 ) @ A )
=> ~ ( ( ord_less_eq_set_v @ B @ A )
=> ~ ( ord_less_eq_set_v @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_542_sup_OboundedE,axiom,
! [B: set_Product_prod_v_v,C2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B @ C2 ) @ A )
=> ~ ( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ~ ( ord_le7336532860387713383od_v_v @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_543_sup__absorb2,axiom,
! [X: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X @ Y2 )
=> ( ( sup_sup_set_v @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_544_sup__absorb2,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
=> ( ( sup_su414716646722978715od_v_v @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_545_sup__absorb1,axiom,
! [Y2: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y2 @ X )
=> ( ( sup_sup_set_v @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_546_sup__absorb1,axiom,
! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y2 @ X )
=> ( ( sup_su414716646722978715od_v_v @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_547_sup_Oabsorb2,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( sup_sup_set_v @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_548_sup_Oabsorb2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( sup_su414716646722978715od_v_v @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_549_sup_Oabsorb1,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( sup_sup_set_v @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_550_sup_Oabsorb1,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( sup_su414716646722978715od_v_v @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_551_sup__unique,axiom,
! [F: set_v > set_v > set_v,X: set_v,Y2: set_v] :
( ! [X3: set_v,Y: set_v] : ( ord_less_eq_set_v @ X3 @ ( F @ X3 @ Y ) )
=> ( ! [X3: set_v,Y: set_v] : ( ord_less_eq_set_v @ Y @ ( F @ X3 @ Y ) )
=> ( ! [X3: set_v,Y: set_v,Z3: set_v] :
( ( ord_less_eq_set_v @ Y @ X3 )
=> ( ( ord_less_eq_set_v @ Z3 @ X3 )
=> ( ord_less_eq_set_v @ ( F @ Y @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_set_v @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_552_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,Y2: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X3 @ ( F @ X3 @ Y ) )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y @ ( F @ X3 @ Y ) )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X3 )
=> ( ( ord_le7336532860387713383od_v_v @ Z3 @ X3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ Y @ Z3 ) @ X3 ) ) )
=> ( ( sup_su414716646722978715od_v_v @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_553_sup_OorderI,axiom,
! [A: set_v,B: set_v] :
( ( A
= ( sup_sup_set_v @ A @ B ) )
=> ( ord_less_eq_set_v @ B @ A ) ) ).
% sup.orderI
thf(fact_554_sup_OorderI,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A
= ( sup_su414716646722978715od_v_v @ A @ B ) )
=> ( ord_le7336532860387713383od_v_v @ B @ A ) ) ).
% sup.orderI
thf(fact_555_sup_OorderE,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( A
= ( sup_sup_set_v @ A @ B ) ) ) ).
% sup.orderE
thf(fact_556_sup_OorderE,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( A
= ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% sup.orderE
thf(fact_557_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_558_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_559_sup__least,axiom,
! [Y2: set_v,X: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ Y2 @ X )
=> ( ( ord_less_eq_set_v @ Z @ X )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ Y2 @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_560_sup__least,axiom,
! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y2 @ X )
=> ( ( ord_le7336532860387713383od_v_v @ Z @ X )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_561_sup__mono,axiom,
! [A: set_v,C2: set_v,B: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A @ C2 )
=> ( ( ord_less_eq_set_v @ B @ D )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B ) @ ( sup_sup_set_v @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_562_sup__mono,axiom,
! [A: 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 @ A @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ ( sup_su414716646722978715od_v_v @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_563_sup_Omono,axiom,
! [C2: set_v,A: set_v,D: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C2 @ A )
=> ( ( ord_less_eq_set_v @ D @ B )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ C2 @ D ) @ ( sup_sup_set_v @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_564_sup_Omono,axiom,
! [C2: set_Product_prod_v_v,A: set_Product_prod_v_v,D: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ D @ B )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ C2 @ D ) @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_565_le__supI2,axiom,
! [X: set_v,B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ X @ B )
=> ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ A @ B ) ) ) ).
% le_supI2
thf(fact_566_le__supI2,axiom,
! [X: set_Product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ B )
=> ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% le_supI2
thf(fact_567_le__supI1,axiom,
! [X: set_v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ X @ A )
=> ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ A @ B ) ) ) ).
% le_supI1
thf(fact_568_le__supI1,axiom,
! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ A )
=> ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% le_supI1
thf(fact_569_sup__ge2,axiom,
! [Y2: set_v,X: set_v] : ( ord_less_eq_set_v @ Y2 @ ( sup_sup_set_v @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_570_sup__ge2,axiom,
! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y2 @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_571_sup__ge1,axiom,
! [X: set_v,Y2: set_v] : ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_572_sup__ge1,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_573_le__supI,axiom,
! [A: set_v,X: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ X )
=> ( ( ord_less_eq_set_v @ B @ X )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_574_le__supI,axiom,
! [A: set_Product_prod_v_v,X: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ X )
=> ( ( ord_le7336532860387713383od_v_v @ B @ X )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_575_le__supE,axiom,
! [A: set_v,B: set_v,X: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_set_v @ A @ X )
=> ~ ( ord_less_eq_set_v @ B @ X ) ) ) ).
% le_supE
thf(fact_576_le__supE,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ X )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A @ X )
=> ~ ( ord_le7336532860387713383od_v_v @ B @ X ) ) ) ).
% le_supE
thf(fact_577_inf__sup__ord_I3_J,axiom,
! [X: set_v,Y2: set_v] : ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_578_inf__sup__ord_I3_J,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_579_inf__sup__ord_I4_J,axiom,
! [Y2: set_v,X: set_v] : ( ord_less_eq_set_v @ Y2 @ ( sup_sup_set_v @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_580_inf__sup__ord_I4_J,axiom,
! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y2 @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_581_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_582_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_583_distrib__imp1,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ! [X3: set_v,Y: set_v,Z3: set_v] :
( ( inf_inf_set_v @ X3 @ ( sup_sup_set_v @ Y @ Z3 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X3 @ Y ) @ ( inf_inf_set_v @ X3 @ Z3 ) ) )
=> ( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y2 ) @ ( sup_sup_set_v @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_584_distrib__imp1,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ Y @ Z3 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X3 @ Y ) @ ( inf_in6271465464967711157od_v_v @ X3 @ Z3 ) ) )
=> ( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_585_distrib__imp2,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ! [X3: set_v,Y: set_v,Z3: set_v] :
( ( sup_sup_set_v @ X3 @ ( inf_inf_set_v @ Y @ Z3 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X3 @ Y ) @ ( sup_sup_set_v @ X3 @ Z3 ) ) )
=> ( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y2 @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ ( inf_inf_set_v @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_586_distrib__imp2,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X3 @ ( inf_in6271465464967711157od_v_v @ Y @ Z3 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X3 @ Y ) @ ( sup_su414716646722978715od_v_v @ X3 @ Z3 ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_587_inf__sup__distrib1,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y2 @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_588_inf__sup__distrib1,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_589_inf__sup__distrib2,axiom,
! [Y2: set_v,Z: set_v,X: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ Y2 @ Z ) @ X )
= ( sup_sup_set_v @ ( inf_inf_set_v @ Y2 @ X ) @ ( inf_inf_set_v @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_590_inf__sup__distrib2,axiom,
! [Y2: 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 @ Y2 @ Z ) @ X )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y2 @ X ) @ ( inf_in6271465464967711157od_v_v @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_591_sup__inf__distrib1,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y2 ) @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_592_sup__inf__distrib1,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_593_sup__inf__distrib2,axiom,
! [Y2: set_v,Z: set_v,X: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ Y2 @ Z ) @ X )
= ( inf_inf_set_v @ ( sup_sup_set_v @ Y2 @ X ) @ ( sup_sup_set_v @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_594_sup__inf__distrib2,axiom,
! [Y2: 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 @ Y2 @ Z ) @ X )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ X ) @ ( sup_su414716646722978715od_v_v @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_595_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y2 @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_596_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_597_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y2 ) @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_598_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_599_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y2: set_v,Z: set_v,X: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ Y2 @ Z ) @ X )
= ( sup_sup_set_v @ ( inf_inf_set_v @ Y2 @ X ) @ ( inf_inf_set_v @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_600_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y2: 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 @ Y2 @ Z ) @ X )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y2 @ X ) @ ( inf_in6271465464967711157od_v_v @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_601_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y2: set_v,Z: set_v,X: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ Y2 @ Z ) @ X )
= ( inf_inf_set_v @ ( sup_sup_set_v @ Y2 @ X ) @ ( sup_sup_set_v @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_602_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y2: 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 @ Y2 @ Z ) @ X )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y2 @ X ) @ ( sup_su414716646722978715od_v_v @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_603_Un__empty__left,axiom,
! [B2: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_604_Un__empty__left,axiom,
! [B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_605_Un__empty__right,axiom,
! [A3: set_v] :
( ( sup_sup_set_v @ A3 @ bot_bot_set_v )
= A3 ) ).
% Un_empty_right
thf(fact_606_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_607_Un__mono,axiom,
! [A3: set_v,C: set_v,B2: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A3 @ C )
=> ( ( ord_less_eq_set_v @ B2 @ D2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A3 @ B2 ) @ ( sup_sup_set_v @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_608_Un__mono,axiom,
! [A3: set_Product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) @ ( sup_su414716646722978715od_v_v @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_609_Un__least,axiom,
! [A3: set_v,C: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A3 @ C )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A3 @ B2 ) @ C ) ) ) ).
% Un_least
thf(fact_610_Un__least,axiom,
! [A3: set_Product_prod_v_v,C: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) @ C ) ) ) ).
% Un_least
thf(fact_611_Un__upper1,axiom,
! [A3: set_v,B2: set_v] : ( ord_less_eq_set_v @ A3 @ ( sup_sup_set_v @ A3 @ B2 ) ) ).
% Un_upper1
thf(fact_612_Un__upper1,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) ) ).
% Un_upper1
thf(fact_613_Un__upper2,axiom,
! [B2: set_v,A3: set_v] : ( ord_less_eq_set_v @ B2 @ ( sup_sup_set_v @ A3 @ B2 ) ) ).
% Un_upper2
thf(fact_614_Un__upper2,axiom,
! [B2: set_Product_prod_v_v,A3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B2 @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) ) ).
% Un_upper2
thf(fact_615_Un__absorb1,axiom,
! [A3: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A3 @ B2 )
=> ( ( sup_sup_set_v @ A3 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_616_Un__absorb1,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
=> ( ( sup_su414716646722978715od_v_v @ A3 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_617_Un__absorb2,axiom,
! [B2: set_v,A3: set_v] :
( ( ord_less_eq_set_v @ B2 @ A3 )
=> ( ( sup_sup_set_v @ A3 @ B2 )
= A3 ) ) ).
% Un_absorb2
thf(fact_618_Un__absorb2,axiom,
! [B2: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A3 )
=> ( ( sup_su414716646722978715od_v_v @ A3 @ B2 )
= A3 ) ) ).
% Un_absorb2
thf(fact_619_subset__UnE,axiom,
! [C: set_v,A3: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A3 @ B2 ) )
=> ~ ! [A6: set_v] :
( ( ord_less_eq_set_v @ A6 @ A3 )
=> ! [B6: set_v] :
( ( ord_less_eq_set_v @ B6 @ B2 )
=> ( C
!= ( sup_sup_set_v @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_620_subset__UnE,axiom,
! [C: set_Product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) )
=> ~ ! [A6: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A6 @ A3 )
=> ! [B6: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B6 @ B2 )
=> ( C
!= ( sup_su414716646722978715od_v_v @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_621_subset__Un__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B3: set_v] :
( ( sup_sup_set_v @ A4 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_622_subset__Un__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_623_Un__Int__crazy,axiom,
! [A3: set_v,B2: set_v,C: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ A3 @ B2 ) @ ( inf_inf_set_v @ B2 @ C ) ) @ ( inf_inf_set_v @ C @ A3 ) )
= ( inf_inf_set_v @ ( inf_inf_set_v @ ( sup_sup_set_v @ A3 @ B2 ) @ ( sup_sup_set_v @ B2 @ C ) ) @ ( sup_sup_set_v @ C @ A3 ) ) ) ).
% Un_Int_crazy
thf(fact_624_Un__Int__crazy,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) ) @ ( inf_in6271465464967711157od_v_v @ C @ A3 ) )
= ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) @ ( sup_su414716646722978715od_v_v @ B2 @ C ) ) @ ( sup_su414716646722978715od_v_v @ C @ A3 ) ) ) ).
% Un_Int_crazy
thf(fact_625_Int__Un__distrib,axiom,
! [A3: set_v,B2: set_v,C: set_v] :
( ( inf_inf_set_v @ A3 @ ( sup_sup_set_v @ B2 @ C ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ A3 @ B2 ) @ ( inf_inf_set_v @ A3 @ C ) ) ) ).
% Int_Un_distrib
thf(fact_626_Int__Un__distrib,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B2 @ C ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) @ ( inf_in6271465464967711157od_v_v @ A3 @ C ) ) ) ).
% Int_Un_distrib
thf(fact_627_Un__Int__distrib,axiom,
! [A3: set_v,B2: set_v,C: set_v] :
( ( sup_sup_set_v @ A3 @ ( inf_inf_set_v @ B2 @ C ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ A3 @ B2 ) @ ( sup_sup_set_v @ A3 @ C ) ) ) ).
% Un_Int_distrib
thf(fact_628_Un__Int__distrib,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) @ ( sup_su414716646722978715od_v_v @ A3 @ C ) ) ) ).
% Un_Int_distrib
thf(fact_629_Int__Un__distrib2,axiom,
! [B2: set_v,C: set_v,A3: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ B2 @ C ) @ A3 )
= ( sup_sup_set_v @ ( inf_inf_set_v @ B2 @ A3 ) @ ( inf_inf_set_v @ C @ A3 ) ) ) ).
% Int_Un_distrib2
thf(fact_630_Int__Un__distrib2,axiom,
! [B2: set_Product_prod_v_v,C: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ C ) @ A3 )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B2 @ A3 ) @ ( inf_in6271465464967711157od_v_v @ C @ A3 ) ) ) ).
% Int_Un_distrib2
thf(fact_631_Un__Int__distrib2,axiom,
! [B2: set_v,C: set_v,A3: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ B2 @ C ) @ A3 )
= ( inf_inf_set_v @ ( sup_sup_set_v @ B2 @ A3 ) @ ( sup_sup_set_v @ C @ A3 ) ) ) ).
% Un_Int_distrib2
thf(fact_632_Un__Int__distrib2,axiom,
! [B2: set_Product_prod_v_v,C: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) @ A3 )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ A3 ) @ ( sup_su414716646722978715od_v_v @ C @ A3 ) ) ) ).
% Un_Int_distrib2
thf(fact_633_graph_Odfss_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,X: produc5741669702376414499t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ~ ! [V2: v,E7: sCC_Bl1394983891496994913t_unit] :
( X
!= ( produc3862955338007567901t_unit @ V2 @ E7 ) ) ) ).
% graph.dfss.cases
thf(fact_634_distrib__inf__le,axiom,
! [X: set_v,Y2: set_v,Z: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y2 ) @ ( inf_inf_set_v @ X @ Z ) ) @ ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y2 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_635_distrib__inf__le,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y2 ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) @ ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y2 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_636_distrib__sup__le,axiom,
! [X: set_v,Y2: set_v,Z: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y2 @ Z ) ) @ ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y2 ) @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_637_distrib__sup__le,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y2 @ Z ) ) @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y2 ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_638_insert__is__Un,axiom,
( insert_v2
= ( ^ [A5: v] : ( sup_sup_set_v @ ( insert_v2 @ A5 @ bot_bot_set_v ) ) ) ) ).
% insert_is_Un
thf(fact_639_insert__is__Un,axiom,
( insert1338601472111419319od_v_v
= ( ^ [A5: product_prod_v_v] : ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A5 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% insert_is_Un
thf(fact_640_Un__singleton__iff,axiom,
! [A3: set_v,B2: set_v,X: v] :
( ( ( sup_sup_set_v @ A3 @ B2 )
= ( insert_v2 @ X @ bot_bot_set_v ) )
= ( ( ( A3 = bot_bot_set_v )
& ( B2
= ( insert_v2 @ X @ bot_bot_set_v ) ) )
| ( ( A3
= ( insert_v2 @ X @ bot_bot_set_v ) )
& ( B2 = bot_bot_set_v ) )
| ( ( A3
= ( insert_v2 @ X @ bot_bot_set_v ) )
& ( B2
= ( insert_v2 @ X @ bot_bot_set_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_641_Un__singleton__iff,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A3 @ B2 )
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= ( ( ( A3 = bot_bo723834152578015283od_v_v )
& ( B2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A3
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B2 = bot_bo723834152578015283od_v_v ) )
| ( ( A3
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_642_singleton__Un__iff,axiom,
! [X: v,A3: set_v,B2: set_v] :
( ( ( insert_v2 @ X @ bot_bot_set_v )
= ( sup_sup_set_v @ A3 @ B2 ) )
= ( ( ( A3 = bot_bot_set_v )
& ( B2
= ( insert_v2 @ X @ bot_bot_set_v ) ) )
| ( ( A3
= ( insert_v2 @ X @ bot_bot_set_v ) )
& ( B2 = bot_bot_set_v ) )
| ( ( A3
= ( insert_v2 @ X @ bot_bot_set_v ) )
& ( B2
= ( insert_v2 @ X @ bot_bot_set_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_643_singleton__Un__iff,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v )
= ( sup_su414716646722978715od_v_v @ A3 @ B2 ) )
= ( ( ( A3 = bot_bo723834152578015283od_v_v )
& ( B2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A3
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B2 = bot_bo723834152578015283od_v_v ) )
| ( ( A3
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_644_Un__Int__assoc__eq,axiom,
! [A3: set_v,B2: set_v,C: set_v] :
( ( ( sup_sup_set_v @ ( inf_inf_set_v @ A3 @ B2 ) @ C )
= ( inf_inf_set_v @ A3 @ ( sup_sup_set_v @ B2 @ C ) ) )
= ( ord_less_eq_set_v @ C @ A3 ) ) ).
% Un_Int_assoc_eq
thf(fact_645_Un__Int__assoc__eq,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) @ C )
= ( inf_in6271465464967711157od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B2 @ C ) ) )
= ( ord_le7336532860387713383od_v_v @ C @ A3 ) ) ).
% Un_Int_assoc_eq
thf(fact_646_is__singletonI_H,axiom,
! [A3: set_v] :
( ( A3 != bot_bot_set_v )
=> ( ! [X3: v,Y: v] :
( ( member_v @ X3 @ A3 )
=> ( ( member_v @ Y @ A3 )
=> ( X3 = Y ) ) )
=> ( is_singleton_v @ A3 ) ) ) ).
% is_singletonI'
thf(fact_647_is__singletonI_H,axiom,
! [A3: set_Product_prod_v_v] :
( ( A3 != bot_bo723834152578015283od_v_v )
=> ( ! [X3: product_prod_v_v,Y: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ( ( member7453568604450474000od_v_v @ Y @ A3 )
=> ( X3 = Y ) ) )
=> ( is_sin9198872032823709915od_v_v @ A3 ) ) ) ).
% is_singletonI'
thf(fact_648_graph_Ora__add__edge,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y2: v,E4: set_Product_prod_v_v,V: v,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E4 )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ V @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ W @ Y2 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ).
% graph.ra_add_edge
thf(fact_649_is__singletonE,axiom,
! [A3: set_v] :
( ( is_singleton_v @ A3 )
=> ~ ! [X3: v] :
( A3
!= ( insert_v2 @ X3 @ bot_bot_set_v ) ) ) ).
% is_singletonE
thf(fact_650_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_651_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_652_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_653_graph_Oavoiding__explored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,X: v,Y2: v,E4: set_Product_prod_v_v,W: v,V: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ E4 )
=> ( ~ ( member_v @ Y2 @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y2 @ ( sup_su414716646722978715od_v_v @ E4 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ) ).
% graph.avoiding_explored
thf(fact_654_subrelI,axiom,
! [R: set_Pr6425124735969554649t_unit,S5: set_Pr6425124735969554649t_unit] :
( ! [X3: v,Y: sCC_Bl1394983891496994913t_unit] :
( ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X3 @ Y ) @ R )
=> ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X3 @ Y ) @ S5 ) )
=> ( ord_le7290744839000465721t_unit @ R @ S5 ) ) ).
% subrelI
thf(fact_655_subrelI,axiom,
! [R: set_Product_prod_v_v,S5: set_Product_prod_v_v] :
( ! [X3: v,Y: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y ) @ R )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y ) @ S5 ) )
=> ( ord_le7336532860387713383od_v_v @ R @ S5 ) ) ).
% subrelI
thf(fact_656_vfin,axiom,
finite_finite_v @ vertices ).
% vfin
thf(fact_657_insert__subsetI,axiom,
! [X: v,A3: set_v,X5: set_v] :
( ( member_v @ X @ A3 )
=> ( ( ord_less_eq_set_v @ X5 @ A3 )
=> ( ord_less_eq_set_v @ ( insert_v2 @ X @ X5 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_658_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_659_subset__emptyI,axiom,
! [A3: set_v] :
( ! [X3: v] :
~ ( member_v @ X3 @ A3 )
=> ( ord_less_eq_set_v @ A3 @ bot_bot_set_v ) ) ).
% subset_emptyI
thf(fact_660_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_661__C2_C,axiom,
accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ v2 @ e ) ) ).
% "2"
thf(fact_662_Field__insert,axiom,
! [A: v,B: v,R: set_Product_prod_v_v] :
( ( field_v @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ A @ B ) @ R ) )
= ( sup_sup_set_v @ ( insert_v2 @ A @ ( insert_v2 @ B @ bot_bot_set_v ) ) @ ( field_v @ R ) ) ) ).
% Field_insert
thf(fact_663_Field__insert,axiom,
! [A: product_prod_v_v,B: product_prod_v_v,R: set_Pr2149350503807050951od_v_v] :
( ( field_7153129647634986036od_v_v @ ( insert5641704497130386615od_v_v @ ( produc4031800376763917143od_v_v @ A @ B ) @ R ) )
= ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) ) @ ( field_7153129647634986036od_v_v @ R ) ) ) ).
% Field_insert
thf(fact_664_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_665_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_666_dfss_Ocases,axiom,
! [X: produc5741669702376414499t_unit] :
~ ! [V2: v,E7: sCC_Bl1394983891496994913t_unit] :
( X
!= ( produc3862955338007567901t_unit @ V2 @ E7 ) ) ).
% dfss.cases
thf(fact_667_Field__empty,axiom,
( ( field_7153129647634986036od_v_v @ bot_bo3282589961317712691od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Field_empty
thf(fact_668_Field__empty,axiom,
( ( field_v @ bot_bo723834152578015283od_v_v )
= bot_bot_set_v ) ).
% Field_empty
thf(fact_669_refl__on__Int,axiom,
! [A3: set_v,R: set_Product_prod_v_v,B2: set_v,S5: set_Product_prod_v_v] :
( ( refl_on_v @ A3 @ R )
=> ( ( refl_on_v @ B2 @ S5 )
=> ( refl_on_v @ ( inf_inf_set_v @ A3 @ B2 ) @ ( inf_in6271465464967711157od_v_v @ R @ S5 ) ) ) ) ).
% refl_on_Int
thf(fact_670_graph_Ovfin,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( finite_finite_v @ Vertices ) ) ).
% graph.vfin
thf(fact_671_mono__Field,axiom,
! [R: set_Pr2149350503807050951od_v_v,S5: set_Pr2149350503807050951od_v_v] :
( ( ord_le6241436655786843239od_v_v @ R @ S5 )
=> ( ord_le7336532860387713383od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ ( field_7153129647634986036od_v_v @ S5 ) ) ) ).
% mono_Field
thf(fact_672_mono__Field,axiom,
! [R: set_Product_prod_v_v,S5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ R @ S5 )
=> ( ord_less_eq_set_v @ ( field_v @ R ) @ ( field_v @ S5 ) ) ) ).
% mono_Field
thf(fact_673_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_674_SCC__Bloemen__Sequential_Ograph__def,axiom,
( sCC_Bloemen_graph_v
= ( ^ [Vertices2: set_v,Successors2: v > set_v] :
( ( finite_finite_v @ Vertices2 )
& ! [X2: v] :
( ( member_v @ X2 @ Vertices2 )
=> ( ord_less_eq_set_v @ ( Successors2 @ X2 ) @ Vertices2 ) ) ) ) ) ).
% SCC_Bloemen_Sequential.graph_def
thf(fact_675_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_676_graph_Ointro,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( finite_finite_v @ Vertices )
=> ( ! [X3: v] :
( ( member_v @ X3 @ Vertices )
=> ( ord_less_eq_set_v @ ( Successors @ X3 ) @ Vertices ) )
=> ( sCC_Bloemen_graph_v @ Vertices @ Successors ) ) ) ).
% graph.intro
thf(fact_677_refl__on__empty,axiom,
refl_on_v @ bot_bot_set_v @ bot_bo723834152578015283od_v_v ).
% refl_on_empty
thf(fact_678_refl__on__empty,axiom,
refl_o4548774019903118566od_v_v @ bot_bo723834152578015283od_v_v @ bot_bo3282589961317712691od_v_v ).
% refl_on_empty
thf(fact_679_finite__Int,axiom,
! [F2: set_v,G: set_v] :
( ( ( finite_finite_v @ F2 )
| ( finite_finite_v @ G ) )
=> ( finite_finite_v @ ( inf_inf_set_v @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_680_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_681_finite__insert,axiom,
! [A: v,A3: set_v] :
( ( finite_finite_v @ ( insert_v2 @ A @ A3 ) )
= ( finite_finite_v @ A3 ) ) ).
% finite_insert
thf(fact_682_finite__subset__induct_H,axiom,
! [F2: set_v,A3: set_v,P: set_v > $o] :
( ( finite_finite_v @ F2 )
=> ( ( ord_less_eq_set_v @ F2 @ A3 )
=> ( ( P @ bot_bot_set_v )
=> ( ! [A7: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ( member_v @ A7 @ A3 )
=> ( ( ord_less_eq_set_v @ F3 @ A3 )
=> ( ~ ( member_v @ A7 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v2 @ A7 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_683_finite__subset__induct_H,axiom,
! [F2: set_Product_prod_v_v,A3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F2 )
=> ( ( ord_le7336532860387713383od_v_v @ F2 @ A3 )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [A7: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( member7453568604450474000od_v_v @ A7 @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ F3 @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ A7 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ A7 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_684_finite__subset__induct,axiom,
! [F2: set_v,A3: set_v,P: set_v > $o] :
( ( finite_finite_v @ F2 )
=> ( ( ord_less_eq_set_v @ F2 @ A3 )
=> ( ( P @ bot_bot_set_v )
=> ( ! [A7: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ( member_v @ A7 @ A3 )
=> ( ~ ( member_v @ A7 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v2 @ A7 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_685_finite__subset__induct,axiom,
! [F2: set_Product_prod_v_v,A3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F2 )
=> ( ( ord_le7336532860387713383od_v_v @ F2 @ A3 )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [A7: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( member7453568604450474000od_v_v @ A7 @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ A7 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ A7 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_686_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_687_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_688_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_689_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_690_finite_OemptyI,axiom,
finite_finite_v @ bot_bot_set_v ).
% finite.emptyI
thf(fact_691_finite_OemptyI,axiom,
finite3348123685078250256od_v_v @ bot_bo723834152578015283od_v_v ).
% finite.emptyI
thf(fact_692_infinite__imp__nonempty,axiom,
! [S: set_v] :
( ~ ( finite_finite_v @ S )
=> ( S != bot_bot_set_v ) ) ).
% infinite_imp_nonempty
thf(fact_693_infinite__imp__nonempty,axiom,
! [S: set_Product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ S )
=> ( S != bot_bo723834152578015283od_v_v ) ) ).
% infinite_imp_nonempty
thf(fact_694_finite__subset,axiom,
! [A3: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A3 @ B2 )
=> ( ( finite_finite_v @ B2 )
=> ( finite_finite_v @ A3 ) ) ) ).
% finite_subset
thf(fact_695_finite__subset,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
=> ( ( finite3348123685078250256od_v_v @ B2 )
=> ( finite3348123685078250256od_v_v @ A3 ) ) ) ).
% finite_subset
thf(fact_696_infinite__super,axiom,
! [S: set_v,T2: set_v] :
( ( ord_less_eq_set_v @ S @ T2 )
=> ( ~ ( finite_finite_v @ S )
=> ~ ( finite_finite_v @ T2 ) ) ) ).
% infinite_super
thf(fact_697_infinite__super,axiom,
! [S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ S @ T2 )
=> ( ~ ( finite3348123685078250256od_v_v @ S )
=> ~ ( finite3348123685078250256od_v_v @ T2 ) ) ) ).
% infinite_super
thf(fact_698_rev__finite__subset,axiom,
! [B2: set_v,A3: set_v] :
( ( finite_finite_v @ B2 )
=> ( ( ord_less_eq_set_v @ A3 @ B2 )
=> ( finite_finite_v @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_699_rev__finite__subset,axiom,
! [B2: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
=> ( finite3348123685078250256od_v_v @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_700_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_701_finite_OinsertI,axiom,
! [A3: set_v,A: v] :
( ( finite_finite_v @ A3 )
=> ( finite_finite_v @ ( insert_v2 @ A @ A3 ) ) ) ).
% finite.insertI
thf(fact_702_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_703_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_704_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_705_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_706_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_707_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_708_finite_Osimps,axiom,
( finite_finite_v
= ( ^ [A5: set_v] :
( ( A5 = bot_bot_set_v )
| ? [A4: set_v,B5: v] :
( ( A5
= ( insert_v2 @ B5 @ A4 ) )
& ( finite_finite_v @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_709_finite_Osimps,axiom,
( finite3348123685078250256od_v_v
= ( ^ [A5: set_Product_prod_v_v] :
( ( A5 = bot_bo723834152578015283od_v_v )
| ? [A4: set_Product_prod_v_v,B5: product_prod_v_v] :
( ( A5
= ( insert1338601472111419319od_v_v @ B5 @ A4 ) )
& ( finite3348123685078250256od_v_v @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_710_finite__induct,axiom,
! [F2: set_v,P: set_v > $o] :
( ( finite_finite_v @ F2 )
=> ( ( P @ bot_bot_set_v )
=> ( ! [X3: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ~ ( member_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v2 @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_711_finite__induct,axiom,
! [F2: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F2 )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [X3: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ~ ( member7453568604450474000od_v_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_712_finite__ne__induct,axiom,
! [F2: set_v,P: set_v > $o] :
( ( finite_finite_v @ F2 )
=> ( ( F2 != bot_bot_set_v )
=> ( ! [X3: v] : ( P @ ( insert_v2 @ X3 @ bot_bot_set_v ) )
=> ( ! [X3: v,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ( F3 != bot_bot_set_v )
=> ( ~ ( member_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v2 @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_713_finite__ne__induct,axiom,
! [F2: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F2 )
=> ( ( F2 != bot_bo723834152578015283od_v_v )
=> ( ! [X3: product_prod_v_v] : ( P @ ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) )
=> ( ! [X3: product_prod_v_v,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( F3 != bot_bo723834152578015283od_v_v )
=> ( ~ ( member7453568604450474000od_v_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_714_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,F3: set_v] :
( ( finite_finite_v @ F3 )
=> ( ~ ( member_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_v2 @ X3 @ F3 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_715_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,F3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ~ ( member7453568604450474000od_v_v @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert1338601472111419319od_v_v @ X3 @ F3 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_716_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_717_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_718_Set_Ois__empty__def,axiom,
( is_empty_v
= ( ^ [A4: set_v] : ( A4 = bot_bot_set_v ) ) ) ).
% Set.is_empty_def
thf(fact_719_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_720_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_721_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_722_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_723_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_724_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_725_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_726_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_727_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_728_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 ) )
=> ! [A9: set_v] :
( ( member_set_v @ A9 @ A3 )
=> ( ord_less_eq_set_v @ X @ A9 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_729_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 ) )
=> ! [A9: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A9 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ X @ A9 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_730_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_731_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_732_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 )
=> ! [A9: set_v] :
( ( member_set_v @ A9 @ A3 )
=> ( ord_less_eq_set_v @ A9 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_733_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 )
=> ! [A9: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A9 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ A9 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_734_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_735_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_736_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_737_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_738_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_739_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_740_Sup__fin_Osubset__imp,axiom,
! [A3: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ B2 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( finite_finite_set_v @ B2 )
=> ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A3 ) @ ( lattic2918178447194608042_set_v @ B2 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_741_Sup__fin_Osubset__imp,axiom,
! [A3: set_se8455005133513928103od_v_v,B2: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A3 @ B2 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( finite6084192165098772208od_v_v @ B2 )
=> ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ ( lattic5151207300795964030od_v_v @ B2 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_742_Inf__fin_Osubset__imp,axiom,
! [A3: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ B2 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( finite_finite_set_v @ B2 )
=> ( ord_less_eq_set_v @ ( lattic8209813555532694032_set_v @ B2 ) @ ( lattic8209813555532694032_set_v @ A3 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_743_Inf__fin_Osubset__imp,axiom,
! [A3: set_se8455005133513928103od_v_v,B2: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A3 @ B2 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( finite6084192165098772208od_v_v @ B2 )
=> ( ord_le7336532860387713383od_v_v @ ( lattic4767070952889939172od_v_v @ B2 ) @ ( lattic4767070952889939172od_v_v @ A3 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_744_Sup__fin_Osubset,axiom,
! [A3: set_se8455005133513928103od_v_v,B2: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( B2 != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le4714265922333009223od_v_v @ B2 @ A3 )
=> ( ( sup_su414716646722978715od_v_v @ ( lattic5151207300795964030od_v_v @ B2 ) @ ( lattic5151207300795964030od_v_v @ A3 ) )
= ( lattic5151207300795964030od_v_v @ A3 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_745_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_746_Sup__fin_Oclosed,axiom,
! [A3: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( member8406446414694345712od_v_v @ ( sup_su414716646722978715od_v_v @ X3 @ Y ) @ ( insert7504383016908236695od_v_v @ X3 @ ( insert7504383016908236695od_v_v @ Y @ bot_bo3497076220358800403od_v_v ) ) )
=> ( member8406446414694345712od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ A3 ) ) ) ) ).
% Sup_fin.closed
thf(fact_747_Inf__fin_Osubset,axiom,
! [A3: set_set_v,B2: set_set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( B2 != bot_bot_set_set_v )
=> ( ( ord_le5216385588623774835_set_v @ B2 @ A3 )
=> ( ( inf_inf_set_v @ ( lattic8209813555532694032_set_v @ B2 ) @ ( lattic8209813555532694032_set_v @ A3 ) )
= ( lattic8209813555532694032_set_v @ A3 ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_748_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_749_Inf__fin_Oclosed,axiom,
! [A3: set_set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ! [X3: set_v,Y: set_v] : ( member_set_v @ ( inf_inf_set_v @ X3 @ Y ) @ ( insert_set_v @ X3 @ ( insert_set_v @ Y @ bot_bot_set_set_v ) ) )
=> ( member_set_v @ ( lattic8209813555532694032_set_v @ A3 ) @ A3 ) ) ) ) ).
% Inf_fin.closed
thf(fact_750_Sup__fin_Ounion,axiom,
! [A3: set_se8455005133513928103od_v_v,B2: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( finite6084192165098772208od_v_v @ B2 )
=> ( ( B2 != bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( sup_su335656005089752955od_v_v @ A3 @ B2 ) )
= ( sup_su414716646722978715od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ ( lattic5151207300795964030od_v_v @ B2 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_751_Inf__fin_Ounion,axiom,
! [A3: set_set_v,B2: set_set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( finite_finite_set_v @ B2 )
=> ( ( B2 != bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ ( sup_sup_set_set_v @ A3 @ B2 ) )
= ( inf_inf_set_v @ ( lattic8209813555532694032_set_v @ A3 ) @ ( lattic8209813555532694032_set_v @ B2 ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_752_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_753_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_754_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_755_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_756_Diff__iff,axiom,
! [C2: v,A3: set_v,B2: set_v] :
( ( member_v @ C2 @ ( minus_minus_set_v @ A3 @ B2 ) )
= ( ( member_v @ C2 @ A3 )
& ~ ( member_v @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_757_Diff__iff,axiom,
! [C2: product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( minus_4183494784930505774od_v_v @ A3 @ B2 ) )
= ( ( member7453568604450474000od_v_v @ C2 @ A3 )
& ~ ( member7453568604450474000od_v_v @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_758_DiffI,axiom,
! [C2: v,A3: set_v,B2: set_v] :
( ( member_v @ C2 @ A3 )
=> ( ~ ( member_v @ C2 @ B2 )
=> ( member_v @ C2 @ ( minus_minus_set_v @ A3 @ B2 ) ) ) ) ).
% DiffI
thf(fact_759_DiffI,axiom,
! [C2: product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ C2 @ B2 )
=> ( member7453568604450474000od_v_v @ C2 @ ( minus_4183494784930505774od_v_v @ A3 @ B2 ) ) ) ) ).
% DiffI
thf(fact_760_Diff__empty,axiom,
! [A3: set_v] :
( ( minus_minus_set_v @ A3 @ bot_bot_set_v )
= A3 ) ).
% Diff_empty
thf(fact_761_Diff__empty,axiom,
! [A3: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
= A3 ) ).
% Diff_empty
thf(fact_762_empty__Diff,axiom,
! [A3: set_v] :
( ( minus_minus_set_v @ bot_bot_set_v @ A3 )
= bot_bot_set_v ) ).
% empty_Diff
thf(fact_763_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_764_Diff__cancel,axiom,
! [A3: set_v] :
( ( minus_minus_set_v @ A3 @ A3 )
= bot_bot_set_v ) ).
% Diff_cancel
thf(fact_765_Diff__cancel,axiom,
! [A3: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ A3 )
= bot_bo723834152578015283od_v_v ) ).
% Diff_cancel
thf(fact_766_Diff__insert0,axiom,
! [X: v,A3: set_v,B2: set_v] :
( ~ ( member_v @ X @ A3 )
=> ( ( minus_minus_set_v @ A3 @ ( insert_v2 @ X @ B2 ) )
= ( minus_minus_set_v @ A3 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_767_Diff__insert0,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ B2 ) )
= ( minus_4183494784930505774od_v_v @ A3 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_768_insert__Diff1,axiom,
! [X: v,B2: set_v,A3: set_v] :
( ( member_v @ X @ B2 )
=> ( ( minus_minus_set_v @ ( insert_v2 @ X @ A3 ) @ B2 )
= ( minus_minus_set_v @ A3 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_769_insert__Diff1,axiom,
! [X: product_prod_v_v,B2: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ B2 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A3 ) @ B2 )
= ( minus_4183494784930505774od_v_v @ A3 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_770_Un__Diff__cancel2,axiom,
! [B2: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ B2 @ A3 ) @ A3 )
= ( sup_su414716646722978715od_v_v @ B2 @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_771_Un__Diff__cancel,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B2 @ A3 ) )
= ( sup_su414716646722978715od_v_v @ A3 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_772_Diff__eq__empty__iff,axiom,
! [A3: set_v,B2: set_v] :
( ( ( minus_minus_set_v @ A3 @ B2 )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ A3 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_773_Diff__eq__empty__iff,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ A3 @ B2 )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ A3 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_774_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_775_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_776_finite__Diff__insert,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B2 ) ) )
= ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_777_finite__Diff__insert,axiom,
! [A3: set_v,A: v,B2: set_v] :
( ( finite_finite_v @ ( minus_minus_set_v @ A3 @ ( insert_v2 @ A @ B2 ) ) )
= ( finite_finite_v @ ( minus_minus_set_v @ A3 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_778_Diff__disjoint,axiom,
! [A3: set_v,B2: set_v] :
( ( inf_inf_set_v @ A3 @ ( minus_minus_set_v @ B2 @ A3 ) )
= bot_bot_set_v ) ).
% Diff_disjoint
thf(fact_779_Diff__disjoint,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B2 @ A3 ) )
= bot_bo723834152578015283od_v_v ) ).
% Diff_disjoint
thf(fact_780_double__diff,axiom,
! [A3: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A3 @ B2 )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ( minus_minus_set_v @ B2 @ ( minus_minus_set_v @ C @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_781_double__diff,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ( minus_4183494784930505774od_v_v @ B2 @ ( minus_4183494784930505774od_v_v @ C @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_782_Diff__subset,axiom,
! [A3: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ B2 ) @ A3 ) ).
% Diff_subset
thf(fact_783_Diff__subset,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B2 ) @ A3 ) ).
% Diff_subset
thf(fact_784_Diff__mono,axiom,
! [A3: set_v,C: set_v,D2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A3 @ C )
=> ( ( ord_less_eq_set_v @ D2 @ B2 )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ B2 ) @ ( minus_minus_set_v @ C @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_785_Diff__mono,axiom,
! [A3: set_Product_prod_v_v,C: set_Product_prod_v_v,D2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ C )
=> ( ( ord_le7336532860387713383od_v_v @ D2 @ B2 )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B2 ) @ ( minus_4183494784930505774od_v_v @ C @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_786_insert__Diff__if,axiom,
! [X: v,B2: set_v,A3: set_v] :
( ( ( member_v @ X @ B2 )
=> ( ( minus_minus_set_v @ ( insert_v2 @ X @ A3 ) @ B2 )
= ( minus_minus_set_v @ A3 @ B2 ) ) )
& ( ~ ( member_v @ X @ B2 )
=> ( ( minus_minus_set_v @ ( insert_v2 @ X @ A3 ) @ B2 )
= ( insert_v2 @ X @ ( minus_minus_set_v @ A3 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_787_insert__Diff__if,axiom,
! [X: product_prod_v_v,B2: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ X @ B2 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A3 ) @ B2 )
= ( minus_4183494784930505774od_v_v @ A3 @ B2 ) ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ B2 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A3 ) @ B2 )
= ( insert1338601472111419319od_v_v @ X @ ( minus_4183494784930505774od_v_v @ A3 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_788_Un__Diff,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) @ C )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ C ) @ ( minus_4183494784930505774od_v_v @ B2 @ C ) ) ) ).
% Un_Diff
thf(fact_789_DiffD2,axiom,
! [C2: v,A3: set_v,B2: set_v] :
( ( member_v @ C2 @ ( minus_minus_set_v @ A3 @ B2 ) )
=> ~ ( member_v @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_790_DiffD2,axiom,
! [C2: product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( minus_4183494784930505774od_v_v @ A3 @ B2 ) )
=> ~ ( member7453568604450474000od_v_v @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_791_DiffD1,axiom,
! [C2: v,A3: set_v,B2: set_v] :
( ( member_v @ C2 @ ( minus_minus_set_v @ A3 @ B2 ) )
=> ( member_v @ C2 @ A3 ) ) ).
% DiffD1
thf(fact_792_DiffD1,axiom,
! [C2: product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( minus_4183494784930505774od_v_v @ A3 @ B2 ) )
=> ( member7453568604450474000od_v_v @ C2 @ A3 ) ) ).
% DiffD1
thf(fact_793_DiffE,axiom,
! [C2: v,A3: set_v,B2: set_v] :
( ( member_v @ C2 @ ( minus_minus_set_v @ A3 @ B2 ) )
=> ~ ( ( member_v @ C2 @ A3 )
=> ( member_v @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_794_DiffE,axiom,
! [C2: product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( minus_4183494784930505774od_v_v @ A3 @ B2 ) )
=> ~ ( ( member7453568604450474000od_v_v @ C2 @ A3 )
=> ( member7453568604450474000od_v_v @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_795_Int__Diff,axiom,
! [A3: set_v,B2: set_v,C: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ B2 ) @ C )
= ( inf_inf_set_v @ A3 @ ( minus_minus_set_v @ B2 @ C ) ) ) ).
% Int_Diff
thf(fact_796_Diff__Int2,axiom,
! [A3: set_v,C: set_v,B2: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ C ) @ ( inf_inf_set_v @ B2 @ C ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ C ) @ B2 ) ) ).
% Diff_Int2
thf(fact_797_Diff__Diff__Int,axiom,
! [A3: set_v,B2: set_v] :
( ( minus_minus_set_v @ A3 @ ( minus_minus_set_v @ A3 @ B2 ) )
= ( inf_inf_set_v @ A3 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_798_Diff__Int__distrib,axiom,
! [C: set_v,A3: set_v,B2: set_v] :
( ( inf_inf_set_v @ C @ ( minus_minus_set_v @ A3 @ B2 ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ C @ A3 ) @ ( inf_inf_set_v @ C @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_799_Diff__Int__distrib2,axiom,
! [A3: set_v,B2: set_v,C: set_v] :
( ( inf_inf_set_v @ ( minus_minus_set_v @ A3 @ B2 ) @ C )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ C ) @ ( inf_inf_set_v @ B2 @ C ) ) ) ).
% Diff_Int_distrib2
thf(fact_800_diff__shunt__var,axiom,
! [X: set_v,Y2: set_v] :
( ( ( minus_minus_set_v @ X @ Y2 )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ X @ Y2 ) ) ).
% diff_shunt_var
thf(fact_801_diff__shunt__var,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ X @ Y2 )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ X @ Y2 ) ) ).
% diff_shunt_var
thf(fact_802_Diff__insert,axiom,
! [A3: set_v,A: v,B2: set_v] :
( ( minus_minus_set_v @ A3 @ ( insert_v2 @ A @ B2 ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A3 @ B2 ) @ ( insert_v2 @ A @ bot_bot_set_v ) ) ) ).
% Diff_insert
thf(fact_803_Diff__insert,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B2 ) @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ).
% Diff_insert
thf(fact_804_insert__Diff,axiom,
! [A: v,A3: set_v] :
( ( member_v @ A @ A3 )
=> ( ( insert_v2 @ A @ ( minus_minus_set_v @ A3 @ ( insert_v2 @ A @ bot_bot_set_v ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_805_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_806_Diff__insert2,axiom,
! [A3: set_v,A: v,B2: set_v] :
( ( minus_minus_set_v @ A3 @ ( insert_v2 @ A @ B2 ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A3 @ ( insert_v2 @ A @ bot_bot_set_v ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_807_Diff__insert2,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_808_Diff__insert__absorb,axiom,
! [X: v,A3: set_v] :
( ~ ( member_v @ X @ A3 )
=> ( ( minus_minus_set_v @ ( insert_v2 @ X @ A3 ) @ ( insert_v2 @ X @ bot_bot_set_v ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_809_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_810_subset__Diff__insert,axiom,
! [A3: set_v,B2: set_v,X: v,C: set_v] :
( ( ord_less_eq_set_v @ A3 @ ( minus_minus_set_v @ B2 @ ( insert_v2 @ X @ C ) ) )
= ( ( ord_less_eq_set_v @ A3 @ ( minus_minus_set_v @ B2 @ C ) )
& ~ ( member_v @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_811_subset__Diff__insert,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,X: product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ X @ C ) ) )
= ( ( ord_le7336532860387713383od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B2 @ C ) )
& ~ ( member7453568604450474000od_v_v @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_812_Diff__triv,axiom,
! [A3: set_v,B2: set_v] :
( ( ( inf_inf_set_v @ A3 @ B2 )
= bot_bot_set_v )
=> ( ( minus_minus_set_v @ A3 @ B2 )
= A3 ) ) ).
% Diff_triv
thf(fact_813_Diff__triv,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A3 @ B2 )
= bot_bo723834152578015283od_v_v )
=> ( ( minus_4183494784930505774od_v_v @ A3 @ B2 )
= A3 ) ) ).
% Diff_triv
thf(fact_814_Int__Diff__disjoint,axiom,
! [A3: set_v,B2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A3 @ B2 ) @ ( minus_minus_set_v @ A3 @ B2 ) )
= bot_bot_set_v ) ).
% Int_Diff_disjoint
thf(fact_815_Int__Diff__disjoint,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) @ ( minus_4183494784930505774od_v_v @ A3 @ B2 ) )
= bot_bo723834152578015283od_v_v ) ).
% Int_Diff_disjoint
thf(fact_816_Diff__subset__conv,axiom,
! [A3: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ B2 ) @ C )
= ( ord_less_eq_set_v @ A3 @ ( sup_sup_set_v @ B2 @ C ) ) ) ).
% Diff_subset_conv
thf(fact_817_Diff__subset__conv,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B2 ) @ C )
= ( ord_le7336532860387713383od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B2 @ C ) ) ) ).
% Diff_subset_conv
thf(fact_818_Diff__partition,axiom,
! [A3: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A3 @ B2 )
=> ( ( sup_sup_set_v @ A3 @ ( minus_minus_set_v @ B2 @ A3 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_819_Diff__partition,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
=> ( ( sup_su414716646722978715od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B2 @ A3 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_820_Un__Diff__Int,axiom,
! [A3: set_v,B2: set_v] :
( ( sup_sup_set_v @ ( minus_minus_set_v @ A3 @ B2 ) @ ( inf_inf_set_v @ A3 @ B2 ) )
= A3 ) ).
% Un_Diff_Int
thf(fact_821_Un__Diff__Int,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B2 ) @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) )
= A3 ) ).
% Un_Diff_Int
thf(fact_822_Int__Diff__Un,axiom,
! [A3: set_v,B2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ A3 @ B2 ) @ ( minus_minus_set_v @ A3 @ B2 ) )
= A3 ) ).
% Int_Diff_Un
thf(fact_823_Int__Diff__Un,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B2 ) @ ( minus_4183494784930505774od_v_v @ A3 @ B2 ) )
= A3 ) ).
% Int_Diff_Un
thf(fact_824_Diff__Int,axiom,
! [A3: set_v,B2: set_v,C: set_v] :
( ( minus_minus_set_v @ A3 @ ( inf_inf_set_v @ B2 @ C ) )
= ( sup_sup_set_v @ ( minus_minus_set_v @ A3 @ B2 ) @ ( minus_minus_set_v @ A3 @ C ) ) ) ).
% Diff_Int
thf(fact_825_Diff__Int,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ ( inf_in6271465464967711157od_v_v @ B2 @ C ) )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B2 ) @ ( minus_4183494784930505774od_v_v @ A3 @ C ) ) ) ).
% Diff_Int
thf(fact_826_Diff__Un,axiom,
! [A3: set_v,B2: set_v,C: set_v] :
( ( minus_minus_set_v @ A3 @ ( sup_sup_set_v @ B2 @ C ) )
= ( inf_inf_set_v @ ( minus_minus_set_v @ A3 @ B2 ) @ ( minus_minus_set_v @ A3 @ C ) ) ) ).
% Diff_Un
thf(fact_827_Diff__Un,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B2 @ C ) )
= ( inf_in6271465464967711157od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B2 ) @ ( minus_4183494784930505774od_v_v @ A3 @ C ) ) ) ).
% Diff_Un
thf(fact_828_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_v @ 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_829_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_830_infinite__coinduct,axiom,
! [X5: set_v > $o,A3: set_v] :
( ( X5 @ A3 )
=> ( ! [A8: set_v] :
( ( X5 @ A8 )
=> ? [X4: v] :
( ( member_v @ 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_831_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_832_infinite__remove,axiom,
! [S: set_v,A: v] :
( ~ ( finite_finite_v @ S )
=> ~ ( finite_finite_v @ ( minus_minus_set_v @ S @ ( insert_v2 @ A @ bot_bot_set_v ) ) ) ) ).
% infinite_remove
thf(fact_833_infinite__remove,axiom,
! [S: set_Product_prod_v_v,A: product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ S )
=> ~ ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ S @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% infinite_remove
thf(fact_834_subset__insert__iff,axiom,
! [A3: set_v,X: v,B2: set_v] :
( ( ord_less_eq_set_v @ A3 @ ( insert_v2 @ X @ B2 ) )
= ( ( ( member_v @ X @ A3 )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ ( insert_v2 @ X @ bot_bot_set_v ) ) @ B2 ) )
& ( ~ ( member_v @ X @ A3 )
=> ( ord_less_eq_set_v @ A3 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_835_subset__insert__iff,axiom,
! [A3: set_Product_prod_v_v,X: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ B2 ) )
= ( ( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B2 ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_836_Diff__single__insert,axiom,
! [A3: set_v,X: v,B2: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ ( insert_v2 @ X @ bot_bot_set_v ) ) @ B2 )
=> ( ord_less_eq_set_v @ A3 @ ( insert_v2 @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_837_Diff__single__insert,axiom,
! [A3: set_Product_prod_v_v,X: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B2 )
=> ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_838_remove__induct,axiom,
! [P: set_v > $o,B2: set_v] :
( ( P @ bot_bot_set_v )
=> ( ( ~ ( finite_finite_v @ B2 )
=> ( P @ B2 ) )
=> ( ! [A8: set_v] :
( ( finite_finite_v @ A8 )
=> ( ( A8 != bot_bot_set_v )
=> ( ( ord_less_eq_set_v @ A8 @ B2 )
=> ( ! [X4: v] :
( ( member_v @ X4 @ A8 )
=> ( P @ ( minus_minus_set_v @ A8 @ ( insert_v2 @ X4 @ bot_bot_set_v ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_839_remove__induct,axiom,
! [P: set_Product_prod_v_v > $o,B2: set_Product_prod_v_v] :
( ( P @ bot_bo723834152578015283od_v_v )
=> ( ( ~ ( finite3348123685078250256od_v_v @ B2 )
=> ( P @ B2 ) )
=> ( ! [A8: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A8 )
=> ( ( A8 != bot_bo723834152578015283od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ A8 @ B2 )
=> ( ! [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 @ B2 ) ) ) ) ).
% remove_induct
thf(fact_840_finite__remove__induct,axiom,
! [B2: set_v,P: set_v > $o] :
( ( finite_finite_v @ B2 )
=> ( ( P @ bot_bot_set_v )
=> ( ! [A8: set_v] :
( ( finite_finite_v @ A8 )
=> ( ( A8 != bot_bot_set_v )
=> ( ( ord_less_eq_set_v @ A8 @ B2 )
=> ( ! [X4: v] :
( ( member_v @ X4 @ A8 )
=> ( P @ ( minus_minus_set_v @ A8 @ ( insert_v2 @ X4 @ bot_bot_set_v ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_841_finite__remove__induct,axiom,
! [B2: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ B2 )
=> ( ( 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 @ B2 )
=> ( ! [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 @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_842_reachable__visited,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V: v,W: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ V @ W )
=> ( ! [X3: v] :
( ( member_v @ X3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ! [Xa2: v] :
( ( member_v @ Xa2 @ ( minus_minus_set_v @ ( successors @ X3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X3 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ V @ X3 )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Xa2 @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ).
% reachable_visited
thf(fact_843_graph_Oreachable__visited,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V: v,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ V @ W )
=> ( ! [X3: v] :
( ( member_v @ X3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ! [Xa2: v] :
( ( member_v @ Xa2 @ ( minus_minus_set_v @ ( Successors @ X3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X3 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ V @ X3 )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Xa2 @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ) ).
% graph.reachable_visited
thf(fact_844_unite__sub__env,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5768913643336123637t_unit @ E @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ).
% unite_sub_env
thf(fact_845_unite__wf__env,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl9196236973127232072t_unit @ successors @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ).
% unite_wf_env
thf(fact_846_graph_Ounite__sub__env,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( sCC_Bl7963838319573962697t_unit @ E @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_847_graph_Ounite__sub__env,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5768913643336123637t_unit @ E @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_848_graph_Ounite__wf__env,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( sCC_Bl7798947040364291444t_unit @ Successors @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_wf_env
thf(fact_849_graph_Ounite__wf__env,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl9196236973127232072t_unit @ Successors @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_wf_env
thf(fact_850_unite__subscc,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5398416737448265317bscc_v @ successors @ ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ) ) ).
% unite_subscc
thf(fact_851_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_852_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_853_stack__class,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v,M: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) )
=> ( member_v @ M @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ) ).
% stack_class
thf(fact_854_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_855_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_856_stack__unexplored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ N @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ).
% stack_unexplored
thf(fact_857_member__remove,axiom,
! [X: v,Y2: v,A3: set_v] :
( ( member_v @ X @ ( remove_v @ Y2 @ A3 ) )
= ( ( member_v @ X @ A3 )
& ( X != Y2 ) ) ) ).
% member_remove
thf(fact_858_member__remove,axiom,
! [X: product_prod_v_v,Y2: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( remove5001965847480235980od_v_v @ Y2 @ A3 ) )
= ( ( member7453568604450474000od_v_v @ X @ A3 )
& ( X != Y2 ) ) ) ).
% member_remove
thf(fact_859_stack__visited,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( member_v @ N @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ).
% stack_visited
thf(fact_860_visited__unexplored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,M: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ M @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ M @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ~ ! [N2: v] :
( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) ) ) ) ) ) ).
% visited_unexplored
thf(fact_861_graph_Ostack__unexplored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ N @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ).
% graph.stack_unexplored
thf(fact_862_graph_Ostack__visited,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( member_v @ N @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ).
% graph.stack_visited
thf(fact_863_lnear__order__on__empty,axiom,
order_8768733634509060168r_on_v @ bot_bot_set_v @ bot_bo723834152578015283od_v_v ).
% lnear_order_on_empty
thf(fact_864_lnear__order__on__empty,axiom,
order_6462556390437124636od_v_v @ bot_bo723834152578015283od_v_v @ bot_bo3282589961317712691od_v_v ).
% lnear_order_on_empty
thf(fact_865_graph_Ovisited__unexplored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,M: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ M @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ M @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ~ ! [N2: v] :
( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) ) ) ) ) ) ) ).
% graph.visited_unexplored
thf(fact_866_graph_Ostack__class,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v,M: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) )
=> ( member_v @ M @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ) ) ).
% graph.stack_class
thf(fact_867_graph_Ounite__subscc,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( sCC_Bl2301996248249672505od_v_v @ Successors @ ( sCC_Bl8440648026628373538t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) @ ( hd_Product_prod_v_v @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_subscc
thf(fact_868_graph_Ounite__subscc,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5398416737448265317bscc_v @ Successors @ ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_subscc
thf(fact_869_pre__dfs__def,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl36166008131615352t_unit @ successors @ V @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
& ~ ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( sCC_Bl649662514949026229able_v @ successors @ ( sCC_Bl1090238580953940555t_unit @ E ) @ V )
& ( ( sCC_Bl3795065053823578884t_unit @ E @ V )
= bot_bot_set_v )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V ) ) ) ) ).
% pre_dfs_def
thf(fact_870_unite__S__tl,axiom,
! [E: sCC_Bl1394983891496994913t_unit,W: v,V: v,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ N @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ N )
= ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ) ) ) ) ).
% unite_S_tl
thf(fact_871_graph_Opre__dfs__def,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl36166008131615352t_unit @ Successors @ V @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
& ~ ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( sCC_Bl649662514949026229able_v @ Successors @ ( sCC_Bl1090238580953940555t_unit @ E ) @ V )
& ( ( sCC_Bl3795065053823578884t_unit @ E @ V )
= bot_bot_set_v )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ V ) ) ) ) ) ).
% graph.pre_dfs_def
thf(fact_872_notempty,axiom,
( ( sCC_Bl8828226123343373779t_unit @ e2 )
= ( cons_v @ v2 @ ( sCC_Bl8828226123343373779t_unit @ e ) ) ) ).
% notempty
thf(fact_873_graph_Ounite__S__tl,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v,V: product_prod_v_v,N: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl7798947040364291444t_unit @ Successors @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( ( member7453568604450474000od_v_v @ N @ ( set_Product_prod_v_v2 @ ( tl_Product_prod_v_v @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) ) ) ) )
=> ( ( sCC_Bl8440648026628373538t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) @ N )
= ( sCC_Bl8440648026628373538t_unit @ E @ N ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_tl
thf(fact_874_graph_Ounite__S__tl,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,W: v,V: v,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ N @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ N )
= ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_tl
thf(fact_875_list_Osimps_I15_J,axiom,
! [X21: product_prod_v_v,X22: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X22 ) )
= ( insert1338601472111419319od_v_v @ X21 @ ( set_Product_prod_v_v2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_876_list_Osimps_I15_J,axiom,
! [X21: v,X22: list_v] :
( ( set_v2 @ ( cons_v @ X21 @ X22 ) )
= ( insert_v2 @ X21 @ ( set_v2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_877_cst_H,axiom,
( ( sCC_Bl9201514103433284750t_unit @ e2 )
= ( cons_v @ v2 @ ( sCC_Bl9201514103433284750t_unit @ e ) ) ) ).
% cst'
thf(fact_878_pre__dfss__def,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
& ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
& ( member_v @ V @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( member_v @ X2 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E ) @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V ) )
& ? [Ns: list_v] :
( ( sCC_Bl9201514103433284750t_unit @ E )
= ( cons_v @ V @ Ns ) ) ) ) ).
% pre_dfss_def
thf(fact_879_set__subset__Cons,axiom,
! [Xs: list_v,X: v] : ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ ( cons_v @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_880_set__subset__Cons,axiom,
! [Xs: list_P7986770385144383213od_v_v,X: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_881_set__empty2,axiom,
! [Xs: list_v] :
( ( bot_bot_set_v
= ( set_v2 @ Xs ) )
= ( Xs = nil_v ) ) ).
% set_empty2
thf(fact_882_set__empty2,axiom,
! [Xs: list_P7986770385144383213od_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( set_Product_prod_v_v2 @ Xs ) )
= ( Xs = nil_Product_prod_v_v ) ) ).
% set_empty2
thf(fact_883_set__empty,axiom,
! [Xs: list_v] :
( ( ( set_v2 @ Xs )
= bot_bot_set_v )
= ( Xs = nil_v ) ) ).
% set_empty
thf(fact_884_set__empty,axiom,
! [Xs: list_P7986770385144383213od_v_v] :
( ( ( set_Product_prod_v_v2 @ Xs )
= bot_bo723834152578015283od_v_v )
= ( Xs = nil_Product_prod_v_v ) ) ).
% set_empty
thf(fact_885_empty__set,axiom,
( bot_bot_set_v
= ( set_v2 @ nil_v ) ) ).
% empty_set
thf(fact_886_empty__set,axiom,
( bot_bo723834152578015283od_v_v
= ( set_Product_prod_v_v2 @ nil_Product_prod_v_v ) ) ).
% empty_set
thf(fact_887_subset__code_I1_J,axiom,
! [Xs: list_v,B2: set_v] :
( ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ B2 )
= ( ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
=> ( member_v @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_888_subset__code_I1_J,axiom,
! [Xs: list_P7986770385144383213od_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ B2 )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( member7453568604450474000od_v_v @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_889_graph_Opre__dfss__def,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
& ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
& ( member_v @ V @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( member_v @ X2 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E ) @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ V ) )
& ? [Ns: list_v] :
( ( sCC_Bl9201514103433284750t_unit @ E )
= ( cons_v @ V @ Ns ) ) ) ) ) ).
% graph.pre_dfss_def
thf(fact_890_dfs__S__hd__stack_I1_J,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ) ) ).
% dfs_S_hd_stack(1)
thf(fact_891_dfs__S__hd__stack_I2_J,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) ) ) ) ).
% dfs_S_hd_stack(2)
thf(fact_892_dfs__S__tl__stack_I2_J,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ! [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X4 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X4 ) ) ) ) ) ).
% dfs_S_tl_stack(2)
thf(fact_893_dfs__S__tl__stack_I1_J,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ).
% dfs_S_tl_stack(1)
thf(fact_894_graph_Odfs__S__tl__stack_I1_J,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ) ).
% graph.dfs_S_tl_stack(1)
thf(fact_895_graph_Odfs__S__tl__stack_I2_J,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ! [X4: v] :
( ( member_v @ X4 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X4 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X4 ) ) ) ) ) ) ).
% graph.dfs_S_tl_stack(2)
thf(fact_896_graph_Odfs__S__hd__stack_I2_J,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) ) ) ) ) ).
% graph.dfs_S_hd_stack(2)
thf(fact_897_graph_Odfs__S__hd__stack_I1_J,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ) ) ) ).
% graph.dfs_S_hd_stack(1)
thf(fact_898_post__dfss__def,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl6082031138996704384t_unit @ successors @ V @ E @ E2 )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
& ( ( sCC_Bl3795065053823578884t_unit @ E2 @ V )
= ( successors @ V ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v2 @ V @ bot_bot_set_v ) ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ E2 @ X2 )
= ( sCC_Bl3795065053823578884t_unit @ E @ X2 ) ) )
& ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
& ! [X2: v] :
( ( member_v @ X2 @ ( successors @ V ) )
=> ( member_v @ X2 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E2 ) @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v )
& ? [Ns: list_v] :
( ( sCC_Bl8828226123343373779t_unit @ E )
= ( append_v @ Ns @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
& ( member_v @ V @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X2 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X2 ) ) )
& ( ( ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) )
= V )
=> ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ V @ X2 ) ) )
& ( ( sCC_Bl9201514103433284750t_unit @ E2 )
= ( sCC_Bl9201514103433284750t_unit @ E ) ) ) ) ).
% post_dfss_def
thf(fact_899_post__dfs__def,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ E2 )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
& ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
& ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
& ( ( sCC_Bl3795065053823578884t_unit @ E2 @ V )
= ( successors @ V ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ E2 @ X2 )
= ( sCC_Bl3795065053823578884t_unit @ E @ X2 ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V ) )
& ? [Ns: list_v] :
( ( sCC_Bl8828226123343373779t_unit @ E )
= ( append_v @ Ns @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
& ( ( ( member_v @ V @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E2 )
= ( sCC_Bl8828226123343373779t_unit @ E ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X2 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X2 ) ) ) )
| ( ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v )
& ( member_v @ V @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X2 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X2 ) ) ) ) )
& ( ( sCC_Bl9201514103433284750t_unit @ E2 )
= ( sCC_Bl9201514103433284750t_unit @ E ) ) ) ) ).
% post_dfs_def
thf(fact_900_equality,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit] :
( ( ( sCC_Bl1090238580953940555t_unit @ R )
= ( sCC_Bl1090238580953940555t_unit @ R2 ) )
=> ( ( ( sCC_Bl1280885523602775798t_unit @ R )
= ( sCC_Bl1280885523602775798t_unit @ R2 ) )
=> ( ( ( sCC_Bl157864678168468314t_unit @ R )
= ( sCC_Bl157864678168468314t_unit @ R2 ) )
=> ( ( ( sCC_Bl4645233313691564917t_unit @ R )
= ( sCC_Bl4645233313691564917t_unit @ R2 ) )
=> ( ( ( sCC_Bl3795065053823578884t_unit @ R )
= ( sCC_Bl3795065053823578884t_unit @ R2 ) )
=> ( ( ( sCC_Bl2536197123907397897t_unit @ R )
= ( sCC_Bl2536197123907397897t_unit @ R2 ) )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ R )
= ( sCC_Bl8828226123343373779t_unit @ R2 ) )
=> ( ( ( sCC_Bl9201514103433284750t_unit @ R )
= ( sCC_Bl9201514103433284750t_unit @ R2 ) )
=> ( ( ( sCC_Bl3567736435408124606t_unit @ R )
= ( sCC_Bl3567736435408124606t_unit @ R2 ) )
=> ( R = R2 ) ) ) ) ) ) ) ) ) ) ).
% equality
thf(fact_901_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_902_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_903_split__list__precedes,axiom,
! [Y2: product_prod_v_v,Ys: list_P7986770385144383213od_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y2 @ ( set_Product_prod_v_v2 @ ( append2138873909117096322od_v_v @ Ys @ ( cons_P4120604216776828829od_v_v @ X @ nil_Product_prod_v_v ) ) ) )
=> ( sCC_Bl2026170059108282219od_v_v @ Y2 @ X @ ( append2138873909117096322od_v_v @ Ys @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) ) ) ).
% split_list_precedes
thf(fact_904_split__list__precedes,axiom,
! [Y2: v,Ys: list_v,X: v,Xs: list_v] :
( ( member_v @ Y2 @ ( set_v2 @ ( append_v @ Ys @ ( cons_v @ X @ nil_v ) ) ) )
=> ( sCC_Bl4022239298816431255edes_v @ Y2 @ X @ ( append_v @ Ys @ ( cons_v @ X @ Xs ) ) ) ) ).
% split_list_precedes
thf(fact_905_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_906_precedes__refl,axiom,
! [X: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ X @ Xs )
= ( member_v @ X @ ( set_v2 @ Xs ) ) ) ).
% precedes_refl
thf(fact_907_precedes__in__tail,axiom,
! [X: v,Z: v,Y2: v,Zs: list_v] :
( ( X != Z )
=> ( ( sCC_Bl4022239298816431255edes_v @ X @ Y2 @ ( cons_v @ Z @ Zs ) )
= ( sCC_Bl4022239298816431255edes_v @ X @ Y2 @ Zs ) ) ) ).
% precedes_in_tail
thf(fact_908_precedes__mem_I1_J,axiom,
! [X: product_prod_v_v,Y2: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X @ Y2 @ Xs )
=> ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_mem(1)
thf(fact_909_precedes__mem_I1_J,axiom,
! [X: v,Y2: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y2 @ Xs )
=> ( member_v @ X @ ( set_v2 @ Xs ) ) ) ).
% precedes_mem(1)
thf(fact_910_precedes__mem_I2_J,axiom,
! [X: product_prod_v_v,Y2: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X @ Y2 @ Xs )
=> ( member7453568604450474000od_v_v @ Y2 @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_mem(2)
thf(fact_911_precedes__mem_I2_J,axiom,
! [X: v,Y2: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y2 @ Xs )
=> ( member_v @ Y2 @ ( set_v2 @ Xs ) ) ) ).
% precedes_mem(2)
thf(fact_912_tail__not__precedes,axiom,
! [Y2: product_prod_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ Y2 @ X @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) )
=> ( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( X = Y2 ) ) ) ).
% tail_not_precedes
thf(fact_913_tail__not__precedes,axiom,
! [Y2: v,X: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ Y2 @ X @ ( cons_v @ X @ Xs ) )
=> ( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( X = Y2 ) ) ) ).
% tail_not_precedes
thf(fact_914_head__precedes,axiom,
! [Y2: product_prod_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y2 @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) )
=> ( sCC_Bl2026170059108282219od_v_v @ X @ Y2 @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) ) ).
% head_precedes
thf(fact_915_head__precedes,axiom,
! [Y2: v,X: v,Xs: list_v] :
( ( member_v @ Y2 @ ( set_v2 @ ( cons_v @ X @ Xs ) ) )
=> ( sCC_Bl4022239298816431255edes_v @ X @ Y2 @ ( cons_v @ X @ Xs ) ) ) ).
% head_precedes
thf(fact_916_precedes__append__left__iff,axiom,
! [X: product_prod_v_v,Ys: list_P7986770385144383213od_v_v,Y2: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Ys ) )
=> ( ( sCC_Bl2026170059108282219od_v_v @ X @ Y2 @ ( append2138873909117096322od_v_v @ Ys @ Xs ) )
= ( sCC_Bl2026170059108282219od_v_v @ X @ Y2 @ Xs ) ) ) ).
% precedes_append_left_iff
thf(fact_917_precedes__append__left__iff,axiom,
! [X: v,Ys: list_v,Y2: v,Xs: list_v] :
( ~ ( member_v @ X @ ( set_v2 @ Ys ) )
=> ( ( sCC_Bl4022239298816431255edes_v @ X @ Y2 @ ( append_v @ Ys @ Xs ) )
= ( sCC_Bl4022239298816431255edes_v @ X @ Y2 @ Xs ) ) ) ).
% precedes_append_left_iff
thf(fact_918_precedes__append__right__iff,axiom,
! [Y2: product_prod_v_v,Ys: list_P7986770385144383213od_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ Y2 @ ( set_Product_prod_v_v2 @ Ys ) )
=> ( ( sCC_Bl2026170059108282219od_v_v @ X @ Y2 @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
= ( sCC_Bl2026170059108282219od_v_v @ X @ Y2 @ Xs ) ) ) ).
% precedes_append_right_iff
thf(fact_919_precedes__append__right__iff,axiom,
! [Y2: v,Ys: list_v,X: v,Xs: list_v] :
( ~ ( member_v @ Y2 @ ( set_v2 @ Ys ) )
=> ( ( sCC_Bl4022239298816431255edes_v @ X @ Y2 @ ( append_v @ Xs @ Ys ) )
= ( sCC_Bl4022239298816431255edes_v @ X @ Y2 @ Xs ) ) ) ).
% precedes_append_right_iff
thf(fact_920_precedes__def,axiom,
( sCC_Bl2026170059108282219od_v_v
= ( ^ [X2: product_prod_v_v,Y3: product_prod_v_v,Xs2: list_P7986770385144383213od_v_v] :
? [L: list_P7986770385144383213od_v_v,R3: list_P7986770385144383213od_v_v] :
( ( Xs2
= ( append2138873909117096322od_v_v @ L @ ( cons_P4120604216776828829od_v_v @ X2 @ R3 ) ) )
& ( member7453568604450474000od_v_v @ Y3 @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X2 @ R3 ) ) ) ) ) ) ).
% precedes_def
thf(fact_921_precedes__def,axiom,
( sCC_Bl4022239298816431255edes_v
= ( ^ [X2: v,Y3: v,Xs2: list_v] :
? [L: list_v,R3: list_v] :
( ( Xs2
= ( append_v @ L @ ( cons_v @ X2 @ R3 ) ) )
& ( member_v @ Y3 @ ( set_v2 @ ( cons_v @ X2 @ R3 ) ) ) ) ) ) ).
% precedes_def
thf(fact_922_surjective,axiom,
! [R: sCC_Bl1394983891496994913t_unit] :
( R
= ( sCC_Bl8064756265740546429t_unit @ ( sCC_Bl1090238580953940555t_unit @ R ) @ ( sCC_Bl1280885523602775798t_unit @ R ) @ ( sCC_Bl157864678168468314t_unit @ R ) @ ( sCC_Bl4645233313691564917t_unit @ R ) @ ( sCC_Bl3795065053823578884t_unit @ R ) @ ( sCC_Bl2536197123907397897t_unit @ R ) @ ( sCC_Bl8828226123343373779t_unit @ R ) @ ( sCC_Bl9201514103433284750t_unit @ R ) @ ( sCC_Bl3567736435408124606t_unit @ R ) ) ) ).
% surjective
thf(fact_923_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_v @ ( coset_v @ nil_v ) @ ( set_v2 @ nil_v ) ) ).
% subset_code(3)
thf(fact_924_subset__code_I3_J,axiom,
~ ( ord_le7336532860387713383od_v_v @ ( coset_766761627116920666od_v_v @ nil_Product_prod_v_v ) @ ( set_Product_prod_v_v2 @ nil_Product_prod_v_v ) ) ).
% subset_code(3)
thf(fact_925_select__convs_I8_J,axiom,
! [Root: v,S6: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl9201514103433284750t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S6 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Cstack ) ).
% select_convs(8)
thf(fact_926_select__convs_I7_J,axiom,
! [Root: v,S6: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S6 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Stack ) ).
% select_convs(7)
thf(fact_927_select__convs_I2_J,axiom,
! [Root: v,S6: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S6 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= S6 ) ).
% select_convs(2)
thf(fact_928_select__convs_I5_J,axiom,
! [Root: v,S6: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S6 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Vsuccs ) ).
% select_convs(5)
thf(fact_929_select__convs_I3_J,axiom,
! [Root: v,S6: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl157864678168468314t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S6 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Explored ) ).
% select_convs(3)
thf(fact_930_select__convs_I4_J,axiom,
! [Root: v,S6: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl4645233313691564917t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S6 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Visited ) ).
% select_convs(4)
thf(fact_931_subset__code_I2_J,axiom,
! [A3: set_v,Ys: list_v] :
( ( ord_less_eq_set_v @ A3 @ ( coset_v @ Ys ) )
= ( ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Ys ) )
=> ~ ( member_v @ X2 @ A3 ) ) ) ) ).
% subset_code(2)
thf(fact_932_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_933_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_934_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_935_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_936_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_937_set__remove1__subset,axiom,
! [X: v,Xs: list_v] : ( ord_less_eq_set_v @ ( set_v2 @ ( remove1_v @ X @ Xs ) ) @ ( set_v2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_938_set__remove1__subset,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] : ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ ( remove333779696311199107od_v_v @ X @ Xs ) ) @ ( set_Product_prod_v_v2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_939_set__remove1__eq,axiom,
! [Xs: list_v,X: v] :
( ( distinct_v @ Xs )
=> ( ( set_v2 @ ( remove1_v @ X @ Xs ) )
= ( minus_minus_set_v @ ( set_v2 @ Xs ) @ ( insert_v2 @ X @ bot_bot_set_v ) ) ) ) ).
% set_remove1_eq
thf(fact_940_set__remove1__eq,axiom,
! [Xs: list_P7986770385144383213od_v_v,X: product_prod_v_v] :
( ( distin6159370996967099744od_v_v @ Xs )
=> ( ( set_Product_prod_v_v2 @ ( remove333779696311199107od_v_v @ X @ Xs ) )
= ( minus_4183494784930505774od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% set_remove1_eq
thf(fact_941_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_v @ X2 @ A4 )
& ! [Y3: v] :
( ( member_v @ Y3 @ A4 )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X2 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_942_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_943_min__bot2,axiom,
! [X: set_v] :
( ( ord_min_set_v @ X @ bot_bot_set_v )
= bot_bot_set_v ) ).
% min_bot2
thf(fact_944_min__bot2,axiom,
! [X: set_Product_prod_v_v] :
( ( ord_mi6996445931809003310od_v_v @ X @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% min_bot2
thf(fact_945_min__bot,axiom,
! [X: set_v] :
( ( ord_min_set_v @ bot_bot_set_v @ X )
= bot_bot_set_v ) ).
% min_bot
thf(fact_946_min__bot,axiom,
! [X: set_Product_prod_v_v] :
( ( ord_mi6996445931809003310od_v_v @ bot_bo723834152578015283od_v_v @ X )
= bot_bo723834152578015283od_v_v ) ).
% min_bot
thf(fact_947_distinct__append,axiom,
! [Xs: list_v,Ys: list_v] :
( ( distinct_v @ ( append_v @ Xs @ Ys ) )
= ( ( distinct_v @ Xs )
& ( distinct_v @ Ys )
& ( ( inf_inf_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ Ys ) )
= bot_bot_set_v ) ) ) ).
% distinct_append
thf(fact_948_distinct__append,axiom,
! [Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( distin6159370996967099744od_v_v @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
= ( ( distin6159370996967099744od_v_v @ Xs )
& ( distin6159370996967099744od_v_v @ Ys )
& ( ( inf_in6271465464967711157od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ Ys ) )
= bot_bo723834152578015283od_v_v ) ) ) ).
% distinct_append
thf(fact_949_min__def,axiom,
( ord_min_set_v
= ( ^ [A5: set_v,B5: set_v] : ( if_set_v @ ( ord_less_eq_set_v @ A5 @ B5 ) @ A5 @ B5 ) ) ) ).
% min_def
thf(fact_950_min__def,axiom,
( ord_mi6996445931809003310od_v_v
= ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] : ( if_set4279007504652509325od_v_v @ ( ord_le7336532860387713383od_v_v @ A5 @ B5 ) @ A5 @ B5 ) ) ) ).
% min_def
thf(fact_951_min__absorb1,axiom,
! [X: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X @ Y2 )
=> ( ( ord_min_set_v @ X @ Y2 )
= X ) ) ).
% min_absorb1
thf(fact_952_min__absorb1,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
=> ( ( ord_mi6996445931809003310od_v_v @ X @ Y2 )
= X ) ) ).
% min_absorb1
thf(fact_953_min__absorb2,axiom,
! [Y2: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y2 @ X )
=> ( ( ord_min_set_v @ X @ Y2 )
= Y2 ) ) ).
% min_absorb2
thf(fact_954_min__absorb2,axiom,
! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y2 @ X )
=> ( ( ord_mi6996445931809003310od_v_v @ X @ Y2 )
= Y2 ) ) ).
% min_absorb2
thf(fact_955_well__order__on__empty,axiom,
order_6972113574731384241r_on_v @ bot_bot_set_v @ bot_bo723834152578015283od_v_v ).
% well_order_on_empty
thf(fact_956_well__order__on__empty,axiom,
order_7541072052284126853od_v_v @ bot_bo723834152578015283od_v_v @ bot_bo3282589961317712691od_v_v ).
% well_order_on_empty
thf(fact_957_distinct__concat__iff,axiom,
! [Xs: list_list_v] :
( ( distinct_v @ ( concat_v @ Xs ) )
= ( ( distinct_list_v @ ( removeAll_list_v @ nil_v @ Xs ) )
& ! [Ys2: list_v] :
( ( member_list_v @ Ys2 @ ( set_list_v2 @ Xs ) )
=> ( distinct_v @ Ys2 ) )
& ! [Ys2: list_v,Zs2: list_v] :
( ( ( member_list_v @ Ys2 @ ( set_list_v2 @ Xs ) )
& ( member_list_v @ Zs2 @ ( set_list_v2 @ Xs ) )
& ( Ys2 != Zs2 ) )
=> ( ( inf_inf_set_v @ ( set_v2 @ Ys2 ) @ ( set_v2 @ Zs2 ) )
= bot_bot_set_v ) ) ) ) ).
% distinct_concat_iff
thf(fact_958_distinct__concat__iff,axiom,
! [Xs: list_l4795378083388841843od_v_v] :
( ( distin6159370996967099744od_v_v @ ( concat2875663619778446888od_v_v @ Xs ) )
= ( ( distin913317783593574886od_v_v @ ( remove5095778601549809401od_v_v @ nil_Product_prod_v_v @ Xs ) )
& ! [Ys2: list_P7986770385144383213od_v_v] :
( ( member4190458934886417558od_v_v @ Ys2 @ ( set_li2340707408155270402od_v_v @ Xs ) )
=> ( distin6159370996967099744od_v_v @ Ys2 ) )
& ! [Ys2: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
( ( ( member4190458934886417558od_v_v @ Ys2 @ ( set_li2340707408155270402od_v_v @ Xs ) )
& ( member4190458934886417558od_v_v @ Zs2 @ ( set_li2340707408155270402od_v_v @ Xs ) )
& ( Ys2 != Zs2 ) )
=> ( ( inf_in6271465464967711157od_v_v @ ( set_Product_prod_v_v2 @ Ys2 ) @ ( set_Product_prod_v_v2 @ Zs2 ) )
= bot_bo723834152578015283od_v_v ) ) ) ) ).
% distinct_concat_iff
thf(fact_959_underS__incl__iff,axiom,
! [R: set_Product_prod_v_v,A: v,B: v] :
( ( order_8768733634509060168r_on_v @ ( field_v @ R ) @ R )
=> ( ( member_v @ A @ ( field_v @ R ) )
=> ( ( member_v @ B @ ( field_v @ R ) )
=> ( ( ord_less_eq_set_v @ ( order_underS_v @ R @ A ) @ ( order_underS_v @ R @ B ) )
= ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ A @ B ) @ R ) ) ) ) ) ).
% underS_incl_iff
thf(fact_960_underS__incl__iff,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v,B: 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 @ B @ ( field_7153129647634986036od_v_v @ R ) )
=> ( ( ord_le7336532860387713383od_v_v @ ( order_5211820470575790509od_v_v @ R @ A ) @ ( order_5211820470575790509od_v_v @ R @ B ) )
= ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ A @ B ) @ R ) ) ) ) ) ).
% underS_incl_iff
thf(fact_961_underS__empty,axiom,
! [A: v,R: set_Product_prod_v_v] :
( ~ ( member_v @ A @ ( field_v @ R ) )
=> ( ( order_underS_v @ R @ A )
= bot_bot_set_v ) ) ).
% underS_empty
thf(fact_962_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_963_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_964_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_965_distinct__concat,axiom,
! [Xs: list_list_v] :
( ( distinct_list_v @ Xs )
=> ( ! [Ys3: list_v] :
( ( member_list_v @ Ys3 @ ( set_list_v2 @ Xs ) )
=> ( distinct_v @ Ys3 ) )
=> ( ! [Ys3: list_v,Zs3: list_v] :
( ( member_list_v @ Ys3 @ ( set_list_v2 @ Xs ) )
=> ( ( member_list_v @ Zs3 @ ( set_list_v2 @ Xs ) )
=> ( ( Ys3 != Zs3 )
=> ( ( inf_inf_set_v @ ( set_v2 @ Ys3 ) @ ( set_v2 @ Zs3 ) )
= bot_bot_set_v ) ) ) )
=> ( distinct_v @ ( concat_v @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_966_distinct__concat,axiom,
! [Xs: list_l4795378083388841843od_v_v] :
( ( distin913317783593574886od_v_v @ Xs )
=> ( ! [Ys3: list_P7986770385144383213od_v_v] :
( ( member4190458934886417558od_v_v @ Ys3 @ ( set_li2340707408155270402od_v_v @ Xs ) )
=> ( distin6159370996967099744od_v_v @ Ys3 ) )
=> ( ! [Ys3: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
( ( member4190458934886417558od_v_v @ Ys3 @ ( set_li2340707408155270402od_v_v @ Xs ) )
=> ( ( member4190458934886417558od_v_v @ Zs3 @ ( set_li2340707408155270402od_v_v @ Xs ) )
=> ( ( Ys3 != Zs3 )
=> ( ( inf_in6271465464967711157od_v_v @ ( set_Product_prod_v_v2 @ Ys3 ) @ ( set_Product_prod_v_v2 @ Zs3 ) )
= bot_bo723834152578015283od_v_v ) ) ) )
=> ( distin6159370996967099744od_v_v @ ( concat2875663619778446888od_v_v @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_967_Refl__under__underS,axiom,
! [R: set_Product_prod_v_v,A: v] :
( ( refl_on_v @ ( field_v @ R ) @ R )
=> ( ( member_v @ 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_968_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_969_Range__insert,axiom,
! [A: v,B: v,R: set_Product_prod_v_v] :
( ( range_v_v @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ A @ B ) @ R ) )
= ( insert_v2 @ B @ ( range_v_v @ R ) ) ) ).
% Range_insert
thf(fact_970_Range__insert,axiom,
! [A: v,B: sCC_Bl1394983891496994913t_unit,R: set_Pr6425124735969554649t_unit] :
( ( range_8313874519294158527t_unit @ ( insert6563505994279474003t_unit @ ( produc3862955338007567901t_unit @ A @ B ) @ R ) )
= ( insert3615455318616551057t_unit @ B @ ( range_8313874519294158527t_unit @ R ) ) ) ).
% Range_insert
thf(fact_971_distinct__disjoint__shuffles,axiom,
! [Xs: list_v,Ys: list_v,Zs: list_v] :
( ( distinct_v @ Xs )
=> ( ( distinct_v @ Ys )
=> ( ( ( inf_inf_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ Ys ) )
= bot_bot_set_v )
=> ( ( member_list_v @ Zs @ ( shuffles_v @ Xs @ Ys ) )
=> ( distinct_v @ Zs ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_972_distinct__disjoint__shuffles,axiom,
! [Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v,Zs: list_P7986770385144383213od_v_v] :
( ( distin6159370996967099744od_v_v @ Xs )
=> ( ( distin6159370996967099744od_v_v @ Ys )
=> ( ( ( inf_in6271465464967711157od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ Ys ) )
= bot_bo723834152578015283od_v_v )
=> ( ( member4190458934886417558od_v_v @ Zs @ ( shuffl71542398924059522od_v_v @ Xs @ Ys ) )
=> ( distin6159370996967099744od_v_v @ Zs ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_973_Range__empty,axiom,
( ( range_v_v @ bot_bo723834152578015283od_v_v )
= bot_bot_set_v ) ).
% Range_empty
thf(fact_974_Range__mono,axiom,
! [R: set_Product_prod_v_v,S5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ R @ S5 )
=> ( ord_less_eq_set_v @ ( range_v_v @ R ) @ ( range_v_v @ S5 ) ) ) ).
% Range_mono
thf(fact_975_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_976_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_977_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_978_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_979_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_980_Domain__insert,axiom,
! [A: v,B: v,R: set_Product_prod_v_v] :
( ( domain_v_v @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ A @ B ) @ R ) )
= ( insert_v2 @ A @ ( domain_v_v @ R ) ) ) ).
% Domain_insert
thf(fact_981_Domain__insert,axiom,
! [A: v,B: sCC_Bl1394983891496994913t_unit,R: set_Pr6425124735969554649t_unit] :
( ( domain2560787710516352232t_unit @ ( insert6563505994279474003t_unit @ ( produc3862955338007567901t_unit @ A @ B ) @ R ) )
= ( insert_v2 @ A @ ( domain2560787710516352232t_unit @ R ) ) ) ).
% Domain_insert
thf(fact_982_Inf__fin_Oset__eq__fold,axiom,
! [X: set_v,Xs: list_set_v] :
( ( lattic8209813555532694032_set_v @ ( set_set_v2 @ ( cons_set_v @ X @ Xs ) ) )
= ( fold_set_v_set_v @ inf_inf_set_v @ Xs @ X ) ) ).
% Inf_fin.set_eq_fold
thf(fact_983_Domain__empty,axiom,
( ( domain_v_v @ bot_bo723834152578015283od_v_v )
= bot_bot_set_v ) ).
% Domain_empty
thf(fact_984_Domain__mono,axiom,
! [R: set_Product_prod_v_v,S5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ R @ S5 )
=> ( ord_less_eq_set_v @ ( domain_v_v @ R ) @ ( domain_v_v @ S5 ) ) ) ).
% Domain_mono
thf(fact_985_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_986_union__fold__insert,axiom,
! [A3: set_v,B2: set_v] :
( ( finite_finite_v @ A3 )
=> ( ( sup_sup_set_v @ A3 @ B2 )
= ( finite_fold_v_set_v @ insert_v2 @ B2 @ A3 ) ) ) ).
% union_fold_insert
thf(fact_987_union__fold__insert,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A3 )
=> ( ( sup_su414716646722978715od_v_v @ A3 @ B2 )
= ( finite6851115414092367464od_v_v @ insert1338601472111419319od_v_v @ B2 @ A3 ) ) ) ).
% union_fold_insert
thf(fact_988_union__set__fold,axiom,
! [Xs: list_v,A3: set_v] :
( ( sup_sup_set_v @ ( set_v2 @ Xs ) @ A3 )
= ( fold_v_set_v @ insert_v2 @ Xs @ A3 ) ) ).
% union_set_fold
thf(fact_989_union__set__fold,axiom,
! [Xs: list_P7986770385144383213od_v_v,A3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ A3 )
= ( fold_P138721258069746347od_v_v @ insert1338601472111419319od_v_v @ Xs @ A3 ) ) ).
% union_set_fold
thf(fact_990_inter__coset__fold,axiom,
! [A3: set_v,Xs: list_v] :
( ( inf_inf_set_v @ A3 @ ( coset_v @ Xs ) )
= ( fold_v_set_v @ remove_v @ Xs @ A3 ) ) ).
% inter_coset_fold
thf(fact_991_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_992_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_993_lists__empty,axiom,
( ( lists_v @ bot_bot_set_v )
= ( insert_list_v @ nil_v @ bot_bot_set_list_v ) ) ).
% lists_empty
thf(fact_994_lists__empty,axiom,
( ( lists_5865669170805476827od_v_v @ bot_bo723834152578015283od_v_v )
= ( insert4087971119735676093od_v_v @ nil_Product_prod_v_v @ bot_bo54012148666785209od_v_v ) ) ).
% lists_empty
thf(fact_995_lists__Int__eq,axiom,
! [A3: set_v,B2: set_v] :
( ( lists_v @ ( inf_inf_set_v @ A3 @ B2 ) )
= ( inf_inf_set_list_v @ ( lists_v @ A3 ) @ ( lists_v @ B2 ) ) ) ).
% lists_Int_eq
thf(fact_996_lists__mono,axiom,
! [A3: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A3 @ B2 )
=> ( ord_le1129530298279361049list_v @ ( lists_v @ A3 ) @ ( lists_v @ B2 ) ) ) ).
% lists_mono
thf(fact_997_lists__mono,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
=> ( ord_le5393391283775026413od_v_v @ ( lists_5865669170805476827od_v_v @ A3 ) @ ( lists_5865669170805476827od_v_v @ B2 ) ) ) ).
% lists_mono
thf(fact_998_lists__IntI,axiom,
! [L2: list_v,A3: set_v,B2: set_v] :
( ( member_list_v @ L2 @ ( lists_v @ A3 ) )
=> ( ( member_list_v @ L2 @ ( lists_v @ B2 ) )
=> ( member_list_v @ L2 @ ( lists_v @ ( inf_inf_set_v @ A3 @ B2 ) ) ) ) ) ).
% lists_IntI
thf(fact_999_shuffles_Oelims,axiom,
! [X: list_v,Xa3: list_v,Y2: set_list_v] :
( ( ( shuffles_v @ X @ Xa3 )
= Y2 )
=> ( ( ( X = nil_v )
=> ( Y2
!= ( insert_list_v @ Xa3 @ bot_bot_set_list_v ) ) )
=> ( ( ( Xa3 = nil_v )
=> ( Y2
!= ( insert_list_v @ X @ bot_bot_set_list_v ) ) )
=> ~ ! [X3: v,Xs3: list_v] :
( ( X
= ( cons_v @ X3 @ Xs3 ) )
=> ! [Y: v,Ys3: list_v] :
( ( Xa3
= ( cons_v @ Y @ Ys3 ) )
=> ( Y2
!= ( sup_sup_set_list_v @ ( image_list_v_list_v @ ( cons_v @ X3 ) @ ( shuffles_v @ Xs3 @ ( cons_v @ Y @ Ys3 ) ) ) @ ( image_list_v_list_v @ ( cons_v @ Y ) @ ( shuffles_v @ ( cons_v @ X3 @ Xs3 ) @ Ys3 ) ) ) ) ) ) ) ) ) ).
% shuffles.elims
thf(fact_1000_pairwise__alt,axiom,
( pairwise_v
= ( ^ [R4: v > v > $o,S7: set_v] :
! [X2: v] :
( ( member_v @ X2 @ S7 )
=> ! [Y3: v] :
( ( member_v @ Y3 @ ( minus_minus_set_v @ S7 @ ( insert_v2 @ X2 @ bot_bot_set_v ) ) )
=> ( R4 @ X2 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_1001_pairwise__alt,axiom,
( pairwi5745945156428401490od_v_v
= ( ^ [R4: product_prod_v_v > product_prod_v_v > $o,S7: set_Product_prod_v_v] :
! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ S7 )
=> ! [Y3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ ( minus_4183494784930505774od_v_v @ S7 @ ( insert1338601472111419319od_v_v @ X2 @ bot_bo723834152578015283od_v_v ) ) )
=> ( R4 @ X2 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_1002_psubset__insert__iff,axiom,
! [A3: set_v,X: v,B2: set_v] :
( ( ord_less_set_v @ A3 @ ( insert_v2 @ X @ B2 ) )
= ( ( ( member_v @ X @ B2 )
=> ( ord_less_set_v @ A3 @ B2 ) )
& ( ~ ( member_v @ X @ B2 )
=> ( ( ( member_v @ X @ A3 )
=> ( ord_less_set_v @ ( minus_minus_set_v @ A3 @ ( insert_v2 @ X @ bot_bot_set_v ) ) @ B2 ) )
& ( ~ ( member_v @ X @ A3 )
=> ( ord_less_eq_set_v @ A3 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1003_psubset__insert__iff,axiom,
! [A3: set_Product_prod_v_v,X: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ B2 ) )
= ( ( ( member7453568604450474000od_v_v @ X @ B2 )
=> ( ord_le4186455585809229939od_v_v @ A3 @ B2 ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ B2 )
=> ( ( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ord_le4186455585809229939od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B2 ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1004_image__eqI,axiom,
! [B: v,F: v > v,X: v,A3: set_v] :
( ( B
= ( F @ X ) )
=> ( ( member_v @ X @ A3 )
=> ( member_v @ B @ ( image_v_v @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1005_image__eqI,axiom,
! [B: product_prod_v_v,F: v > product_prod_v_v,X: v,A3: set_v] :
( ( B
= ( F @ X ) )
=> ( ( member_v @ X @ A3 )
=> ( member7453568604450474000od_v_v @ B @ ( image_9222788639401671577od_v_v @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1006_image__eqI,axiom,
! [B: v,F: product_prod_v_v > v,X: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( B
= ( F @ X ) )
=> ( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( member_v @ B @ ( image_6152814753742948081_v_v_v @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1007_image__eqI,axiom,
! [B: product_prod_v_v,F: product_prod_v_v > product_prod_v_v,X: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( B
= ( F @ X ) )
=> ( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( member7453568604450474000od_v_v @ B @ ( image_781944334261467077od_v_v @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1008_image__empty,axiom,
! [F: v > v] :
( ( image_v_v @ F @ bot_bot_set_v )
= bot_bot_set_v ) ).
% image_empty
thf(fact_1009_image__empty,axiom,
! [F: v > product_prod_v_v] :
( ( image_9222788639401671577od_v_v @ F @ bot_bot_set_v )
= bot_bo723834152578015283od_v_v ) ).
% image_empty
thf(fact_1010_image__empty,axiom,
! [F: product_prod_v_v > v] :
( ( image_6152814753742948081_v_v_v @ F @ bot_bo723834152578015283od_v_v )
= bot_bot_set_v ) ).
% image_empty
thf(fact_1011_image__empty,axiom,
! [F: product_prod_v_v > product_prod_v_v] :
( ( image_781944334261467077od_v_v @ F @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% image_empty
thf(fact_1012_empty__is__image,axiom,
! [F: v > v,A3: set_v] :
( ( bot_bot_set_v
= ( image_v_v @ F @ A3 ) )
= ( A3 = bot_bot_set_v ) ) ).
% empty_is_image
thf(fact_1013_empty__is__image,axiom,
! [F: product_prod_v_v > v,A3: set_Product_prod_v_v] :
( ( bot_bot_set_v
= ( image_6152814753742948081_v_v_v @ F @ A3 ) )
= ( A3 = bot_bo723834152578015283od_v_v ) ) ).
% empty_is_image
thf(fact_1014_empty__is__image,axiom,
! [F: v > product_prod_v_v,A3: set_v] :
( ( bot_bo723834152578015283od_v_v
= ( image_9222788639401671577od_v_v @ F @ A3 ) )
= ( A3 = bot_bot_set_v ) ) ).
% empty_is_image
thf(fact_1015_empty__is__image,axiom,
! [F: product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( image_781944334261467077od_v_v @ F @ A3 ) )
= ( A3 = bot_bo723834152578015283od_v_v ) ) ).
% empty_is_image
thf(fact_1016_image__is__empty,axiom,
! [F: v > v,A3: set_v] :
( ( ( image_v_v @ F @ A3 )
= bot_bot_set_v )
= ( A3 = bot_bot_set_v ) ) ).
% image_is_empty
thf(fact_1017_image__is__empty,axiom,
! [F: product_prod_v_v > v,A3: set_Product_prod_v_v] :
( ( ( image_6152814753742948081_v_v_v @ F @ A3 )
= bot_bot_set_v )
= ( A3 = bot_bo723834152578015283od_v_v ) ) ).
% image_is_empty
thf(fact_1018_image__is__empty,axiom,
! [F: v > product_prod_v_v,A3: set_v] :
( ( ( image_9222788639401671577od_v_v @ F @ A3 )
= bot_bo723834152578015283od_v_v )
= ( A3 = bot_bot_set_v ) ) ).
% image_is_empty
thf(fact_1019_image__is__empty,axiom,
! [F: product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ( image_781944334261467077od_v_v @ F @ A3 )
= bot_bo723834152578015283od_v_v )
= ( A3 = bot_bo723834152578015283od_v_v ) ) ).
% image_is_empty
thf(fact_1020_image__insert,axiom,
! [F: v > v,A: v,B2: set_v] :
( ( image_v_v @ F @ ( insert_v2 @ A @ B2 ) )
= ( insert_v2 @ ( F @ A ) @ ( image_v_v @ F @ B2 ) ) ) ).
% image_insert
thf(fact_1021_image__insert,axiom,
! [F: v > product_prod_v_v,A: v,B2: set_v] :
( ( image_9222788639401671577od_v_v @ F @ ( insert_v2 @ A @ B2 ) )
= ( insert1338601472111419319od_v_v @ ( F @ A ) @ ( image_9222788639401671577od_v_v @ F @ B2 ) ) ) ).
% image_insert
thf(fact_1022_image__insert,axiom,
! [F: product_prod_v_v > v,A: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( image_6152814753742948081_v_v_v @ F @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( insert_v2 @ ( F @ A ) @ ( image_6152814753742948081_v_v_v @ F @ B2 ) ) ) ).
% image_insert
thf(fact_1023_image__insert,axiom,
! [F: product_prod_v_v > product_prod_v_v,A: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( image_781944334261467077od_v_v @ F @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( insert1338601472111419319od_v_v @ ( F @ A ) @ ( image_781944334261467077od_v_v @ F @ B2 ) ) ) ).
% image_insert
thf(fact_1024_insert__image,axiom,
! [X: v,A3: set_v,F: v > v] :
( ( member_v @ X @ A3 )
=> ( ( insert_v2 @ ( F @ X ) @ ( image_v_v @ F @ A3 ) )
= ( image_v_v @ F @ A3 ) ) ) ).
% insert_image
thf(fact_1025_insert__image,axiom,
! [X: v,A3: set_v,F: v > product_prod_v_v] :
( ( member_v @ X @ A3 )
=> ( ( insert1338601472111419319od_v_v @ ( F @ X ) @ ( image_9222788639401671577od_v_v @ F @ A3 ) )
= ( image_9222788639401671577od_v_v @ F @ A3 ) ) ) ).
% insert_image
thf(fact_1026_insert__image,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,F: product_prod_v_v > v] :
( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ( insert_v2 @ ( F @ X ) @ ( image_6152814753742948081_v_v_v @ F @ A3 ) )
= ( image_6152814753742948081_v_v_v @ F @ A3 ) ) ) ).
% insert_image
thf(fact_1027_insert__image,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ( insert1338601472111419319od_v_v @ ( F @ X ) @ ( image_781944334261467077od_v_v @ F @ A3 ) )
= ( image_781944334261467077od_v_v @ F @ A3 ) ) ) ).
% insert_image
thf(fact_1028_psubsetI,axiom,
! [A3: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A3 @ B2 )
=> ( ( A3 != B2 )
=> ( ord_less_set_v @ A3 @ B2 ) ) ) ).
% psubsetI
thf(fact_1029_psubsetI,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
=> ( ( A3 != B2 )
=> ( ord_le4186455585809229939od_v_v @ A3 @ B2 ) ) ) ).
% psubsetI
thf(fact_1030_all__finite__subset__image,axiom,
! [F: v > v,A3: set_v,P: set_v > $o] :
( ( ! [B3: set_v] :
( ( ( finite_finite_v @ B3 )
& ( ord_less_eq_set_v @ B3 @ ( image_v_v @ F @ A3 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_v] :
( ( ( finite_finite_v @ B3 )
& ( ord_less_eq_set_v @ B3 @ A3 ) )
=> ( P @ ( image_v_v @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1031_all__finite__subset__image,axiom,
! [F: product_prod_v_v > v,A3: set_Product_prod_v_v,P: set_v > $o] :
( ( ! [B3: set_v] :
( ( ( finite_finite_v @ B3 )
& ( ord_less_eq_set_v @ B3 @ ( image_6152814753742948081_v_v_v @ F @ A3 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_Product_prod_v_v] :
( ( ( finite3348123685078250256od_v_v @ B3 )
& ( ord_le7336532860387713383od_v_v @ B3 @ A3 ) )
=> ( P @ ( image_6152814753742948081_v_v_v @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1032_all__finite__subset__image,axiom,
! [F: v > product_prod_v_v,A3: set_v,P: set_Product_prod_v_v > $o] :
( ( ! [B3: set_Product_prod_v_v] :
( ( ( finite3348123685078250256od_v_v @ B3 )
& ( ord_le7336532860387713383od_v_v @ B3 @ ( image_9222788639401671577od_v_v @ F @ A3 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_v] :
( ( ( finite_finite_v @ B3 )
& ( ord_less_eq_set_v @ B3 @ A3 ) )
=> ( P @ ( image_9222788639401671577od_v_v @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1033_all__finite__subset__image,axiom,
! [F: product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( ! [B3: set_Product_prod_v_v] :
( ( ( finite3348123685078250256od_v_v @ B3 )
& ( ord_le7336532860387713383od_v_v @ B3 @ ( image_781944334261467077od_v_v @ F @ A3 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_Product_prod_v_v] :
( ( ( finite3348123685078250256od_v_v @ B3 )
& ( ord_le7336532860387713383od_v_v @ B3 @ A3 ) )
=> ( P @ ( image_781944334261467077od_v_v @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1034_ex__finite__subset__image,axiom,
! [F: v > v,A3: set_v,P: set_v > $o] :
( ( ? [B3: set_v] :
( ( finite_finite_v @ B3 )
& ( ord_less_eq_set_v @ B3 @ ( image_v_v @ F @ A3 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_v] :
( ( finite_finite_v @ B3 )
& ( ord_less_eq_set_v @ B3 @ A3 )
& ( P @ ( image_v_v @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1035_ex__finite__subset__image,axiom,
! [F: product_prod_v_v > v,A3: set_Product_prod_v_v,P: set_v > $o] :
( ( ? [B3: set_v] :
( ( finite_finite_v @ B3 )
& ( ord_less_eq_set_v @ B3 @ ( image_6152814753742948081_v_v_v @ F @ A3 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ B3 )
& ( ord_le7336532860387713383od_v_v @ B3 @ A3 )
& ( P @ ( image_6152814753742948081_v_v_v @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1036_ex__finite__subset__image,axiom,
! [F: v > product_prod_v_v,A3: set_v,P: set_Product_prod_v_v > $o] :
( ( ? [B3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ B3 )
& ( ord_le7336532860387713383od_v_v @ B3 @ ( image_9222788639401671577od_v_v @ F @ A3 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_v] :
( ( finite_finite_v @ B3 )
& ( ord_less_eq_set_v @ B3 @ A3 )
& ( P @ ( image_9222788639401671577od_v_v @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1037_ex__finite__subset__image,axiom,
! [F: product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( ? [B3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ B3 )
& ( ord_le7336532860387713383od_v_v @ B3 @ ( image_781944334261467077od_v_v @ F @ A3 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ B3 )
& ( ord_le7336532860387713383od_v_v @ B3 @ A3 )
& ( P @ ( image_781944334261467077od_v_v @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1038_finite__subset__image,axiom,
! [B2: set_v,F: v > v,A3: set_v] :
( ( finite_finite_v @ B2 )
=> ( ( ord_less_eq_set_v @ B2 @ ( image_v_v @ F @ A3 ) )
=> ? [C4: set_v] :
( ( ord_less_eq_set_v @ C4 @ A3 )
& ( finite_finite_v @ C4 )
& ( B2
= ( image_v_v @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1039_finite__subset__image,axiom,
! [B2: set_v,F: product_prod_v_v > v,A3: set_Product_prod_v_v] :
( ( finite_finite_v @ B2 )
=> ( ( ord_less_eq_set_v @ B2 @ ( image_6152814753742948081_v_v_v @ F @ A3 ) )
=> ? [C4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C4 @ A3 )
& ( finite3348123685078250256od_v_v @ C4 )
& ( B2
= ( image_6152814753742948081_v_v_v @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1040_finite__subset__image,axiom,
! [B2: set_Product_prod_v_v,F: v > product_prod_v_v,A3: set_v] :
( ( finite3348123685078250256od_v_v @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ ( image_9222788639401671577od_v_v @ F @ A3 ) )
=> ? [C4: set_v] :
( ( ord_less_eq_set_v @ C4 @ A3 )
& ( finite_finite_v @ C4 )
& ( B2
= ( image_9222788639401671577od_v_v @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1041_finite__subset__image,axiom,
! [B2: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ ( image_781944334261467077od_v_v @ F @ A3 ) )
=> ? [C4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C4 @ A3 )
& ( finite3348123685078250256od_v_v @ C4 )
& ( B2
= ( image_781944334261467077od_v_v @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1042_finite__surj,axiom,
! [A3: set_v,B2: set_v,F: v > v] :
( ( finite_finite_v @ A3 )
=> ( ( ord_less_eq_set_v @ B2 @ ( image_v_v @ F @ A3 ) )
=> ( finite_finite_v @ B2 ) ) ) ).
% finite_surj
thf(fact_1043_finite__surj,axiom,
! [A3: set_v,B2: set_Product_prod_v_v,F: v > product_prod_v_v] :
( ( finite_finite_v @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ ( image_9222788639401671577od_v_v @ F @ A3 ) )
=> ( finite3348123685078250256od_v_v @ B2 ) ) ) ).
% finite_surj
thf(fact_1044_sup_Ostrict__coboundedI2,axiom,
! [C2: set_Product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ C2 @ B )
=> ( ord_le4186455585809229939od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_1045_sup_Ostrict__coboundedI1,axiom,
! [C2: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ C2 @ A )
=> ( ord_le4186455585809229939od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_1046_sup_Ostrict__order__iff,axiom,
( ord_le4186455585809229939od_v_v
= ( ^ [B5: set_Product_prod_v_v,A5: set_Product_prod_v_v] :
( ( A5
= ( sup_su414716646722978715od_v_v @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1047_sup_Ostrict__boundedE,axiom,
! [B: set_Product_prod_v_v,C2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ ( sup_su414716646722978715od_v_v @ B @ C2 ) @ A )
=> ~ ( ( ord_le4186455585809229939od_v_v @ B @ A )
=> ~ ( ord_le4186455585809229939od_v_v @ C2 @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_1048_sup_Oabsorb4,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ A @ B )
=> ( ( sup_su414716646722978715od_v_v @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_1049_sup_Oabsorb3,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ B @ A )
=> ( ( sup_su414716646722978715od_v_v @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_1050_less__supI2,axiom,
! [X: set_Product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ X @ B )
=> ( ord_le4186455585809229939od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% less_supI2
thf(fact_1051_less__supI1,axiom,
! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ X @ A )
=> ( ord_le4186455585809229939od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% less_supI1
thf(fact_1052_image__Un,axiom,
! [F: product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( image_781944334261467077od_v_v @ F @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) )
= ( sup_su414716646722978715od_v_v @ ( image_781944334261467077od_v_v @ F @ A3 ) @ ( image_781944334261467077od_v_v @ F @ B2 ) ) ) ).
% image_Un
thf(fact_1053_subset__image__iff,axiom,
! [B2: set_v,F: v > v,A3: set_v] :
( ( ord_less_eq_set_v @ B2 @ ( image_v_v @ F @ A3 ) )
= ( ? [AA: set_v] :
( ( ord_less_eq_set_v @ AA @ A3 )
& ( B2
= ( image_v_v @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1054_subset__image__iff,axiom,
! [B2: set_v,F: product_prod_v_v > v,A3: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ B2 @ ( image_6152814753742948081_v_v_v @ F @ A3 ) )
= ( ? [AA: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ AA @ A3 )
& ( B2
= ( image_6152814753742948081_v_v_v @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1055_subset__image__iff,axiom,
! [B2: set_Product_prod_v_v,F: v > product_prod_v_v,A3: set_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ ( image_9222788639401671577od_v_v @ F @ A3 ) )
= ( ? [AA: set_v] :
( ( ord_less_eq_set_v @ AA @ A3 )
& ( B2
= ( image_9222788639401671577od_v_v @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1056_subset__image__iff,axiom,
! [B2: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ ( image_781944334261467077od_v_v @ F @ A3 ) )
= ( ? [AA: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ AA @ A3 )
& ( B2
= ( image_781944334261467077od_v_v @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1057_subset__imageE,axiom,
! [B2: set_v,F: v > v,A3: set_v] :
( ( ord_less_eq_set_v @ B2 @ ( image_v_v @ F @ A3 ) )
=> ~ ! [C4: set_v] :
( ( ord_less_eq_set_v @ C4 @ A3 )
=> ( B2
!= ( image_v_v @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1058_subset__imageE,axiom,
! [B2: set_v,F: product_prod_v_v > v,A3: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ B2 @ ( image_6152814753742948081_v_v_v @ F @ A3 ) )
=> ~ ! [C4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C4 @ A3 )
=> ( B2
!= ( image_6152814753742948081_v_v_v @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1059_subset__imageE,axiom,
! [B2: set_Product_prod_v_v,F: v > product_prod_v_v,A3: set_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ ( image_9222788639401671577od_v_v @ F @ A3 ) )
=> ~ ! [C4: set_v] :
( ( ord_less_eq_set_v @ C4 @ A3 )
=> ( B2
!= ( image_9222788639401671577od_v_v @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1060_subset__imageE,axiom,
! [B2: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ ( image_781944334261467077od_v_v @ F @ A3 ) )
=> ~ ! [C4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C4 @ A3 )
=> ( B2
!= ( image_781944334261467077od_v_v @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_1061_image__subsetI,axiom,
! [A3: set_v,F: v > v,B2: set_v] :
( ! [X3: v] :
( ( member_v @ X3 @ A3 )
=> ( member_v @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_v @ ( image_v_v @ F @ A3 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1062_image__subsetI,axiom,
! [A3: set_Product_prod_v_v,F: product_prod_v_v > v,B2: set_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ( member_v @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_v @ ( image_6152814753742948081_v_v_v @ F @ A3 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1063_image__subsetI,axiom,
! [A3: set_v,F: v > product_prod_v_v,B2: set_Product_prod_v_v] :
( ! [X3: v] :
( ( member_v @ X3 @ A3 )
=> ( member7453568604450474000od_v_v @ ( F @ X3 ) @ B2 ) )
=> ( ord_le7336532860387713383od_v_v @ ( image_9222788639401671577od_v_v @ F @ A3 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1064_image__subsetI,axiom,
! [A3: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v,B2: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ( member7453568604450474000od_v_v @ ( F @ X3 ) @ B2 ) )
=> ( ord_le7336532860387713383od_v_v @ ( image_781944334261467077od_v_v @ F @ A3 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1065_image__mono,axiom,
! [A3: set_v,B2: set_v,F: v > v] :
( ( ord_less_eq_set_v @ A3 @ B2 )
=> ( ord_less_eq_set_v @ ( image_v_v @ F @ A3 ) @ ( image_v_v @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1066_image__mono,axiom,
! [A3: set_v,B2: set_v,F: v > product_prod_v_v] :
( ( ord_less_eq_set_v @ A3 @ B2 )
=> ( ord_le7336532860387713383od_v_v @ ( image_9222788639401671577od_v_v @ F @ A3 ) @ ( image_9222788639401671577od_v_v @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1067_image__mono,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,F: product_prod_v_v > v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
=> ( ord_less_eq_set_v @ ( image_6152814753742948081_v_v_v @ F @ A3 ) @ ( image_6152814753742948081_v_v_v @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1068_image__mono,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
=> ( ord_le7336532860387713383od_v_v @ ( image_781944334261467077od_v_v @ F @ A3 ) @ ( image_781944334261467077od_v_v @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1069_leD,axiom,
! [Y2: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y2 @ X )
=> ~ ( ord_less_set_v @ X @ Y2 ) ) ).
% leD
thf(fact_1070_leD,axiom,
! [Y2: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y2 @ X )
=> ~ ( ord_le4186455585809229939od_v_v @ X @ Y2 ) ) ).
% leD
thf(fact_1071_nless__le,axiom,
! [A: set_v,B: set_v] :
( ( ~ ( ord_less_set_v @ A @ B ) )
= ( ~ ( ord_less_eq_set_v @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1072_nless__le,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ~ ( ord_le4186455585809229939od_v_v @ A @ B ) )
= ( ~ ( ord_le7336532860387713383od_v_v @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1073_antisym__conv1,axiom,
! [X: set_v,Y2: set_v] :
( ~ ( ord_less_set_v @ X @ Y2 )
=> ( ( ord_less_eq_set_v @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_1074_antisym__conv1,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ~ ( ord_le4186455585809229939od_v_v @ X @ Y2 )
=> ( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_1075_antisym__conv2,axiom,
! [X: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X @ Y2 )
=> ( ( ~ ( ord_less_set_v @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_1076_antisym__conv2,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
=> ( ( ~ ( ord_le4186455585809229939od_v_v @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_1077_less__le__not__le,axiom,
( ord_less_set_v
= ( ^ [X2: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X2 @ Y3 )
& ~ ( ord_less_eq_set_v @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1078_less__le__not__le,axiom,
( ord_le4186455585809229939od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ Y3 )
& ~ ( ord_le7336532860387713383od_v_v @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1079_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_v
= ( ^ [A5: set_v,B5: set_v] :
( ( ord_less_set_v @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1080_order_Oorder__iff__strict,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1081_order_Ostrict__iff__order,axiom,
( ord_less_set_v
= ( ^ [A5: set_v,B5: set_v] :
( ( ord_less_eq_set_v @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1082_order_Ostrict__iff__order,axiom,
( ord_le4186455585809229939od_v_v
= ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1083_order_Ostrict__trans1,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_set_v @ B @ C2 )
=> ( ord_less_set_v @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1084_order_Ostrict__trans1,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le4186455585809229939od_v_v @ B @ C2 )
=> ( ord_le4186455585809229939od_v_v @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1085_order_Ostrict__trans2,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( ord_less_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ord_less_set_v @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1086_order_Ostrict__trans2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ord_le4186455585809229939od_v_v @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1087_order_Ostrict__iff__not,axiom,
( ord_less_set_v
= ( ^ [A5: set_v,B5: set_v] :
( ( ord_less_eq_set_v @ A5 @ B5 )
& ~ ( ord_less_eq_set_v @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1088_order_Ostrict__iff__not,axiom,
( ord_le4186455585809229939od_v_v
= ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A5 @ B5 )
& ~ ( ord_le7336532860387713383od_v_v @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1089_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_v
= ( ^ [B5: set_v,A5: set_v] :
( ( ord_less_set_v @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1090_dual__order_Oorder__iff__strict,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B5: set_Product_prod_v_v,A5: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1091_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_v
= ( ^ [B5: set_v,A5: set_v] :
( ( ord_less_eq_set_v @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1092_dual__order_Ostrict__iff__order,axiom,
( ord_le4186455585809229939od_v_v
= ( ^ [B5: set_Product_prod_v_v,A5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1093_dual__order_Ostrict__trans1,axiom,
! [B: set_v,A: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( ord_less_set_v @ C2 @ B )
=> ( ord_less_set_v @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1094_dual__order_Ostrict__trans1,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( ord_le4186455585809229939od_v_v @ C2 @ B )
=> ( ord_le4186455585809229939od_v_v @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1095_dual__order_Ostrict__trans2,axiom,
! [B: set_v,A: set_v,C2: set_v] :
( ( ord_less_set_v @ B @ A )
=> ( ( ord_less_eq_set_v @ C2 @ B )
=> ( ord_less_set_v @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1096_dual__order_Ostrict__trans2,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ B @ A )
=> ( ( ord_le7336532860387713383od_v_v @ C2 @ B )
=> ( ord_le4186455585809229939od_v_v @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1097_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_v
= ( ^ [B5: set_v,A5: set_v] :
( ( ord_less_eq_set_v @ B5 @ A5 )
& ~ ( ord_less_eq_set_v @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1098_dual__order_Ostrict__iff__not,axiom,
( ord_le4186455585809229939od_v_v
= ( ^ [B5: set_Product_prod_v_v,A5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B5 @ A5 )
& ~ ( ord_le7336532860387713383od_v_v @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1099_order_Ostrict__implies__order,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_set_v @ A @ B )
=> ( ord_less_eq_set_v @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1100_order_Ostrict__implies__order,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ A @ B )
=> ( ord_le7336532860387713383od_v_v @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1101_dual__order_Ostrict__implies__order,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_set_v @ B @ A )
=> ( ord_less_eq_set_v @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1102_dual__order_Ostrict__implies__order,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ B @ A )
=> ( ord_le7336532860387713383od_v_v @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1103_order__le__less,axiom,
( ord_less_eq_set_v
= ( ^ [X2: set_v,Y3: set_v] :
( ( ord_less_set_v @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1104_order__le__less,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1105_order__less__le,axiom,
( ord_less_set_v
= ( ^ [X2: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1106_order__less__le,axiom,
( ord_le4186455585809229939od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1107_order__less__imp__le,axiom,
! [X: set_v,Y2: set_v] :
( ( ord_less_set_v @ X @ Y2 )
=> ( ord_less_eq_set_v @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_1108_order__less__imp__le,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ X @ Y2 )
=> ( ord_le7336532860387713383od_v_v @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_1109_order__le__neq__trans,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_v @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1110_order__le__neq__trans,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( A != B )
=> ( ord_le4186455585809229939od_v_v @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1111_order__neq__le__trans,axiom,
! [A: set_v,B: set_v] :
( ( A != B )
=> ( ( ord_less_eq_set_v @ A @ B )
=> ( ord_less_set_v @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1112_order__neq__le__trans,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A != B )
=> ( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ord_le4186455585809229939od_v_v @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1113_order__le__less__trans,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X @ Y2 )
=> ( ( ord_less_set_v @ Y2 @ Z )
=> ( ord_less_set_v @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1114_order__le__less__trans,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
=> ( ( ord_le4186455585809229939od_v_v @ Y2 @ Z )
=> ( ord_le4186455585809229939od_v_v @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1115_order__less__le__trans,axiom,
! [X: set_v,Y2: set_v,Z: set_v] :
( ( ord_less_set_v @ X @ Y2 )
=> ( ( ord_less_eq_set_v @ Y2 @ Z )
=> ( ord_less_set_v @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1116_order__less__le__trans,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ X @ Y2 )
=> ( ( ord_le7336532860387713383od_v_v @ Y2 @ Z )
=> ( ord_le4186455585809229939od_v_v @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1117_order__le__less__subst2,axiom,
! [A: set_v,B: set_v,F: set_v > set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_set_v @ ( F @ B ) @ C2 )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_v @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1118_order__le__less__subst2,axiom,
! [A: set_v,B: set_v,F: set_v > set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_le4186455585809229939od_v_v @ ( F @ B ) @ C2 )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le4186455585809229939od_v_v @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1119_order__le__less__subst2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C2: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_less_set_v @ ( F @ B ) @ C2 )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_v @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1120_order__le__less__subst2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le4186455585809229939od_v_v @ ( F @ B ) @ C2 )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le4186455585809229939od_v_v @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1121_order__less__le__subst1,axiom,
! [A: set_v,F: set_v > set_v,B: set_v,C2: set_v] :
( ( ord_less_set_v @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_v @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1122_order__less__le__subst1,axiom,
! [A: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B: set_v,C2: set_v] :
( ( ord_le4186455585809229939od_v_v @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_v @ B @ C2 )
=> ( ! [X3: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le4186455585809229939od_v_v @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1123_order__less__le__subst1,axiom,
! [A: set_v,F: set_Product_prod_v_v > set_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_less_set_v @ A @ ( F @ B ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_v @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1124_order__less__le__subst1,axiom,
! [A: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ A @ ( F @ B ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C2 )
=> ( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le4186455585809229939od_v_v @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1125_order__le__imp__less__or__eq,axiom,
! [X: set_v,Y2: set_v] :
( ( ord_less_eq_set_v @ X @ Y2 )
=> ( ( ord_less_set_v @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1126_order__le__imp__less__or__eq,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y2 )
=> ( ( ord_le4186455585809229939od_v_v @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1127_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B3: set_v] :
( ( ord_less_set_v @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1128_subset__iff__psubset__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1129_subset__psubset__trans,axiom,
! [A3: set_v,B2: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A3 @ B2 )
=> ( ( ord_less_set_v @ B2 @ C )
=> ( ord_less_set_v @ A3 @ C ) ) ) ).
% subset_psubset_trans
thf(fact_1130_subset__psubset__trans,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
=> ( ( ord_le4186455585809229939od_v_v @ B2 @ C )
=> ( ord_le4186455585809229939od_v_v @ A3 @ C ) ) ) ).
% subset_psubset_trans
thf(fact_1131_subset__not__subset__eq,axiom,
( ord_less_set_v
= ( ^ [A4: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ A4 @ B3 )
& ~ ( ord_less_eq_set_v @ B3 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1132_subset__not__subset__eq,axiom,
( ord_le4186455585809229939od_v_v
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B3 )
& ~ ( ord_le7336532860387713383od_v_v @ B3 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1133_psubset__subset__trans,axiom,
! [A3: set_v,B2: set_v,C: set_v] :
( ( ord_less_set_v @ A3 @ B2 )
=> ( ( ord_less_eq_set_v @ B2 @ C )
=> ( ord_less_set_v @ A3 @ C ) ) ) ).
% psubset_subset_trans
thf(fact_1134_psubset__subset__trans,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ A3 @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ C )
=> ( ord_le4186455585809229939od_v_v @ A3 @ C ) ) ) ).
% psubset_subset_trans
thf(fact_1135_psubset__imp__subset,axiom,
! [A3: set_v,B2: set_v] :
( ( ord_less_set_v @ A3 @ B2 )
=> ( ord_less_eq_set_v @ A3 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_1136_psubset__imp__subset,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ A3 @ B2 )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_1137_psubset__eq,axiom,
( ord_less_set_v
= ( ^ [A4: set_v,B3: set_v] :
( ( ord_less_eq_set_v @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_1138_psubset__eq,axiom,
( ord_le4186455585809229939od_v_v
= ( ^ [A4: set_Product_prod_v_v,B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_1139_psubsetE,axiom,
! [A3: set_v,B2: set_v] :
( ( ord_less_set_v @ A3 @ B2 )
=> ~ ( ( ord_less_eq_set_v @ A3 @ B2 )
=> ( ord_less_eq_set_v @ B2 @ A3 ) ) ) ).
% psubsetE
thf(fact_1140_psubsetE,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ A3 @ B2 )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
=> ( ord_le7336532860387713383od_v_v @ B2 @ A3 ) ) ) ).
% psubsetE
thf(fact_1141_rev__image__eqI,axiom,
! [X: v,A3: set_v,B: v,F: v > v] :
( ( member_v @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_v @ B @ ( image_v_v @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_1142_rev__image__eqI,axiom,
! [X: v,A3: set_v,B: product_prod_v_v,F: v > product_prod_v_v] :
( ( member_v @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member7453568604450474000od_v_v @ B @ ( image_9222788639401671577od_v_v @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_1143_rev__image__eqI,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,B: v,F: product_prod_v_v > v] :
( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member_v @ B @ ( image_6152814753742948081_v_v_v @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_1144_rev__image__eqI,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,B: product_prod_v_v,F: product_prod_v_v > product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ( B
= ( F @ X ) )
=> ( member7453568604450474000od_v_v @ B @ ( image_781944334261467077od_v_v @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_1145_pairwiseI,axiom,
! [S: set_v,R5: v > v > $o] :
( ! [X3: v,Y: v] :
( ( member_v @ X3 @ S )
=> ( ( member_v @ Y @ S )
=> ( ( X3 != Y )
=> ( R5 @ X3 @ Y ) ) ) )
=> ( pairwise_v @ R5 @ S ) ) ).
% pairwiseI
thf(fact_1146_pairwiseI,axiom,
! [S: set_Product_prod_v_v,R5: product_prod_v_v > product_prod_v_v > $o] :
( ! [X3: product_prod_v_v,Y: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ S )
=> ( ( member7453568604450474000od_v_v @ Y @ S )
=> ( ( X3 != Y )
=> ( R5 @ X3 @ Y ) ) ) )
=> ( pairwi5745945156428401490od_v_v @ R5 @ S ) ) ).
% pairwiseI
thf(fact_1147_pairwiseD,axiom,
! [R5: v > v > $o,S: set_v,X: v,Y2: v] :
( ( pairwise_v @ R5 @ S )
=> ( ( member_v @ X @ S )
=> ( ( member_v @ Y2 @ S )
=> ( ( X != Y2 )
=> ( R5 @ X @ Y2 ) ) ) ) ) ).
% pairwiseD
thf(fact_1148_pairwiseD,axiom,
! [R5: product_prod_v_v > product_prod_v_v > $o,S: set_Product_prod_v_v,X: product_prod_v_v,Y2: product_prod_v_v] :
( ( pairwi5745945156428401490od_v_v @ R5 @ S )
=> ( ( member7453568604450474000od_v_v @ X @ S )
=> ( ( member7453568604450474000od_v_v @ Y2 @ S )
=> ( ( X != Y2 )
=> ( R5 @ X @ Y2 ) ) ) ) ) ).
% pairwiseD
thf(fact_1149_psubsetD,axiom,
! [A3: set_v,B2: set_v,C2: v] :
( ( ord_less_set_v @ A3 @ B2 )
=> ( ( member_v @ C2 @ A3 )
=> ( member_v @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_1150_psubsetD,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ A3 @ B2 )
=> ( ( member7453568604450474000od_v_v @ C2 @ A3 )
=> ( member7453568604450474000od_v_v @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_1151_imageI,axiom,
! [X: v,A3: set_v,F: v > v] :
( ( member_v @ X @ A3 )
=> ( member_v @ ( F @ X ) @ ( image_v_v @ F @ A3 ) ) ) ).
% imageI
thf(fact_1152_imageI,axiom,
! [X: v,A3: set_v,F: v > product_prod_v_v] :
( ( member_v @ X @ A3 )
=> ( member7453568604450474000od_v_v @ ( F @ X ) @ ( image_9222788639401671577od_v_v @ F @ A3 ) ) ) ).
% imageI
thf(fact_1153_imageI,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,F: product_prod_v_v > v] :
( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( member_v @ ( F @ X ) @ ( image_6152814753742948081_v_v_v @ F @ A3 ) ) ) ).
% imageI
thf(fact_1154_imageI,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,F: product_prod_v_v > product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( member7453568604450474000od_v_v @ ( F @ X ) @ ( image_781944334261467077od_v_v @ F @ A3 ) ) ) ).
% imageI
thf(fact_1155_all__subset__image,axiom,
! [F: v > v,A3: set_v,P: set_v > $o] :
( ( ! [B3: set_v] :
( ( ord_less_eq_set_v @ B3 @ ( image_v_v @ F @ A3 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_v] :
( ( ord_less_eq_set_v @ B3 @ A3 )
=> ( P @ ( image_v_v @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_1156_all__subset__image,axiom,
! [F: product_prod_v_v > v,A3: set_Product_prod_v_v,P: set_v > $o] :
( ( ! [B3: set_v] :
( ( ord_less_eq_set_v @ B3 @ ( image_6152814753742948081_v_v_v @ F @ A3 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B3 @ A3 )
=> ( P @ ( image_6152814753742948081_v_v_v @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_1157_all__subset__image,axiom,
! [F: v > product_prod_v_v,A3: set_v,P: set_Product_prod_v_v > $o] :
( ( ! [B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B3 @ ( image_9222788639401671577od_v_v @ F @ A3 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_v] :
( ( ord_less_eq_set_v @ B3 @ A3 )
=> ( P @ ( image_9222788639401671577od_v_v @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_1158_all__subset__image,axiom,
! [F: product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( ! [B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B3 @ ( image_781944334261467077od_v_v @ F @ A3 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B3 @ A3 )
=> ( P @ ( image_781944334261467077od_v_v @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_1159_less__infI1,axiom,
! [A: set_v,X: set_v,B: set_v] :
( ( ord_less_set_v @ A @ X )
=> ( ord_less_set_v @ ( inf_inf_set_v @ A @ B ) @ X ) ) ).
% less_infI1
thf(fact_1160_less__infI2,axiom,
! [B: set_v,X: set_v,A: set_v] :
( ( ord_less_set_v @ B @ X )
=> ( ord_less_set_v @ ( inf_inf_set_v @ A @ B ) @ X ) ) ).
% less_infI2
thf(fact_1161_inf_Oabsorb3,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_set_v @ A @ B )
=> ( ( inf_inf_set_v @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_1162_inf_Oabsorb4,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_set_v @ B @ A )
=> ( ( inf_inf_set_v @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_1163_inf_Ostrict__boundedE,axiom,
! [A: set_v,B: set_v,C2: set_v] :
( ( ord_less_set_v @ A @ ( inf_inf_set_v @ B @ C2 ) )
=> ~ ( ( ord_less_set_v @ A @ B )
=> ~ ( ord_less_set_v @ A @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_1164_inf_Ostrict__order__iff,axiom,
( ord_less_set_v
= ( ^ [A5: set_v,B5: set_v] :
( ( A5
= ( inf_inf_set_v @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_1165_inf_Ostrict__coboundedI1,axiom,
! [A: set_v,C2: set_v,B: set_v] :
( ( ord_less_set_v @ A @ C2 )
=> ( ord_less_set_v @ ( inf_inf_set_v @ A @ B ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_1166_inf_Ostrict__coboundedI2,axiom,
! [B: set_v,C2: set_v,A: set_v] :
( ( ord_less_set_v @ B @ C2 )
=> ( ord_less_set_v @ ( inf_inf_set_v @ A @ B ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_1167_pairwise__subset,axiom,
! [P: v > v > $o,S: set_v,T2: set_v] :
( ( pairwise_v @ P @ S )
=> ( ( ord_less_eq_set_v @ T2 @ S )
=> ( pairwise_v @ P @ T2 ) ) ) ).
% pairwise_subset
thf(fact_1168_pairwise__subset,axiom,
! [P: product_prod_v_v > product_prod_v_v > $o,S: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( pairwi5745945156428401490od_v_v @ P @ S )
=> ( ( ord_le7336532860387713383od_v_v @ T2 @ S )
=> ( pairwi5745945156428401490od_v_v @ P @ T2 ) ) ) ).
% pairwise_subset
thf(fact_1169_pairwise__mono,axiom,
! [P: v > v > $o,A3: set_v,Q: v > v > $o,B2: set_v] :
( ( pairwise_v @ P @ A3 )
=> ( ! [X3: v,Y: v] :
( ( P @ X3 @ Y )
=> ( Q @ X3 @ Y ) )
=> ( ( ord_less_eq_set_v @ B2 @ A3 )
=> ( pairwise_v @ Q @ B2 ) ) ) ) ).
% pairwise_mono
thf(fact_1170_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,B2: set_Product_prod_v_v] :
( ( pairwi5745945156428401490od_v_v @ P @ A3 )
=> ( ! [X3: product_prod_v_v,Y: product_prod_v_v] :
( ( P @ X3 @ Y )
=> ( Q @ X3 @ Y ) )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ A3 )
=> ( pairwi5745945156428401490od_v_v @ Q @ B2 ) ) ) ) ).
% pairwise_mono
thf(fact_1171_image__Int__subset,axiom,
! [F: v > v,A3: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( image_v_v @ F @ ( inf_inf_set_v @ A3 @ B2 ) ) @ ( inf_inf_set_v @ ( image_v_v @ F @ A3 ) @ ( image_v_v @ F @ B2 ) ) ) ).
% image_Int_subset
thf(fact_1172_image__Int__subset,axiom,
! [F: v > product_prod_v_v,A3: set_v,B2: set_v] : ( ord_le7336532860387713383od_v_v @ ( image_9222788639401671577od_v_v @ F @ ( inf_inf_set_v @ A3 @ B2 ) ) @ ( inf_in6271465464967711157od_v_v @ ( image_9222788639401671577od_v_v @ F @ A3 ) @ ( image_9222788639401671577od_v_v @ F @ B2 ) ) ) ).
% image_Int_subset
thf(fact_1173_pairwise__empty,axiom,
! [P: v > v > $o] : ( pairwise_v @ P @ bot_bot_set_v ) ).
% pairwise_empty
thf(fact_1174_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_1175_pairwise__insert,axiom,
! [R: v > v > $o,X: v,S5: set_v] :
( ( pairwise_v @ R @ ( insert_v2 @ X @ S5 ) )
= ( ! [Y3: v] :
( ( ( member_v @ Y3 @ S5 )
& ( Y3 != X ) )
=> ( ( R @ X @ Y3 )
& ( R @ Y3 @ X ) ) )
& ( pairwise_v @ R @ S5 ) ) ) ).
% pairwise_insert
thf(fact_1176_pairwise__insert,axiom,
! [R: product_prod_v_v > product_prod_v_v > $o,X: product_prod_v_v,S5: set_Product_prod_v_v] :
( ( pairwi5745945156428401490od_v_v @ R @ ( insert1338601472111419319od_v_v @ X @ S5 ) )
= ( ! [Y3: product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ Y3 @ S5 )
& ( Y3 != X ) )
=> ( ( R @ X @ Y3 )
& ( R @ Y3 @ X ) ) )
& ( pairwi5745945156428401490od_v_v @ R @ S5 ) ) ) ).
% pairwise_insert
thf(fact_1177_not__psubset__empty,axiom,
! [A3: set_v] :
~ ( ord_less_set_v @ A3 @ bot_bot_set_v ) ).
% not_psubset_empty
thf(fact_1178_not__psubset__empty,axiom,
! [A3: set_Product_prod_v_v] :
~ ( ord_le4186455585809229939od_v_v @ A3 @ bot_bo723834152578015283od_v_v ) ).
% not_psubset_empty
thf(fact_1179_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_1180_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_1181_bot_Oextremum__strict,axiom,
! [A: set_v] :
~ ( ord_less_set_v @ A @ bot_bot_set_v ) ).
% bot.extremum_strict
thf(fact_1182_bot_Oextremum__strict,axiom,
! [A: set_Product_prod_v_v] :
~ ( ord_le4186455585809229939od_v_v @ A @ bot_bo723834152578015283od_v_v ) ).
% bot.extremum_strict
thf(fact_1183_psubset__imp__ex__mem,axiom,
! [A3: set_v,B2: set_v] :
( ( ord_less_set_v @ A3 @ B2 )
=> ? [B7: v] : ( member_v @ B7 @ ( minus_minus_set_v @ B2 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1184_psubset__imp__ex__mem,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ A3 @ B2 )
=> ? [B7: product_prod_v_v] : ( member7453568604450474000od_v_v @ B7 @ ( minus_4183494784930505774od_v_v @ B2 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1185_Cons__shuffles__subset1,axiom,
! [X: v,Xs: list_v,Ys: list_v] : ( ord_le1129530298279361049list_v @ ( image_list_v_list_v @ ( cons_v @ X ) @ ( shuffles_v @ Xs @ Ys ) ) @ ( shuffles_v @ ( cons_v @ X @ Xs ) @ Ys ) ) ).
% Cons_shuffles_subset1
thf(fact_1186_Cons__shuffles__subset2,axiom,
! [Y2: v,Xs: list_v,Ys: list_v] : ( ord_le1129530298279361049list_v @ ( image_list_v_list_v @ ( cons_v @ Y2 ) @ ( shuffles_v @ Xs @ Ys ) ) @ ( shuffles_v @ Xs @ ( cons_v @ Y2 @ Ys ) ) ) ).
% Cons_shuffles_subset2
thf(fact_1187_pairwise__singleton,axiom,
! [P: v > v > $o,A3: v] : ( pairwise_v @ P @ ( insert_v2 @ A3 @ bot_bot_set_v ) ) ).
% pairwise_singleton
thf(fact_1188_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_1189_underS__Field3,axiom,
! [R: set_Product_prod_v_v,A: v] :
( ( ( field_v @ R )
!= bot_bot_set_v )
=> ( ord_less_set_v @ ( order_underS_v @ R @ A ) @ ( field_v @ R ) ) ) ).
% underS_Field3
thf(fact_1190_underS__Field3,axiom,
! [R: set_Pr2149350503807050951od_v_v,A: product_prod_v_v] :
( ( ( field_7153129647634986036od_v_v @ R )
!= bot_bo723834152578015283od_v_v )
=> ( ord_le4186455585809229939od_v_v @ ( order_5211820470575790509od_v_v @ R @ A ) @ ( field_7153129647634986036od_v_v @ R ) ) ) ).
% underS_Field3
thf(fact_1191_Sup__fin_Ohom__commute,axiom,
! [H: set_Product_prod_v_v > set_Product_prod_v_v,N3: set_se8455005133513928103od_v_v] :
( ! [X3: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( H @ ( sup_su414716646722978715od_v_v @ X3 @ Y ) )
= ( sup_su414716646722978715od_v_v @ ( H @ X3 ) @ ( H @ Y ) ) )
=> ( ( finite6084192165098772208od_v_v @ N3 )
=> ( ( N3 != bot_bo3497076220358800403od_v_v )
=> ( ( H @ ( lattic5151207300795964030od_v_v @ N3 ) )
= ( lattic5151207300795964030od_v_v @ ( image_5212826947168092101od_v_v @ H @ N3 ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_1192_Inf__fin_Ohom__commute,axiom,
! [H: set_v > set_v,N3: set_set_v] :
( ! [X3: set_v,Y: set_v] :
( ( H @ ( inf_inf_set_v @ X3 @ Y ) )
= ( inf_inf_set_v @ ( H @ X3 ) @ ( H @ Y ) ) )
=> ( ( finite_finite_set_v @ N3 )
=> ( ( N3 != bot_bot_set_set_v )
=> ( ( H @ ( lattic8209813555532694032_set_v @ N3 ) )
= ( lattic8209813555532694032_set_v @ ( image_set_v_set_v @ H @ N3 ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_1193_finite__induct__select,axiom,
! [S: set_v,P: set_v > $o] :
( ( finite_finite_v @ S )
=> ( ( P @ bot_bot_set_v )
=> ( ! [T3: set_v] :
( ( ord_less_set_v @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X4: v] :
( ( member_v @ X4 @ ( minus_minus_set_v @ S @ T3 ) )
& ( P @ ( insert_v2 @ X4 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1194_finite__induct__select,axiom,
! [S: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ S )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [T3: set_Product_prod_v_v] :
( ( ord_le4186455585809229939od_v_v @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ ( minus_4183494784930505774od_v_v @ S @ T3 ) )
& ( P @ ( insert1338601472111419319od_v_v @ X4 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1195_in__image__insert__iff,axiom,
! [B2: set_set_v,X: v,A3: set_v] :
( ! [C4: set_v] :
( ( member_set_v @ C4 @ B2 )
=> ~ ( member_v @ X @ C4 ) )
=> ( ( member_set_v @ A3 @ ( image_set_v_set_v @ ( insert_v2 @ X ) @ B2 ) )
= ( ( member_v @ X @ A3 )
& ( member_set_v @ ( minus_minus_set_v @ A3 @ ( insert_v2 @ X @ bot_bot_set_v ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1196_in__image__insert__iff,axiom,
! [B2: set_se8455005133513928103od_v_v,X: product_prod_v_v,A3: set_Product_prod_v_v] :
( ! [C4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ C4 @ B2 )
=> ~ ( member7453568604450474000od_v_v @ X @ C4 ) )
=> ( ( member8406446414694345712od_v_v @ A3 @ ( image_5212826947168092101od_v_v @ ( insert1338601472111419319od_v_v @ X ) @ B2 ) )
= ( ( member7453568604450474000od_v_v @ X @ A3 )
& ( member8406446414694345712od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1197_verit__comp__simplify1_I2_J,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1198_verit__comp__simplify1_I2_J,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1199_shuffles_Opelims,axiom,
! [X: list_v,Xa3: list_v,Y2: set_list_v] :
( ( ( shuffles_v @ X @ Xa3 )
= Y2 )
=> ( ( accp_P8827133495749382256list_v @ shuffles_rel_v @ ( produc6795410681906604247list_v @ X @ Xa3 ) )
=> ( ( ( X = nil_v )
=> ( ( Y2
= ( insert_list_v @ Xa3 @ bot_bot_set_list_v ) )
=> ~ ( accp_P8827133495749382256list_v @ shuffles_rel_v @ ( produc6795410681906604247list_v @ nil_v @ Xa3 ) ) ) )
=> ( ( ( Xa3 = nil_v )
=> ( ( Y2
= ( insert_list_v @ X @ bot_bot_set_list_v ) )
=> ~ ( accp_P8827133495749382256list_v @ shuffles_rel_v @ ( produc6795410681906604247list_v @ X @ nil_v ) ) ) )
=> ~ ! [X3: v,Xs3: list_v] :
( ( X
= ( cons_v @ X3 @ Xs3 ) )
=> ! [Y: v,Ys3: list_v] :
( ( Xa3
= ( cons_v @ Y @ Ys3 ) )
=> ( ( Y2
= ( sup_sup_set_list_v @ ( image_list_v_list_v @ ( cons_v @ X3 ) @ ( shuffles_v @ Xs3 @ ( cons_v @ Y @ Ys3 ) ) ) @ ( image_list_v_list_v @ ( cons_v @ Y ) @ ( shuffles_v @ ( cons_v @ X3 @ Xs3 ) @ Ys3 ) ) ) )
=> ~ ( accp_P8827133495749382256list_v @ shuffles_rel_v @ ( produc6795410681906604247list_v @ ( cons_v @ X3 @ Xs3 ) @ ( cons_v @ Y @ Ys3 ) ) ) ) ) ) ) ) ) ) ).
% shuffles.pelims
thf(fact_1200_Fpow__mono,axiom,
! [A3: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A3 @ B2 )
=> ( ord_le5216385588623774835_set_v @ ( finite_Fpow_v @ A3 ) @ ( finite_Fpow_v @ B2 ) ) ) ).
% Fpow_mono
thf(fact_1201_Fpow__mono,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
=> ( ord_le4714265922333009223od_v_v @ ( finite315275465967970893od_v_v @ A3 ) @ ( finite315275465967970893od_v_v @ B2 ) ) ) ).
% Fpow_mono
thf(fact_1202_empty__in__Fpow,axiom,
! [A3: set_v] : ( member_set_v @ bot_bot_set_v @ ( finite_Fpow_v @ A3 ) ) ).
% empty_in_Fpow
thf(fact_1203_empty__in__Fpow,axiom,
! [A3: set_Product_prod_v_v] : ( member8406446414694345712od_v_v @ bot_bo723834152578015283od_v_v @ ( finite315275465967970893od_v_v @ A3 ) ) ).
% empty_in_Fpow
thf(fact_1204_subset__subseqs,axiom,
! [X5: set_v,Xs: list_v] :
( ( ord_less_eq_set_v @ X5 @ ( set_v2 @ Xs ) )
=> ( member_set_v @ X5 @ ( image_list_v_set_v @ set_v2 @ ( set_list_v2 @ ( subseqs_v @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_1205_subset__subseqs,axiom,
! [X5: set_Product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( ord_le7336532860387713383od_v_v @ X5 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( member8406446414694345712od_v_v @ X5 @ ( image_2941599411333419807od_v_v @ set_Product_prod_v_v2 @ ( set_li2340707408155270402od_v_v @ ( subseq9186754075894343740od_v_v @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_1206_Inf__fin_Osemilattice__order__set__axioms,axiom,
lattic8986249360443818957_set_v @ inf_inf_set_v @ ord_less_eq_set_v @ ord_less_set_v ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_1207_Inf__fin_Osemilattice__order__set__axioms,axiom,
lattic3284118369257910689od_v_v @ inf_in6271465464967711157od_v_v @ ord_le7336532860387713383od_v_v @ ord_le4186455585809229939od_v_v ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_1208_semilattice__order__set_Osubset__imp,axiom,
! [F: v > v > v,Less_eq: v > v > $o,Less: v > v > $o,A3: set_v,B2: set_v] :
( ( lattic5078705180708912365_set_v @ F @ Less_eq @ Less )
=> ( ( ord_less_eq_set_v @ A3 @ B2 )
=> ( ( A3 != bot_bot_set_v )
=> ( ( finite_finite_v @ B2 )
=> ( Less_eq @ ( lattic5116578512385870317ce_F_v @ F @ B2 ) @ ( lattic5116578512385870317ce_F_v @ F @ A3 ) ) ) ) ) ) ).
% semilattice_order_set.subset_imp
thf(fact_1209_semilattice__order__set_Osubset__imp,axiom,
! [F: product_prod_v_v > product_prod_v_v > product_prod_v_v,Less_eq: product_prod_v_v > product_prod_v_v > $o,Less: product_prod_v_v > product_prod_v_v > $o,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( lattic3597912463104479681od_v_v @ F @ Less_eq @ Less )
=> ( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
=> ( ( A3 != bot_bo723834152578015283od_v_v )
=> ( ( finite3348123685078250256od_v_v @ B2 )
=> ( Less_eq @ ( lattic8975544278839647937od_v_v @ F @ B2 ) @ ( lattic8975544278839647937od_v_v @ F @ A3 ) ) ) ) ) ) ).
% semilattice_order_set.subset_imp
thf(fact_1210_PowI,axiom,
! [A3: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A3 @ B2 )
=> ( member_set_v @ A3 @ ( pow_v @ B2 ) ) ) ).
% PowI
thf(fact_1211_PowI,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
=> ( member8406446414694345712od_v_v @ A3 @ ( pow_Product_prod_v_v @ B2 ) ) ) ).
% PowI
thf(fact_1212_Pow__iff,axiom,
! [A3: set_v,B2: set_v] :
( ( member_set_v @ A3 @ ( pow_v @ B2 ) )
= ( ord_less_eq_set_v @ A3 @ B2 ) ) ).
% Pow_iff
thf(fact_1213_Pow__iff,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A3 @ ( pow_Product_prod_v_v @ B2 ) )
= ( ord_le7336532860387713383od_v_v @ A3 @ B2 ) ) ).
% Pow_iff
thf(fact_1214_Pow__Int__eq,axiom,
! [A3: set_v,B2: set_v] :
( ( pow_v @ ( inf_inf_set_v @ A3 @ B2 ) )
= ( inf_inf_set_set_v @ ( pow_v @ A3 ) @ ( pow_v @ B2 ) ) ) ).
% Pow_Int_eq
thf(fact_1215_Pow__empty,axiom,
( ( pow_v @ bot_bot_set_v )
= ( insert_set_v @ bot_bot_set_v @ bot_bot_set_set_v ) ) ).
% Pow_empty
thf(fact_1216_Pow__empty,axiom,
( ( pow_Product_prod_v_v @ bot_bo723834152578015283od_v_v )
= ( insert7504383016908236695od_v_v @ bot_bo723834152578015283od_v_v @ bot_bo3497076220358800403od_v_v ) ) ).
% Pow_empty
thf(fact_1217_Pow__singleton__iff,axiom,
! [X5: set_v,Y5: set_v] :
( ( ( pow_v @ X5 )
= ( insert_set_v @ Y5 @ bot_bot_set_set_v ) )
= ( ( X5 = bot_bot_set_v )
& ( Y5 = bot_bot_set_v ) ) ) ).
% Pow_singleton_iff
thf(fact_1218_Pow__singleton__iff,axiom,
! [X5: set_Product_prod_v_v,Y5: set_Product_prod_v_v] :
( ( ( pow_Product_prod_v_v @ X5 )
= ( insert7504383016908236695od_v_v @ Y5 @ bot_bo3497076220358800403od_v_v ) )
= ( ( X5 = bot_bo723834152578015283od_v_v )
& ( Y5 = bot_bo723834152578015283od_v_v ) ) ) ).
% Pow_singleton_iff
thf(fact_1219_Pow__bottom,axiom,
! [B2: set_v] : ( member_set_v @ bot_bot_set_v @ ( pow_v @ B2 ) ) ).
% Pow_bottom
thf(fact_1220_Pow__bottom,axiom,
! [B2: set_Product_prod_v_v] : ( member8406446414694345712od_v_v @ bot_bo723834152578015283od_v_v @ ( pow_Product_prod_v_v @ B2 ) ) ).
% Pow_bottom
thf(fact_1221_Pow__insert,axiom,
! [A: v,A3: set_v] :
( ( pow_v @ ( insert_v2 @ A @ A3 ) )
= ( sup_sup_set_set_v @ ( pow_v @ A3 ) @ ( image_set_v_set_v @ ( insert_v2 @ A ) @ ( pow_v @ A3 ) ) ) ) ).
% Pow_insert
thf(fact_1222_Pow__insert,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( pow_Product_prod_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) )
= ( sup_su335656005089752955od_v_v @ ( pow_Product_prod_v_v @ A3 ) @ ( image_5212826947168092101od_v_v @ ( insert1338601472111419319od_v_v @ A ) @ ( pow_Product_prod_v_v @ A3 ) ) ) ) ).
% Pow_insert
thf(fact_1223_Inf__fin__def,axiom,
( lattic8209813555532694032_set_v
= ( lattic2714821108077596877_set_v @ inf_inf_set_v ) ) ).
% Inf_fin_def
thf(fact_1224_PowD,axiom,
! [A3: set_v,B2: set_v] :
( ( member_set_v @ A3 @ ( pow_v @ B2 ) )
=> ( ord_less_eq_set_v @ A3 @ B2 ) ) ).
% PowD
thf(fact_1225_PowD,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A3 @ ( pow_Product_prod_v_v @ B2 ) )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B2 ) ) ).
% PowD
thf(fact_1226_Pow__mono,axiom,
! [A3: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A3 @ B2 )
=> ( ord_le5216385588623774835_set_v @ ( pow_v @ A3 ) @ ( pow_v @ B2 ) ) ) ).
% Pow_mono
thf(fact_1227_Pow__mono,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B2 )
=> ( ord_le4714265922333009223od_v_v @ ( pow_Product_prod_v_v @ A3 ) @ ( pow_Product_prod_v_v @ B2 ) ) ) ).
% Pow_mono
thf(fact_1228_Un__Pow__subset,axiom,
! [A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le4714265922333009223od_v_v @ ( sup_su335656005089752955od_v_v @ ( pow_Product_prod_v_v @ A3 ) @ ( pow_Product_prod_v_v @ B2 ) ) @ ( pow_Product_prod_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) ) ) ).
% Un_Pow_subset
thf(fact_1229_semilattice__order__set_OboundedE,axiom,
! [F: v > v > v,Less_eq: v > v > $o,Less: v > v > $o,A3: set_v,X: v] :
( ( lattic5078705180708912365_set_v @ F @ Less_eq @ Less )
=> ( ( finite_finite_v @ A3 )
=> ( ( A3 != bot_bot_set_v )
=> ( ( Less_eq @ X @ ( lattic5116578512385870317ce_F_v @ F @ A3 ) )
=> ! [A9: v] :
( ( member_v @ A9 @ A3 )
=> ( Less_eq @ X @ A9 ) ) ) ) ) ) ).
% semilattice_order_set.boundedE
thf(fact_1230_semilattice__order__set_OboundedE,axiom,
! [F: product_prod_v_v > product_prod_v_v > product_prod_v_v,Less_eq: product_prod_v_v > product_prod_v_v > $o,Less: product_prod_v_v > product_prod_v_v > $o,A3: set_Product_prod_v_v,X: product_prod_v_v] :
( ( lattic3597912463104479681od_v_v @ F @ Less_eq @ Less )
=> ( ( finite3348123685078250256od_v_v @ A3 )
=> ( ( A3 != bot_bo723834152578015283od_v_v )
=> ( ( Less_eq @ X @ ( lattic8975544278839647937od_v_v @ F @ A3 ) )
=> ! [A9: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A9 @ A3 )
=> ( Less_eq @ X @ A9 ) ) ) ) ) ) ).
% semilattice_order_set.boundedE
thf(fact_1231_semilattice__order__set_OboundedI,axiom,
! [F: v > v > v,Less_eq: v > v > $o,Less: v > v > $o,A3: set_v,X: v] :
( ( lattic5078705180708912365_set_v @ F @ Less_eq @ Less )
=> ( ( finite_finite_v @ A3 )
=> ( ( A3 != bot_bot_set_v )
=> ( ! [A7: v] :
( ( member_v @ A7 @ A3 )
=> ( Less_eq @ X @ A7 ) )
=> ( Less_eq @ X @ ( lattic5116578512385870317ce_F_v @ F @ A3 ) ) ) ) ) ) ).
% semilattice_order_set.boundedI
thf(fact_1232_semilattice__order__set_OboundedI,axiom,
! [F: product_prod_v_v > product_prod_v_v > product_prod_v_v,Less_eq: product_prod_v_v > product_prod_v_v > $o,Less: product_prod_v_v > product_prod_v_v > $o,A3: set_Product_prod_v_v,X: product_prod_v_v] :
( ( lattic3597912463104479681od_v_v @ F @ Less_eq @ Less )
=> ( ( finite3348123685078250256od_v_v @ A3 )
=> ( ( A3 != bot_bo723834152578015283od_v_v )
=> ( ! [A7: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A7 @ A3 )
=> ( Less_eq @ X @ A7 ) )
=> ( Less_eq @ X @ ( lattic8975544278839647937od_v_v @ F @ A3 ) ) ) ) ) ) ).
% semilattice_order_set.boundedI
thf(fact_1233_semilattice__order__set_Obounded__iff,axiom,
! [F: v > v > v,Less_eq: v > v > $o,Less: v > v > $o,A3: set_v,X: v] :
( ( lattic5078705180708912365_set_v @ F @ Less_eq @ Less )
=> ( ( finite_finite_v @ A3 )
=> ( ( A3 != bot_bot_set_v )
=> ( ( Less_eq @ X @ ( lattic5116578512385870317ce_F_v @ F @ A3 ) )
= ( ! [X2: v] :
( ( member_v @ X2 @ A3 )
=> ( Less_eq @ X @ X2 ) ) ) ) ) ) ) ).
% semilattice_order_set.bounded_iff
thf(fact_1234_semilattice__order__set_Obounded__iff,axiom,
! [F: product_prod_v_v > product_prod_v_v > product_prod_v_v,Less_eq: product_prod_v_v > product_prod_v_v > $o,Less: product_prod_v_v > product_prod_v_v > $o,A3: set_Product_prod_v_v,X: product_prod_v_v] :
( ( lattic3597912463104479681od_v_v @ F @ Less_eq @ Less )
=> ( ( finite3348123685078250256od_v_v @ A3 )
=> ( ( A3 != bot_bo723834152578015283od_v_v )
=> ( ( Less_eq @ X @ ( lattic8975544278839647937od_v_v @ F @ A3 ) )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ( Less_eq @ X @ X2 ) ) ) ) ) ) ) ).
% semilattice_order_set.bounded_iff
thf(fact_1235_Pow__set_I1_J,axiom,
( ( pow_v @ ( set_v2 @ nil_v ) )
= ( insert_set_v @ bot_bot_set_v @ bot_bot_set_set_v ) ) ).
% Pow_set(1)
thf(fact_1236_Pow__set_I1_J,axiom,
( ( pow_Product_prod_v_v @ ( set_Product_prod_v_v2 @ nil_Product_prod_v_v ) )
= ( insert7504383016908236695od_v_v @ bot_bo723834152578015283od_v_v @ bot_bo3497076220358800403od_v_v ) ) ).
% Pow_set(1)
thf(fact_1237_semilattice__set_Oremove,axiom,
! [F: v > v > v,A3: set_v,X: v] :
( ( lattic5961991414251573153_set_v @ F )
=> ( ( finite_finite_v @ A3 )
=> ( ( member_v @ X @ A3 )
=> ( ( ( ( minus_minus_set_v @ A3 @ ( insert_v2 @ X @ bot_bot_set_v ) )
= bot_bot_set_v )
=> ( ( lattic5116578512385870317ce_F_v @ F @ A3 )
= X ) )
& ( ( ( minus_minus_set_v @ A3 @ ( insert_v2 @ X @ bot_bot_set_v ) )
!= bot_bot_set_v )
=> ( ( lattic5116578512385870317ce_F_v @ F @ A3 )
= ( F @ X @ ( lattic5116578512385870317ce_F_v @ F @ ( minus_minus_set_v @ A3 @ ( insert_v2 @ X @ bot_bot_set_v ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.remove
thf(fact_1238_semilattice__set_Oremove,axiom,
! [F: product_prod_v_v > product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v,X: product_prod_v_v] :
( ( lattic4743553628190527605od_v_v @ F )
=> ( ( finite3348123685078250256od_v_v @ A3 )
=> ( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ( ( ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= bot_bo723834152578015283od_v_v )
=> ( ( lattic8975544278839647937od_v_v @ F @ A3 )
= X ) )
& ( ( ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
!= bot_bo723834152578015283od_v_v )
=> ( ( lattic8975544278839647937od_v_v @ F @ A3 )
= ( F @ X @ ( lattic8975544278839647937od_v_v @ F @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.remove
thf(fact_1239_semilattice__set_Oinsert__remove,axiom,
! [F: v > v > v,A3: set_v,X: v] :
( ( lattic5961991414251573153_set_v @ F )
=> ( ( finite_finite_v @ A3 )
=> ( ( ( ( minus_minus_set_v @ A3 @ ( insert_v2 @ X @ bot_bot_set_v ) )
= bot_bot_set_v )
=> ( ( lattic5116578512385870317ce_F_v @ F @ ( insert_v2 @ X @ A3 ) )
= X ) )
& ( ( ( minus_minus_set_v @ A3 @ ( insert_v2 @ X @ bot_bot_set_v ) )
!= bot_bot_set_v )
=> ( ( lattic5116578512385870317ce_F_v @ F @ ( insert_v2 @ X @ A3 ) )
= ( F @ X @ ( lattic5116578512385870317ce_F_v @ F @ ( minus_minus_set_v @ A3 @ ( insert_v2 @ X @ bot_bot_set_v ) ) ) ) ) ) ) ) ) ).
% semilattice_set.insert_remove
thf(fact_1240_semilattice__set_Oinsert__remove,axiom,
! [F: product_prod_v_v > product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v,X: product_prod_v_v] :
( ( lattic4743553628190527605od_v_v @ F )
=> ( ( finite3348123685078250256od_v_v @ A3 )
=> ( ( ( ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= bot_bo723834152578015283od_v_v )
=> ( ( lattic8975544278839647937od_v_v @ F @ ( insert1338601472111419319od_v_v @ X @ A3 ) )
= X ) )
& ( ( ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
!= bot_bo723834152578015283od_v_v )
=> ( ( lattic8975544278839647937od_v_v @ F @ ( insert1338601472111419319od_v_v @ X @ A3 ) )
= ( F @ X @ ( lattic8975544278839647937od_v_v @ F @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ) ) ).
% semilattice_set.insert_remove
thf(fact_1241_Inf__fin_Osemilattice__set__axioms,axiom,
lattic1258622430249649793_set_v @ inf_inf_set_v ).
% Inf_fin.semilattice_set_axioms
thf(fact_1242_semilattice__set_Osingleton,axiom,
! [F: v > v > v,X: v] :
( ( lattic5961991414251573153_set_v @ F )
=> ( ( lattic5116578512385870317ce_F_v @ F @ ( insert_v2 @ X @ bot_bot_set_v ) )
= X ) ) ).
% semilattice_set.singleton
thf(fact_1243_semilattice__set_Osingleton,axiom,
! [F: product_prod_v_v > product_prod_v_v > product_prod_v_v,X: product_prod_v_v] :
( ( lattic4743553628190527605od_v_v @ F )
=> ( ( lattic8975544278839647937od_v_v @ F @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= X ) ) ).
% semilattice_set.singleton
thf(fact_1244_semilattice__set_Ohom__commute,axiom,
! [F: v > v > v,H: v > v,N3: set_v] :
( ( lattic5961991414251573153_set_v @ F )
=> ( ! [X3: v,Y: v] :
( ( H @ ( F @ X3 @ Y ) )
= ( F @ ( H @ X3 ) @ ( H @ Y ) ) )
=> ( ( finite_finite_v @ N3 )
=> ( ( N3 != bot_bot_set_v )
=> ( ( H @ ( lattic5116578512385870317ce_F_v @ F @ N3 ) )
= ( lattic5116578512385870317ce_F_v @ F @ ( image_v_v @ H @ N3 ) ) ) ) ) ) ) ).
% semilattice_set.hom_commute
thf(fact_1245_semilattice__set_Ohom__commute,axiom,
! [F: product_prod_v_v > product_prod_v_v > product_prod_v_v,H: product_prod_v_v > product_prod_v_v,N3: set_Product_prod_v_v] :
( ( lattic4743553628190527605od_v_v @ F )
=> ( ! [X3: product_prod_v_v,Y: product_prod_v_v] :
( ( H @ ( F @ X3 @ Y ) )
= ( F @ ( H @ X3 ) @ ( H @ Y ) ) )
=> ( ( finite3348123685078250256od_v_v @ N3 )
=> ( ( N3 != bot_bo723834152578015283od_v_v )
=> ( ( H @ ( lattic8975544278839647937od_v_v @ F @ N3 ) )
= ( lattic8975544278839647937od_v_v @ F @ ( image_781944334261467077od_v_v @ H @ N3 ) ) ) ) ) ) ) ).
% semilattice_set.hom_commute
thf(fact_1246_semilattice__set_Osubset,axiom,
! [F: v > v > v,A3: set_v,B2: set_v] :
( ( lattic5961991414251573153_set_v @ F )
=> ( ( finite_finite_v @ A3 )
=> ( ( B2 != bot_bot_set_v )
=> ( ( ord_less_eq_set_v @ B2 @ A3 )
=> ( ( F @ ( lattic5116578512385870317ce_F_v @ F @ B2 ) @ ( lattic5116578512385870317ce_F_v @ F @ A3 ) )
= ( lattic5116578512385870317ce_F_v @ F @ A3 ) ) ) ) ) ) ).
% semilattice_set.subset
thf(fact_1247_semilattice__set_Osubset,axiom,
! [F: product_prod_v_v > product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( lattic4743553628190527605od_v_v @ F )
=> ( ( finite3348123685078250256od_v_v @ A3 )
=> ( ( B2 != bot_bo723834152578015283od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ A3 )
=> ( ( F @ ( lattic8975544278839647937od_v_v @ F @ B2 ) @ ( lattic8975544278839647937od_v_v @ F @ A3 ) )
= ( lattic8975544278839647937od_v_v @ F @ A3 ) ) ) ) ) ) ).
% semilattice_set.subset
thf(fact_1248_semilattice__set_Oclosed,axiom,
! [F: v > v > v,A3: set_v] :
( ( lattic5961991414251573153_set_v @ F )
=> ( ( finite_finite_v @ A3 )
=> ( ( A3 != bot_bot_set_v )
=> ( ! [X3: v,Y: v] : ( member_v @ ( F @ X3 @ Y ) @ ( insert_v2 @ X3 @ ( insert_v2 @ Y @ bot_bot_set_v ) ) )
=> ( member_v @ ( lattic5116578512385870317ce_F_v @ F @ A3 ) @ A3 ) ) ) ) ) ).
% semilattice_set.closed
thf(fact_1249_semilattice__set_Oclosed,axiom,
! [F: product_prod_v_v > product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v] :
( ( lattic4743553628190527605od_v_v @ F )
=> ( ( finite3348123685078250256od_v_v @ A3 )
=> ( ( A3 != bot_bo723834152578015283od_v_v )
=> ( ! [X3: product_prod_v_v,Y: product_prod_v_v] : ( member7453568604450474000od_v_v @ ( F @ X3 @ Y ) @ ( insert1338601472111419319od_v_v @ X3 @ ( insert1338601472111419319od_v_v @ Y @ bot_bo723834152578015283od_v_v ) ) )
=> ( member7453568604450474000od_v_v @ ( lattic8975544278839647937od_v_v @ F @ A3 ) @ A3 ) ) ) ) ) ).
% semilattice_set.closed
thf(fact_1250_semilattice__set_Oinsert,axiom,
! [F: v > v > v,A3: set_v,X: v] :
( ( lattic5961991414251573153_set_v @ F )
=> ( ( finite_finite_v @ A3 )
=> ( ( A3 != bot_bot_set_v )
=> ( ( lattic5116578512385870317ce_F_v @ F @ ( insert_v2 @ X @ A3 ) )
= ( F @ X @ ( lattic5116578512385870317ce_F_v @ F @ A3 ) ) ) ) ) ) ).
% semilattice_set.insert
thf(fact_1251_semilattice__set_Oinsert,axiom,
! [F: product_prod_v_v > product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v,X: product_prod_v_v] :
( ( lattic4743553628190527605od_v_v @ F )
=> ( ( finite3348123685078250256od_v_v @ A3 )
=> ( ( A3 != bot_bo723834152578015283od_v_v )
=> ( ( lattic8975544278839647937od_v_v @ F @ ( insert1338601472111419319od_v_v @ X @ A3 ) )
= ( F @ X @ ( lattic8975544278839647937od_v_v @ F @ A3 ) ) ) ) ) ) ).
% semilattice_set.insert
thf(fact_1252_semilattice__set_Oinsert__not__elem,axiom,
! [F: v > v > v,A3: set_v,X: v] :
( ( lattic5961991414251573153_set_v @ F )
=> ( ( finite_finite_v @ A3 )
=> ( ~ ( member_v @ X @ A3 )
=> ( ( A3 != bot_bot_set_v )
=> ( ( lattic5116578512385870317ce_F_v @ F @ ( insert_v2 @ X @ A3 ) )
= ( F @ X @ ( lattic5116578512385870317ce_F_v @ F @ A3 ) ) ) ) ) ) ) ).
% semilattice_set.insert_not_elem
thf(fact_1253_semilattice__set_Oinsert__not__elem,axiom,
! [F: product_prod_v_v > product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v,X: product_prod_v_v] :
( ( lattic4743553628190527605od_v_v @ F )
=> ( ( finite3348123685078250256od_v_v @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ( A3 != bot_bo723834152578015283od_v_v )
=> ( ( lattic8975544278839647937od_v_v @ F @ ( insert1338601472111419319od_v_v @ X @ A3 ) )
= ( F @ X @ ( lattic8975544278839647937od_v_v @ F @ A3 ) ) ) ) ) ) ) ).
% semilattice_set.insert_not_elem
thf(fact_1254_semilattice__set_Ounion,axiom,
! [F: v > v > v,A3: set_v,B2: set_v] :
( ( lattic5961991414251573153_set_v @ F )
=> ( ( finite_finite_v @ A3 )
=> ( ( A3 != bot_bot_set_v )
=> ( ( finite_finite_v @ B2 )
=> ( ( B2 != bot_bot_set_v )
=> ( ( lattic5116578512385870317ce_F_v @ F @ ( sup_sup_set_v @ A3 @ B2 ) )
= ( F @ ( lattic5116578512385870317ce_F_v @ F @ A3 ) @ ( lattic5116578512385870317ce_F_v @ F @ B2 ) ) ) ) ) ) ) ) ).
% semilattice_set.union
thf(fact_1255_semilattice__set_Ounion,axiom,
! [F: product_prod_v_v > product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( lattic4743553628190527605od_v_v @ F )
=> ( ( finite3348123685078250256od_v_v @ A3 )
=> ( ( A3 != bot_bo723834152578015283od_v_v )
=> ( ( finite3348123685078250256od_v_v @ B2 )
=> ( ( B2 != bot_bo723834152578015283od_v_v )
=> ( ( lattic8975544278839647937od_v_v @ F @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) )
= ( F @ ( lattic8975544278839647937od_v_v @ F @ A3 ) @ ( lattic8975544278839647937od_v_v @ F @ B2 ) ) ) ) ) ) ) ) ).
% semilattice_set.union
thf(fact_1256_semilattice__set_Oeq__fold,axiom,
! [F: product_prod_v_v > product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v,X: product_prod_v_v] :
( ( lattic4743553628190527605od_v_v @ F )
=> ( ( finite3348123685078250256od_v_v @ A3 )
=> ( ( lattic8975544278839647937od_v_v @ F @ ( insert1338601472111419319od_v_v @ X @ A3 ) )
= ( finite1005027896263410312od_v_v @ F @ X @ A3 ) ) ) ) ).
% semilattice_set.eq_fold
thf(fact_1257_semilattice__set_Oeq__fold,axiom,
! [F: v > v > v,A3: set_v,X: v] :
( ( lattic5961991414251573153_set_v @ F )
=> ( ( finite_finite_v @ A3 )
=> ( ( lattic5116578512385870317ce_F_v @ F @ ( insert_v2 @ X @ A3 ) )
= ( finite_fold_v_v @ F @ X @ A3 ) ) ) ) ).
% semilattice_set.eq_fold
thf(fact_1258_chains__extend,axiom,
! [C2: set_set_v,S: set_set_v,Z: set_v] :
( ( member_set_set_v @ C2 @ ( chains_v @ S ) )
=> ( ( member_set_v @ Z @ S )
=> ( ! [X3: set_v] :
( ( member_set_v @ X3 @ C2 )
=> ( ord_less_eq_set_v @ X3 @ Z ) )
=> ( member_set_set_v @ ( sup_sup_set_set_v @ ( insert_set_v @ Z @ bot_bot_set_set_v ) @ C2 ) @ ( chains_v @ S ) ) ) ) ) ).
% chains_extend
thf(fact_1259_chains__extend,axiom,
! [C2: set_se8455005133513928103od_v_v,S: set_se8455005133513928103od_v_v,Z: set_Product_prod_v_v] :
( ( member5511408251247217616od_v_v @ C2 @ ( chains2766520108962750135od_v_v @ S ) )
=> ( ( member8406446414694345712od_v_v @ Z @ S )
=> ( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ C2 )
=> ( ord_le7336532860387713383od_v_v @ X3 @ Z ) )
=> ( member5511408251247217616od_v_v @ ( sup_su335656005089752955od_v_v @ ( insert7504383016908236695od_v_v @ Z @ bot_bo3497076220358800403od_v_v ) @ C2 ) @ ( chains2766520108962750135od_v_v @ S ) ) ) ) ) ).
% chains_extend
thf(fact_1260_inf__img__fin__domE_H,axiom,
! [F: v > v,A3: set_v] :
( ( finite_finite_v @ ( image_v_v @ F @ A3 ) )
=> ( ~ ( finite_finite_v @ A3 )
=> ~ ! [Y: v] :
( ( member_v @ Y @ ( image_v_v @ F @ A3 ) )
=> ( finite_finite_v @ ( inf_inf_set_v @ ( vimage_v_v @ F @ ( insert_v2 @ Y @ bot_bot_set_v ) ) @ A3 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_1261_inf__img__fin__domE_H,axiom,
! [F: v > product_prod_v_v,A3: set_v] :
( ( finite3348123685078250256od_v_v @ ( image_9222788639401671577od_v_v @ F @ A3 ) )
=> ( ~ ( finite_finite_v @ A3 )
=> ~ ! [Y: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y @ ( image_9222788639401671577od_v_v @ F @ A3 ) )
=> ( finite_finite_v @ ( inf_inf_set_v @ ( vimage6966088051850144591od_v_v @ F @ ( insert1338601472111419319od_v_v @ Y @ bot_bo723834152578015283od_v_v ) ) @ A3 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_1262_vimageI,axiom,
! [F: v > v,A: v,B: v,B2: set_v] :
( ( ( F @ A )
= B )
=> ( ( member_v @ B @ B2 )
=> ( member_v @ A @ ( vimage_v_v @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_1263_vimageI,axiom,
! [F: product_prod_v_v > v,A: product_prod_v_v,B: v,B2: set_v] :
( ( ( F @ A )
= B )
=> ( ( member_v @ B @ B2 )
=> ( member7453568604450474000od_v_v @ A @ ( vimage3896114166191421095_v_v_v @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_1264_vimageI,axiom,
! [F: v > product_prod_v_v,A: v,B: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( F @ A )
= B )
=> ( ( member7453568604450474000od_v_v @ B @ B2 )
=> ( member_v @ A @ ( vimage6966088051850144591od_v_v @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_1265_vimageI,axiom,
! [F: product_prod_v_v > product_prod_v_v,A: product_prod_v_v,B: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( F @ A )
= B )
=> ( ( member7453568604450474000od_v_v @ B @ B2 )
=> ( member7453568604450474000od_v_v @ A @ ( vimage6257782490871955835od_v_v @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_1266_vimage__eq,axiom,
! [A: v,F: v > v,B2: set_v] :
( ( member_v @ A @ ( vimage_v_v @ F @ B2 ) )
= ( member_v @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_1267_vimage__eq,axiom,
! [A: v,F: v > product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member_v @ A @ ( vimage6966088051850144591od_v_v @ F @ B2 ) )
= ( member7453568604450474000od_v_v @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_1268_vimage__eq,axiom,
! [A: product_prod_v_v,F: product_prod_v_v > v,B2: set_v] :
( ( member7453568604450474000od_v_v @ A @ ( vimage3896114166191421095_v_v_v @ F @ B2 ) )
= ( member_v @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_1269_vimage__eq,axiom,
! [A: product_prod_v_v,F: product_prod_v_v > product_prod_v_v,B2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( vimage6257782490871955835od_v_v @ F @ B2 ) )
= ( member7453568604450474000od_v_v @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_1270_vimage__empty,axiom,
! [F: v > v] :
( ( vimage_v_v @ F @ bot_bot_set_v )
= bot_bot_set_v ) ).
% vimage_empty
thf(fact_1271_vimage__empty,axiom,
! [F: product_prod_v_v > v] :
( ( vimage3896114166191421095_v_v_v @ F @ bot_bot_set_v )
= bot_bo723834152578015283od_v_v ) ).
% vimage_empty
thf(fact_1272_vimage__empty,axiom,
! [F: v > product_prod_v_v] :
( ( vimage6966088051850144591od_v_v @ F @ bot_bo723834152578015283od_v_v )
= bot_bot_set_v ) ).
% vimage_empty
thf(fact_1273_vimage__empty,axiom,
! [F: product_prod_v_v > product_prod_v_v] :
( ( vimage6257782490871955835od_v_v @ F @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% vimage_empty
thf(fact_1274_vimage__Un,axiom,
! [F: product_prod_v_v > product_prod_v_v,A3: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( vimage6257782490871955835od_v_v @ F @ ( sup_su414716646722978715od_v_v @ A3 @ B2 ) )
= ( sup_su414716646722978715od_v_v @ ( vimage6257782490871955835od_v_v @ F @ A3 ) @ ( vimage6257782490871955835od_v_v @ F @ B2 ) ) ) ).
% vimage_Un
thf(fact_1275_vimage__Int,axiom,
! [F: v > v,A3: set_v,B2: set_v] :
( ( vimage_v_v @ F @ ( inf_inf_set_v @ A3 @ B2 ) )
= ( inf_inf_set_v @ ( vimage_v_v @ F @ A3 ) @ ( vimage_v_v @ F @ B2 ) ) ) ).
% vimage_Int
thf(fact_1276_vimage__insert,axiom,
! [F: product_prod_v_v > v,A: v,B2: set_v] :
( ( vimage3896114166191421095_v_v_v @ F @ ( insert_v2 @ A @ B2 ) )
= ( sup_su414716646722978715od_v_v @ ( vimage3896114166191421095_v_v_v @ F @ ( insert_v2 @ A @ bot_bot_set_v ) ) @ ( vimage3896114166191421095_v_v_v @ F @ B2 ) ) ) ).
% vimage_insert
thf(fact_1277_vimage__insert,axiom,
! [F: product_prod_v_v > product_prod_v_v,A: product_prod_v_v,B2: set_Product_prod_v_v] :
( ( vimage6257782490871955835od_v_v @ F @ ( insert1338601472111419319od_v_v @ A @ B2 ) )
= ( sup_su414716646722978715od_v_v @ ( vimage6257782490871955835od_v_v @ F @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) @ ( vimage6257782490871955835od_v_v @ F @ B2 ) ) ) ).
% vimage_insert
thf(fact_1278_vimage__singleton__eq,axiom,
! [A: product_prod_v_v,F: product_prod_v_v > product_prod_v_v,B: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( vimage6257782490871955835od_v_v @ F @ ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) ) )
= ( ( F @ A )
= B ) ) ).
% vimage_singleton_eq
% Helper facts (5)
thf(help_If_2_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X: set_v,Y2: set_v] :
( ( if_set_v @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X: set_v,Y2: set_v] :
( ( if_set_v @ $true @ X @ Y2 )
= X ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_T,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( if_set4279007504652509325od_v_v @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_T,axiom,
! [X: set_Product_prod_v_v,Y2: set_Product_prod_v_v] :
( ( if_set4279007504652509325od_v_v @ $true @ X @ Y2 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member_v @ v2 @ ( sCC_Bl157864678168468314t_unit @ e2 ) ).
%------------------------------------------------------------------------------