TPTP Problem File: SLH0823^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Undirected_Graph_Theory/0016_Undirected_Graph_Walks/prob_00050_001820__13157844_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1544 ( 592 unt; 253 typ; 0 def)
% Number of atoms : 3830 (1617 equ; 0 cnn)
% Maximal formula atoms : 18 ( 2 avg)
% Number of connectives : 11966 ( 554 ~; 69 |; 397 &;9254 @)
% ( 0 <=>;1692 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 781 ( 781 >; 0 *; 0 +; 0 <<)
% Number of symbols : 241 ( 238 usr; 15 con; 0-4 aty)
% Number of variables : 3744 ( 150 ^;3346 !; 248 ?;3744 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:33:07.696
%------------------------------------------------------------------------------
% Could-be-implicit typings (15)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_J_J,type,
set_se433578199980034526_set_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_J,type,
list_list_set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_J,type,
set_list_set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_J,type,
set_set_set_set_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
list_list_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
set_list_set_a: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
list_set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
list_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (238)
thf(sy_c_Finite__Set_OFpow_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
finite3131931315176300580_set_a: set_set_set_a > set_set_set_set_a ).
thf(sy_c_Finite__Set_OFpow_001t__Set__Oset_Itf__a_J,type,
finite_Fpow_set_a: set_set_a > set_set_set_a ).
thf(sy_c_Finite__Set_OFpow_001tf__a,type,
finite_Fpow_a: set_a > set_set_a ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
finite5318320746233006407_set_a: set_set_set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
finite7209287970140883943_set_a: set_set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
minus_3359197881701045381_set_a: set_set_set_a > set_set_set_a > set_set_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
minus_5736297505244876581_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Omonoid_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
monoid_list_set_a: ( list_set_a > list_set_a > list_set_a ) > list_set_a > $o ).
thf(sy_c_Groups_Omonoid_001t__List__Olist_Itf__a_J,type,
monoid_list_a: ( list_a > list_a > list_a ) > list_a > $o ).
thf(sy_c_Groups_Omonoid_001t__Set__Oset_Itf__a_J,type,
monoid_set_a: ( set_a > set_a > set_a ) > set_a > $o ).
thf(sy_c_If_001t__List__Olist_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
if_list_set_set_a: $o > list_set_set_a > list_set_set_a > list_set_set_a ).
thf(sy_c_If_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
if_list_set_a: $o > list_set_a > list_set_a > list_set_a ).
thf(sy_c_If_001t__List__Olist_Itf__a_J,type,
if_list_a: $o > list_a > list_a > list_a ).
thf(sy_c_If_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
if_set_set_set_a: $o > set_set_set_a > set_set_set_a > set_set_set_a ).
thf(sy_c_If_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
if_set_set_a: $o > set_set_a > set_set_a > set_set_a ).
thf(sy_c_If_001t__Set__Oset_Itf__a_J,type,
if_set_a: $o > set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
sup_su5748565005391983768_set_a: set_list_set_a > set_list_set_a > set_list_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
sup_sup_set_list_a: set_list_a > set_list_a > set_list_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_J,type,
sup_su6872963709084814930_set_a: set_set_set_set_a > set_set_set_set_a > set_set_set_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
sup_su2076012971530813682_set_a: set_set_set_a > set_set_set_a > set_set_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
sup_sup_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
lattic5824591902637136597_set_a: set_set_set_set_a > set_set_set_a ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
lattic338143333561554293_set_a: set_set_set_a > set_set_a ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_Itf__a_J,type,
lattic2918178356826803221_set_a: set_set_a > set_a ).
thf(sy_c_List_Oappend_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
append_list_set_a: list_list_set_a > list_list_set_a > list_list_set_a ).
thf(sy_c_List_Oappend_001t__List__Olist_Itf__a_J,type,
append_list_a: list_list_a > list_list_a > list_list_a ).
thf(sy_c_List_Oappend_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
append_set_set_a: list_set_set_a > list_set_set_a > list_set_set_a ).
thf(sy_c_List_Oappend_001t__Set__Oset_Itf__a_J,type,
append_set_a: list_set_a > list_set_a > list_set_a ).
thf(sy_c_List_Oappend_001tf__a,type,
append_a: list_a > list_a > list_a ).
thf(sy_c_List_Obind_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
bind_set_a_set_a: list_set_a > ( set_a > list_set_a ) > list_set_a ).
thf(sy_c_List_Obind_001t__Set__Oset_Itf__a_J_001tf__a,type,
bind_set_a_a: list_set_a > ( set_a > list_a ) > list_a ).
thf(sy_c_List_Obind_001tf__a_001t__Set__Oset_Itf__a_J,type,
bind_a_set_a: list_a > ( a > list_set_a ) > list_set_a ).
thf(sy_c_List_Obind_001tf__a_001tf__a,type,
bind_a_a: list_a > ( a > list_a ) > list_a ).
thf(sy_c_List_Obutlast_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
butlast_set_set_a: list_set_set_a > list_set_set_a ).
thf(sy_c_List_Obutlast_001t__Set__Oset_Itf__a_J,type,
butlast_set_a: list_set_a > list_set_a ).
thf(sy_c_List_Obutlast_001tf__a,type,
butlast_a: list_a > list_a ).
thf(sy_c_List_Ocan__select_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
can_select_set_set_a: ( set_set_a > $o ) > set_set_set_a > $o ).
thf(sy_c_List_Ocan__select_001t__Set__Oset_Itf__a_J,type,
can_select_set_a: ( set_a > $o ) > set_set_a > $o ).
thf(sy_c_List_Ocan__select_001tf__a,type,
can_select_a: ( a > $o ) > set_a > $o ).
thf(sy_c_List_Oconcat_001t__Set__Oset_Itf__a_J,type,
concat_set_a: list_list_set_a > list_set_a ).
thf(sy_c_List_Oconcat_001tf__a,type,
concat_a: list_list_a > list_a ).
thf(sy_c_List_Ocoset_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
coset_set_set_a: list_set_set_a > set_set_set_a ).
thf(sy_c_List_Ocoset_001t__Set__Oset_Itf__a_J,type,
coset_set_a: list_set_a > set_set_a ).
thf(sy_c_List_Ocoset_001tf__a,type,
coset_a: list_a > set_a ).
thf(sy_c_List_Odistinct__adj_001t__Set__Oset_Itf__a_J,type,
distinct_adj_set_a: list_set_a > $o ).
thf(sy_c_List_Odistinct__adj_001tf__a,type,
distinct_adj_a: list_a > $o ).
thf(sy_c_List_Oinsert_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
insert_set_set_a: set_set_a > list_set_set_a > list_set_set_a ).
thf(sy_c_List_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > list_set_a > list_set_a ).
thf(sy_c_List_Oinsert_001tf__a,type,
insert_a: a > list_a > list_a ).
thf(sy_c_List_Olast_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
last_set_set_a: list_set_set_a > set_set_a ).
thf(sy_c_List_Olast_001t__Set__Oset_Itf__a_J,type,
last_set_a: list_set_a > set_a ).
thf(sy_c_List_Olast_001tf__a,type,
last_a: list_a > a ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
cons_list_set_a: list_set_a > list_list_set_a > list_list_set_a ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__a_J,type,
cons_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Olist_OCons_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
cons_set_set_a: set_set_a > list_set_set_a > list_set_set_a ).
thf(sy_c_List_Olist_OCons_001t__Set__Oset_Itf__a_J,type,
cons_set_a: set_a > list_set_a > list_set_a ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
nil_list_set_a: list_list_set_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
nil_list_a: list_list_a ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
nil_set_set_a: list_set_set_a ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_Itf__a_J,type,
nil_set_a: list_set_a ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Ohd_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
hd_list_set_a: list_list_set_a > list_set_a ).
thf(sy_c_List_Olist_Ohd_001t__List__Olist_Itf__a_J,type,
hd_list_a: list_list_a > list_a ).
thf(sy_c_List_Olist_Ohd_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
hd_set_set_a: list_set_set_a > set_set_a ).
thf(sy_c_List_Olist_Ohd_001t__Set__Oset_Itf__a_J,type,
hd_set_a: list_set_a > set_a ).
thf(sy_c_List_Olist_Ohd_001tf__a,type,
hd_a: list_a > a ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_list_set_set_a2: list_list_set_set_a > set_list_set_set_a ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
set_list_set_a2: list_list_set_a > set_list_set_a ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__a_J,type,
set_list_a2: list_list_a > set_list_a ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_set_a2: list_set_set_a > set_set_set_a ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_Itf__a_J,type,
set_set_a2: list_set_a > set_set_a ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist_Otl_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
tl_set_set_a: list_set_set_a > list_set_set_a ).
thf(sy_c_List_Olist_Otl_001t__Set__Oset_Itf__a_J,type,
tl_set_a: list_set_a > list_set_a ).
thf(sy_c_List_Olist_Otl_001tf__a,type,
tl_a: list_a > list_a ).
thf(sy_c_List_Olist__ex1_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
list_ex1_set_set_a: ( set_set_a > $o ) > list_set_set_a > $o ).
thf(sy_c_List_Olist__ex1_001t__Set__Oset_Itf__a_J,type,
list_ex1_set_a: ( set_a > $o ) > list_set_a > $o ).
thf(sy_c_List_Olist__ex1_001tf__a,type,
list_ex1_a: ( a > $o ) > list_a > $o ).
thf(sy_c_List_Olists_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
lists_set_set_a: set_set_set_a > set_list_set_set_a ).
thf(sy_c_List_Olists_001t__Set__Oset_Itf__a_J,type,
lists_set_a: set_set_a > set_list_set_a ).
thf(sy_c_List_Olists_001tf__a,type,
lists_a: set_a > set_list_a ).
thf(sy_c_List_Olistset_001t__Set__Oset_Itf__a_J,type,
listset_set_a: list_set_set_a > set_list_set_a ).
thf(sy_c_List_Olistset_001tf__a,type,
listset_a: list_set_a > set_list_a ).
thf(sy_c_List_Omaps_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
maps_set_a_set_a: ( set_a > list_set_a ) > list_set_a > list_set_a ).
thf(sy_c_List_Omaps_001t__Set__Oset_Itf__a_J_001tf__a,type,
maps_set_a_a: ( set_a > list_a ) > list_set_a > list_a ).
thf(sy_c_List_Omaps_001tf__a_001t__Set__Oset_Itf__a_J,type,
maps_a_set_a: ( a > list_set_a ) > list_a > list_set_a ).
thf(sy_c_List_Omaps_001tf__a_001tf__a,type,
maps_a_a: ( a > list_a ) > list_a > list_a ).
thf(sy_c_List_Omember_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: list_set_set_a > set_set_a > $o ).
thf(sy_c_List_Omember_001t__Set__Oset_Itf__a_J,type,
member_set_a: list_set_a > set_a > $o ).
thf(sy_c_List_Omember_001tf__a,type,
member_a: list_a > a > $o ).
thf(sy_c_List_Onull_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
null_set_set_a: list_set_set_a > $o ).
thf(sy_c_List_Onull_001t__Set__Oset_Itf__a_J,type,
null_set_a: list_set_a > $o ).
thf(sy_c_List_Onull_001tf__a,type,
null_a: list_a > $o ).
thf(sy_c_List_Oproduct__lists_001t__Set__Oset_Itf__a_J,type,
product_lists_set_a: list_list_set_a > list_list_set_a ).
thf(sy_c_List_Oproduct__lists_001tf__a,type,
product_lists_a: list_list_a > list_list_a ).
thf(sy_c_List_Oremdups__adj_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
remdup1882599702278213626_set_a: list_set_set_a > list_set_set_a ).
thf(sy_c_List_Oremdups__adj_001t__Set__Oset_Itf__a_J,type,
remdups_adj_set_a: list_set_a > list_set_a ).
thf(sy_c_List_Oremdups__adj_001tf__a,type,
remdups_adj_a: list_a > list_a ).
thf(sy_c_List_Oremdups__adj__rel_001t__Set__Oset_Itf__a_J,type,
remdup6457802342601013479_set_a: list_set_a > list_set_a > $o ).
thf(sy_c_List_Oremdups__adj__rel_001tf__a,type,
remdups_adj_rel_a: list_a > list_a > $o ).
thf(sy_c_List_Oremove1_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
remove1_set_set_a: set_set_a > list_set_set_a > list_set_set_a ).
thf(sy_c_List_Oremove1_001t__Set__Oset_Itf__a_J,type,
remove1_set_a: set_a > list_set_a > list_set_a ).
thf(sy_c_List_Oremove1_001tf__a,type,
remove1_a: a > list_a > list_a ).
thf(sy_c_List_OremoveAll_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
removeAll_set_set_a: set_set_a > list_set_set_a > list_set_set_a ).
thf(sy_c_List_OremoveAll_001t__Set__Oset_Itf__a_J,type,
removeAll_set_a: set_a > list_set_a > list_set_a ).
thf(sy_c_List_OremoveAll_001tf__a,type,
removeAll_a: a > list_a > list_a ).
thf(sy_c_List_Orotate1_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
rotate1_set_set_a: list_set_set_a > list_set_set_a ).
thf(sy_c_List_Orotate1_001t__Set__Oset_Itf__a_J,type,
rotate1_set_a: list_set_a > list_set_a ).
thf(sy_c_List_Orotate1_001tf__a,type,
rotate1_a: list_a > list_a ).
thf(sy_c_List_Oset__Cons_001tf__a,type,
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member6684481465865166061_set_a: list_set_set_a > set_list_set_set_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
member_list_set_a: list_set_a > set_list_set_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_J,type,
member7634106644413650855_set_a: set_set_set_set_a > set_se433578199980034526_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
member_set_set_set_a: set_set_set_a > set_set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a2: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a2: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a2: a > set_a > $o ).
thf(sy_v_xs,type,
xs: list_a ).
thf(sy_v_ys,type,
ys: list_a ).
% Relevant facts (1277)
thf(fact_0_walk__edges__append__ss1,axiom,
! [Ys: list_a,Xs: list_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Ys ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ Ys ) ) ) ) ).
% walk_edges_append_ss1
thf(fact_1_comp__sgraph_Owalk__edges__append__ss1,axiom,
! [Ys: list_set_a,Xs: list_set_a] : ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ Ys ) ) @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ Ys ) ) ) ) ).
% comp_sgraph.walk_edges_append_ss1
thf(fact_2_comp__sgraph_Owalk__edges__append__ss1,axiom,
! [Ys: list_a,Xs: list_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Ys ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ Ys ) ) ) ) ).
% comp_sgraph.walk_edges_append_ss1
thf(fact_3_walk__edges__tl__ss,axiom,
! [Xs: list_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( tl_a @ Xs ) ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Xs ) ) ) ).
% walk_edges_tl_ss
thf(fact_4_append_Oassoc,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( append_a @ ( append_a @ A @ B ) @ C )
= ( append_a @ A @ ( append_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_5_append_Oassoc,axiom,
! [A: list_set_a,B: list_set_a,C: list_set_a] :
( ( append_set_a @ ( append_set_a @ A @ B ) @ C )
= ( append_set_a @ A @ ( append_set_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_6_append__assoc,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( append_a @ ( append_a @ Xs @ Ys ) @ Zs )
= ( append_a @ Xs @ ( append_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_7_append__assoc,axiom,
! [Xs: list_set_a,Ys: list_set_a,Zs: list_set_a] :
( ( append_set_a @ ( append_set_a @ Xs @ Ys ) @ Zs )
= ( append_set_a @ Xs @ ( append_set_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_8_append__same__eq,axiom,
! [Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( append_a @ Ys @ Xs )
= ( append_a @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_9_append__same__eq,axiom,
! [Ys: list_set_a,Xs: list_set_a,Zs: list_set_a] :
( ( ( append_set_a @ Ys @ Xs )
= ( append_set_a @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_10_same__append__eq,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_11_same__append__eq,axiom,
! [Xs: list_set_a,Ys: list_set_a,Zs: list_set_a] :
( ( ( append_set_a @ Xs @ Ys )
= ( append_set_a @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_12_subsetI,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ! [X: set_set_a] :
( ( member_set_set_a2 @ X @ A2 )
=> ( member_set_set_a2 @ X @ B2 ) )
=> ( ord_le5722252365846178494_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_13_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X: a] :
( ( member_a2 @ X @ A2 )
=> ( member_a2 @ X @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_14_subsetI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ! [X: set_a] :
( ( member_set_a2 @ X @ A2 )
=> ( member_set_a2 @ X @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_15_subset__antisym,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( ( ord_le5722252365846178494_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_16_subset__antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_17_subset__antisym,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_18_order__refl,axiom,
! [X2: set_set_set_a] : ( ord_le5722252365846178494_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_19_order__refl,axiom,
! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_20_order__refl,axiom,
! [X2: set_set_a] : ( ord_le3724670747650509150_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_21_dual__order_Orefl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_22_dual__order_Orefl,axiom,
! [A: set_set_set_a] : ( ord_le5722252365846178494_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_23_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_24_subset__code_I1_J,axiom,
! [Xs: list_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ B2 )
= ( ! [X3: set_a] :
( ( member_set_a2 @ X3 @ ( set_set_a2 @ Xs ) )
=> ( member_set_a2 @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_25_subset__code_I1_J,axiom,
! [Xs: list_set_set_a,B2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ Xs ) @ B2 )
= ( ! [X3: set_set_a] :
( ( member_set_set_a2 @ X3 @ ( set_set_set_a2 @ Xs ) )
=> ( member_set_set_a2 @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_26_subset__code_I1_J,axiom,
! [Xs: list_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B2 )
= ( ! [X3: a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
=> ( member_a2 @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_27_ulgraph_Owalk__edges__append__ss1,axiom,
! [Vertices: set_a,Edges: set_set_a,Ys: list_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Ys ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ Ys ) ) ) ) ) ).
% ulgraph.walk_edges_append_ss1
thf(fact_28_ulgraph_Owalk__edges__append__ss1,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Ys: list_set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ Ys ) ) @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ Ys ) ) ) ) ) ).
% ulgraph.walk_edges_append_ss1
thf(fact_29_comp__sgraph_Owalk__edges__tl__ss,axiom,
! [Xs: list_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( tl_a @ Xs ) ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Xs ) ) ) ).
% comp_sgraph.walk_edges_tl_ss
thf(fact_30_comp__sgraph_Owalk__edges__tl__ss,axiom,
! [Xs: list_set_a] : ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( tl_set_a @ Xs ) ) ) @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ Xs ) ) ) ).
% comp_sgraph.walk_edges_tl_ss
thf(fact_31_ulgraph_Owalk__edges__tl__ss,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( tl_a @ Xs ) ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ Xs ) ) ) ) ).
% ulgraph.walk_edges_tl_ss
thf(fact_32_ulgraph_Owalk__edges__tl__ss,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ ( tl_set_a @ Xs ) ) ) @ ( set_set_set_a2 @ ( undire6234387080713648494_set_a @ Xs ) ) ) ) ).
% ulgraph.walk_edges_tl_ss
thf(fact_33_order__antisym__conv,axiom,
! [Y: set_set_a,X2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X2 )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_34_order__antisym__conv,axiom,
! [Y: set_set_set_a,X2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ Y @ X2 )
=> ( ( ord_le5722252365846178494_set_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_35_order__antisym__conv,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ( ( ord_less_eq_set_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_36_ord__le__eq__subst,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_37_ord__le__eq__subst,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > set_set_set_a,C: set_set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y2 )
=> ( ord_le5722252365846178494_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le5722252365846178494_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_38_ord__le__eq__subst,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_39_ord__le__eq__subst,axiom,
! [A: set_set_set_a,B: set_set_set_a,F: set_set_set_a > set_set_a,C: set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_set_set_a,Y2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_40_ord__le__eq__subst,axiom,
! [A: set_set_set_a,B: set_set_set_a,F: set_set_set_a > set_set_set_a,C: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_set_set_a,Y2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X @ Y2 )
=> ( ord_le5722252365846178494_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le5722252365846178494_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_41_ord__le__eq__subst,axiom,
! [A: set_set_set_a,B: set_set_set_a,F: set_set_set_a > set_a,C: set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_set_set_a,Y2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_42_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_set_a,C: set_set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_43_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_set_set_a,C: set_set_set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_le5722252365846178494_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le5722252365846178494_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_44_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_45_ord__eq__le__subst,axiom,
! [A: set_set_a,F: set_set_a > set_set_a,B: set_set_a,C: set_set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ! [X: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_46_ord__eq__le__subst,axiom,
! [A: set_set_set_a,F: set_set_a > set_set_set_a,B: set_set_a,C: set_set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ! [X: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y2 )
=> ( ord_le5722252365846178494_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le5722252365846178494_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_47_ord__eq__le__subst,axiom,
! [A: set_a,F: set_set_a > set_a,B: set_set_a,C: set_set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ! [X: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_48_ord__eq__le__subst,axiom,
! [A: set_set_a,F: set_set_set_a > set_set_a,B: set_set_set_a,C: set_set_set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5722252365846178494_set_a @ B @ C )
=> ( ! [X: set_set_set_a,Y2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_49_ord__eq__le__subst,axiom,
! [A: set_set_set_a,F: set_set_set_a > set_set_set_a,B: set_set_set_a,C: set_set_set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5722252365846178494_set_a @ B @ C )
=> ( ! [X: set_set_set_a,Y2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X @ Y2 )
=> ( ord_le5722252365846178494_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le5722252365846178494_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_50_ord__eq__le__subst,axiom,
! [A: set_a,F: set_set_set_a > set_a,B: set_set_set_a,C: set_set_set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5722252365846178494_set_a @ B @ C )
=> ( ! [X: set_set_set_a,Y2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_51_ord__eq__le__subst,axiom,
! [A: set_set_a,F: set_a > set_set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_52_ord__eq__le__subst,axiom,
! [A: set_set_set_a,F: set_a > set_set_set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_le5722252365846178494_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le5722252365846178494_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_53_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_54_order__eq__refl,axiom,
! [X2: set_set_a,Y: set_set_a] :
( ( X2 = Y )
=> ( ord_le3724670747650509150_set_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_55_order__eq__refl,axiom,
! [X2: set_set_set_a,Y: set_set_set_a] :
( ( X2 = Y )
=> ( ord_le5722252365846178494_set_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_56_order__eq__refl,axiom,
! [X2: set_a,Y: set_a] :
( ( X2 = Y )
=> ( ord_less_eq_set_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_57_order__subst2,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_58_order__subst2,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > set_set_set_a,C: set_set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le5722252365846178494_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y2 )
=> ( ord_le5722252365846178494_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le5722252365846178494_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_59_order__subst2,axiom,
! [A: set_set_a,B: set_set_a,F: set_set_a > set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_60_order__subst2,axiom,
! [A: set_set_set_a,B: set_set_set_a,F: set_set_set_a > set_set_a,C: set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_set_set_a,Y2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_61_order__subst2,axiom,
! [A: set_set_set_a,B: set_set_set_a,F: set_set_set_a > set_set_set_a,C: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( ord_le5722252365846178494_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_set_set_a,Y2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X @ Y2 )
=> ( ord_le5722252365846178494_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le5722252365846178494_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_62_order__subst2,axiom,
! [A: set_set_set_a,B: set_set_set_a,F: set_set_set_a > set_a,C: set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_set_set_a,Y2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_63_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_set_a,C: set_set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_64_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_set_set_a,C: set_set_set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_le5722252365846178494_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_le5722252365846178494_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le5722252365846178494_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_65_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_66_order__subst1,axiom,
! [A: set_set_a,F: set_set_a > set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ! [X: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_67_order__subst1,axiom,
! [A: set_set_a,F: set_set_set_a > set_set_a,B: set_set_set_a,C: set_set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( F @ B ) )
=> ( ( ord_le5722252365846178494_set_a @ B @ C )
=> ( ! [X: set_set_set_a,Y2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_68_order__subst1,axiom,
! [A: set_set_a,F: set_a > set_set_a,B: set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_69_order__subst1,axiom,
! [A: set_set_set_a,F: set_set_a > set_set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ! [X: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y2 )
=> ( ord_le5722252365846178494_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le5722252365846178494_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_70_order__subst1,axiom,
! [A: set_set_set_a,F: set_set_set_a > set_set_set_a,B: set_set_set_a,C: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ ( F @ B ) )
=> ( ( ord_le5722252365846178494_set_a @ B @ C )
=> ( ! [X: set_set_set_a,Y2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X @ Y2 )
=> ( ord_le5722252365846178494_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le5722252365846178494_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_71_order__subst1,axiom,
! [A: set_set_set_a,F: set_a > set_set_set_a,B: set_a,C: set_a] :
( ( ord_le5722252365846178494_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_le5722252365846178494_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le5722252365846178494_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_72_order__subst1,axiom,
! [A: set_a,F: set_set_a > set_a,B: set_set_a,C: set_set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ! [X: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_73_order__subst1,axiom,
! [A: set_a,F: set_set_set_a > set_a,B: set_set_set_a,C: set_set_set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_le5722252365846178494_set_a @ B @ C )
=> ( ! [X: set_set_set_a,Y2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_74_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_75_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_set_a,Z: set_set_a] : ( Y3 = Z ) )
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_76_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_set_set_a,Z: set_set_set_a] : ( Y3 = Z ) )
= ( ^ [A3: set_set_set_a,B3: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A3 @ B3 )
& ( ord_le5722252365846178494_set_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_77_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_78_antisym,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_79_antisym,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( ord_le5722252365846178494_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_80_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_81_dual__order_Otrans,axiom,
! [B: set_set_a,A: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( ord_le3724670747650509150_set_a @ C @ B )
=> ( ord_le3724670747650509150_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_82_dual__order_Otrans,axiom,
! [B: set_set_set_a,A: set_set_set_a,C: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B @ A )
=> ( ( ord_le5722252365846178494_set_a @ C @ B )
=> ( ord_le5722252365846178494_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_83_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_84_dual__order_Oantisym,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_85_dual__order_Oantisym,axiom,
! [B: set_set_set_a,A: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B @ A )
=> ( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_86_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_87_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_set_a,Z: set_set_a] : ( Y3 = Z ) )
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A3 )
& ( ord_le3724670747650509150_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_88_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_set_set_a,Z: set_set_set_a] : ( Y3 = Z ) )
= ( ^ [A3: set_set_set_a,B3: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B3 @ A3 )
& ( ord_le5722252365846178494_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_89_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_90_order__trans,axiom,
! [X2: set_set_a,Y: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ Z2 )
=> ( ord_le3724670747650509150_set_a @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_91_order__trans,axiom,
! [X2: set_set_set_a,Y: set_set_set_a,Z2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X2 @ Y )
=> ( ( ord_le5722252365846178494_set_a @ Y @ Z2 )
=> ( ord_le5722252365846178494_set_a @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_92_order__trans,axiom,
! [X2: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_eq_set_a @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_93_order_Otrans,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_94_order_Otrans,axiom,
! [A: set_set_set_a,B: set_set_set_a,C: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( ord_le5722252365846178494_set_a @ B @ C )
=> ( ord_le5722252365846178494_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_95_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_96_order__antisym,axiom,
! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ( ord_le3724670747650509150_set_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_97_order__antisym,axiom,
! [X2: set_set_set_a,Y: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X2 @ Y )
=> ( ( ord_le5722252365846178494_set_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_98_order__antisym,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_99_ord__le__eq__trans,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_100_ord__le__eq__trans,axiom,
! [A: set_set_set_a,B: set_set_set_a,C: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_le5722252365846178494_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_101_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_102_ord__eq__le__trans,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( A = B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_103_ord__eq__le__trans,axiom,
! [A: set_set_set_a,B: set_set_set_a,C: set_set_set_a] :
( ( A = B )
=> ( ( ord_le5722252365846178494_set_a @ B @ C )
=> ( ord_le5722252365846178494_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_104_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_105_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_set_a,Z: set_set_a] : ( Y3 = Z ) )
= ( ^ [X3: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y4 )
& ( ord_le3724670747650509150_set_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_106_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_set_set_a,Z: set_set_set_a] : ( Y3 = Z ) )
= ( ^ [X3: set_set_set_a,Y4: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X3 @ Y4 )
& ( ord_le5722252365846178494_set_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_107_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_108_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X3: set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_109_Collect__mono__iff,axiom,
! [P: set_set_a > $o,Q: set_set_a > $o] :
( ( ord_le5722252365846178494_set_a @ ( collect_set_set_a @ P ) @ ( collect_set_set_a @ Q ) )
= ( ! [X3: set_set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_110_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_111_set__eq__subset,axiom,
( ( ^ [Y3: set_set_a,Z: set_set_a] : ( Y3 = Z ) )
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_112_set__eq__subset,axiom,
( ( ^ [Y3: set_set_set_a,Z: set_set_set_a] : ( Y3 = Z ) )
= ( ^ [A4: set_set_set_a,B4: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A4 @ B4 )
& ( ord_le5722252365846178494_set_a @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_113_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_114_subset__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_115_subset__trans,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,C2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( ( ord_le5722252365846178494_set_a @ B2 @ C2 )
=> ( ord_le5722252365846178494_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_116_subset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_117_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X: set_a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_118_Collect__mono,axiom,
! [P: set_set_a > $o,Q: set_set_a > $o] :
( ! [X: set_set_a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le5722252365846178494_set_a @ ( collect_set_set_a @ P ) @ ( collect_set_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_119_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_120_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a2 @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_121_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a2 @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_122_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a2 @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_123_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a2 @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_124_subset__refl,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_125_subset__refl,axiom,
! [A2: set_set_set_a] : ( ord_le5722252365846178494_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_126_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_127_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
! [T: set_a] :
( ( member_set_a2 @ T @ A4 )
=> ( member_set_a2 @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_128_subset__iff,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [A4: set_set_set_a,B4: set_set_set_a] :
! [T: set_set_a] :
( ( member_set_set_a2 @ T @ A4 )
=> ( member_set_set_a2 @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_129_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [T: a] :
( ( member_a2 @ T @ A4 )
=> ( member_a2 @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_130_equalityD2,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 = B2 )
=> ( ord_le3724670747650509150_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_131_equalityD2,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( A2 = B2 )
=> ( ord_le5722252365846178494_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_132_equalityD2,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_133_equalityD1,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 = B2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_134_equalityD1,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( A2 = B2 )
=> ( ord_le5722252365846178494_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_135_equalityD1,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_136_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
! [X3: set_a] :
( ( member_set_a2 @ X3 @ A4 )
=> ( member_set_a2 @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_137_subset__eq,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [A4: set_set_set_a,B4: set_set_set_a] :
! [X3: set_set_a] :
( ( member_set_set_a2 @ X3 @ A4 )
=> ( member_set_set_a2 @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_138_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [X3: a] :
( ( member_a2 @ X3 @ A4 )
=> ( member_a2 @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_139_equalityE,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ~ ( ord_le3724670747650509150_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_140_equalityE,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ~ ( ord_le5722252365846178494_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_141_equalityE,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_142_subsetD,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_set_a2 @ C @ A2 )
=> ( member_set_a2 @ C @ B2 ) ) ) ).
% subsetD
thf(fact_143_subsetD,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,C: set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( ( member_set_set_a2 @ C @ A2 )
=> ( member_set_set_a2 @ C @ B2 ) ) ) ).
% subsetD
thf(fact_144_subsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a2 @ C @ A2 )
=> ( member_a2 @ C @ B2 ) ) ) ).
% subsetD
thf(fact_145_in__mono,axiom,
! [A2: set_set_a,B2: set_set_a,X2: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_set_a2 @ X2 @ A2 )
=> ( member_set_a2 @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_146_in__mono,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,X2: set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( ( member_set_set_a2 @ X2 @ A2 )
=> ( member_set_set_a2 @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_147_in__mono,axiom,
! [A2: set_a,B2: set_a,X2: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a2 @ X2 @ A2 )
=> ( member_a2 @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_148_append__eq__append__conv2,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Ts ) )
= ( ? [Us: list_a] :
( ( ( Xs
= ( append_a @ Zs @ Us ) )
& ( ( append_a @ Us @ Ys )
= Ts ) )
| ( ( ( append_a @ Xs @ Us )
= Zs )
& ( Ys
= ( append_a @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_149_append__eq__append__conv2,axiom,
! [Xs: list_set_a,Ys: list_set_a,Zs: list_set_a,Ts: list_set_a] :
( ( ( append_set_a @ Xs @ Ys )
= ( append_set_a @ Zs @ Ts ) )
= ( ? [Us: list_set_a] :
( ( ( Xs
= ( append_set_a @ Zs @ Us ) )
& ( ( append_set_a @ Us @ Ys )
= Ts ) )
| ( ( ( append_set_a @ Xs @ Us )
= Zs )
& ( Ys
= ( append_set_a @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_150_append__eq__appendI,axiom,
! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us2: list_a] :
( ( ( append_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_a @ Xs1 @ Us2 ) )
=> ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_151_append__eq__appendI,axiom,
! [Xs: list_set_a,Xs1: list_set_a,Zs: list_set_a,Ys: list_set_a,Us2: list_set_a] :
( ( ( append_set_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_set_a @ Xs1 @ Us2 ) )
=> ( ( append_set_a @ Xs @ Ys )
= ( append_set_a @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_152_list__set__tl,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ ( tl_a @ Xs ) ) )
=> ( member_a2 @ X2 @ ( set_a2 @ Xs ) ) ) ).
% list_set_tl
thf(fact_153_list__set__tl,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a2 @ X2 @ ( set_set_a2 @ ( tl_set_a @ Xs ) ) )
=> ( member_set_a2 @ X2 @ ( set_set_a2 @ Xs ) ) ) ).
% list_set_tl
thf(fact_154_list__set__tl,axiom,
! [X2: set_set_a,Xs: list_set_set_a] :
( ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ ( tl_set_set_a @ Xs ) ) )
=> ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ Xs ) ) ) ).
% list_set_tl
thf(fact_155_Greatest__equality,axiom,
! [P: set_set_a > $o,X2: set_set_a] :
( ( P @ X2 )
=> ( ! [Y2: set_set_a] :
( ( P @ Y2 )
=> ( ord_le3724670747650509150_set_a @ Y2 @ X2 ) )
=> ( ( order_3565860530148683671_set_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_156_Greatest__equality,axiom,
! [P: set_set_set_a > $o,X2: set_set_set_a] :
( ( P @ X2 )
=> ( ! [Y2: set_set_set_a] :
( ( P @ Y2 )
=> ( ord_le5722252365846178494_set_a @ Y2 @ X2 ) )
=> ( ( order_7241106464633449719_set_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_157_Greatest__equality,axiom,
! [P: set_a > $o,X2: set_a] :
( ( P @ X2 )
=> ( ! [Y2: set_a] :
( ( P @ Y2 )
=> ( ord_less_eq_set_a @ Y2 @ X2 ) )
=> ( ( order_Greatest_set_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_158_GreatestI2__order,axiom,
! [P: set_set_a > $o,X2: set_set_a,Q: set_set_a > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_set_a] :
( ( P @ Y2 )
=> ( ord_le3724670747650509150_set_a @ Y2 @ X2 ) )
=> ( ! [X: set_set_a] :
( ( P @ X )
=> ( ! [Y5: set_set_a] :
( ( P @ Y5 )
=> ( ord_le3724670747650509150_set_a @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_3565860530148683671_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_159_GreatestI2__order,axiom,
! [P: set_set_set_a > $o,X2: set_set_set_a,Q: set_set_set_a > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_set_set_a] :
( ( P @ Y2 )
=> ( ord_le5722252365846178494_set_a @ Y2 @ X2 ) )
=> ( ! [X: set_set_set_a] :
( ( P @ X )
=> ( ! [Y5: set_set_set_a] :
( ( P @ Y5 )
=> ( ord_le5722252365846178494_set_a @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_7241106464633449719_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_160_GreatestI2__order,axiom,
! [P: set_a > $o,X2: set_a,Q: set_a > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_a] :
( ( P @ Y2 )
=> ( ord_less_eq_set_a @ Y2 @ X2 ) )
=> ( ! [X: set_a] :
( ( P @ X )
=> ( ! [Y5: set_a] :
( ( P @ Y5 )
=> ( ord_less_eq_set_a @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_161_subset__code_I2_J,axiom,
! [A2: set_set_a,Ys: list_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( coset_set_a @ Ys ) )
= ( ! [X3: set_a] :
( ( member_set_a2 @ X3 @ ( set_set_a2 @ Ys ) )
=> ~ ( member_set_a2 @ X3 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_162_subset__code_I2_J,axiom,
! [A2: set_set_set_a,Ys: list_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ ( coset_set_set_a @ Ys ) )
= ( ! [X3: set_set_a] :
( ( member_set_set_a2 @ X3 @ ( set_set_set_a2 @ Ys ) )
=> ~ ( member_set_set_a2 @ X3 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_163_subset__code_I2_J,axiom,
! [A2: set_a,Ys: list_a] :
( ( ord_less_eq_set_a @ A2 @ ( coset_a @ Ys ) )
= ( ! [X3: a] :
( ( member_a2 @ X3 @ ( set_a2 @ Ys ) )
=> ~ ( member_a2 @ X3 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_164_in__set__member,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
= ( member_a @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_165_in__set__member,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a2 @ X2 @ ( set_set_a2 @ Xs ) )
= ( member_set_a @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_166_in__set__member,axiom,
! [X2: set_set_a,Xs: list_set_set_a] :
( ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ Xs ) )
= ( member_set_set_a @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_167_le__rel__bool__arg__iff,axiom,
( ord_le2411852534652195563_set_a
= ( ^ [X4: $o > set_set_a,Y6: $o > set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le3724670747650509150_set_a @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_168_le__rel__bool__arg__iff,axiom,
( ord_le4018019852466371659_set_a
= ( ^ [X4: $o > set_set_set_a,Y6: $o > set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le5722252365846178494_set_a @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_169_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_set_a
= ( ^ [X4: $o > set_a,Y6: $o > set_a] :
( ( ord_less_eq_set_a @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_set_a @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_170_verit__comp__simplify1_I2_J,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_171_verit__comp__simplify1_I2_J,axiom,
! [A: set_set_set_a] : ( ord_le5722252365846178494_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_172_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_173_tl__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_174_tl__append2,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( Xs != nil_set_a )
=> ( ( tl_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( append_set_a @ ( tl_set_a @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_175_in__set__insert,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_176_in__set__insert,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a2 @ X2 @ ( set_set_a2 @ Xs ) )
=> ( ( insert_set_a @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_177_in__set__insert,axiom,
! [X2: set_set_a,Xs: list_set_set_a] :
( ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ Xs ) )
=> ( ( insert_set_set_a @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_178_list__ex1__iff,axiom,
( list_ex1_a
= ( ^ [P2: a > $o,Xs2: list_a] :
? [X3: a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs2 ) )
& ( P2 @ X3 )
& ! [Y4: a] :
( ( ( member_a2 @ Y4 @ ( set_a2 @ Xs2 ) )
& ( P2 @ Y4 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_179_list__ex1__iff,axiom,
( list_ex1_set_a
= ( ^ [P2: set_a > $o,Xs2: list_set_a] :
? [X3: set_a] :
( ( member_set_a2 @ X3 @ ( set_set_a2 @ Xs2 ) )
& ( P2 @ X3 )
& ! [Y4: set_a] :
( ( ( member_set_a2 @ Y4 @ ( set_set_a2 @ Xs2 ) )
& ( P2 @ Y4 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_180_list__ex1__iff,axiom,
( list_ex1_set_set_a
= ( ^ [P2: set_set_a > $o,Xs2: list_set_set_a] :
? [X3: set_set_a] :
( ( member_set_set_a2 @ X3 @ ( set_set_set_a2 @ Xs2 ) )
& ( P2 @ X3 )
& ! [Y4: set_set_a] :
( ( ( member_set_set_a2 @ Y4 @ ( set_set_set_a2 @ Xs2 ) )
& ( P2 @ Y4 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_181_subgraph__axioms__def,axiom,
( undire7690874192998179526_set_a
= ( ^ [V_H: set_set_a,E_H: set_set_set_a,V_G: set_set_a,E_G: set_set_set_a] :
( ( ord_le3724670747650509150_set_a @ V_H @ V_G )
& ( ord_le5722252365846178494_set_a @ E_H @ E_G ) ) ) ) ).
% subgraph_axioms_def
thf(fact_182_subgraph__axioms__def,axiom,
( undire3972223832325498918_set_a
= ( ^ [V_H: set_set_set_a,E_H: set_set_set_set_a,V_G: set_set_set_a,E_G: set_set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ V_H @ V_G )
& ( ord_le8049040685576063006_set_a @ E_H @ E_G ) ) ) ) ).
% subgraph_axioms_def
thf(fact_183_subgraph__axioms__def,axiom,
( undire4675926955456076134ioms_a
= ( ^ [V_H: set_a,E_H: set_set_a,V_G: set_a,E_G: set_set_a] :
( ( ord_less_eq_set_a @ V_H @ V_G )
& ( ord_le3724670747650509150_set_a @ E_H @ E_G ) ) ) ) ).
% subgraph_axioms_def
thf(fact_184_walk__edges_Osimps_I1_J,axiom,
( ( undire7337870655677353998dges_a @ nil_a )
= nil_set_a ) ).
% walk_edges.simps(1)
thf(fact_185_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_186_append_Oright__neutral,axiom,
! [A: list_set_a] :
( ( append_set_a @ A @ nil_set_a )
= A ) ).
% append.right_neutral
thf(fact_187_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_188_append__Nil2,axiom,
! [Xs: list_set_a] :
( ( append_set_a @ Xs @ nil_set_a )
= Xs ) ).
% append_Nil2
thf(fact_189_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_190_append__self__conv,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( append_set_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_set_a ) ) ).
% append_self_conv
thf(fact_191_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_192_self__append__conv,axiom,
! [Y: list_set_a,Ys: list_set_a] :
( ( Y
= ( append_set_a @ Y @ Ys ) )
= ( Ys = nil_set_a ) ) ).
% self_append_conv
thf(fact_193_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_194_append__self__conv2,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( append_set_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_set_a ) ) ).
% append_self_conv2
thf(fact_195_self__append__conv2,axiom,
! [Y: list_a,Xs: list_a] :
( ( Y
= ( append_a @ Xs @ Y ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_196_self__append__conv2,axiom,
! [Y: list_set_a,Xs: list_set_a] :
( ( Y
= ( append_set_a @ Xs @ Y ) )
= ( Xs = nil_set_a ) ) ).
% self_append_conv2
thf(fact_197_Nil__is__append__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( nil_a
= ( append_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_198_Nil__is__append__conv,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( nil_set_a
= ( append_set_a @ Xs @ Ys ) )
= ( ( Xs = nil_set_a )
& ( Ys = nil_set_a ) ) ) ).
% Nil_is_append_conv
thf(fact_199_append__is__Nil__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_200_append__is__Nil__conv,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( append_set_a @ Xs @ Ys )
= nil_set_a )
= ( ( Xs = nil_set_a )
& ( Ys = nil_set_a ) ) ) ).
% append_is_Nil_conv
thf(fact_201_list__ex1__simps_I1_J,axiom,
! [P: a > $o] :
~ ( list_ex1_a @ P @ nil_a ) ).
% list_ex1_simps(1)
thf(fact_202_list__ex1__simps_I1_J,axiom,
! [P: set_a > $o] :
~ ( list_ex1_set_a @ P @ nil_set_a ) ).
% list_ex1_simps(1)
thf(fact_203_member__rec_I2_J,axiom,
! [Y: a] :
~ ( member_a @ nil_a @ Y ) ).
% member_rec(2)
thf(fact_204_member__rec_I2_J,axiom,
! [Y: set_a] :
~ ( member_set_a @ nil_set_a @ Y ) ).
% member_rec(2)
thf(fact_205_comp__sgraph_Owalk__edges_Osimps_I1_J,axiom,
( ( undire7337870655677353998dges_a @ nil_a )
= nil_set_a ) ).
% comp_sgraph.walk_edges.simps(1)
thf(fact_206_comp__sgraph_Owalk__edges_Osimps_I1_J,axiom,
( ( undire6234387080713648494_set_a @ nil_set_a )
= nil_set_set_a ) ).
% comp_sgraph.walk_edges.simps(1)
thf(fact_207_ulgraph_Owalk__edges_Osimps_I1_J,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire7337870655677353998dges_a @ nil_a )
= nil_set_a ) ) ).
% ulgraph.walk_edges.simps(1)
thf(fact_208_ulgraph_Owalk__edges_Osimps_I1_J,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire6234387080713648494_set_a @ nil_set_a )
= nil_set_set_a ) ) ).
% ulgraph.walk_edges.simps(1)
thf(fact_209_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_210_append__Nil,axiom,
! [Ys: list_set_a] :
( ( append_set_a @ nil_set_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_211_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_212_append_Oleft__neutral,axiom,
! [A: list_set_a] :
( ( append_set_a @ nil_set_a @ A )
= A ) ).
% append.left_neutral
thf(fact_213_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_214_eq__Nil__appendI,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_set_a @ nil_set_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_215_subset__code_I3_J,axiom,
~ ( ord_le3724670747650509150_set_a @ ( coset_set_a @ nil_set_a ) @ ( set_set_a2 @ nil_set_a ) ) ).
% subset_code(3)
thf(fact_216_subset__code_I3_J,axiom,
~ ( ord_le5722252365846178494_set_a @ ( coset_set_set_a @ nil_set_set_a ) @ ( set_set_set_a2 @ nil_set_set_a ) ) ).
% subset_code(3)
thf(fact_217_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_a @ ( coset_a @ nil_a ) @ ( set_a2 @ nil_a ) ) ).
% subset_code(3)
thf(fact_218_list_Osel_I2_J,axiom,
( ( tl_a @ nil_a )
= nil_a ) ).
% list.sel(2)
thf(fact_219_list_Osel_I2_J,axiom,
( ( tl_set_a @ nil_set_a )
= nil_set_a ) ).
% list.sel(2)
thf(fact_220_list_Oset__sel_I2_J,axiom,
! [A: list_a,X2: a] :
( ( A != nil_a )
=> ( ( member_a2 @ X2 @ ( set_a2 @ ( tl_a @ A ) ) )
=> ( member_a2 @ X2 @ ( set_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_221_list_Oset__sel_I2_J,axiom,
! [A: list_set_a,X2: set_a] :
( ( A != nil_set_a )
=> ( ( member_set_a2 @ X2 @ ( set_set_a2 @ ( tl_set_a @ A ) ) )
=> ( member_set_a2 @ X2 @ ( set_set_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_222_list_Oset__sel_I2_J,axiom,
! [A: list_set_set_a,X2: set_set_a] :
( ( A != nil_set_set_a )
=> ( ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ ( tl_set_set_a @ A ) ) )
=> ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_223_tl__append__if,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( tl_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_224_tl__append__if,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( Xs = nil_set_a )
=> ( ( tl_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( tl_set_a @ Ys ) ) )
& ( ( Xs != nil_set_a )
=> ( ( tl_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( append_set_a @ ( tl_set_a @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_225_subgraph__axioms_Ointro,axiom,
! [V_H2: set_set_a,V_G2: set_set_a,E_H2: set_set_set_a,E_G2: set_set_set_a] :
( ( ord_le3724670747650509150_set_a @ V_H2 @ V_G2 )
=> ( ( ord_le5722252365846178494_set_a @ E_H2 @ E_G2 )
=> ( undire7690874192998179526_set_a @ V_H2 @ E_H2 @ V_G2 @ E_G2 ) ) ) ).
% subgraph_axioms.intro
thf(fact_226_subgraph__axioms_Ointro,axiom,
! [V_H2: set_set_set_a,V_G2: set_set_set_a,E_H2: set_set_set_set_a,E_G2: set_set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ V_H2 @ V_G2 )
=> ( ( ord_le8049040685576063006_set_a @ E_H2 @ E_G2 )
=> ( undire3972223832325498918_set_a @ V_H2 @ E_H2 @ V_G2 @ E_G2 ) ) ) ).
% subgraph_axioms.intro
thf(fact_227_subgraph__axioms_Ointro,axiom,
! [V_H2: set_a,V_G2: set_a,E_H2: set_set_a,E_G2: set_set_a] :
( ( ord_less_eq_set_a @ V_H2 @ V_G2 )
=> ( ( ord_le3724670747650509150_set_a @ E_H2 @ E_G2 )
=> ( undire4675926955456076134ioms_a @ V_H2 @ E_H2 @ V_G2 @ E_G2 ) ) ) ).
% subgraph_axioms.intro
thf(fact_228_can__select__set__list__ex1,axiom,
! [P: set_a > $o,A2: list_set_a] :
( ( can_select_set_a @ P @ ( set_set_a2 @ A2 ) )
= ( list_ex1_set_a @ P @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_229_can__select__set__list__ex1,axiom,
! [P: set_set_a > $o,A2: list_set_set_a] :
( ( can_select_set_set_a @ P @ ( set_set_set_a2 @ A2 ) )
= ( list_ex1_set_set_a @ P @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_230_walk__edges_Osimps_I2_J,axiom,
! [X2: a] :
( ( undire7337870655677353998dges_a @ ( cons_a @ X2 @ nil_a ) )
= nil_set_a ) ).
% walk_edges.simps(2)
thf(fact_231_ulgraph_Owalk__edges_Osimps_I2_J,axiom,
! [Vertices: set_a,Edges: set_set_a,X2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire7337870655677353998dges_a @ ( cons_a @ X2 @ nil_a ) )
= nil_set_a ) ) ).
% ulgraph.walk_edges.simps(2)
thf(fact_232_ulgraph_Owalk__edges_Osimps_I2_J,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X2: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire6234387080713648494_set_a @ ( cons_set_a @ X2 @ nil_set_a ) )
= nil_set_set_a ) ) ).
% ulgraph.walk_edges.simps(2)
thf(fact_233_bind__simps_I1_J,axiom,
! [F: a > list_a] :
( ( bind_a_a @ nil_a @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_234_bind__simps_I1_J,axiom,
! [F: a > list_set_a] :
( ( bind_a_set_a @ nil_a @ F )
= nil_set_a ) ).
% bind_simps(1)
thf(fact_235_bind__simps_I1_J,axiom,
! [F: set_a > list_a] :
( ( bind_set_a_a @ nil_set_a @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_236_bind__simps_I1_J,axiom,
! [F: set_a > list_set_a] :
( ( bind_set_a_set_a @ nil_set_a @ F )
= nil_set_a ) ).
% bind_simps(1)
thf(fact_237_List_Oset__insert,axiom,
! [X2: a,Xs: list_a] :
( ( set_a2 @ ( insert_a @ X2 @ Xs ) )
= ( insert_a2 @ X2 @ ( set_a2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_238_List_Oset__insert,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( set_set_a2 @ ( insert_set_a @ X2 @ Xs ) )
= ( insert_set_a2 @ X2 @ ( set_set_a2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_239_List_Oset__insert,axiom,
! [X2: set_set_a,Xs: list_set_set_a] :
( ( set_set_set_a2 @ ( insert_set_set_a @ X2 @ Xs ) )
= ( insert_set_set_a2 @ X2 @ ( set_set_set_a2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_240_append_Omonoid__axioms,axiom,
monoid_list_a @ append_a @ nil_a ).
% append.monoid_axioms
thf(fact_241_append_Omonoid__axioms,axiom,
monoid_list_set_a @ append_set_a @ nil_set_a ).
% append.monoid_axioms
thf(fact_242_not__in__set__insert,axiom,
! [X2: set_set_a,Xs: list_set_set_a] :
( ~ ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ Xs ) )
=> ( ( insert_set_set_a @ X2 @ Xs )
= ( cons_set_set_a @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_243_not__in__set__insert,axiom,
! [X2: a,Xs: list_a] :
( ~ ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X2 @ Xs )
= ( cons_a @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_244_not__in__set__insert,axiom,
! [X2: set_a,Xs: list_set_a] :
( ~ ( member_set_a2 @ X2 @ ( set_set_a2 @ Xs ) )
=> ( ( insert_set_a @ X2 @ Xs )
= ( cons_set_a @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_245_walk__edges_Ocases,axiom,
! [X2: list_a] :
( ( X2 != nil_a )
=> ( ! [X: a] :
( X2
!= ( cons_a @ X @ nil_a ) )
=> ~ ! [X: a,Y2: a,Ys2: list_a] :
( X2
!= ( cons_a @ X @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ).
% walk_edges.cases
thf(fact_246_list_Oinject,axiom,
! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X22 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_247_list_Oinject,axiom,
! [X21: set_a,X22: list_set_a,Y21: set_a,Y22: list_set_a] :
( ( ( cons_set_a @ X21 @ X22 )
= ( cons_set_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_248_insert__absorb2,axiom,
! [X2: a,A2: set_a] :
( ( insert_a2 @ X2 @ ( insert_a2 @ X2 @ A2 ) )
= ( insert_a2 @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_249_insert__iff,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a2 @ A @ ( insert_a2 @ B @ A2 ) )
= ( ( A = B )
| ( member_a2 @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_250_insert__iff,axiom,
! [A: set_a,B: set_a,A2: set_set_a] :
( ( member_set_a2 @ A @ ( insert_set_a2 @ B @ A2 ) )
= ( ( A = B )
| ( member_set_a2 @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_251_insertCI,axiom,
! [A: a,B2: set_a,B: a] :
( ( ~ ( member_a2 @ A @ B2 )
=> ( A = B ) )
=> ( member_a2 @ A @ ( insert_a2 @ B @ B2 ) ) ) ).
% insertCI
thf(fact_252_insertCI,axiom,
! [A: set_a,B2: set_set_a,B: set_a] :
( ( ~ ( member_set_a2 @ A @ B2 )
=> ( A = B ) )
=> ( member_set_a2 @ A @ ( insert_set_a2 @ B @ B2 ) ) ) ).
% insertCI
thf(fact_253_member__remove,axiom,
! [X2: a,Y: a,A2: set_a] :
( ( member_a2 @ X2 @ ( remove_a @ Y @ A2 ) )
= ( ( member_a2 @ X2 @ A2 )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_254_member__remove,axiom,
! [X2: set_a,Y: set_a,A2: set_set_a] :
( ( member_set_a2 @ X2 @ ( remove_set_a @ Y @ A2 ) )
= ( ( member_set_a2 @ X2 @ A2 )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_255_insert__subset,axiom,
! [X2: set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a2 @ X2 @ A2 ) @ B2 )
= ( ( member_set_a2 @ X2 @ B2 )
& ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_256_insert__subset,axiom,
! [X2: set_set_a,A2: set_set_set_a,B2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( insert_set_set_a2 @ X2 @ A2 ) @ B2 )
= ( ( member_set_set_a2 @ X2 @ B2 )
& ( ord_le5722252365846178494_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_257_insert__subset,axiom,
! [X2: a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( insert_a2 @ X2 @ A2 ) @ B2 )
= ( ( member_a2 @ X2 @ B2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_258_list_Osimps_I15_J,axiom,
! [X21: set_set_a,X22: list_set_set_a] :
( ( set_set_set_a2 @ ( cons_set_set_a @ X21 @ X22 ) )
= ( insert_set_set_a2 @ X21 @ ( set_set_set_a2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_259_list_Osimps_I15_J,axiom,
! [X21: a,X22: list_a] :
( ( set_a2 @ ( cons_a @ X21 @ X22 ) )
= ( insert_a2 @ X21 @ ( set_a2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_260_list_Osimps_I15_J,axiom,
! [X21: set_a,X22: list_set_a] :
( ( set_set_a2 @ ( cons_set_a @ X21 @ X22 ) )
= ( insert_set_a2 @ X21 @ ( set_set_a2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_261_append1__eq__conv,axiom,
! [Xs: list_a,X2: a,Ys: list_a,Y: a] :
( ( ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X2 = Y ) ) ) ).
% append1_eq_conv
thf(fact_262_append1__eq__conv,axiom,
! [Xs: list_set_a,X2: set_a,Ys: list_set_a,Y: set_a] :
( ( ( append_set_a @ Xs @ ( cons_set_a @ X2 @ nil_set_a ) )
= ( append_set_a @ Ys @ ( cons_set_a @ Y @ nil_set_a ) ) )
= ( ( Xs = Ys )
& ( X2 = Y ) ) ) ).
% append1_eq_conv
thf(fact_263_insert__Nil,axiom,
! [X2: a] :
( ( insert_a @ X2 @ nil_a )
= ( cons_a @ X2 @ nil_a ) ) ).
% insert_Nil
thf(fact_264_insert__Nil,axiom,
! [X2: set_a] :
( ( insert_set_a @ X2 @ nil_set_a )
= ( cons_set_a @ X2 @ nil_set_a ) ) ).
% insert_Nil
thf(fact_265_bind__simps_I2_J,axiom,
! [X2: a,Xs: list_a,F: a > list_a] :
( ( bind_a_a @ ( cons_a @ X2 @ Xs ) @ F )
= ( append_a @ ( F @ X2 ) @ ( bind_a_a @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_266_bind__simps_I2_J,axiom,
! [X2: a,Xs: list_a,F: a > list_set_a] :
( ( bind_a_set_a @ ( cons_a @ X2 @ Xs ) @ F )
= ( append_set_a @ ( F @ X2 ) @ ( bind_a_set_a @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_267_bind__simps_I2_J,axiom,
! [X2: set_a,Xs: list_set_a,F: set_a > list_a] :
( ( bind_set_a_a @ ( cons_set_a @ X2 @ Xs ) @ F )
= ( append_a @ ( F @ X2 ) @ ( bind_set_a_a @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_268_bind__simps_I2_J,axiom,
! [X2: set_a,Xs: list_set_a,F: set_a > list_set_a] :
( ( bind_set_a_set_a @ ( cons_set_a @ X2 @ Xs ) @ F )
= ( append_set_a @ ( F @ X2 ) @ ( bind_set_a_set_a @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_269_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a2 @ A @ A2 )
=> ? [B5: set_a] :
( ( A2
= ( insert_a2 @ A @ B5 ) )
& ~ ( member_a2 @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_270_mk__disjoint__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a2 @ A @ A2 )
=> ? [B5: set_set_a] :
( ( A2
= ( insert_set_a2 @ A @ B5 ) )
& ~ ( member_set_a2 @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_271_not__Cons__self2,axiom,
! [X2: a,Xs: list_a] :
( ( cons_a @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_272_not__Cons__self2,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( cons_set_a @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_273_insert__commute,axiom,
! [X2: a,Y: a,A2: set_a] :
( ( insert_a2 @ X2 @ ( insert_a2 @ Y @ A2 ) )
= ( insert_a2 @ Y @ ( insert_a2 @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_274_insert__eq__iff,axiom,
! [A: a,A2: set_a,B: a,B2: set_a] :
( ~ ( member_a2 @ A @ A2 )
=> ( ~ ( member_a2 @ B @ B2 )
=> ( ( ( insert_a2 @ A @ A2 )
= ( insert_a2 @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_a] :
( ( A2
= ( insert_a2 @ B @ C3 ) )
& ~ ( member_a2 @ B @ C3 )
& ( B2
= ( insert_a2 @ A @ C3 ) )
& ~ ( member_a2 @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_275_insert__eq__iff,axiom,
! [A: set_a,A2: set_set_a,B: set_a,B2: set_set_a] :
( ~ ( member_set_a2 @ A @ A2 )
=> ( ~ ( member_set_a2 @ B @ B2 )
=> ( ( ( insert_set_a2 @ A @ A2 )
= ( insert_set_a2 @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_set_a] :
( ( A2
= ( insert_set_a2 @ B @ C3 ) )
& ~ ( member_set_a2 @ B @ C3 )
& ( B2
= ( insert_set_a2 @ A @ C3 ) )
& ~ ( member_set_a2 @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_276_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a2 @ A @ A2 )
=> ( ( insert_a2 @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_277_insert__absorb,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a2 @ A @ A2 )
=> ( ( insert_set_a2 @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_278_insert__ident,axiom,
! [X2: a,A2: set_a,B2: set_a] :
( ~ ( member_a2 @ X2 @ A2 )
=> ( ~ ( member_a2 @ X2 @ B2 )
=> ( ( ( insert_a2 @ X2 @ A2 )
= ( insert_a2 @ X2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_279_insert__ident,axiom,
! [X2: set_a,A2: set_set_a,B2: set_set_a] :
( ~ ( member_set_a2 @ X2 @ A2 )
=> ( ~ ( member_set_a2 @ X2 @ B2 )
=> ( ( ( insert_set_a2 @ X2 @ A2 )
= ( insert_set_a2 @ X2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_280_Set_Oset__insert,axiom,
! [X2: a,A2: set_a] :
( ( member_a2 @ X2 @ A2 )
=> ~ ! [B5: set_a] :
( ( A2
= ( insert_a2 @ X2 @ B5 ) )
=> ( member_a2 @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_281_Set_Oset__insert,axiom,
! [X2: set_a,A2: set_set_a] :
( ( member_set_a2 @ X2 @ A2 )
=> ~ ! [B5: set_set_a] :
( ( A2
= ( insert_set_a2 @ X2 @ B5 ) )
=> ( member_set_a2 @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_282_insertI2,axiom,
! [A: a,B2: set_a,B: a] :
( ( member_a2 @ A @ B2 )
=> ( member_a2 @ A @ ( insert_a2 @ B @ B2 ) ) ) ).
% insertI2
thf(fact_283_insertI2,axiom,
! [A: set_a,B2: set_set_a,B: set_a] :
( ( member_set_a2 @ A @ B2 )
=> ( member_set_a2 @ A @ ( insert_set_a2 @ B @ B2 ) ) ) ).
% insertI2
thf(fact_284_insertI1,axiom,
! [A: a,B2: set_a] : ( member_a2 @ A @ ( insert_a2 @ A @ B2 ) ) ).
% insertI1
thf(fact_285_insertI1,axiom,
! [A: set_a,B2: set_set_a] : ( member_set_a2 @ A @ ( insert_set_a2 @ A @ B2 ) ) ).
% insertI1
thf(fact_286_insertE,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a2 @ A @ ( insert_a2 @ B @ A2 ) )
=> ( ( A != B )
=> ( member_a2 @ A @ A2 ) ) ) ).
% insertE
thf(fact_287_insertE,axiom,
! [A: set_a,B: set_a,A2: set_set_a] :
( ( member_set_a2 @ A @ ( insert_set_a2 @ B @ A2 ) )
=> ( ( A != B )
=> ( member_set_a2 @ A @ A2 ) ) ) ).
% insertE
thf(fact_288_transpose_Ocases,axiom,
! [X2: list_list_a] :
( ( X2 != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X2
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X: a,Xs3: list_a,Xss: list_list_a] :
( X2
!= ( cons_list_a @ ( cons_a @ X @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_289_transpose_Ocases,axiom,
! [X2: list_list_set_a] :
( ( X2 != nil_list_set_a )
=> ( ! [Xss: list_list_set_a] :
( X2
!= ( cons_list_set_a @ nil_set_a @ Xss ) )
=> ~ ! [X: set_a,Xs3: list_set_a,Xss: list_list_set_a] :
( X2
!= ( cons_list_set_a @ ( cons_set_a @ X @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_290_subset__insertI2,axiom,
! [A2: set_set_a,B2: set_set_a,B: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a2 @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_291_subset__insertI2,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,B: set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( ord_le5722252365846178494_set_a @ A2 @ ( insert_set_set_a2 @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_292_subset__insertI2,axiom,
! [A2: set_a,B2: set_a,B: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a2 @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_293_subset__insertI,axiom,
! [B2: set_set_a,A: set_a] : ( ord_le3724670747650509150_set_a @ B2 @ ( insert_set_a2 @ A @ B2 ) ) ).
% subset_insertI
thf(fact_294_subset__insertI,axiom,
! [B2: set_set_set_a,A: set_set_a] : ( ord_le5722252365846178494_set_a @ B2 @ ( insert_set_set_a2 @ A @ B2 ) ) ).
% subset_insertI
thf(fact_295_subset__insertI,axiom,
! [B2: set_a,A: a] : ( ord_less_eq_set_a @ B2 @ ( insert_a2 @ A @ B2 ) ) ).
% subset_insertI
thf(fact_296_subset__insert,axiom,
! [X2: set_a,A2: set_set_a,B2: set_set_a] :
( ~ ( member_set_a2 @ X2 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a2 @ X2 @ B2 ) )
= ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_297_subset__insert,axiom,
! [X2: set_set_a,A2: set_set_set_a,B2: set_set_set_a] :
( ~ ( member_set_set_a2 @ X2 @ A2 )
=> ( ( ord_le5722252365846178494_set_a @ A2 @ ( insert_set_set_a2 @ X2 @ B2 ) )
= ( ord_le5722252365846178494_set_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_298_subset__insert,axiom,
! [X2: a,A2: set_a,B2: set_a] :
( ~ ( member_a2 @ X2 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a2 @ X2 @ B2 ) )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_299_insert__mono,axiom,
! [C2: set_set_a,D: set_set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ D )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a2 @ A @ C2 ) @ ( insert_set_a2 @ A @ D ) ) ) ).
% insert_mono
thf(fact_300_insert__mono,axiom,
! [C2: set_set_set_a,D: set_set_set_a,A: set_set_a] :
( ( ord_le5722252365846178494_set_a @ C2 @ D )
=> ( ord_le5722252365846178494_set_a @ ( insert_set_set_a2 @ A @ C2 ) @ ( insert_set_set_a2 @ A @ D ) ) ) ).
% insert_mono
thf(fact_301_insert__mono,axiom,
! [C2: set_a,D: set_a,A: a] :
( ( ord_less_eq_set_a @ C2 @ D )
=> ( ord_less_eq_set_a @ ( insert_a2 @ A @ C2 ) @ ( insert_a2 @ A @ D ) ) ) ).
% insert_mono
thf(fact_302_comp__sgraph_Owalk__edges_Ocases,axiom,
! [X2: list_a] :
( ( X2 != nil_a )
=> ( ! [X: a] :
( X2
!= ( cons_a @ X @ nil_a ) )
=> ~ ! [X: a,Y2: a,Ys2: list_a] :
( X2
!= ( cons_a @ X @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ).
% comp_sgraph.walk_edges.cases
thf(fact_303_comp__sgraph_Owalk__edges_Ocases,axiom,
! [X2: list_set_a] :
( ( X2 != nil_set_a )
=> ( ! [X: set_a] :
( X2
!= ( cons_set_a @ X @ nil_set_a ) )
=> ~ ! [X: set_a,Y2: set_a,Ys2: list_set_a] :
( X2
!= ( cons_set_a @ X @ ( cons_set_a @ Y2 @ Ys2 ) ) ) ) ) ).
% comp_sgraph.walk_edges.cases
thf(fact_304_list__exhaust3,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ! [X: a] :
( Xs
!= ( cons_a @ X @ nil_a ) )
=> ~ ! [X: a,Y2: a,Ys2: list_a] :
( Xs
!= ( cons_a @ X @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ).
% list_exhaust3
thf(fact_305_list__exhaust3,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
=> ( ! [X: set_a] :
( Xs
!= ( cons_set_a @ X @ nil_set_a ) )
=> ~ ! [X: set_a,Y2: set_a,Ys2: list_set_a] :
( Xs
!= ( cons_set_a @ X @ ( cons_set_a @ Y2 @ Ys2 ) ) ) ) ) ).
% list_exhaust3
thf(fact_306_list__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X: a] : ( P @ ( cons_a @ X @ nil_a ) )
=> ( ! [X: a,Xs3: list_a] :
( ( Xs3 != nil_a )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_a @ X @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_307_list__nonempty__induct,axiom,
! [Xs: list_set_a,P: list_set_a > $o] :
( ( Xs != nil_set_a )
=> ( ! [X: set_a] : ( P @ ( cons_set_a @ X @ nil_set_a ) )
=> ( ! [X: set_a,Xs3: list_set_a] :
( ( Xs3 != nil_set_a )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_set_a @ X @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_308_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X: a,Xs3: list_a] : ( P @ ( cons_a @ X @ Xs3 ) @ nil_a )
=> ( ! [Y2: a,Ys2: list_a] : ( P @ nil_a @ ( cons_a @ Y2 @ Ys2 ) )
=> ( ! [X: a,Xs3: list_a,Y2: a,Ys2: list_a] :
( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_a @ X @ Xs3 ) @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_309_list__induct2_H,axiom,
! [P: list_a > list_set_a > $o,Xs: list_a,Ys: list_set_a] :
( ( P @ nil_a @ nil_set_a )
=> ( ! [X: a,Xs3: list_a] : ( P @ ( cons_a @ X @ Xs3 ) @ nil_set_a )
=> ( ! [Y2: set_a,Ys2: list_set_a] : ( P @ nil_a @ ( cons_set_a @ Y2 @ Ys2 ) )
=> ( ! [X: a,Xs3: list_a,Y2: set_a,Ys2: list_set_a] :
( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_a @ X @ Xs3 ) @ ( cons_set_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_310_list__induct2_H,axiom,
! [P: list_set_a > list_a > $o,Xs: list_set_a,Ys: list_a] :
( ( P @ nil_set_a @ nil_a )
=> ( ! [X: set_a,Xs3: list_set_a] : ( P @ ( cons_set_a @ X @ Xs3 ) @ nil_a )
=> ( ! [Y2: a,Ys2: list_a] : ( P @ nil_set_a @ ( cons_a @ Y2 @ Ys2 ) )
=> ( ! [X: set_a,Xs3: list_set_a,Y2: a,Ys2: list_a] :
( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_set_a @ X @ Xs3 ) @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_311_list__induct2_H,axiom,
! [P: list_set_a > list_set_a > $o,Xs: list_set_a,Ys: list_set_a] :
( ( P @ nil_set_a @ nil_set_a )
=> ( ! [X: set_a,Xs3: list_set_a] : ( P @ ( cons_set_a @ X @ Xs3 ) @ nil_set_a )
=> ( ! [Y2: set_a,Ys2: list_set_a] : ( P @ nil_set_a @ ( cons_set_a @ Y2 @ Ys2 ) )
=> ( ! [X: set_a,Xs3: list_set_a,Y2: set_a,Ys2: list_set_a] :
( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_set_a @ X @ Xs3 ) @ ( cons_set_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_312_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y4: a,Ys3: list_a] :
( Xs
= ( cons_a @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_313_neq__Nil__conv,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
= ( ? [Y4: set_a,Ys3: list_set_a] :
( Xs
= ( cons_set_a @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_314_min__list_Ocases,axiom,
! [X2: list_set_a] :
( ! [X: set_a,Xs3: list_set_a] :
( X2
!= ( cons_set_a @ X @ Xs3 ) )
=> ( X2 = nil_set_a ) ) ).
% min_list.cases
thf(fact_315_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_316_list_Oexhaust,axiom,
! [Y: list_set_a] :
( ( Y != nil_set_a )
=> ~ ! [X212: set_a,X222: list_set_a] :
( Y
!= ( cons_set_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_317_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_318_list_OdiscI,axiom,
! [List: list_set_a,X21: set_a,X22: list_set_a] :
( ( List
= ( cons_set_a @ X21 @ X22 ) )
=> ( List != nil_set_a ) ) ).
% list.discI
thf(fact_319_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_320_list_Odistinct_I1_J,axiom,
! [X21: set_a,X22: list_set_a] :
( nil_set_a
!= ( cons_set_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_321_set__ConsD,axiom,
! [Y: set_set_a,X2: set_set_a,Xs: list_set_set_a] :
( ( member_set_set_a2 @ Y @ ( set_set_set_a2 @ ( cons_set_set_a @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_set_set_a2 @ Y @ ( set_set_set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_322_set__ConsD,axiom,
! [Y: a,X2: a,Xs: list_a] :
( ( member_a2 @ Y @ ( set_a2 @ ( cons_a @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_a2 @ Y @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_323_set__ConsD,axiom,
! [Y: set_a,X2: set_a,Xs: list_set_a] :
( ( member_set_a2 @ Y @ ( set_set_a2 @ ( cons_set_a @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_set_a2 @ Y @ ( set_set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_324_list_Oset__cases,axiom,
! [E: set_set_a,A: list_set_set_a] :
( ( member_set_set_a2 @ E @ ( set_set_set_a2 @ A ) )
=> ( ! [Z22: list_set_set_a] :
( A
!= ( cons_set_set_a @ E @ Z22 ) )
=> ~ ! [Z1: set_set_a,Z22: list_set_set_a] :
( ( A
= ( cons_set_set_a @ Z1 @ Z22 ) )
=> ~ ( member_set_set_a2 @ E @ ( set_set_set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_325_list_Oset__cases,axiom,
! [E: a,A: list_a] :
( ( member_a2 @ E @ ( set_a2 @ A ) )
=> ( ! [Z22: list_a] :
( A
!= ( cons_a @ E @ Z22 ) )
=> ~ ! [Z1: a,Z22: list_a] :
( ( A
= ( cons_a @ Z1 @ Z22 ) )
=> ~ ( member_a2 @ E @ ( set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_326_list_Oset__cases,axiom,
! [E: set_a,A: list_set_a] :
( ( member_set_a2 @ E @ ( set_set_a2 @ A ) )
=> ( ! [Z22: list_set_a] :
( A
!= ( cons_set_a @ E @ Z22 ) )
=> ~ ! [Z1: set_a,Z22: list_set_a] :
( ( A
= ( cons_set_a @ Z1 @ Z22 ) )
=> ~ ( member_set_a2 @ E @ ( set_set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_327_list_Oset__intros_I1_J,axiom,
! [X21: set_set_a,X22: list_set_set_a] : ( member_set_set_a2 @ X21 @ ( set_set_set_a2 @ ( cons_set_set_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_328_list_Oset__intros_I1_J,axiom,
! [X21: a,X22: list_a] : ( member_a2 @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_329_list_Oset__intros_I1_J,axiom,
! [X21: set_a,X22: list_set_a] : ( member_set_a2 @ X21 @ ( set_set_a2 @ ( cons_set_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_330_list_Oset__intros_I2_J,axiom,
! [Y: set_set_a,X22: list_set_set_a,X21: set_set_a] :
( ( member_set_set_a2 @ Y @ ( set_set_set_a2 @ X22 ) )
=> ( member_set_set_a2 @ Y @ ( set_set_set_a2 @ ( cons_set_set_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_331_list_Oset__intros_I2_J,axiom,
! [Y: a,X22: list_a,X21: a] :
( ( member_a2 @ Y @ ( set_a2 @ X22 ) )
=> ( member_a2 @ Y @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_332_list_Oset__intros_I2_J,axiom,
! [Y: set_a,X22: list_set_a,X21: set_a] :
( ( member_set_a2 @ Y @ ( set_set_a2 @ X22 ) )
=> ( member_set_a2 @ Y @ ( set_set_a2 @ ( cons_set_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_333_can__select__def,axiom,
( can_select_a
= ( ^ [P2: a > $o,A4: set_a] :
? [X3: a] :
( ( member_a2 @ X3 @ A4 )
& ( P2 @ X3 )
& ! [Y4: a] :
( ( ( member_a2 @ Y4 @ A4 )
& ( P2 @ Y4 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_334_can__select__def,axiom,
( can_select_set_a
= ( ^ [P2: set_a > $o,A4: set_set_a] :
? [X3: set_a] :
( ( member_set_a2 @ X3 @ A4 )
& ( P2 @ X3 )
& ! [Y4: set_a] :
( ( ( member_set_a2 @ Y4 @ A4 )
& ( P2 @ Y4 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_335_Cons__eq__appendI,axiom,
! [X2: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( cons_a @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X2 @ Xs )
= ( append_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_336_Cons__eq__appendI,axiom,
! [X2: set_a,Xs1: list_set_a,Ys: list_set_a,Xs: list_set_a,Zs: list_set_a] :
( ( ( cons_set_a @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_set_a @ Xs1 @ Zs ) )
=> ( ( cons_set_a @ X2 @ Xs )
= ( append_set_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_337_append__Cons,axiom,
! [X2: a,Xs: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X2 @ Xs ) @ Ys )
= ( cons_a @ X2 @ ( append_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_338_append__Cons,axiom,
! [X2: set_a,Xs: list_set_a,Ys: list_set_a] :
( ( append_set_a @ ( cons_set_a @ X2 @ Xs ) @ Ys )
= ( cons_set_a @ X2 @ ( append_set_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_339_list_Osel_I3_J,axiom,
! [X21: a,X22: list_a] :
( ( tl_a @ ( cons_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_340_list_Osel_I3_J,axiom,
! [X21: set_a,X22: list_set_a] :
( ( tl_set_a @ ( cons_set_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_341_member__rec_I1_J,axiom,
! [X2: a,Xs: list_a,Y: a] :
( ( member_a @ ( cons_a @ X2 @ Xs ) @ Y )
= ( ( X2 = Y )
| ( member_a @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_342_member__rec_I1_J,axiom,
! [X2: set_a,Xs: list_set_a,Y: set_a] :
( ( member_set_a @ ( cons_set_a @ X2 @ Xs ) @ Y )
= ( ( X2 = Y )
| ( member_set_a @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_343_set__subset__Cons,axiom,
! [Xs: list_set_a,X2: set_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ ( set_set_a2 @ ( cons_set_a @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_344_set__subset__Cons,axiom,
! [Xs: list_set_set_a,X2: set_set_a] : ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ Xs ) @ ( set_set_set_a2 @ ( cons_set_set_a @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_345_set__subset__Cons,axiom,
! [Xs: list_a,X2: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_346_rev__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ( P @ nil_a )
=> ( ! [X: a,Xs3: list_a] :
( ( P @ Xs3 )
=> ( P @ ( append_a @ Xs3 @ ( cons_a @ X @ nil_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_347_rev__induct,axiom,
! [P: list_set_a > $o,Xs: list_set_a] :
( ( P @ nil_set_a )
=> ( ! [X: set_a,Xs3: list_set_a] :
( ( P @ Xs3 )
=> ( P @ ( append_set_a @ Xs3 @ ( cons_set_a @ X @ nil_set_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_348_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys2: list_a,Y2: a] :
( Xs
!= ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_349_rev__exhaust,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
=> ~ ! [Ys2: list_set_a,Y2: set_a] :
( Xs
!= ( append_set_a @ Ys2 @ ( cons_set_a @ Y2 @ nil_set_a ) ) ) ) ).
% rev_exhaust
thf(fact_350_Cons__eq__append__conv,axiom,
! [X2: a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X2 @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X2 @ Xs )
= Zs ) )
| ? [Ys4: list_a] :
( ( ( cons_a @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append_a @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_351_Cons__eq__append__conv,axiom,
! [X2: set_a,Xs: list_set_a,Ys: list_set_a,Zs: list_set_a] :
( ( ( cons_set_a @ X2 @ Xs )
= ( append_set_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_set_a )
& ( ( cons_set_a @ X2 @ Xs )
= Zs ) )
| ? [Ys4: list_set_a] :
( ( ( cons_set_a @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append_set_a @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_352_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X2: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X2 @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X2 @ Xs ) ) )
| ? [Ys4: list_a] :
( ( Ys
= ( cons_a @ X2 @ Ys4 ) )
& ( ( append_a @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_353_append__eq__Cons__conv,axiom,
! [Ys: list_set_a,Zs: list_set_a,X2: set_a,Xs: list_set_a] :
( ( ( append_set_a @ Ys @ Zs )
= ( cons_set_a @ X2 @ Xs ) )
= ( ( ( Ys = nil_set_a )
& ( Zs
= ( cons_set_a @ X2 @ Xs ) ) )
| ? [Ys4: list_set_a] :
( ( Ys
= ( cons_set_a @ X2 @ Ys4 ) )
& ( ( append_set_a @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_354_rev__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X: a] : ( P @ ( cons_a @ X @ nil_a ) )
=> ( ! [X: a,Xs3: list_a] :
( ( Xs3 != nil_a )
=> ( ( P @ Xs3 )
=> ( P @ ( append_a @ Xs3 @ ( cons_a @ X @ nil_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_355_rev__nonempty__induct,axiom,
! [Xs: list_set_a,P: list_set_a > $o] :
( ( Xs != nil_set_a )
=> ( ! [X: set_a] : ( P @ ( cons_set_a @ X @ nil_set_a ) )
=> ( ! [X: set_a,Xs3: list_set_a] :
( ( Xs3 != nil_set_a )
=> ( ( P @ Xs3 )
=> ( P @ ( append_set_a @ Xs3 @ ( cons_set_a @ X @ nil_set_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_356_split__list__first__prop__iff,axiom,
! [Xs: list_set_set_a,P: set_set_a > $o] :
( ( ? [X3: set_set_a] :
( ( member_set_set_a2 @ X3 @ ( set_set_set_a2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_set_set_a,X3: set_set_a] :
( ? [Zs2: list_set_set_a] :
( Xs
= ( append_set_set_a @ Ys3 @ ( cons_set_set_a @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y4: set_set_a] :
( ( member_set_set_a2 @ Y4 @ ( set_set_set_a2 @ Ys3 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_357_split__list__first__prop__iff,axiom,
! [Xs: list_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_a,X3: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y4: a] :
( ( member_a2 @ Y4 @ ( set_a2 @ Ys3 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_358_split__list__first__prop__iff,axiom,
! [Xs: list_set_a,P: set_a > $o] :
( ( ? [X3: set_a] :
( ( member_set_a2 @ X3 @ ( set_set_a2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_set_a,X3: set_a] :
( ? [Zs2: list_set_a] :
( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y4: set_a] :
( ( member_set_a2 @ Y4 @ ( set_set_a2 @ Ys3 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_359_split__list__last__prop__iff,axiom,
! [Xs: list_set_set_a,P: set_set_a > $o] :
( ( ? [X3: set_set_a] :
( ( member_set_set_a2 @ X3 @ ( set_set_set_a2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_set_set_a,X3: set_set_a,Zs2: list_set_set_a] :
( ( Xs
= ( append_set_set_a @ Ys3 @ ( cons_set_set_a @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y4: set_set_a] :
( ( member_set_set_a2 @ Y4 @ ( set_set_set_a2 @ Zs2 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_360_split__list__last__prop__iff,axiom,
! [Xs: list_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_a,X3: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y4: a] :
( ( member_a2 @ Y4 @ ( set_a2 @ Zs2 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_361_split__list__last__prop__iff,axiom,
! [Xs: list_set_a,P: set_a > $o] :
( ( ? [X3: set_a] :
( ( member_set_a2 @ X3 @ ( set_set_a2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_set_a,X3: set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y4: set_a] :
( ( member_set_a2 @ Y4 @ ( set_set_a2 @ Zs2 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_362_in__set__conv__decomp__first,axiom,
! [X2: set_set_a,Xs: list_set_set_a] :
( ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ Xs ) )
= ( ? [Ys3: list_set_set_a,Zs2: list_set_set_a] :
( ( Xs
= ( append_set_set_a @ Ys3 @ ( cons_set_set_a @ X2 @ Zs2 ) ) )
& ~ ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_363_in__set__conv__decomp__first,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs2 ) ) )
& ~ ( member_a2 @ X2 @ ( set_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_364_in__set__conv__decomp__first,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a2 @ X2 @ ( set_set_a2 @ Xs ) )
= ( ? [Ys3: list_set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X2 @ Zs2 ) ) )
& ~ ( member_set_a2 @ X2 @ ( set_set_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_365_in__set__conv__decomp__last,axiom,
! [X2: set_set_a,Xs: list_set_set_a] :
( ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ Xs ) )
= ( ? [Ys3: list_set_set_a,Zs2: list_set_set_a] :
( ( Xs
= ( append_set_set_a @ Ys3 @ ( cons_set_set_a @ X2 @ Zs2 ) ) )
& ~ ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_366_in__set__conv__decomp__last,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs2 ) ) )
& ~ ( member_a2 @ X2 @ ( set_a2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_367_in__set__conv__decomp__last,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a2 @ X2 @ ( set_set_a2 @ Xs ) )
= ( ? [Ys3: list_set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X2 @ Zs2 ) ) )
& ~ ( member_set_a2 @ X2 @ ( set_set_a2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_368_split__list__first__propE,axiom,
! [Xs: list_set_set_a,P: set_set_a > $o] :
( ? [X5: set_set_a] :
( ( member_set_set_a2 @ X5 @ ( set_set_set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_set_set_a,X: set_set_a] :
( ? [Zs3: list_set_set_a] :
( Xs
= ( append_set_set_a @ Ys2 @ ( cons_set_set_a @ X @ Zs3 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa: set_set_a] :
( ( member_set_set_a2 @ Xa @ ( set_set_set_a2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_369_split__list__first__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a2 @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_a,X: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs3 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa: a] :
( ( member_a2 @ Xa @ ( set_a2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_370_split__list__first__propE,axiom,
! [Xs: list_set_a,P: set_a > $o] :
( ? [X5: set_a] :
( ( member_set_a2 @ X5 @ ( set_set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_set_a,X: set_a] :
( ? [Zs3: list_set_a] :
( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X @ Zs3 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa: set_a] :
( ( member_set_a2 @ Xa @ ( set_set_a2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_371_split__list__last__propE,axiom,
! [Xs: list_set_set_a,P: set_set_a > $o] :
( ? [X5: set_set_a] :
( ( member_set_set_a2 @ X5 @ ( set_set_set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_set_set_a,X: set_set_a,Zs3: list_set_set_a] :
( ( Xs
= ( append_set_set_a @ Ys2 @ ( cons_set_set_a @ X @ Zs3 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa: set_set_a] :
( ( member_set_set_a2 @ Xa @ ( set_set_set_a2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_372_split__list__last__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a2 @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_a,X: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs3 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa: a] :
( ( member_a2 @ Xa @ ( set_a2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_373_split__list__last__propE,axiom,
! [Xs: list_set_a,P: set_a > $o] :
( ? [X5: set_a] :
( ( member_set_a2 @ X5 @ ( set_set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_set_a,X: set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X @ Zs3 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa: set_a] :
( ( member_set_a2 @ Xa @ ( set_set_a2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_374_split__list__first__prop,axiom,
! [Xs: list_set_set_a,P: set_set_a > $o] :
( ? [X5: set_set_a] :
( ( member_set_set_a2 @ X5 @ ( set_set_set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_set_set_a,X: set_set_a] :
( ? [Zs3: list_set_set_a] :
( Xs
= ( append_set_set_a @ Ys2 @ ( cons_set_set_a @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Xa: set_set_a] :
( ( member_set_set_a2 @ Xa @ ( set_set_set_a2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_375_split__list__first__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a2 @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_a,X: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Xa: a] :
( ( member_a2 @ Xa @ ( set_a2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_376_split__list__first__prop,axiom,
! [Xs: list_set_a,P: set_a > $o] :
( ? [X5: set_a] :
( ( member_set_a2 @ X5 @ ( set_set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_set_a,X: set_a] :
( ? [Zs3: list_set_a] :
( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Xa: set_a] :
( ( member_set_a2 @ Xa @ ( set_set_a2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_377_split__list__last__prop,axiom,
! [Xs: list_set_set_a,P: set_set_a > $o] :
( ? [X5: set_set_a] :
( ( member_set_set_a2 @ X5 @ ( set_set_set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_set_set_a,X: set_set_a,Zs3: list_set_set_a] :
( ( Xs
= ( append_set_set_a @ Ys2 @ ( cons_set_set_a @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Xa: set_set_a] :
( ( member_set_set_a2 @ Xa @ ( set_set_set_a2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_378_split__list__last__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a2 @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_a,X: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Xa: a] :
( ( member_a2 @ Xa @ ( set_a2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_379_split__list__last__prop,axiom,
! [Xs: list_set_a,P: set_a > $o] :
( ? [X5: set_a] :
( ( member_set_a2 @ X5 @ ( set_set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_set_a,X: set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Xa: set_a] :
( ( member_set_a2 @ Xa @ ( set_set_a2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_380_in__set__conv__decomp,axiom,
! [X2: set_set_a,Xs: list_set_set_a] :
( ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ Xs ) )
= ( ? [Ys3: list_set_set_a,Zs2: list_set_set_a] :
( Xs
= ( append_set_set_a @ Ys3 @ ( cons_set_set_a @ X2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_381_in__set__conv__decomp,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_382_in__set__conv__decomp,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a2 @ X2 @ ( set_set_a2 @ Xs ) )
= ( ? [Ys3: list_set_a,Zs2: list_set_a] :
( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_383_append__Cons__eq__iff,axiom,
! [X2: set_set_a,Xs: list_set_set_a,Ys: list_set_set_a,Xs4: list_set_set_a,Ys5: list_set_set_a] :
( ~ ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ Xs ) )
=> ( ~ ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ Ys ) )
=> ( ( ( append_set_set_a @ Xs @ ( cons_set_set_a @ X2 @ Ys ) )
= ( append_set_set_a @ Xs4 @ ( cons_set_set_a @ X2 @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_384_append__Cons__eq__iff,axiom,
! [X2: a,Xs: list_a,Ys: list_a,Xs4: list_a,Ys5: list_a] :
( ~ ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ~ ( member_a2 @ X2 @ ( set_a2 @ Ys ) )
=> ( ( ( append_a @ Xs @ ( cons_a @ X2 @ Ys ) )
= ( append_a @ Xs4 @ ( cons_a @ X2 @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_385_append__Cons__eq__iff,axiom,
! [X2: set_a,Xs: list_set_a,Ys: list_set_a,Xs4: list_set_a,Ys5: list_set_a] :
( ~ ( member_set_a2 @ X2 @ ( set_set_a2 @ Xs ) )
=> ( ~ ( member_set_a2 @ X2 @ ( set_set_a2 @ Ys ) )
=> ( ( ( append_set_a @ Xs @ ( cons_set_a @ X2 @ Ys ) )
= ( append_set_a @ Xs4 @ ( cons_set_a @ X2 @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_386_split__list__propE,axiom,
! [Xs: list_set_set_a,P: set_set_a > $o] :
( ? [X5: set_set_a] :
( ( member_set_set_a2 @ X5 @ ( set_set_set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_set_set_a,X: set_set_a] :
( ? [Zs3: list_set_set_a] :
( Xs
= ( append_set_set_a @ Ys2 @ ( cons_set_set_a @ X @ Zs3 ) ) )
=> ~ ( P @ X ) ) ) ).
% split_list_propE
thf(fact_387_split__list__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a2 @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_a,X: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs3 ) ) )
=> ~ ( P @ X ) ) ) ).
% split_list_propE
thf(fact_388_split__list__propE,axiom,
! [Xs: list_set_a,P: set_a > $o] :
( ? [X5: set_a] :
( ( member_set_a2 @ X5 @ ( set_set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_set_a,X: set_a] :
( ? [Zs3: list_set_a] :
( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X @ Zs3 ) ) )
=> ~ ( P @ X ) ) ) ).
% split_list_propE
thf(fact_389_split__list__first,axiom,
! [X2: set_set_a,Xs: list_set_set_a] :
( ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ Xs ) )
=> ? [Ys2: list_set_set_a,Zs3: list_set_set_a] :
( ( Xs
= ( append_set_set_a @ Ys2 @ ( cons_set_set_a @ X2 @ Zs3 ) ) )
& ~ ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_390_split__list__first,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs3 ) ) )
& ~ ( member_a2 @ X2 @ ( set_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_391_split__list__first,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a2 @ X2 @ ( set_set_a2 @ Xs ) )
=> ? [Ys2: list_set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X2 @ Zs3 ) ) )
& ~ ( member_set_a2 @ X2 @ ( set_set_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_392_split__list__prop,axiom,
! [Xs: list_set_set_a,P: set_set_a > $o] :
( ? [X5: set_set_a] :
( ( member_set_set_a2 @ X5 @ ( set_set_set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_set_set_a,X: set_set_a] :
( ? [Zs3: list_set_set_a] :
( Xs
= ( append_set_set_a @ Ys2 @ ( cons_set_set_a @ X @ Zs3 ) ) )
& ( P @ X ) ) ) ).
% split_list_prop
thf(fact_393_split__list__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a2 @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_a,X: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs3 ) ) )
& ( P @ X ) ) ) ).
% split_list_prop
thf(fact_394_split__list__prop,axiom,
! [Xs: list_set_a,P: set_a > $o] :
( ? [X5: set_a] :
( ( member_set_a2 @ X5 @ ( set_set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_set_a,X: set_a] :
( ? [Zs3: list_set_a] :
( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X @ Zs3 ) ) )
& ( P @ X ) ) ) ).
% split_list_prop
thf(fact_395_split__list__last,axiom,
! [X2: set_set_a,Xs: list_set_set_a] :
( ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ Xs ) )
=> ? [Ys2: list_set_set_a,Zs3: list_set_set_a] :
( ( Xs
= ( append_set_set_a @ Ys2 @ ( cons_set_set_a @ X2 @ Zs3 ) ) )
& ~ ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_396_split__list__last,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs3 ) ) )
& ~ ( member_a2 @ X2 @ ( set_a2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_397_split__list__last,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a2 @ X2 @ ( set_set_a2 @ Xs ) )
=> ? [Ys2: list_set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X2 @ Zs3 ) ) )
& ~ ( member_set_a2 @ X2 @ ( set_set_a2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_398_split__list,axiom,
! [X2: set_set_a,Xs: list_set_set_a] :
( ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ Xs ) )
=> ? [Ys2: list_set_set_a,Zs3: list_set_set_a] :
( Xs
= ( append_set_set_a @ Ys2 @ ( cons_set_set_a @ X2 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_399_split__list,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs3: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_400_split__list,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a2 @ X2 @ ( set_set_a2 @ Xs ) )
=> ? [Ys2: list_set_a,Zs3: list_set_a] :
( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X2 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_401_ulgraph_Owalk__edges_Ocases,axiom,
! [Vertices: set_a,Edges: set_set_a,X2: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( X2 != nil_a )
=> ( ! [X: a] :
( X2
!= ( cons_a @ X @ nil_a ) )
=> ~ ! [X: a,Y2: a,Ys2: list_a] :
( X2
!= ( cons_a @ X @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ) ).
% ulgraph.walk_edges.cases
thf(fact_402_ulgraph_Owalk__edges_Ocases,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X2: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( X2 != nil_set_a )
=> ( ! [X: set_a] :
( X2
!= ( cons_set_a @ X @ nil_set_a ) )
=> ~ ! [X: set_a,Y2: set_a,Ys2: list_set_a] :
( X2
!= ( cons_set_a @ X @ ( cons_set_a @ Y2 @ Ys2 ) ) ) ) ) ) ).
% ulgraph.walk_edges.cases
thf(fact_403_Nil__tl,axiom,
! [Xs: list_a] :
( ( nil_a
= ( tl_a @ Xs ) )
= ( ( Xs = nil_a )
| ? [X3: a] :
( Xs
= ( cons_a @ X3 @ nil_a ) ) ) ) ).
% Nil_tl
thf(fact_404_Nil__tl,axiom,
! [Xs: list_set_a] :
( ( nil_set_a
= ( tl_set_a @ Xs ) )
= ( ( Xs = nil_set_a )
| ? [X3: set_a] :
( Xs
= ( cons_set_a @ X3 @ nil_set_a ) ) ) ) ).
% Nil_tl
thf(fact_405_tl__Nil,axiom,
! [Xs: list_a] :
( ( ( tl_a @ Xs )
= nil_a )
= ( ( Xs = nil_a )
| ? [X3: a] :
( Xs
= ( cons_a @ X3 @ nil_a ) ) ) ) ).
% tl_Nil
thf(fact_406_tl__Nil,axiom,
! [Xs: list_set_a] :
( ( ( tl_set_a @ Xs )
= nil_set_a )
= ( ( Xs = nil_set_a )
| ? [X3: set_a] :
( Xs
= ( cons_set_a @ X3 @ nil_set_a ) ) ) ) ).
% tl_Nil
thf(fact_407_comp__sgraph_Owalk__edges_Osimps_I2_J,axiom,
! [X2: a] :
( ( undire7337870655677353998dges_a @ ( cons_a @ X2 @ nil_a ) )
= nil_set_a ) ).
% comp_sgraph.walk_edges.simps(2)
thf(fact_408_comp__sgraph_Owalk__edges_Osimps_I2_J,axiom,
! [X2: set_a] :
( ( undire6234387080713648494_set_a @ ( cons_set_a @ X2 @ nil_set_a ) )
= nil_set_set_a ) ).
% comp_sgraph.walk_edges.simps(2)
thf(fact_409_List_Oinsert__def,axiom,
( insert_set_set_a
= ( ^ [X3: set_set_a,Xs2: list_set_set_a] : ( if_list_set_set_a @ ( member_set_set_a2 @ X3 @ ( set_set_set_a2 @ Xs2 ) ) @ Xs2 @ ( cons_set_set_a @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_410_List_Oinsert__def,axiom,
( insert_a
= ( ^ [X3: a,Xs2: list_a] : ( if_list_a @ ( member_a2 @ X3 @ ( set_a2 @ Xs2 ) ) @ Xs2 @ ( cons_a @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_411_List_Oinsert__def,axiom,
( insert_set_a
= ( ^ [X3: set_a,Xs2: list_set_a] : ( if_list_set_a @ ( member_set_a2 @ X3 @ ( set_set_a2 @ Xs2 ) ) @ Xs2 @ ( cons_set_a @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_412_walk__edges__app,axiom,
! [Xs: list_a,Y: a,X2: a] :
( ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( cons_a @ Y @ ( cons_a @ X2 @ nil_a ) ) ) )
= ( append_set_a @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( cons_a @ Y @ nil_a ) ) ) @ ( cons_set_a @ ( insert_a2 @ Y @ ( insert_a2 @ X2 @ bot_bot_set_a ) ) @ nil_set_a ) ) ) ).
% walk_edges_app
thf(fact_413_walk__edges_Oelims,axiom,
! [X2: list_a,Y: list_set_a] :
( ( ( undire7337870655677353998dges_a @ X2 )
= Y )
=> ( ( ( X2 = nil_a )
=> ( Y != nil_set_a ) )
=> ( ( ? [X: a] :
( X2
= ( cons_a @ X @ nil_a ) )
=> ( Y != nil_set_a ) )
=> ~ ! [X: a,Y2: a,Ys2: list_a] :
( ( X2
= ( cons_a @ X @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( Y
!= ( cons_set_a @ ( insert_a2 @ X @ ( insert_a2 @ Y2 @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ) ) ) ).
% walk_edges.elims
thf(fact_414_walk__edges_Osimps_I3_J,axiom,
! [X2: a,Y: a,Ys: list_a] :
( ( undire7337870655677353998dges_a @ ( cons_a @ X2 @ ( cons_a @ Y @ Ys ) ) )
= ( cons_set_a @ ( insert_a2 @ X2 @ ( insert_a2 @ Y @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ ( cons_a @ Y @ Ys ) ) ) ) ).
% walk_edges.simps(3)
thf(fact_415_ulgraph_Owalk__edges__app,axiom,
! [Vertices: set_a,Edges: set_set_a,Xs: list_a,Y: a,X2: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( cons_a @ Y @ ( cons_a @ X2 @ nil_a ) ) ) )
= ( append_set_a @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( cons_a @ Y @ nil_a ) ) ) @ ( cons_set_a @ ( insert_a2 @ Y @ ( insert_a2 @ X2 @ bot_bot_set_a ) ) @ nil_set_a ) ) ) ) ).
% ulgraph.walk_edges_app
thf(fact_416_ulgraph_Owalk__edges__app,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,Xs: list_set_a,Y: set_a,X2: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ ( cons_set_a @ Y @ ( cons_set_a @ X2 @ nil_set_a ) ) ) )
= ( append_set_set_a @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ ( cons_set_a @ Y @ nil_set_a ) ) ) @ ( cons_set_set_a @ ( insert_set_a2 @ Y @ ( insert_set_a2 @ X2 @ bot_bot_set_set_a ) ) @ nil_set_set_a ) ) ) ) ).
% ulgraph.walk_edges_app
thf(fact_417_the__elem__set,axiom,
! [X2: set_set_a] :
( ( the_elem_set_set_a @ ( set_set_set_a2 @ ( cons_set_set_a @ X2 @ nil_set_set_a ) ) )
= X2 ) ).
% the_elem_set
thf(fact_418_the__elem__set,axiom,
! [X2: a] :
( ( the_elem_a @ ( set_a2 @ ( cons_a @ X2 @ nil_a ) ) )
= X2 ) ).
% the_elem_set
thf(fact_419_the__elem__set,axiom,
! [X2: set_a] :
( ( the_elem_set_a @ ( set_set_a2 @ ( cons_set_a @ X2 @ nil_set_a ) ) )
= X2 ) ).
% the_elem_set
thf(fact_420_ulgraph_Owalk__edges_Oelims,axiom,
! [Vertices: set_a,Edges: set_set_a,X2: list_a,Y: list_set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( ( undire7337870655677353998dges_a @ X2 )
= Y )
=> ( ( ( X2 = nil_a )
=> ( Y != nil_set_a ) )
=> ( ( ? [X: a] :
( X2
= ( cons_a @ X @ nil_a ) )
=> ( Y != nil_set_a ) )
=> ~ ! [X: a,Y2: a,Ys2: list_a] :
( ( X2
= ( cons_a @ X @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( Y
!= ( cons_set_a @ ( insert_a2 @ X @ ( insert_a2 @ Y2 @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ) ) ) ) ).
% ulgraph.walk_edges.elims
thf(fact_421_ulgraph_Owalk__edges_Oelims,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X2: list_set_a,Y: list_set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( ( undire6234387080713648494_set_a @ X2 )
= Y )
=> ( ( ( X2 = nil_set_a )
=> ( Y != nil_set_set_a ) )
=> ( ( ? [X: set_a] :
( X2
= ( cons_set_a @ X @ nil_set_a ) )
=> ( Y != nil_set_set_a ) )
=> ~ ! [X: set_a,Y2: set_a,Ys2: list_set_a] :
( ( X2
= ( cons_set_a @ X @ ( cons_set_a @ Y2 @ Ys2 ) ) )
=> ( Y
!= ( cons_set_set_a @ ( insert_set_a2 @ X @ ( insert_set_a2 @ Y2 @ bot_bot_set_set_a ) ) @ ( undire6234387080713648494_set_a @ ( cons_set_a @ Y2 @ Ys2 ) ) ) ) ) ) ) ) ) ).
% ulgraph.walk_edges.elims
thf(fact_422_comp__sgraph_Owalk__edges__app,axiom,
! [Xs: list_a,Y: a,X2: a] :
( ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( cons_a @ Y @ ( cons_a @ X2 @ nil_a ) ) ) )
= ( append_set_a @ ( undire7337870655677353998dges_a @ ( append_a @ Xs @ ( cons_a @ Y @ nil_a ) ) ) @ ( cons_set_a @ ( insert_a2 @ Y @ ( insert_a2 @ X2 @ bot_bot_set_a ) ) @ nil_set_a ) ) ) ).
% comp_sgraph.walk_edges_app
thf(fact_423_comp__sgraph_Owalk__edges__app,axiom,
! [Xs: list_set_a,Y: set_a,X2: set_a] :
( ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ ( cons_set_a @ Y @ ( cons_set_a @ X2 @ nil_set_a ) ) ) )
= ( append_set_set_a @ ( undire6234387080713648494_set_a @ ( append_set_a @ Xs @ ( cons_set_a @ Y @ nil_set_a ) ) ) @ ( cons_set_set_a @ ( insert_set_a2 @ Y @ ( insert_set_a2 @ X2 @ bot_bot_set_set_a ) ) @ nil_set_set_a ) ) ) ).
% comp_sgraph.walk_edges_app
thf(fact_424_insert__subsetI,axiom,
! [X2: set_a,A2: set_set_a,X6: set_set_a] :
( ( member_set_a2 @ X2 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ X6 @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a2 @ X2 @ X6 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_425_insert__subsetI,axiom,
! [X2: set_set_a,A2: set_set_set_a,X6: set_set_set_a] :
( ( member_set_set_a2 @ X2 @ A2 )
=> ( ( ord_le5722252365846178494_set_a @ X6 @ A2 )
=> ( ord_le5722252365846178494_set_a @ ( insert_set_set_a2 @ X2 @ X6 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_426_insert__subsetI,axiom,
! [X2: a,A2: set_a,X6: set_a] :
( ( member_a2 @ X2 @ A2 )
=> ( ( ord_less_eq_set_a @ X6 @ A2 )
=> ( ord_less_eq_set_a @ ( insert_a2 @ X2 @ X6 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_427_comp__sgraph_Owalk__edges_Oelims,axiom,
! [X2: list_a,Y: list_set_a] :
( ( ( undire7337870655677353998dges_a @ X2 )
= Y )
=> ( ( ( X2 = nil_a )
=> ( Y != nil_set_a ) )
=> ( ( ? [X: a] :
( X2
= ( cons_a @ X @ nil_a ) )
=> ( Y != nil_set_a ) )
=> ~ ! [X: a,Y2: a,Ys2: list_a] :
( ( X2
= ( cons_a @ X @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( Y
!= ( cons_set_a @ ( insert_a2 @ X @ ( insert_a2 @ Y2 @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ) ) ) ).
% comp_sgraph.walk_edges.elims
thf(fact_428_comp__sgraph_Owalk__edges_Oelims,axiom,
! [X2: list_set_a,Y: list_set_set_a] :
( ( ( undire6234387080713648494_set_a @ X2 )
= Y )
=> ( ( ( X2 = nil_set_a )
=> ( Y != nil_set_set_a ) )
=> ( ( ? [X: set_a] :
( X2
= ( cons_set_a @ X @ nil_set_a ) )
=> ( Y != nil_set_set_a ) )
=> ~ ! [X: set_a,Y2: set_a,Ys2: list_set_a] :
( ( X2
= ( cons_set_a @ X @ ( cons_set_a @ Y2 @ Ys2 ) ) )
=> ( Y
!= ( cons_set_set_a @ ( insert_set_a2 @ X @ ( insert_set_a2 @ Y2 @ bot_bot_set_set_a ) ) @ ( undire6234387080713648494_set_a @ ( cons_set_a @ Y2 @ Ys2 ) ) ) ) ) ) ) ) ).
% comp_sgraph.walk_edges.elims
thf(fact_429_maps__simps_I1_J,axiom,
! [F: a > list_a,X2: a,Xs: list_a] :
( ( maps_a_a @ F @ ( cons_a @ X2 @ Xs ) )
= ( append_a @ ( F @ X2 ) @ ( maps_a_a @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_430_maps__simps_I1_J,axiom,
! [F: a > list_set_a,X2: a,Xs: list_a] :
( ( maps_a_set_a @ F @ ( cons_a @ X2 @ Xs ) )
= ( append_set_a @ ( F @ X2 ) @ ( maps_a_set_a @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_431_maps__simps_I1_J,axiom,
! [F: set_a > list_a,X2: set_a,Xs: list_set_a] :
( ( maps_set_a_a @ F @ ( cons_set_a @ X2 @ Xs ) )
= ( append_a @ ( F @ X2 ) @ ( maps_set_a_a @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_432_maps__simps_I1_J,axiom,
! [F: set_a > list_set_a,X2: set_a,Xs: list_set_a] :
( ( maps_set_a_set_a @ F @ ( cons_set_a @ X2 @ Xs ) )
= ( append_set_a @ ( F @ X2 ) @ ( maps_set_a_set_a @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_433_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a2 @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_434_empty__iff,axiom,
! [C: a] :
~ ( member_a2 @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_435_all__not__in__conv,axiom,
! [A2: set_set_a] :
( ( ! [X3: set_a] :
~ ( member_set_a2 @ X3 @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_436_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X3: a] :
~ ( member_a2 @ X3 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_437_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_438_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_439_is__edge__between__def,axiom,
( undire8544646567961481629ween_a
= ( ^ [X4: set_a,Y6: set_a,E2: set_a] :
? [X3: a,Y4: a] :
( ( E2
= ( insert_a2 @ X3 @ ( insert_a2 @ Y4 @ bot_bot_set_a ) ) )
& ( member_a2 @ X3 @ X4 )
& ( member_a2 @ Y4 @ Y6 ) ) ) ) ).
% is_edge_between_def
thf(fact_440_subset__empty,axiom,
! [A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a )
= ( A2 = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_441_subset__empty,axiom,
! [A2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ bot_bo3380559777022489994_set_a )
= ( A2 = bot_bo3380559777022489994_set_a ) ) ).
% subset_empty
thf(fact_442_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_443_empty__subsetI,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A2 ) ).
% empty_subsetI
thf(fact_444_empty__subsetI,axiom,
! [A2: set_set_set_a] : ( ord_le5722252365846178494_set_a @ bot_bo3380559777022489994_set_a @ A2 ) ).
% empty_subsetI
thf(fact_445_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_446_singletonI,axiom,
! [A: set_a] : ( member_set_a2 @ A @ ( insert_set_a2 @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_447_singletonI,axiom,
! [A: a] : ( member_a2 @ A @ ( insert_a2 @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_448_singleton__insert__inj__eq_H,axiom,
! [A: set_a,A2: set_set_a,B: set_a] :
( ( ( insert_set_a2 @ A @ A2 )
= ( insert_set_a2 @ B @ bot_bot_set_set_a ) )
= ( ( A = B )
& ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a2 @ B @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_449_singleton__insert__inj__eq_H,axiom,
! [A: set_set_a,A2: set_set_set_a,B: set_set_a] :
( ( ( insert_set_set_a2 @ A @ A2 )
= ( insert_set_set_a2 @ B @ bot_bo3380559777022489994_set_a ) )
= ( ( A = B )
& ( ord_le5722252365846178494_set_a @ A2 @ ( insert_set_set_a2 @ B @ bot_bo3380559777022489994_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_450_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B: a] :
( ( ( insert_a2 @ A @ A2 )
= ( insert_a2 @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a2 @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_451_singleton__insert__inj__eq,axiom,
! [B: set_a,A: set_a,A2: set_set_a] :
( ( ( insert_set_a2 @ B @ bot_bot_set_set_a )
= ( insert_set_a2 @ A @ A2 ) )
= ( ( A = B )
& ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a2 @ B @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_452_singleton__insert__inj__eq,axiom,
! [B: set_set_a,A: set_set_a,A2: set_set_set_a] :
( ( ( insert_set_set_a2 @ B @ bot_bo3380559777022489994_set_a )
= ( insert_set_set_a2 @ A @ A2 ) )
= ( ( A = B )
& ( ord_le5722252365846178494_set_a @ A2 @ ( insert_set_set_a2 @ B @ bot_bo3380559777022489994_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_453_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A2: set_a] :
( ( ( insert_a2 @ B @ bot_bot_set_a )
= ( insert_a2 @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a2 @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_454_set__empty2,axiom,
! [Xs: list_set_a] :
( ( bot_bot_set_set_a
= ( set_set_a2 @ Xs ) )
= ( Xs = nil_set_a ) ) ).
% set_empty2
thf(fact_455_set__empty2,axiom,
! [Xs: list_set_set_a] :
( ( bot_bo3380559777022489994_set_a
= ( set_set_set_a2 @ Xs ) )
= ( Xs = nil_set_set_a ) ) ).
% set_empty2
thf(fact_456_set__empty2,axiom,
! [Xs: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% set_empty2
thf(fact_457_set__empty,axiom,
! [Xs: list_set_a] :
( ( ( set_set_a2 @ Xs )
= bot_bot_set_set_a )
= ( Xs = nil_set_a ) ) ).
% set_empty
thf(fact_458_set__empty,axiom,
! [Xs: list_set_set_a] :
( ( ( set_set_set_a2 @ Xs )
= bot_bo3380559777022489994_set_a )
= ( Xs = nil_set_set_a ) ) ).
% set_empty
thf(fact_459_set__empty,axiom,
! [Xs: list_a] :
( ( ( set_a2 @ Xs )
= bot_bot_set_a )
= ( Xs = nil_a ) ) ).
% set_empty
thf(fact_460_the__elem__eq,axiom,
! [X2: a] :
( ( the_elem_a @ ( insert_a2 @ X2 @ bot_bot_set_a ) )
= X2 ) ).
% the_elem_eq
thf(fact_461_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a2 @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_462_emptyE,axiom,
! [A: a] :
~ ( member_a2 @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_463_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a2 @ A @ A2 ) ) ).
% equals0D
thf(fact_464_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a2 @ A @ A2 ) ) ).
% equals0D
thf(fact_465_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y2: set_a] :
~ ( member_set_a2 @ Y2 @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_466_equals0I,axiom,
! [A2: set_a] :
( ! [Y2: a] :
~ ( member_a2 @ Y2 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_467_ex__in__conv,axiom,
! [A2: set_set_a] :
( ( ? [X3: set_a] : ( member_set_a2 @ X3 @ A2 ) )
= ( A2 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_468_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X3: a] : ( member_a2 @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_469_subset__emptyI,axiom,
! [A2: set_set_a] :
( ! [X: set_a] :
~ ( member_set_a2 @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a ) ) ).
% subset_emptyI
thf(fact_470_subset__emptyI,axiom,
! [A2: set_set_set_a] :
( ! [X: set_set_a] :
~ ( member_set_set_a2 @ X @ A2 )
=> ( ord_le5722252365846178494_set_a @ A2 @ bot_bo3380559777022489994_set_a ) ) ).
% subset_emptyI
thf(fact_471_subset__emptyI,axiom,
! [A2: set_a] :
( ! [X: a] :
~ ( member_a2 @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_472_bot_Oextremum,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% bot.extremum
thf(fact_473_bot_Oextremum,axiom,
! [A: set_set_set_a] : ( ord_le5722252365846178494_set_a @ bot_bo3380559777022489994_set_a @ A ) ).
% bot.extremum
thf(fact_474_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_475_bot_Oextremum__unique,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% bot.extremum_unique
thf(fact_476_bot_Oextremum__unique,axiom,
! [A: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ bot_bo3380559777022489994_set_a )
= ( A = bot_bo3380559777022489994_set_a ) ) ).
% bot.extremum_unique
thf(fact_477_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_478_bot_Oextremum__uniqueI,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
=> ( A = bot_bot_set_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_479_bot_Oextremum__uniqueI,axiom,
! [A: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ bot_bo3380559777022489994_set_a )
=> ( A = bot_bo3380559777022489994_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_480_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_481_singletonD,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a2 @ B @ ( insert_set_a2 @ A @ bot_bot_set_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_482_singletonD,axiom,
! [B: a,A: a] :
( ( member_a2 @ B @ ( insert_a2 @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_483_singleton__iff,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a2 @ B @ ( insert_set_a2 @ A @ bot_bot_set_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_484_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a2 @ B @ ( insert_a2 @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_485_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ( insert_a2 @ A @ ( insert_a2 @ B @ bot_bot_set_a ) )
= ( insert_a2 @ C @ ( insert_a2 @ D2 @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_486_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a2 @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_487_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a2 @ A @ bot_bot_set_a )
= ( insert_a2 @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_488_ulgraph_Oempty__not__edge,axiom,
! [Vertices: set_a,Edges: set_set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ~ ( member_set_a2 @ bot_bot_set_a @ Edges ) ) ).
% ulgraph.empty_not_edge
thf(fact_489_subset__singleton__iff,axiom,
! [X6: set_set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ X6 @ ( insert_set_a2 @ A @ bot_bot_set_set_a ) )
= ( ( X6 = bot_bot_set_set_a )
| ( X6
= ( insert_set_a2 @ A @ bot_bot_set_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_490_subset__singleton__iff,axiom,
! [X6: set_set_set_a,A: set_set_a] :
( ( ord_le5722252365846178494_set_a @ X6 @ ( insert_set_set_a2 @ A @ bot_bo3380559777022489994_set_a ) )
= ( ( X6 = bot_bo3380559777022489994_set_a )
| ( X6
= ( insert_set_set_a2 @ A @ bot_bo3380559777022489994_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_491_subset__singleton__iff,axiom,
! [X6: set_a,A: a] :
( ( ord_less_eq_set_a @ X6 @ ( insert_a2 @ A @ bot_bot_set_a ) )
= ( ( X6 = bot_bot_set_a )
| ( X6
= ( insert_a2 @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_492_subset__singletonD,axiom,
! [A2: set_set_a,X2: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a2 @ X2 @ bot_bot_set_set_a ) )
=> ( ( A2 = bot_bot_set_set_a )
| ( A2
= ( insert_set_a2 @ X2 @ bot_bot_set_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_493_subset__singletonD,axiom,
! [A2: set_set_set_a,X2: set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ ( insert_set_set_a2 @ X2 @ bot_bo3380559777022489994_set_a ) )
=> ( ( A2 = bot_bo3380559777022489994_set_a )
| ( A2
= ( insert_set_set_a2 @ X2 @ bot_bo3380559777022489994_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_494_subset__singletonD,axiom,
! [A2: set_a,X2: a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a2 @ X2 @ bot_bot_set_a ) )
=> ( ( A2 = bot_bot_set_a )
| ( A2
= ( insert_a2 @ X2 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_495_empty__set,axiom,
( bot_bot_set_set_a
= ( set_set_a2 @ nil_set_a ) ) ).
% empty_set
thf(fact_496_empty__set,axiom,
( bot_bo3380559777022489994_set_a
= ( set_set_set_a2 @ nil_set_set_a ) ) ).
% empty_set
thf(fact_497_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_498_maps__simps_I2_J,axiom,
! [F: a > list_a] :
( ( maps_a_a @ F @ nil_a )
= nil_a ) ).
% maps_simps(2)
thf(fact_499_maps__simps_I2_J,axiom,
! [F: a > list_set_a] :
( ( maps_a_set_a @ F @ nil_a )
= nil_set_a ) ).
% maps_simps(2)
thf(fact_500_maps__simps_I2_J,axiom,
! [F: set_a > list_a] :
( ( maps_set_a_a @ F @ nil_set_a )
= nil_a ) ).
% maps_simps(2)
thf(fact_501_maps__simps_I2_J,axiom,
! [F: set_a > list_set_a] :
( ( maps_set_a_set_a @ F @ nil_set_a )
= nil_set_a ) ).
% maps_simps(2)
thf(fact_502_comp__sgraph_Owalk__edges_Osimps_I3_J,axiom,
! [X2: a,Y: a,Ys: list_a] :
( ( undire7337870655677353998dges_a @ ( cons_a @ X2 @ ( cons_a @ Y @ Ys ) ) )
= ( cons_set_a @ ( insert_a2 @ X2 @ ( insert_a2 @ Y @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ ( cons_a @ Y @ Ys ) ) ) ) ).
% comp_sgraph.walk_edges.simps(3)
thf(fact_503_comp__sgraph_Owalk__edges_Osimps_I3_J,axiom,
! [X2: set_a,Y: set_a,Ys: list_set_a] :
( ( undire6234387080713648494_set_a @ ( cons_set_a @ X2 @ ( cons_set_a @ Y @ Ys ) ) )
= ( cons_set_set_a @ ( insert_set_a2 @ X2 @ ( insert_set_a2 @ Y @ bot_bot_set_set_a ) ) @ ( undire6234387080713648494_set_a @ ( cons_set_a @ Y @ Ys ) ) ) ) ).
% comp_sgraph.walk_edges.simps(3)
thf(fact_504_ulgraph_Owalk__edges_Osimps_I3_J,axiom,
! [Vertices: set_a,Edges: set_set_a,X2: a,Y: a,Ys: list_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire7337870655677353998dges_a @ ( cons_a @ X2 @ ( cons_a @ Y @ Ys ) ) )
= ( cons_set_a @ ( insert_a2 @ X2 @ ( insert_a2 @ Y @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ ( cons_a @ Y @ Ys ) ) ) ) ) ).
% ulgraph.walk_edges.simps(3)
thf(fact_505_ulgraph_Owalk__edges_Osimps_I3_J,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X2: set_a,Y: set_a,Ys: list_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire6234387080713648494_set_a @ ( cons_set_a @ X2 @ ( cons_set_a @ Y @ Ys ) ) )
= ( cons_set_set_a @ ( insert_set_a2 @ X2 @ ( insert_set_a2 @ Y @ bot_bot_set_set_a ) ) @ ( undire6234387080713648494_set_a @ ( cons_set_a @ Y @ Ys ) ) ) ) ) ).
% ulgraph.walk_edges.simps(3)
thf(fact_506_ulgraph_Ois__edge__between__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X6: set_set_a,Y7: set_set_a,E: set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire2578756059399487229_set_a @ X6 @ Y7 @ E )
= ( ? [X3: set_a,Y4: set_a] :
( ( E
= ( insert_set_a2 @ X3 @ ( insert_set_a2 @ Y4 @ bot_bot_set_set_a ) ) )
& ( member_set_a2 @ X3 @ X6 )
& ( member_set_a2 @ Y4 @ Y7 ) ) ) ) ) ).
% ulgraph.is_edge_between_def
thf(fact_507_ulgraph_Ois__edge__between__def,axiom,
! [Vertices: set_a,Edges: set_set_a,X6: set_a,Y7: set_a,E: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8544646567961481629ween_a @ X6 @ Y7 @ E )
= ( ? [X3: a,Y4: a] :
( ( E
= ( insert_a2 @ X3 @ ( insert_a2 @ Y4 @ bot_bot_set_a ) ) )
& ( member_a2 @ X3 @ X6 )
& ( member_a2 @ Y4 @ Y7 ) ) ) ) ) ).
% ulgraph.is_edge_between_def
thf(fact_508_walk__edges_Opelims,axiom,
! [X2: list_a,Y: list_set_a] :
( ( ( undire7337870655677353998dges_a @ X2 )
= Y )
=> ( ( accp_list_a @ undire7966302452035489203_rel_a @ X2 )
=> ( ( ( X2 = nil_a )
=> ( ( Y = nil_set_a )
=> ~ ( accp_list_a @ undire7966302452035489203_rel_a @ nil_a ) ) )
=> ( ! [X: a] :
( ( X2
= ( cons_a @ X @ nil_a ) )
=> ( ( Y = nil_set_a )
=> ~ ( accp_list_a @ undire7966302452035489203_rel_a @ ( cons_a @ X @ nil_a ) ) ) )
=> ~ ! [X: a,Y2: a,Ys2: list_a] :
( ( X2
= ( cons_a @ X @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( ( Y
= ( cons_set_a @ ( insert_a2 @ X @ ( insert_a2 @ Y2 @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ ( cons_a @ Y2 @ Ys2 ) ) ) )
=> ~ ( accp_list_a @ undire7966302452035489203_rel_a @ ( cons_a @ X @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ) ) ) ) ).
% walk_edges.pelims
thf(fact_509_comp__sgraph_Ois__edge__between__def,axiom,
( undire2578756059399487229_set_a
= ( ^ [X4: set_set_a,Y6: set_set_a,E2: set_set_a] :
? [X3: set_a,Y4: set_a] :
( ( E2
= ( insert_set_a2 @ X3 @ ( insert_set_a2 @ Y4 @ bot_bot_set_set_a ) ) )
& ( member_set_a2 @ X3 @ X4 )
& ( member_set_a2 @ Y4 @ Y6 ) ) ) ) ).
% comp_sgraph.is_edge_between_def
thf(fact_510_comp__sgraph_Ois__edge__between__def,axiom,
( undire8544646567961481629ween_a
= ( ^ [X4: set_a,Y6: set_a,E2: set_a] :
? [X3: a,Y4: a] :
( ( E2
= ( insert_a2 @ X3 @ ( insert_a2 @ Y4 @ bot_bot_set_a ) ) )
& ( member_a2 @ X3 @ X4 )
& ( member_a2 @ Y4 @ Y6 ) ) ) ) ).
% comp_sgraph.is_edge_between_def
thf(fact_511_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A4: set_a] :
( A4
= ( insert_a2 @ ( the_elem_a @ A4 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_512_is__singletonI,axiom,
! [X2: a] : ( is_singleton_a @ ( insert_a2 @ X2 @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_513_product__lists_Osimps_I1_J,axiom,
( ( product_lists_a @ nil_list_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% product_lists.simps(1)
thf(fact_514_product__lists_Osimps_I1_J,axiom,
( ( product_lists_set_a @ nil_list_set_a )
= ( cons_list_set_a @ nil_set_a @ nil_list_set_a ) ) ).
% product_lists.simps(1)
thf(fact_515_ulgraph_Owalk__edges_Opelims,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,X2: list_set_a,Y: list_set_set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( ( undire6234387080713648494_set_a @ X2 )
= Y )
=> ( ( accp_list_set_a @ undire7993746975499772691_set_a @ X2 )
=> ( ( ( X2 = nil_set_a )
=> ( ( Y = nil_set_set_a )
=> ~ ( accp_list_set_a @ undire7993746975499772691_set_a @ nil_set_a ) ) )
=> ( ! [X: set_a] :
( ( X2
= ( cons_set_a @ X @ nil_set_a ) )
=> ( ( Y = nil_set_set_a )
=> ~ ( accp_list_set_a @ undire7993746975499772691_set_a @ ( cons_set_a @ X @ nil_set_a ) ) ) )
=> ~ ! [X: set_a,Y2: set_a,Ys2: list_set_a] :
( ( X2
= ( cons_set_a @ X @ ( cons_set_a @ Y2 @ Ys2 ) ) )
=> ( ( Y
= ( cons_set_set_a @ ( insert_set_a2 @ X @ ( insert_set_a2 @ Y2 @ bot_bot_set_set_a ) ) @ ( undire6234387080713648494_set_a @ ( cons_set_a @ Y2 @ Ys2 ) ) ) )
=> ~ ( accp_list_set_a @ undire7993746975499772691_set_a @ ( cons_set_a @ X @ ( cons_set_a @ Y2 @ Ys2 ) ) ) ) ) ) ) ) ) ) ).
% ulgraph.walk_edges.pelims
thf(fact_516_ulgraph_Owalk__edges_Opelims,axiom,
! [Vertices: set_a,Edges: set_set_a,X2: list_a,Y: list_set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( ( undire7337870655677353998dges_a @ X2 )
= Y )
=> ( ( accp_list_a @ undire7966302452035489203_rel_a @ X2 )
=> ( ( ( X2 = nil_a )
=> ( ( Y = nil_set_a )
=> ~ ( accp_list_a @ undire7966302452035489203_rel_a @ nil_a ) ) )
=> ( ! [X: a] :
( ( X2
= ( cons_a @ X @ nil_a ) )
=> ( ( Y = nil_set_a )
=> ~ ( accp_list_a @ undire7966302452035489203_rel_a @ ( cons_a @ X @ nil_a ) ) ) )
=> ~ ! [X: a,Y2: a,Ys2: list_a] :
( ( X2
= ( cons_a @ X @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( ( Y
= ( cons_set_a @ ( insert_a2 @ X @ ( insert_a2 @ Y2 @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ ( cons_a @ Y2 @ Ys2 ) ) ) )
=> ~ ( accp_list_a @ undire7966302452035489203_rel_a @ ( cons_a @ X @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ) ) ) ) ) ).
% ulgraph.walk_edges.pelims
thf(fact_517_listset_Osimps_I1_J,axiom,
( ( listset_set_a @ nil_set_set_a )
= ( insert_list_set_a @ nil_set_a @ bot_bo4397488018069675312_set_a ) ) ).
% listset.simps(1)
thf(fact_518_listset_Osimps_I1_J,axiom,
( ( listset_a @ nil_set_a )
= ( insert_list_a @ nil_a @ bot_bot_set_list_a ) ) ).
% listset.simps(1)
thf(fact_519_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_520_is__singletonI_H,axiom,
! [A2: set_set_a] :
( ( A2 != bot_bot_set_set_a )
=> ( ! [X: set_a,Y2: set_a] :
( ( member_set_a2 @ X @ A2 )
=> ( ( member_set_a2 @ Y2 @ A2 )
=> ( X = Y2 ) ) )
=> ( is_singleton_set_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_521_is__singletonI_H,axiom,
! [A2: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ! [X: a,Y2: a] :
( ( member_a2 @ X @ A2 )
=> ( ( member_a2 @ Y2 @ A2 )
=> ( X = Y2 ) ) )
=> ( is_singleton_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_522_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A4: set_a] :
? [X3: a] :
( A4
= ( insert_a2 @ X3 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_523_is__singletonE,axiom,
! [A2: set_a] :
( ( is_singleton_a @ A2 )
=> ~ ! [X: a] :
( A2
!= ( insert_a2 @ X @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_524_comp__sgraph_Owalk__edges_Opelims,axiom,
! [X2: list_set_a,Y: list_set_set_a] :
( ( ( undire6234387080713648494_set_a @ X2 )
= Y )
=> ( ( accp_list_set_a @ undire7993746975499772691_set_a @ X2 )
=> ( ( ( X2 = nil_set_a )
=> ( ( Y = nil_set_set_a )
=> ~ ( accp_list_set_a @ undire7993746975499772691_set_a @ nil_set_a ) ) )
=> ( ! [X: set_a] :
( ( X2
= ( cons_set_a @ X @ nil_set_a ) )
=> ( ( Y = nil_set_set_a )
=> ~ ( accp_list_set_a @ undire7993746975499772691_set_a @ ( cons_set_a @ X @ nil_set_a ) ) ) )
=> ~ ! [X: set_a,Y2: set_a,Ys2: list_set_a] :
( ( X2
= ( cons_set_a @ X @ ( cons_set_a @ Y2 @ Ys2 ) ) )
=> ( ( Y
= ( cons_set_set_a @ ( insert_set_a2 @ X @ ( insert_set_a2 @ Y2 @ bot_bot_set_set_a ) ) @ ( undire6234387080713648494_set_a @ ( cons_set_a @ Y2 @ Ys2 ) ) ) )
=> ~ ( accp_list_set_a @ undire7993746975499772691_set_a @ ( cons_set_a @ X @ ( cons_set_a @ Y2 @ Ys2 ) ) ) ) ) ) ) ) ) ).
% comp_sgraph.walk_edges.pelims
thf(fact_525_comp__sgraph_Owalk__edges_Opelims,axiom,
! [X2: list_a,Y: list_set_a] :
( ( ( undire7337870655677353998dges_a @ X2 )
= Y )
=> ( ( accp_list_a @ undire7966302452035489203_rel_a @ X2 )
=> ( ( ( X2 = nil_a )
=> ( ( Y = nil_set_a )
=> ~ ( accp_list_a @ undire7966302452035489203_rel_a @ nil_a ) ) )
=> ( ! [X: a] :
( ( X2
= ( cons_a @ X @ nil_a ) )
=> ( ( Y = nil_set_a )
=> ~ ( accp_list_a @ undire7966302452035489203_rel_a @ ( cons_a @ X @ nil_a ) ) ) )
=> ~ ! [X: a,Y2: a,Ys2: list_a] :
( ( X2
= ( cons_a @ X @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( ( Y
= ( cons_set_a @ ( insert_a2 @ X @ ( insert_a2 @ Y2 @ bot_bot_set_a ) ) @ ( undire7337870655677353998dges_a @ ( cons_a @ Y2 @ Ys2 ) ) ) )
=> ~ ( accp_list_a @ undire7966302452035489203_rel_a @ ( cons_a @ X @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ) ) ) ) ).
% comp_sgraph.walk_edges.pelims
thf(fact_526_listset_Osimps_I2_J,axiom,
! [A2: set_a,As: list_set_a] :
( ( listset_a @ ( cons_set_a @ A2 @ As ) )
= ( set_Cons_a @ A2 @ ( listset_a @ As ) ) ) ).
% listset.simps(2)
thf(fact_527_lists__empty,axiom,
( ( lists_set_a @ bot_bot_set_set_a )
= ( insert_list_set_a @ nil_set_a @ bot_bo4397488018069675312_set_a ) ) ).
% lists_empty
thf(fact_528_lists__empty,axiom,
( ( lists_a @ bot_bot_set_a )
= ( insert_list_a @ nil_a @ bot_bot_set_list_a ) ) ).
% lists_empty
thf(fact_529_subseqs_Osimps_I1_J,axiom,
( ( subseqs_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% subseqs.simps(1)
thf(fact_530_subseqs_Osimps_I1_J,axiom,
( ( subseqs_set_a @ nil_set_a )
= ( cons_list_set_a @ nil_set_a @ nil_list_set_a ) ) ).
% subseqs.simps(1)
thf(fact_531_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A4: set_a] : ( A4 = bot_bot_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_532_concat__eq__append__conv,axiom,
! [Xss2: list_list_a,Ys: list_a,Zs: list_a] :
( ( ( concat_a @ Xss2 )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_list_a )
=> ( ( Ys = nil_a )
& ( Zs = nil_a ) ) )
& ( ( Xss2 != nil_list_a )
=> ? [Xss1: list_list_a,Xs2: list_a,Xs5: list_a,Xss22: list_list_a] :
( ( Xss2
= ( append_list_a @ Xss1 @ ( cons_list_a @ ( append_a @ Xs2 @ Xs5 ) @ Xss22 ) ) )
& ( Ys
= ( append_a @ ( concat_a @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append_a @ Xs5 @ ( concat_a @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_533_concat__eq__append__conv,axiom,
! [Xss2: list_list_set_a,Ys: list_set_a,Zs: list_set_a] :
( ( ( concat_set_a @ Xss2 )
= ( append_set_a @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_list_set_a )
=> ( ( Ys = nil_set_a )
& ( Zs = nil_set_a ) ) )
& ( ( Xss2 != nil_list_set_a )
=> ? [Xss1: list_list_set_a,Xs2: list_set_a,Xs5: list_set_a,Xss22: list_list_set_a] :
( ( Xss2
= ( append_list_set_a @ Xss1 @ ( cons_list_set_a @ ( append_set_a @ Xs2 @ Xs5 ) @ Xss22 ) ) )
& ( Ys
= ( append_set_a @ ( concat_set_a @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append_set_a @ Xs5 @ ( concat_set_a @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_534_butlast__snoc,axiom,
! [Xs: list_a,X2: a] :
( ( butlast_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_535_butlast__snoc,axiom,
! [Xs: list_set_a,X2: set_a] :
( ( butlast_set_a @ ( append_set_a @ Xs @ ( cons_set_a @ X2 @ nil_set_a ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_536_ulgraph_Ohas__loop__def,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3617971648856834880loop_a @ Edges @ V )
= ( member_set_a2 @ ( insert_a2 @ V @ bot_bot_set_a ) @ Edges ) ) ) ).
% ulgraph.has_loop_def
thf(fact_537_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_a] :
( ( nil_a
= ( concat_a @ Xss2 ) )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xss2 ) )
=> ( X3 = nil_a ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_538_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_set_a] :
( ( nil_set_a
= ( concat_set_a @ Xss2 ) )
= ( ! [X3: list_set_a] :
( ( member_list_set_a @ X3 @ ( set_list_set_a2 @ Xss2 ) )
=> ( X3 = nil_set_a ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_539_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_a] :
( ( ( concat_a @ Xss2 )
= nil_a )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xss2 ) )
=> ( X3 = nil_a ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_540_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_set_a] :
( ( ( concat_set_a @ Xss2 )
= nil_set_a )
= ( ! [X3: list_set_a] :
( ( member_list_set_a @ X3 @ ( set_list_set_a2 @ Xss2 ) )
=> ( X3 = nil_set_a ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_541_Cons__in__lists__iff,axiom,
! [X2: a,Xs: list_a,A2: set_a] :
( ( member_list_a @ ( cons_a @ X2 @ Xs ) @ ( lists_a @ A2 ) )
= ( ( member_a2 @ X2 @ A2 )
& ( member_list_a @ Xs @ ( lists_a @ A2 ) ) ) ) ).
% Cons_in_lists_iff
thf(fact_542_Cons__in__lists__iff,axiom,
! [X2: set_a,Xs: list_set_a,A2: set_set_a] :
( ( member_list_set_a @ ( cons_set_a @ X2 @ Xs ) @ ( lists_set_a @ A2 ) )
= ( ( member_set_a2 @ X2 @ A2 )
& ( member_list_set_a @ Xs @ ( lists_set_a @ A2 ) ) ) ) ).
% Cons_in_lists_iff
thf(fact_543_in__listsI,axiom,
! [Xs: list_a,A2: set_a] :
( ! [X: a] :
( ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ( member_a2 @ X @ A2 ) )
=> ( member_list_a @ Xs @ ( lists_a @ A2 ) ) ) ).
% in_listsI
thf(fact_544_in__listsI,axiom,
! [Xs: list_set_a,A2: set_set_a] :
( ! [X: set_a] :
( ( member_set_a2 @ X @ ( set_set_a2 @ Xs ) )
=> ( member_set_a2 @ X @ A2 ) )
=> ( member_list_set_a @ Xs @ ( lists_set_a @ A2 ) ) ) ).
% in_listsI
thf(fact_545_in__listsI,axiom,
! [Xs: list_set_set_a,A2: set_set_set_a] :
( ! [X: set_set_a] :
( ( member_set_set_a2 @ X @ ( set_set_set_a2 @ Xs ) )
=> ( member_set_set_a2 @ X @ A2 ) )
=> ( member6684481465865166061_set_a @ Xs @ ( lists_set_set_a @ A2 ) ) ) ).
% in_listsI
thf(fact_546_append__in__lists__conv,axiom,
! [Xs: list_a,Ys: list_a,A2: set_a] :
( ( member_list_a @ ( append_a @ Xs @ Ys ) @ ( lists_a @ A2 ) )
= ( ( member_list_a @ Xs @ ( lists_a @ A2 ) )
& ( member_list_a @ Ys @ ( lists_a @ A2 ) ) ) ) ).
% append_in_lists_conv
thf(fact_547_append__in__lists__conv,axiom,
! [Xs: list_set_a,Ys: list_set_a,A2: set_set_a] :
( ( member_list_set_a @ ( append_set_a @ Xs @ Ys ) @ ( lists_set_a @ A2 ) )
= ( ( member_list_set_a @ Xs @ ( lists_set_a @ A2 ) )
& ( member_list_set_a @ Ys @ ( lists_set_a @ A2 ) ) ) ) ).
% append_in_lists_conv
thf(fact_548_concat__append,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( concat_a @ ( append_list_a @ Xs @ Ys ) )
= ( append_a @ ( concat_a @ Xs ) @ ( concat_a @ Ys ) ) ) ).
% concat_append
thf(fact_549_concat__append,axiom,
! [Xs: list_list_set_a,Ys: list_list_set_a] :
( ( concat_set_a @ ( append_list_set_a @ Xs @ Ys ) )
= ( append_set_a @ ( concat_set_a @ Xs ) @ ( concat_set_a @ Ys ) ) ) ).
% concat_append
thf(fact_550_lists__mono,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ord_le864617614081865828_set_a @ ( lists_set_a @ A2 ) @ ( lists_set_a @ B2 ) ) ) ).
% lists_mono
thf(fact_551_lists__mono,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( ord_le4590583055503244740_set_a @ ( lists_set_set_a @ A2 ) @ ( lists_set_set_a @ B2 ) ) ) ).
% lists_mono
thf(fact_552_lists__mono,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_le8861187494160871172list_a @ ( lists_a @ A2 ) @ ( lists_a @ B2 ) ) ) ).
% lists_mono
thf(fact_553_lists_OCons,axiom,
! [A: a,A2: set_a,L: list_a] :
( ( member_a2 @ A @ A2 )
=> ( ( member_list_a @ L @ ( lists_a @ A2 ) )
=> ( member_list_a @ ( cons_a @ A @ L ) @ ( lists_a @ A2 ) ) ) ) ).
% lists.Cons
thf(fact_554_lists_OCons,axiom,
! [A: set_a,A2: set_set_a,L: list_set_a] :
( ( member_set_a2 @ A @ A2 )
=> ( ( member_list_set_a @ L @ ( lists_set_a @ A2 ) )
=> ( member_list_set_a @ ( cons_set_a @ A @ L ) @ ( lists_set_a @ A2 ) ) ) ) ).
% lists.Cons
thf(fact_555_listsE,axiom,
! [X2: a,L: list_a,A2: set_a] :
( ( member_list_a @ ( cons_a @ X2 @ L ) @ ( lists_a @ A2 ) )
=> ~ ( ( member_a2 @ X2 @ A2 )
=> ~ ( member_list_a @ L @ ( lists_a @ A2 ) ) ) ) ).
% listsE
thf(fact_556_listsE,axiom,
! [X2: set_a,L: list_set_a,A2: set_set_a] :
( ( member_list_set_a @ ( cons_set_a @ X2 @ L ) @ ( lists_set_a @ A2 ) )
=> ~ ( ( member_set_a2 @ X2 @ A2 )
=> ~ ( member_list_set_a @ L @ ( lists_set_a @ A2 ) ) ) ) ).
% listsE
thf(fact_557_lists_ONil,axiom,
! [A2: set_a] : ( member_list_a @ nil_a @ ( lists_a @ A2 ) ) ).
% lists.Nil
thf(fact_558_lists_ONil,axiom,
! [A2: set_set_a] : ( member_list_set_a @ nil_set_a @ ( lists_set_a @ A2 ) ) ).
% lists.Nil
thf(fact_559_in__listsD,axiom,
! [Xs: list_a,A2: set_a] :
( ( member_list_a @ Xs @ ( lists_a @ A2 ) )
=> ! [X5: a] :
( ( member_a2 @ X5 @ ( set_a2 @ Xs ) )
=> ( member_a2 @ X5 @ A2 ) ) ) ).
% in_listsD
thf(fact_560_in__listsD,axiom,
! [Xs: list_set_a,A2: set_set_a] :
( ( member_list_set_a @ Xs @ ( lists_set_a @ A2 ) )
=> ! [X5: set_a] :
( ( member_set_a2 @ X5 @ ( set_set_a2 @ Xs ) )
=> ( member_set_a2 @ X5 @ A2 ) ) ) ).
% in_listsD
thf(fact_561_in__listsD,axiom,
! [Xs: list_set_set_a,A2: set_set_set_a] :
( ( member6684481465865166061_set_a @ Xs @ ( lists_set_set_a @ A2 ) )
=> ! [X5: set_set_a] :
( ( member_set_set_a2 @ X5 @ ( set_set_set_a2 @ Xs ) )
=> ( member_set_set_a2 @ X5 @ A2 ) ) ) ).
% in_listsD
thf(fact_562_in__lists__conv__set,axiom,
! [Xs: list_a,A2: set_a] :
( ( member_list_a @ Xs @ ( lists_a @ A2 ) )
= ( ! [X3: a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
=> ( member_a2 @ X3 @ A2 ) ) ) ) ).
% in_lists_conv_set
thf(fact_563_in__lists__conv__set,axiom,
! [Xs: list_set_a,A2: set_set_a] :
( ( member_list_set_a @ Xs @ ( lists_set_a @ A2 ) )
= ( ! [X3: set_a] :
( ( member_set_a2 @ X3 @ ( set_set_a2 @ Xs ) )
=> ( member_set_a2 @ X3 @ A2 ) ) ) ) ).
% in_lists_conv_set
thf(fact_564_in__lists__conv__set,axiom,
! [Xs: list_set_set_a,A2: set_set_set_a] :
( ( member6684481465865166061_set_a @ Xs @ ( lists_set_set_a @ A2 ) )
= ( ! [X3: set_set_a] :
( ( member_set_set_a2 @ X3 @ ( set_set_set_a2 @ Xs ) )
=> ( member_set_set_a2 @ X3 @ A2 ) ) ) ) ).
% in_lists_conv_set
thf(fact_565_butlast_Osimps_I1_J,axiom,
( ( butlast_a @ nil_a )
= nil_a ) ).
% butlast.simps(1)
thf(fact_566_butlast_Osimps_I1_J,axiom,
( ( butlast_set_a @ nil_set_a )
= nil_set_a ) ).
% butlast.simps(1)
thf(fact_567_in__set__butlastD,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ ( butlast_a @ Xs ) ) )
=> ( member_a2 @ X2 @ ( set_a2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_568_in__set__butlastD,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a2 @ X2 @ ( set_set_a2 @ ( butlast_set_a @ Xs ) ) )
=> ( member_set_a2 @ X2 @ ( set_set_a2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_569_in__set__butlastD,axiom,
! [X2: set_set_a,Xs: list_set_set_a] :
( ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ ( butlast_set_set_a @ Xs ) ) )
=> ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_570_butlast__tl,axiom,
! [Xs: list_a] :
( ( butlast_a @ ( tl_a @ Xs ) )
= ( tl_a @ ( butlast_a @ Xs ) ) ) ).
% butlast_tl
thf(fact_571_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3617971648856834880loop_a @ Edges @ V )
=> ( member_a2 @ V @ Vertices ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_572_ulgraph_Ohas__loop__in__verts,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire5774735625301615776_set_a @ Edges @ V )
=> ( member_set_a2 @ V @ Vertices ) ) ) ).
% ulgraph.has_loop_in_verts
thf(fact_573_Cons__in__subseqsD,axiom,
! [Y: a,Ys: list_a,Xs: list_a] :
( ( member_list_a @ ( cons_a @ Y @ Ys ) @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) )
=> ( member_list_a @ Ys @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_574_Cons__in__subseqsD,axiom,
! [Y: set_a,Ys: list_set_a,Xs: list_set_a] :
( ( member_list_set_a @ ( cons_set_a @ Y @ Ys ) @ ( set_list_set_a2 @ ( subseqs_set_a @ Xs ) ) )
=> ( member_list_set_a @ Ys @ ( set_list_set_a2 @ ( subseqs_set_a @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_575_lists_Ocases,axiom,
! [A: list_a,A2: set_a] :
( ( member_list_a @ A @ ( lists_a @ A2 ) )
=> ( ( A != nil_a )
=> ~ ! [A5: a,L2: list_a] :
( ( A
= ( cons_a @ A5 @ L2 ) )
=> ( ( member_a2 @ A5 @ A2 )
=> ~ ( member_list_a @ L2 @ ( lists_a @ A2 ) ) ) ) ) ) ).
% lists.cases
thf(fact_576_lists_Ocases,axiom,
! [A: list_set_a,A2: set_set_a] :
( ( member_list_set_a @ A @ ( lists_set_a @ A2 ) )
=> ( ( A != nil_set_a )
=> ~ ! [A5: set_a,L2: list_set_a] :
( ( A
= ( cons_set_a @ A5 @ L2 ) )
=> ( ( member_set_a2 @ A5 @ A2 )
=> ~ ( member_list_set_a @ L2 @ ( lists_set_a @ A2 ) ) ) ) ) ) ).
% lists.cases
thf(fact_577_lists_Osimps,axiom,
! [A: list_a,A2: set_a] :
( ( member_list_a @ A @ ( lists_a @ A2 ) )
= ( ( A = nil_a )
| ? [A3: a,L3: list_a] :
( ( A
= ( cons_a @ A3 @ L3 ) )
& ( member_a2 @ A3 @ A2 )
& ( member_list_a @ L3 @ ( lists_a @ A2 ) ) ) ) ) ).
% lists.simps
thf(fact_578_lists_Osimps,axiom,
! [A: list_set_a,A2: set_set_a] :
( ( member_list_set_a @ A @ ( lists_set_a @ A2 ) )
= ( ( A = nil_set_a )
| ? [A3: set_a,L3: list_set_a] :
( ( A
= ( cons_set_a @ A3 @ L3 ) )
& ( member_set_a2 @ A3 @ A2 )
& ( member_list_set_a @ L3 @ ( lists_set_a @ A2 ) ) ) ) ) ).
% lists.simps
thf(fact_579_concat_Osimps_I1_J,axiom,
( ( concat_a @ nil_list_a )
= nil_a ) ).
% concat.simps(1)
thf(fact_580_concat_Osimps_I1_J,axiom,
( ( concat_set_a @ nil_list_set_a )
= nil_set_a ) ).
% concat.simps(1)
thf(fact_581_concat_Osimps_I2_J,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( concat_a @ ( cons_list_a @ X2 @ Xs ) )
= ( append_a @ X2 @ ( concat_a @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_582_concat_Osimps_I2_J,axiom,
! [X2: list_set_a,Xs: list_list_set_a] :
( ( concat_set_a @ ( cons_list_set_a @ X2 @ Xs ) )
= ( append_set_a @ X2 @ ( concat_set_a @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_583_butlast_Osimps_I2_J,axiom,
! [Xs: list_a,X2: a] :
( ( ( Xs = nil_a )
=> ( ( butlast_a @ ( cons_a @ X2 @ Xs ) )
= nil_a ) )
& ( ( Xs != nil_a )
=> ( ( butlast_a @ ( cons_a @ X2 @ Xs ) )
= ( cons_a @ X2 @ ( butlast_a @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_584_butlast_Osimps_I2_J,axiom,
! [Xs: list_set_a,X2: set_a] :
( ( ( Xs = nil_set_a )
=> ( ( butlast_set_a @ ( cons_set_a @ X2 @ Xs ) )
= nil_set_a ) )
& ( ( Xs != nil_set_a )
=> ( ( butlast_set_a @ ( cons_set_a @ X2 @ Xs ) )
= ( cons_set_a @ X2 @ ( butlast_set_a @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_585_butlast__append,axiom,
! [Ys: list_a,Xs: list_a] :
( ( ( Ys = nil_a )
=> ( ( butlast_a @ ( append_a @ Xs @ Ys ) )
= ( butlast_a @ Xs ) ) )
& ( ( Ys != nil_a )
=> ( ( butlast_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ Xs @ ( butlast_a @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_586_butlast__append,axiom,
! [Ys: list_set_a,Xs: list_set_a] :
( ( ( Ys = nil_set_a )
=> ( ( butlast_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( butlast_set_a @ Xs ) ) )
& ( ( Ys != nil_set_a )
=> ( ( butlast_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( append_set_a @ Xs @ ( butlast_set_a @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_587_in__set__butlast__appendI,axiom,
! [X2: a,Xs: list_a,Ys: list_a] :
( ( ( member_a2 @ X2 @ ( set_a2 @ ( butlast_a @ Xs ) ) )
| ( member_a2 @ X2 @ ( set_a2 @ ( butlast_a @ Ys ) ) ) )
=> ( member_a2 @ X2 @ ( set_a2 @ ( butlast_a @ ( append_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_588_in__set__butlast__appendI,axiom,
! [X2: set_a,Xs: list_set_a,Ys: list_set_a] :
( ( ( member_set_a2 @ X2 @ ( set_set_a2 @ ( butlast_set_a @ Xs ) ) )
| ( member_set_a2 @ X2 @ ( set_set_a2 @ ( butlast_set_a @ Ys ) ) ) )
=> ( member_set_a2 @ X2 @ ( set_set_a2 @ ( butlast_set_a @ ( append_set_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_589_in__set__butlast__appendI,axiom,
! [X2: set_set_a,Xs: list_set_set_a,Ys: list_set_set_a] :
( ( ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ ( butlast_set_set_a @ Xs ) ) )
| ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ ( butlast_set_set_a @ Ys ) ) ) )
=> ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ ( butlast_set_set_a @ ( append_set_set_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_590_concat__eq__appendD,axiom,
! [Xss2: list_list_a,Ys: list_a,Zs: list_a] :
( ( ( concat_a @ Xss2 )
= ( append_a @ Ys @ Zs ) )
=> ( ( Xss2 != nil_list_a )
=> ? [Xss12: list_list_a,Xs3: list_a,Xs6: list_a,Xss23: list_list_a] :
( ( Xss2
= ( append_list_a @ Xss12 @ ( cons_list_a @ ( append_a @ Xs3 @ Xs6 ) @ Xss23 ) ) )
& ( Ys
= ( append_a @ ( concat_a @ Xss12 ) @ Xs3 ) )
& ( Zs
= ( append_a @ Xs6 @ ( concat_a @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_591_concat__eq__appendD,axiom,
! [Xss2: list_list_set_a,Ys: list_set_a,Zs: list_set_a] :
( ( ( concat_set_a @ Xss2 )
= ( append_set_a @ Ys @ Zs ) )
=> ( ( Xss2 != nil_list_set_a )
=> ? [Xss12: list_list_set_a,Xs3: list_set_a,Xs6: list_set_a,Xss23: list_list_set_a] :
( ( Xss2
= ( append_list_set_a @ Xss12 @ ( cons_list_set_a @ ( append_set_a @ Xs3 @ Xs6 ) @ Xss23 ) ) )
& ( Ys
= ( append_set_a @ ( concat_set_a @ Xss12 ) @ Xs3 ) )
& ( Zs
= ( append_set_a @ Xs6 @ ( concat_set_a @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_592_bot__empty__eq,axiom,
( bot_bot_set_a_o
= ( ^ [X3: set_a] : ( member_set_a2 @ X3 @ bot_bot_set_set_a ) ) ) ).
% bot_empty_eq
thf(fact_593_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : ( member_a2 @ X3 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_594_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_595_is__empty__set,axiom,
! [Xs: list_set_a] :
( ( is_empty_set_a @ ( set_set_a2 @ Xs ) )
= ( null_set_a @ Xs ) ) ).
% is_empty_set
thf(fact_596_is__empty__set,axiom,
! [Xs: list_set_set_a] :
( ( is_empty_set_set_a @ ( set_set_set_a2 @ Xs ) )
= ( null_set_set_a @ Xs ) ) ).
% is_empty_set
thf(fact_597_ulgraph_Oincident__loops__simp_I1_J,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire3617971648856834880loop_a @ Edges @ V )
=> ( ( undire4753905205749729249oops_a @ Edges @ V )
= ( insert_set_a2 @ ( insert_a2 @ V @ bot_bot_set_a ) @ bot_bot_set_set_a ) ) ) ) ).
% ulgraph.incident_loops_simp(1)
thf(fact_598_append__butlast__last__id,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( append_a @ ( butlast_a @ Xs ) @ ( cons_a @ ( last_a @ Xs ) @ nil_a ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_599_append__butlast__last__id,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
=> ( ( append_set_a @ ( butlast_set_a @ Xs ) @ ( cons_set_a @ ( last_set_a @ Xs ) @ nil_set_a ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_600_snoc__eq__iff__butlast,axiom,
! [Xs: list_a,X2: a,Ys: list_a] :
( ( ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) )
= Ys )
= ( ( Ys != nil_a )
& ( ( butlast_a @ Ys )
= Xs )
& ( ( last_a @ Ys )
= X2 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_601_snoc__eq__iff__butlast,axiom,
! [Xs: list_set_a,X2: set_a,Ys: list_set_a] :
( ( ( append_set_a @ Xs @ ( cons_set_a @ X2 @ nil_set_a ) )
= Ys )
= ( ( Ys != nil_set_a )
& ( ( butlast_set_a @ Ys )
= Xs )
& ( ( last_set_a @ Ys )
= X2 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_602_remdups__adj__append,axiom,
! [Xs_1: list_a,X2: a,Xs_2: list_a] :
( ( remdups_adj_a @ ( append_a @ Xs_1 @ ( cons_a @ X2 @ Xs_2 ) ) )
= ( append_a @ ( remdups_adj_a @ ( append_a @ Xs_1 @ ( cons_a @ X2 @ nil_a ) ) ) @ ( tl_a @ ( remdups_adj_a @ ( cons_a @ X2 @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_603_remdups__adj__append,axiom,
! [Xs_1: list_set_a,X2: set_a,Xs_2: list_set_a] :
( ( remdups_adj_set_a @ ( append_set_a @ Xs_1 @ ( cons_set_a @ X2 @ Xs_2 ) ) )
= ( append_set_a @ ( remdups_adj_set_a @ ( append_set_a @ Xs_1 @ ( cons_set_a @ X2 @ nil_set_a ) ) ) @ ( tl_set_a @ ( remdups_adj_set_a @ ( cons_set_a @ X2 @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_604_remdups__adj__Nil__iff,axiom,
! [Xs: list_a] :
( ( ( remdups_adj_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% remdups_adj_Nil_iff
thf(fact_605_remdups__adj__Nil__iff,axiom,
! [Xs: list_set_a] :
( ( ( remdups_adj_set_a @ Xs )
= nil_set_a )
= ( Xs = nil_set_a ) ) ).
% remdups_adj_Nil_iff
thf(fact_606_remdups__adj__set,axiom,
! [Xs: list_set_a] :
( ( set_set_a2 @ ( remdups_adj_set_a @ Xs ) )
= ( set_set_a2 @ Xs ) ) ).
% remdups_adj_set
thf(fact_607_remdups__adj__set,axiom,
! [Xs: list_set_set_a] :
( ( set_set_set_a2 @ ( remdup1882599702278213626_set_a @ Xs ) )
= ( set_set_set_a2 @ Xs ) ) ).
% remdups_adj_set
thf(fact_608_last__appendL,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Xs ) ) ) ).
% last_appendL
thf(fact_609_last__appendL,axiom,
! [Ys: list_set_a,Xs: list_set_a] :
( ( Ys = nil_set_a )
=> ( ( last_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( last_set_a @ Xs ) ) ) ).
% last_appendL
thf(fact_610_last__appendR,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Ys ) ) ) ).
% last_appendR
thf(fact_611_last__appendR,axiom,
! [Ys: list_set_a,Xs: list_set_a] :
( ( Ys != nil_set_a )
=> ( ( last_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( last_set_a @ Ys ) ) ) ).
% last_appendR
thf(fact_612_remdups__adj__Cons__alt,axiom,
! [X2: a,Xs: list_a] :
( ( cons_a @ X2 @ ( tl_a @ ( remdups_adj_a @ ( cons_a @ X2 @ Xs ) ) ) )
= ( remdups_adj_a @ ( cons_a @ X2 @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_613_remdups__adj__Cons__alt,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( cons_set_a @ X2 @ ( tl_set_a @ ( remdups_adj_set_a @ ( cons_set_a @ X2 @ Xs ) ) ) )
= ( remdups_adj_set_a @ ( cons_set_a @ X2 @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_614_last__snoc,axiom,
! [Xs: list_a,X2: a] :
( ( last_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) )
= X2 ) ).
% last_snoc
thf(fact_615_last__snoc,axiom,
! [Xs: list_set_a,X2: set_a] :
( ( last_set_a @ ( append_set_a @ Xs @ ( cons_set_a @ X2 @ nil_set_a ) ) )
= X2 ) ).
% last_snoc
thf(fact_616_remdups__adj_Osimps_I3_J,axiom,
! [X2: a,Y: a,Xs: list_a] :
( ( ( X2 = Y )
=> ( ( remdups_adj_a @ ( cons_a @ X2 @ ( cons_a @ Y @ Xs ) ) )
= ( remdups_adj_a @ ( cons_a @ X2 @ Xs ) ) ) )
& ( ( X2 != Y )
=> ( ( remdups_adj_a @ ( cons_a @ X2 @ ( cons_a @ Y @ Xs ) ) )
= ( cons_a @ X2 @ ( remdups_adj_a @ ( cons_a @ Y @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_617_remdups__adj_Osimps_I3_J,axiom,
! [X2: set_a,Y: set_a,Xs: list_set_a] :
( ( ( X2 = Y )
=> ( ( remdups_adj_set_a @ ( cons_set_a @ X2 @ ( cons_set_a @ Y @ Xs ) ) )
= ( remdups_adj_set_a @ ( cons_set_a @ X2 @ Xs ) ) ) )
& ( ( X2 != Y )
=> ( ( remdups_adj_set_a @ ( cons_set_a @ X2 @ ( cons_set_a @ Y @ Xs ) ) )
= ( cons_set_a @ X2 @ ( remdups_adj_set_a @ ( cons_set_a @ Y @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_618_remdups__adj_Osimps_I1_J,axiom,
( ( remdups_adj_a @ nil_a )
= nil_a ) ).
% remdups_adj.simps(1)
thf(fact_619_remdups__adj_Osimps_I1_J,axiom,
( ( remdups_adj_set_a @ nil_set_a )
= nil_set_a ) ).
% remdups_adj.simps(1)
thf(fact_620_remdups__adj_Oelims,axiom,
! [X2: list_a,Y: list_a] :
( ( ( remdups_adj_a @ X2 )
= Y )
=> ( ( ( X2 = nil_a )
=> ( Y != nil_a ) )
=> ( ! [X: a] :
( ( X2
= ( cons_a @ X @ nil_a ) )
=> ( Y
!= ( cons_a @ X @ nil_a ) ) )
=> ~ ! [X: a,Y2: a,Xs3: list_a] :
( ( X2
= ( cons_a @ X @ ( cons_a @ Y2 @ Xs3 ) ) )
=> ~ ( ( ( X = Y2 )
=> ( Y
= ( remdups_adj_a @ ( cons_a @ X @ Xs3 ) ) ) )
& ( ( X != Y2 )
=> ( Y
= ( cons_a @ X @ ( remdups_adj_a @ ( cons_a @ Y2 @ Xs3 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_621_remdups__adj_Oelims,axiom,
! [X2: list_set_a,Y: list_set_a] :
( ( ( remdups_adj_set_a @ X2 )
= Y )
=> ( ( ( X2 = nil_set_a )
=> ( Y != nil_set_a ) )
=> ( ! [X: set_a] :
( ( X2
= ( cons_set_a @ X @ nil_set_a ) )
=> ( Y
!= ( cons_set_a @ X @ nil_set_a ) ) )
=> ~ ! [X: set_a,Y2: set_a,Xs3: list_set_a] :
( ( X2
= ( cons_set_a @ X @ ( cons_set_a @ Y2 @ Xs3 ) ) )
=> ~ ( ( ( X = Y2 )
=> ( Y
= ( remdups_adj_set_a @ ( cons_set_a @ X @ Xs3 ) ) ) )
& ( ( X != Y2 )
=> ( Y
= ( cons_set_a @ X @ ( remdups_adj_set_a @ ( cons_set_a @ Y2 @ Xs3 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_622_remdups__adj_Osimps_I2_J,axiom,
! [X2: a] :
( ( remdups_adj_a @ ( cons_a @ X2 @ nil_a ) )
= ( cons_a @ X2 @ nil_a ) ) ).
% remdups_adj.simps(2)
thf(fact_623_remdups__adj_Osimps_I2_J,axiom,
! [X2: set_a] :
( ( remdups_adj_set_a @ ( cons_set_a @ X2 @ nil_set_a ) )
= ( cons_set_a @ X2 @ nil_set_a ) ) ).
% remdups_adj.simps(2)
thf(fact_624_last__ConsR,axiom,
! [Xs: list_a,X2: a] :
( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X2 @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_625_last__ConsR,axiom,
! [Xs: list_set_a,X2: set_a] :
( ( Xs != nil_set_a )
=> ( ( last_set_a @ ( cons_set_a @ X2 @ Xs ) )
= ( last_set_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_626_last__ConsL,axiom,
! [Xs: list_a,X2: a] :
( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_627_last__ConsL,axiom,
! [Xs: list_set_a,X2: set_a] :
( ( Xs = nil_set_a )
=> ( ( last_set_a @ ( cons_set_a @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_628_last_Osimps,axiom,
! [Xs: list_a,X2: a] :
( ( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X2 @ Xs ) )
= X2 ) )
& ( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X2 @ Xs ) )
= ( last_a @ Xs ) ) ) ) ).
% last.simps
thf(fact_629_last_Osimps,axiom,
! [Xs: list_set_a,X2: set_a] :
( ( ( Xs = nil_set_a )
=> ( ( last_set_a @ ( cons_set_a @ X2 @ Xs ) )
= X2 ) )
& ( ( Xs != nil_set_a )
=> ( ( last_set_a @ ( cons_set_a @ X2 @ Xs ) )
= ( last_set_a @ Xs ) ) ) ) ).
% last.simps
thf(fact_630_last__in__set,axiom,
! [As2: list_a] :
( ( As2 != nil_a )
=> ( member_a2 @ ( last_a @ As2 ) @ ( set_a2 @ As2 ) ) ) ).
% last_in_set
thf(fact_631_last__in__set,axiom,
! [As2: list_set_a] :
( ( As2 != nil_set_a )
=> ( member_set_a2 @ ( last_set_a @ As2 ) @ ( set_set_a2 @ As2 ) ) ) ).
% last_in_set
thf(fact_632_last__in__set,axiom,
! [As2: list_set_set_a] :
( ( As2 != nil_set_set_a )
=> ( member_set_set_a2 @ ( last_set_set_a @ As2 ) @ ( set_set_set_a2 @ As2 ) ) ) ).
% last_in_set
thf(fact_633_last__append,axiom,
! [Ys: list_a,Xs: list_a] :
( ( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Xs ) ) )
& ( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Ys ) ) ) ) ).
% last_append
thf(fact_634_last__append,axiom,
! [Ys: list_set_a,Xs: list_set_a] :
( ( ( Ys = nil_set_a )
=> ( ( last_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( last_set_a @ Xs ) ) )
& ( ( Ys != nil_set_a )
=> ( ( last_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( last_set_a @ Ys ) ) ) ) ).
% last_append
thf(fact_635_longest__common__suffix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ss: list_a,Xs6: list_a,Ys6: list_a] :
( ( Xs
= ( append_a @ Xs6 @ Ss ) )
& ( Ys
= ( append_a @ Ys6 @ Ss ) )
& ( ( Xs6 = nil_a )
| ( Ys6 = nil_a )
| ( ( last_a @ Xs6 )
!= ( last_a @ Ys6 ) ) ) ) ).
% longest_common_suffix
thf(fact_636_longest__common__suffix,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
? [Ss: list_set_a,Xs6: list_set_a,Ys6: list_set_a] :
( ( Xs
= ( append_set_a @ Xs6 @ Ss ) )
& ( Ys
= ( append_set_a @ Ys6 @ Ss ) )
& ( ( Xs6 = nil_set_a )
| ( Ys6 = nil_set_a )
| ( ( last_set_a @ Xs6 )
!= ( last_set_a @ Ys6 ) ) ) ) ).
% longest_common_suffix
thf(fact_637_last__tl,axiom,
! [Xs: list_a] :
( ( ( Xs = nil_a )
| ( ( tl_a @ Xs )
!= nil_a ) )
=> ( ( last_a @ ( tl_a @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_tl
thf(fact_638_last__tl,axiom,
! [Xs: list_set_a] :
( ( ( Xs = nil_set_a )
| ( ( tl_set_a @ Xs )
!= nil_set_a ) )
=> ( ( last_set_a @ ( tl_set_a @ Xs ) )
= ( last_set_a @ Xs ) ) ) ).
% last_tl
thf(fact_639_null__rec_I1_J,axiom,
! [X2: a,Xs: list_a] :
~ ( null_a @ ( cons_a @ X2 @ Xs ) ) ).
% null_rec(1)
thf(fact_640_null__rec_I1_J,axiom,
! [X2: set_a,Xs: list_set_a] :
~ ( null_set_a @ ( cons_set_a @ X2 @ Xs ) ) ).
% null_rec(1)
thf(fact_641_eq__Nil__null,axiom,
! [Xs: list_a] :
( ( Xs = nil_a )
= ( null_a @ Xs ) ) ).
% eq_Nil_null
thf(fact_642_eq__Nil__null,axiom,
! [Xs: list_set_a] :
( ( Xs = nil_set_a )
= ( null_set_a @ Xs ) ) ).
% eq_Nil_null
thf(fact_643_null__rec_I2_J,axiom,
null_a @ nil_a ).
% null_rec(2)
thf(fact_644_null__rec_I2_J,axiom,
null_set_a @ nil_set_a ).
% null_rec(2)
thf(fact_645_remdups__adj__append__two,axiom,
! [Xs: list_a,X2: a,Y: a] :
( ( remdups_adj_a @ ( append_a @ Xs @ ( cons_a @ X2 @ ( cons_a @ Y @ nil_a ) ) ) )
= ( append_a @ ( remdups_adj_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) ) @ ( if_list_a @ ( X2 = Y ) @ nil_a @ ( cons_a @ Y @ nil_a ) ) ) ) ).
% remdups_adj_append_two
thf(fact_646_remdups__adj__append__two,axiom,
! [Xs: list_set_a,X2: set_a,Y: set_a] :
( ( remdups_adj_set_a @ ( append_set_a @ Xs @ ( cons_set_a @ X2 @ ( cons_set_a @ Y @ nil_set_a ) ) ) )
= ( append_set_a @ ( remdups_adj_set_a @ ( append_set_a @ Xs @ ( cons_set_a @ X2 @ nil_set_a ) ) ) @ ( if_list_set_a @ ( X2 = Y ) @ nil_set_a @ ( cons_set_a @ Y @ nil_set_a ) ) ) ) ).
% remdups_adj_append_two
thf(fact_647_remdups__adj_Opelims,axiom,
! [X2: list_set_a,Y: list_set_a] :
( ( ( remdups_adj_set_a @ X2 )
= Y )
=> ( ( accp_list_set_a @ remdup6457802342601013479_set_a @ X2 )
=> ( ( ( X2 = nil_set_a )
=> ( ( Y = nil_set_a )
=> ~ ( accp_list_set_a @ remdup6457802342601013479_set_a @ nil_set_a ) ) )
=> ( ! [X: set_a] :
( ( X2
= ( cons_set_a @ X @ nil_set_a ) )
=> ( ( Y
= ( cons_set_a @ X @ nil_set_a ) )
=> ~ ( accp_list_set_a @ remdup6457802342601013479_set_a @ ( cons_set_a @ X @ nil_set_a ) ) ) )
=> ~ ! [X: set_a,Y2: set_a,Xs3: list_set_a] :
( ( X2
= ( cons_set_a @ X @ ( cons_set_a @ Y2 @ Xs3 ) ) )
=> ( ( ( ( X = Y2 )
=> ( Y
= ( remdups_adj_set_a @ ( cons_set_a @ X @ Xs3 ) ) ) )
& ( ( X != Y2 )
=> ( Y
= ( cons_set_a @ X @ ( remdups_adj_set_a @ ( cons_set_a @ Y2 @ Xs3 ) ) ) ) ) )
=> ~ ( accp_list_set_a @ remdup6457802342601013479_set_a @ ( cons_set_a @ X @ ( cons_set_a @ Y2 @ Xs3 ) ) ) ) ) ) ) ) ) ).
% remdups_adj.pelims
thf(fact_648_remdups__adj_Opelims,axiom,
! [X2: list_a,Y: list_a] :
( ( ( remdups_adj_a @ X2 )
= Y )
=> ( ( accp_list_a @ remdups_adj_rel_a @ X2 )
=> ( ( ( X2 = nil_a )
=> ( ( Y = nil_a )
=> ~ ( accp_list_a @ remdups_adj_rel_a @ nil_a ) ) )
=> ( ! [X: a] :
( ( X2
= ( cons_a @ X @ nil_a ) )
=> ( ( Y
= ( cons_a @ X @ nil_a ) )
=> ~ ( accp_list_a @ remdups_adj_rel_a @ ( cons_a @ X @ nil_a ) ) ) )
=> ~ ! [X: a,Y2: a,Xs3: list_a] :
( ( X2
= ( cons_a @ X @ ( cons_a @ Y2 @ Xs3 ) ) )
=> ( ( ( ( X = Y2 )
=> ( Y
= ( remdups_adj_a @ ( cons_a @ X @ Xs3 ) ) ) )
& ( ( X != Y2 )
=> ( Y
= ( cons_a @ X @ ( remdups_adj_a @ ( cons_a @ Y2 @ Xs3 ) ) ) ) ) )
=> ~ ( accp_list_a @ remdups_adj_rel_a @ ( cons_a @ X @ ( cons_a @ Y2 @ Xs3 ) ) ) ) ) ) ) ) ) ).
% remdups_adj.pelims
thf(fact_649_comp__sgraph_Oincident__loops__simp_I1_J,axiom,
! [S: set_a,V: a] :
( ( undire3617971648856834880loop_a @ ( undire2918257014606996450dges_a @ S ) @ V )
=> ( ( undire4753905205749729249oops_a @ ( undire2918257014606996450dges_a @ S ) @ V )
= ( insert_set_a2 @ ( insert_a2 @ V @ bot_bot_set_a ) @ bot_bot_set_set_a ) ) ) ).
% comp_sgraph.incident_loops_simp(1)
thf(fact_650_Pow__set_I1_J,axiom,
( ( pow_set_a @ ( set_set_a2 @ nil_set_a ) )
= ( insert_set_set_a2 @ bot_bot_set_set_a @ bot_bo3380559777022489994_set_a ) ) ).
% Pow_set(1)
thf(fact_651_Pow__set_I1_J,axiom,
( ( pow_set_set_a @ ( set_set_set_a2 @ nil_set_set_a ) )
= ( insert_set_set_set_a @ bot_bo3380559777022489994_set_a @ bot_bo4178452617224790762_set_a ) ) ).
% Pow_set(1)
thf(fact_652_Pow__set_I1_J,axiom,
( ( pow_a @ ( set_a2 @ nil_a ) )
= ( insert_set_a2 @ bot_bot_set_a @ bot_bot_set_set_a ) ) ).
% Pow_set(1)
thf(fact_653_subset__subseqs,axiom,
! [X6: set_set_a,Xs: list_set_a] :
( ( ord_le3724670747650509150_set_a @ X6 @ ( set_set_a2 @ Xs ) )
=> ( member_set_set_a2 @ X6 @ ( image_8804695481318321887_set_a @ set_set_a2 @ ( set_list_set_a2 @ ( subseqs_set_a @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_654_subset__subseqs,axiom,
! [X6: set_set_set_a,Xs: list_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X6 @ ( set_set_set_a2 @ Xs ) )
=> ( member_set_set_set_a @ X6 @ ( image_4529260449202881247_set_a @ set_set_set_a2 @ ( set_list_set_set_a2 @ ( subseqs_set_set_a @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_655_subset__subseqs,axiom,
! [X6: set_a,Xs: list_a] :
( ( ord_less_eq_set_a @ X6 @ ( set_a2 @ Xs ) )
=> ( member_set_a2 @ X6 @ ( image_list_a_set_a @ set_a2 @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_656_remdups__adj__append_H,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
| ( Ys = nil_a )
| ( ( last_a @ Xs )
!= ( hd_a @ Ys ) ) )
=> ( ( remdups_adj_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( remdups_adj_a @ Xs ) @ ( remdups_adj_a @ Ys ) ) ) ) ).
% remdups_adj_append'
thf(fact_657_remdups__adj__append_H,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( Xs = nil_set_a )
| ( Ys = nil_set_a )
| ( ( last_set_a @ Xs )
!= ( hd_set_a @ Ys ) ) )
=> ( ( remdups_adj_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( append_set_a @ ( remdups_adj_set_a @ Xs ) @ ( remdups_adj_set_a @ Ys ) ) ) ) ).
% remdups_adj_append'
thf(fact_658_Pow__empty,axiom,
( ( pow_a @ bot_bot_set_a )
= ( insert_set_a2 @ bot_bot_set_a @ bot_bot_set_set_a ) ) ).
% Pow_empty
thf(fact_659_image__eqI,axiom,
! [B: a,F: a > a,X2: a,A2: set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_a2 @ X2 @ A2 )
=> ( member_a2 @ B @ ( image_a_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_660_image__eqI,axiom,
! [B: set_a,F: a > set_a,X2: a,A2: set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_a2 @ X2 @ A2 )
=> ( member_set_a2 @ B @ ( image_a_set_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_661_image__eqI,axiom,
! [B: a,F: set_a > a,X2: set_a,A2: set_set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_set_a2 @ X2 @ A2 )
=> ( member_a2 @ B @ ( image_set_a_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_662_image__eqI,axiom,
! [B: set_a,F: set_a > set_a,X2: set_a,A2: set_set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_set_a2 @ X2 @ A2 )
=> ( member_set_a2 @ B @ ( image_set_a_set_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_663_image__empty,axiom,
! [F: a > a] :
( ( image_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_664_empty__is__image,axiom,
! [F: a > a,A2: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_665_image__is__empty,axiom,
! [F: a > a,A2: set_a] :
( ( ( image_a_a @ F @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_666_image__insert,axiom,
! [F: a > a,A: a,B2: set_a] :
( ( image_a_a @ F @ ( insert_a2 @ A @ B2 ) )
= ( insert_a2 @ ( F @ A ) @ ( image_a_a @ F @ B2 ) ) ) ).
% image_insert
thf(fact_667_insert__image,axiom,
! [X2: a,A2: set_a,F: a > a] :
( ( member_a2 @ X2 @ A2 )
=> ( ( insert_a2 @ ( F @ X2 ) @ ( image_a_a @ F @ A2 ) )
= ( image_a_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_668_insert__image,axiom,
! [X2: set_a,A2: set_set_a,F: set_a > a] :
( ( member_set_a2 @ X2 @ A2 )
=> ( ( insert_a2 @ ( F @ X2 ) @ ( image_set_a_a @ F @ A2 ) )
= ( image_set_a_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_669_PowI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( member_set_set_a2 @ A2 @ ( pow_set_a @ B2 ) ) ) ).
% PowI
thf(fact_670_PowI,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( member_set_set_set_a @ A2 @ ( pow_set_set_a @ B2 ) ) ) ).
% PowI
thf(fact_671_PowI,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( member_set_a2 @ A2 @ ( pow_a @ B2 ) ) ) ).
% PowI
thf(fact_672_Pow__iff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( member_set_set_a2 @ A2 @ ( pow_set_a @ B2 ) )
= ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% Pow_iff
thf(fact_673_Pow__iff,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( member_set_set_set_a @ A2 @ ( pow_set_set_a @ B2 ) )
= ( ord_le5722252365846178494_set_a @ A2 @ B2 ) ) ).
% Pow_iff
thf(fact_674_Pow__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( member_set_a2 @ A2 @ ( pow_a @ B2 ) )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% Pow_iff
thf(fact_675_hd__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ).
% hd_append2
thf(fact_676_hd__append2,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( Xs != nil_set_a )
=> ( ( hd_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( hd_set_a @ Xs ) ) ) ).
% hd_append2
thf(fact_677_list_Ocollapse,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_678_list_Ocollapse,axiom,
! [List: list_set_a] :
( ( List != nil_set_a )
=> ( ( cons_set_a @ ( hd_set_a @ List ) @ ( tl_set_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_679_hd__Cons__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( tl_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_680_hd__Cons__tl,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
=> ( ( cons_set_a @ ( hd_set_a @ Xs ) @ ( tl_set_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_681_Pow__singleton__iff,axiom,
! [X6: set_a,Y7: set_a] :
( ( ( pow_a @ X6 )
= ( insert_set_a2 @ Y7 @ bot_bot_set_set_a ) )
= ( ( X6 = bot_bot_set_a )
& ( Y7 = bot_bot_set_a ) ) ) ).
% Pow_singleton_iff
thf(fact_682_all__edges__subset__Pow,axiom,
! [A2: set_a] : ( ord_le3724670747650509150_set_a @ ( undire2918257014606996450dges_a @ A2 ) @ ( pow_a @ A2 ) ) ).
% all_edges_subset_Pow
thf(fact_683_all__edges__subset__Pow,axiom,
! [A2: set_set_a] : ( ord_le5722252365846178494_set_a @ ( undire8247866692393712962_set_a @ A2 ) @ ( pow_set_a @ A2 ) ) ).
% all_edges_subset_Pow
thf(fact_684_imageI,axiom,
! [X2: a,A2: set_a,F: a > a] :
( ( member_a2 @ X2 @ A2 )
=> ( member_a2 @ ( F @ X2 ) @ ( image_a_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_685_imageI,axiom,
! [X2: a,A2: set_a,F: a > set_a] :
( ( member_a2 @ X2 @ A2 )
=> ( member_set_a2 @ ( F @ X2 ) @ ( image_a_set_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_686_imageI,axiom,
! [X2: set_a,A2: set_set_a,F: set_a > a] :
( ( member_set_a2 @ X2 @ A2 )
=> ( member_a2 @ ( F @ X2 ) @ ( image_set_a_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_687_imageI,axiom,
! [X2: set_a,A2: set_set_a,F: set_a > set_a] :
( ( member_set_a2 @ X2 @ A2 )
=> ( member_set_a2 @ ( F @ X2 ) @ ( image_set_a_set_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_688_Pow__top,axiom,
! [A2: set_a] : ( member_set_a2 @ A2 @ ( pow_a @ A2 ) ) ).
% Pow_top
thf(fact_689_rev__image__eqI,axiom,
! [X2: a,A2: set_a,B: a,F: a > a] :
( ( member_a2 @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_a2 @ B @ ( image_a_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_690_rev__image__eqI,axiom,
! [X2: a,A2: set_a,B: set_a,F: a > set_a] :
( ( member_a2 @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_a2 @ B @ ( image_a_set_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_691_rev__image__eqI,axiom,
! [X2: set_a,A2: set_set_a,B: a,F: set_a > a] :
( ( member_set_a2 @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_a2 @ B @ ( image_set_a_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_692_rev__image__eqI,axiom,
! [X2: set_a,A2: set_set_a,B: set_a,F: set_a > set_a] :
( ( member_set_a2 @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_a2 @ B @ ( image_set_a_set_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_693_comp__sgraph_Oe__in__all__edges,axiom,
! [E: set_a,S: set_a] :
( ( member_set_a2 @ E @ ( undire2918257014606996450dges_a @ S ) )
=> ( member_set_a2 @ E @ ( undire2918257014606996450dges_a @ S ) ) ) ).
% comp_sgraph.e_in_all_edges
thf(fact_694_comp__sgraph_Oempty__not__edge,axiom,
! [S: set_a] :
~ ( member_set_a2 @ bot_bot_set_a @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.empty_not_edge
thf(fact_695_comp__sgraph_Owellformed,axiom,
! [E: set_set_a,S: set_set_a] :
( ( member_set_set_a2 @ E @ ( undire8247866692393712962_set_a @ S ) )
=> ( ord_le3724670747650509150_set_a @ E @ S ) ) ).
% comp_sgraph.wellformed
thf(fact_696_comp__sgraph_Owellformed,axiom,
! [E: set_set_set_a,S: set_set_set_a] :
( ( member_set_set_set_a @ E @ ( undire863132691662983842_set_a @ S ) )
=> ( ord_le5722252365846178494_set_a @ E @ S ) ) ).
% comp_sgraph.wellformed
thf(fact_697_comp__sgraph_Owellformed,axiom,
! [E: set_a,S: set_a] :
( ( member_set_a2 @ E @ ( undire2918257014606996450dges_a @ S ) )
=> ( ord_less_eq_set_a @ E @ S ) ) ).
% comp_sgraph.wellformed
thf(fact_698_comp__sgraph_Oe__in__all__edges__ss,axiom,
! [E: set_set_a,S: set_set_a,V2: set_set_a] :
( ( member_set_set_a2 @ E @ ( undire8247866692393712962_set_a @ S ) )
=> ( ( ord_le3724670747650509150_set_a @ E @ V2 )
=> ( ( ord_le3724670747650509150_set_a @ V2 @ S )
=> ( member_set_set_a2 @ E @ ( undire8247866692393712962_set_a @ V2 ) ) ) ) ) ).
% comp_sgraph.e_in_all_edges_ss
thf(fact_699_comp__sgraph_Oe__in__all__edges__ss,axiom,
! [E: set_set_set_a,S: set_set_set_a,V2: set_set_set_a] :
( ( member_set_set_set_a @ E @ ( undire863132691662983842_set_a @ S ) )
=> ( ( ord_le5722252365846178494_set_a @ E @ V2 )
=> ( ( ord_le5722252365846178494_set_a @ V2 @ S )
=> ( member_set_set_set_a @ E @ ( undire863132691662983842_set_a @ V2 ) ) ) ) ) ).
% comp_sgraph.e_in_all_edges_ss
thf(fact_700_comp__sgraph_Oe__in__all__edges__ss,axiom,
! [E: set_a,S: set_a,V2: set_a] :
( ( member_set_a2 @ E @ ( undire2918257014606996450dges_a @ S ) )
=> ( ( ord_less_eq_set_a @ E @ V2 )
=> ( ( ord_less_eq_set_a @ V2 @ S )
=> ( member_set_a2 @ E @ ( undire2918257014606996450dges_a @ V2 ) ) ) ) ) ).
% comp_sgraph.e_in_all_edges_ss
thf(fact_701_image__mono,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_a > set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ A2 ) @ ( image_set_a_set_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_702_image__mono,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_a > set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ord_le5722252365846178494_set_a @ ( image_4955109552351689957_set_a @ F @ A2 ) @ ( image_4955109552351689957_set_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_703_image__mono,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_a > a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A2 ) @ ( image_set_a_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_704_image__mono,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,F: set_set_a > set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( image_6061375613820669477_set_a @ F @ A2 ) @ ( image_6061375613820669477_set_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_705_image__mono,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,F: set_set_a > set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( ord_le5722252365846178494_set_a @ ( image_1042221919965026181_set_a @ F @ A2 ) @ ( image_1042221919965026181_set_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_706_image__mono,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,F: set_set_a > a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( image_set_set_a_a @ F @ A2 ) @ ( image_set_set_a_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_707_image__mono,axiom,
! [A2: set_a,B2: set_a,F: a > set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A2 ) @ ( image_a_set_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_708_image__mono,axiom,
! [A2: set_a,B2: set_a,F: a > set_set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_le5722252365846178494_set_a @ ( image_a_set_set_a @ F @ A2 ) @ ( image_a_set_set_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_709_image__mono,axiom,
! [A2: set_a,B2: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_710_image__subsetI,axiom,
! [A2: set_a,F: a > set_a,B2: set_set_a] :
( ! [X: a] :
( ( member_a2 @ X @ A2 )
=> ( member_set_a2 @ ( F @ X ) @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_711_image__subsetI,axiom,
! [A2: set_set_a,F: set_a > set_a,B2: set_set_a] :
( ! [X: set_a] :
( ( member_set_a2 @ X @ A2 )
=> ( member_set_a2 @ ( F @ X ) @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_712_image__subsetI,axiom,
! [A2: set_a,F: a > set_set_a,B2: set_set_set_a] :
( ! [X: a] :
( ( member_a2 @ X @ A2 )
=> ( member_set_set_a2 @ ( F @ X ) @ B2 ) )
=> ( ord_le5722252365846178494_set_a @ ( image_a_set_set_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_713_image__subsetI,axiom,
! [A2: set_set_a,F: set_a > set_set_a,B2: set_set_set_a] :
( ! [X: set_a] :
( ( member_set_a2 @ X @ A2 )
=> ( member_set_set_a2 @ ( F @ X ) @ B2 ) )
=> ( ord_le5722252365846178494_set_a @ ( image_4955109552351689957_set_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_714_image__subsetI,axiom,
! [A2: set_a,F: a > a,B2: set_a] :
( ! [X: a] :
( ( member_a2 @ X @ A2 )
=> ( member_a2 @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_715_image__subsetI,axiom,
! [A2: set_set_a,F: set_a > a,B2: set_a] :
( ! [X: set_a] :
( ( member_set_a2 @ X @ A2 )
=> ( member_a2 @ ( F @ X ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_716_subset__imageE,axiom,
! [B2: set_set_a,F: set_a > set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ ( image_set_a_set_a @ F @ A2 ) )
=> ~ ! [C4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C4 @ A2 )
=> ( B2
!= ( image_set_a_set_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_717_subset__imageE,axiom,
! [B2: set_set_a,F: set_set_a > set_a,A2: set_set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ ( image_6061375613820669477_set_a @ F @ A2 ) )
=> ~ ! [C4: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ C4 @ A2 )
=> ( B2
!= ( image_6061375613820669477_set_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_718_subset__imageE,axiom,
! [B2: set_set_a,F: a > set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ ( image_a_set_a @ F @ A2 ) )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A2 )
=> ( B2
!= ( image_a_set_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_719_subset__imageE,axiom,
! [B2: set_set_set_a,F: set_a > set_set_a,A2: set_set_a] :
( ( ord_le5722252365846178494_set_a @ B2 @ ( image_4955109552351689957_set_a @ F @ A2 ) )
=> ~ ! [C4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C4 @ A2 )
=> ( B2
!= ( image_4955109552351689957_set_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_720_subset__imageE,axiom,
! [B2: set_set_set_a,F: set_set_a > set_set_a,A2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B2 @ ( image_1042221919965026181_set_a @ F @ A2 ) )
=> ~ ! [C4: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ C4 @ A2 )
=> ( B2
!= ( image_1042221919965026181_set_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_721_subset__imageE,axiom,
! [B2: set_set_set_a,F: a > set_set_a,A2: set_a] :
( ( ord_le5722252365846178494_set_a @ B2 @ ( image_a_set_set_a @ F @ A2 ) )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A2 )
=> ( B2
!= ( image_a_set_set_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_722_subset__imageE,axiom,
! [B2: set_a,F: set_a > a,A2: set_set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_set_a_a @ F @ A2 ) )
=> ~ ! [C4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C4 @ A2 )
=> ( B2
!= ( image_set_a_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_723_subset__imageE,axiom,
! [B2: set_a,F: set_set_a > a,A2: set_set_set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_set_set_a_a @ F @ A2 ) )
=> ~ ! [C4: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ C4 @ A2 )
=> ( B2
!= ( image_set_set_a_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_724_subset__imageE,axiom,
! [B2: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A2 )
=> ( B2
!= ( image_a_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_725_subset__image__iff,axiom,
! [B2: set_set_a,F: set_a > set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ ( image_set_a_set_a @ F @ A2 ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A2 )
& ( B2
= ( image_set_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_726_subset__image__iff,axiom,
! [B2: set_set_a,F: set_set_a > set_a,A2: set_set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ ( image_6061375613820669477_set_a @ F @ A2 ) )
= ( ? [AA: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ AA @ A2 )
& ( B2
= ( image_6061375613820669477_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_727_subset__image__iff,axiom,
! [B2: set_set_a,F: a > set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ ( image_a_set_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B2
= ( image_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_728_subset__image__iff,axiom,
! [B2: set_set_set_a,F: set_a > set_set_a,A2: set_set_a] :
( ( ord_le5722252365846178494_set_a @ B2 @ ( image_4955109552351689957_set_a @ F @ A2 ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A2 )
& ( B2
= ( image_4955109552351689957_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_729_subset__image__iff,axiom,
! [B2: set_set_set_a,F: set_set_a > set_set_a,A2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B2 @ ( image_1042221919965026181_set_a @ F @ A2 ) )
= ( ? [AA: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ AA @ A2 )
& ( B2
= ( image_1042221919965026181_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_730_subset__image__iff,axiom,
! [B2: set_set_set_a,F: a > set_set_a,A2: set_a] :
( ( ord_le5722252365846178494_set_a @ B2 @ ( image_a_set_set_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B2
= ( image_a_set_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_731_subset__image__iff,axiom,
! [B2: set_a,F: set_a > a,A2: set_set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_set_a_a @ F @ A2 ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A2 )
& ( B2
= ( image_set_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_732_subset__image__iff,axiom,
! [B2: set_a,F: set_set_a > a,A2: set_set_set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_set_set_a_a @ F @ A2 ) )
= ( ? [AA: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ AA @ A2 )
& ( B2
= ( image_set_set_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_733_subset__image__iff,axiom,
! [B2: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B2
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_734_list_Osel_I1_J,axiom,
! [X21: a,X22: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_735_list_Osel_I1_J,axiom,
! [X21: set_a,X22: list_set_a] :
( ( hd_set_a @ ( cons_set_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_736_Pow__bottom,axiom,
! [B2: set_a] : ( member_set_a2 @ bot_bot_set_a @ ( pow_a @ B2 ) ) ).
% Pow_bottom
thf(fact_737_comp__sgraph_Owellformed__all__edges,axiom,
! [S: set_a] : ( ord_le3724670747650509150_set_a @ ( undire2918257014606996450dges_a @ S ) @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.wellformed_all_edges
thf(fact_738_comp__sgraph_Owellformed__all__edges,axiom,
! [S: set_set_a] : ( ord_le5722252365846178494_set_a @ ( undire8247866692393712962_set_a @ S ) @ ( undire8247866692393712962_set_a @ S ) ) ).
% comp_sgraph.wellformed_all_edges
thf(fact_739_PowD,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( member_set_set_a2 @ A2 @ ( pow_set_a @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% PowD
thf(fact_740_PowD,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( member_set_set_set_a @ A2 @ ( pow_set_set_a @ B2 ) )
=> ( ord_le5722252365846178494_set_a @ A2 @ B2 ) ) ).
% PowD
thf(fact_741_PowD,axiom,
! [A2: set_a,B2: set_a] :
( ( member_set_a2 @ A2 @ ( pow_a @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% PowD
thf(fact_742_comp__sgraph_Ono__loops,axiom,
! [V: a,S: set_a] :
( ( member_a2 @ V @ S )
=> ~ ( undire3617971648856834880loop_a @ ( undire2918257014606996450dges_a @ S ) @ V ) ) ).
% comp_sgraph.no_loops
thf(fact_743_comp__sgraph_Ono__loops,axiom,
! [V: set_a,S: set_set_a] :
( ( member_set_a2 @ V @ S )
=> ~ ( undire5774735625301615776_set_a @ ( undire8247866692393712962_set_a @ S ) @ V ) ) ).
% comp_sgraph.no_loops
thf(fact_744_comp__sgraph_Ohas__loop__in__verts,axiom,
! [S: set_a,V: a] :
( ( undire3617971648856834880loop_a @ ( undire2918257014606996450dges_a @ S ) @ V )
=> ( member_a2 @ V @ S ) ) ).
% comp_sgraph.has_loop_in_verts
thf(fact_745_comp__sgraph_Ohas__loop__in__verts,axiom,
! [S: set_set_a,V: set_a] :
( ( undire5774735625301615776_set_a @ ( undire8247866692393712962_set_a @ S ) @ V )
=> ( member_set_a2 @ V @ S ) ) ).
% comp_sgraph.has_loop_in_verts
thf(fact_746_hd__concat,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( ( hd_list_a @ Xs )
!= nil_a )
=> ( ( hd_a @ ( concat_a @ Xs ) )
= ( hd_a @ ( hd_list_a @ Xs ) ) ) ) ) ).
% hd_concat
thf(fact_747_hd__concat,axiom,
! [Xs: list_list_set_a] :
( ( Xs != nil_list_set_a )
=> ( ( ( hd_list_set_a @ Xs )
!= nil_set_a )
=> ( ( hd_set_a @ ( concat_set_a @ Xs ) )
= ( hd_set_a @ ( hd_list_set_a @ Xs ) ) ) ) ) ).
% hd_concat
thf(fact_748_subseqs__powset,axiom,
! [Xs: list_set_a] :
( ( image_8804695481318321887_set_a @ set_set_a2 @ ( set_list_set_a2 @ ( subseqs_set_a @ Xs ) ) )
= ( pow_set_a @ ( set_set_a2 @ Xs ) ) ) ).
% subseqs_powset
thf(fact_749_subseqs__powset,axiom,
! [Xs: list_set_set_a] :
( ( image_4529260449202881247_set_a @ set_set_set_a2 @ ( set_list_set_set_a2 @ ( subseqs_set_set_a @ Xs ) ) )
= ( pow_set_set_a @ ( set_set_set_a2 @ Xs ) ) ) ).
% subseqs_powset
thf(fact_750_comp__sgraph_Owellformed__alt__snd,axiom,
! [X2: set_a,Y: set_a,S: set_set_a] :
( ( member_set_set_a2 @ ( insert_set_a2 @ X2 @ ( insert_set_a2 @ Y @ bot_bot_set_set_a ) ) @ ( undire8247866692393712962_set_a @ S ) )
=> ( member_set_a2 @ Y @ S ) ) ).
% comp_sgraph.wellformed_alt_snd
thf(fact_751_comp__sgraph_Owellformed__alt__snd,axiom,
! [X2: a,Y: a,S: set_a] :
( ( member_set_a2 @ ( insert_a2 @ X2 @ ( insert_a2 @ Y @ bot_bot_set_a ) ) @ ( undire2918257014606996450dges_a @ S ) )
=> ( member_a2 @ Y @ S ) ) ).
% comp_sgraph.wellformed_alt_snd
thf(fact_752_comp__sgraph_Owellformed__alt__fst,axiom,
! [X2: set_a,Y: set_a,S: set_set_a] :
( ( member_set_set_a2 @ ( insert_set_a2 @ X2 @ ( insert_set_a2 @ Y @ bot_bot_set_set_a ) ) @ ( undire8247866692393712962_set_a @ S ) )
=> ( member_set_a2 @ X2 @ S ) ) ).
% comp_sgraph.wellformed_alt_fst
thf(fact_753_comp__sgraph_Owellformed__alt__fst,axiom,
! [X2: a,Y: a,S: set_a] :
( ( member_set_a2 @ ( insert_a2 @ X2 @ ( insert_a2 @ Y @ bot_bot_set_a ) ) @ ( undire2918257014606996450dges_a @ S ) )
=> ( member_a2 @ X2 @ S ) ) ).
% comp_sgraph.wellformed_alt_fst
thf(fact_754_comp__sgraph_Osingleton__not__edge,axiom,
! [X2: a,S: set_a] :
~ ( member_set_a2 @ ( insert_a2 @ X2 @ bot_bot_set_a ) @ ( undire2918257014606996450dges_a @ S ) ) ).
% comp_sgraph.singleton_not_edge
thf(fact_755_all__edges__mono,axiom,
! [Vs: set_set_a,Ws: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Vs @ Ws )
=> ( ord_le5722252365846178494_set_a @ ( undire8247866692393712962_set_a @ Vs ) @ ( undire8247866692393712962_set_a @ Ws ) ) ) ).
% all_edges_mono
thf(fact_756_all__edges__mono,axiom,
! [Vs: set_set_set_a,Ws: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ Vs @ Ws )
=> ( ord_le8049040685576063006_set_a @ ( undire863132691662983842_set_a @ Vs ) @ ( undire863132691662983842_set_a @ Ws ) ) ) ).
% all_edges_mono
thf(fact_757_all__edges__mono,axiom,
! [Vs: set_a,Ws: set_a] :
( ( ord_less_eq_set_a @ Vs @ Ws )
=> ( ord_le3724670747650509150_set_a @ ( undire2918257014606996450dges_a @ Vs ) @ ( undire2918257014606996450dges_a @ Ws ) ) ) ).
% all_edges_mono
thf(fact_758_hd__in__set,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( member_a2 @ ( hd_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_759_hd__in__set,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
=> ( member_set_a2 @ ( hd_set_a @ Xs ) @ ( set_set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_760_hd__in__set,axiom,
! [Xs: list_set_set_a] :
( ( Xs != nil_set_set_a )
=> ( member_set_set_a2 @ ( hd_set_set_a @ Xs ) @ ( set_set_set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_761_list_Oset__sel_I1_J,axiom,
! [A: list_a] :
( ( A != nil_a )
=> ( member_a2 @ ( hd_a @ A ) @ ( set_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_762_list_Oset__sel_I1_J,axiom,
! [A: list_set_a] :
( ( A != nil_set_a )
=> ( member_set_a2 @ ( hd_set_a @ A ) @ ( set_set_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_763_list_Oset__sel_I1_J,axiom,
! [A: list_set_set_a] :
( ( A != nil_set_set_a )
=> ( member_set_set_a2 @ ( hd_set_set_a @ A ) @ ( set_set_set_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_764_longest__common__prefix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ps: list_a,Xs6: list_a,Ys6: list_a] :
( ( Xs
= ( append_a @ Ps @ Xs6 ) )
& ( Ys
= ( append_a @ Ps @ Ys6 ) )
& ( ( Xs6 = nil_a )
| ( Ys6 = nil_a )
| ( ( hd_a @ Xs6 )
!= ( hd_a @ Ys6 ) ) ) ) ).
% longest_common_prefix
thf(fact_765_longest__common__prefix,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
? [Ps: list_set_a,Xs6: list_set_a,Ys6: list_set_a] :
( ( Xs
= ( append_set_a @ Ps @ Xs6 ) )
& ( Ys
= ( append_set_a @ Ps @ Ys6 ) )
& ( ( Xs6 = nil_set_a )
| ( Ys6 = nil_set_a )
| ( ( hd_set_a @ Xs6 )
!= ( hd_set_a @ Ys6 ) ) ) ) ).
% longest_common_prefix
thf(fact_766_hd__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ) ).
% hd_append
thf(fact_767_hd__append,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( Xs = nil_set_a )
=> ( ( hd_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( hd_set_a @ Ys ) ) )
& ( ( Xs != nil_set_a )
=> ( ( hd_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( hd_set_a @ Xs ) ) ) ) ).
% hd_append
thf(fact_768_list_Oexpand,axiom,
! [List: list_a,List2: list_a] :
( ( ( List = nil_a )
= ( List2 = nil_a ) )
=> ( ( ( List != nil_a )
=> ( ( List2 != nil_a )
=> ( ( ( hd_a @ List )
= ( hd_a @ List2 ) )
& ( ( tl_a @ List )
= ( tl_a @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_769_list_Oexpand,axiom,
! [List: list_set_a,List2: list_set_a] :
( ( ( List = nil_set_a )
= ( List2 = nil_set_a ) )
=> ( ( ( List != nil_set_a )
=> ( ( List2 != nil_set_a )
=> ( ( ( hd_set_a @ List )
= ( hd_set_a @ List2 ) )
& ( ( tl_set_a @ List )
= ( tl_set_a @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_770_hd__Nil__eq__last,axiom,
( ( hd_a @ nil_a )
= ( last_a @ nil_a ) ) ).
% hd_Nil_eq_last
thf(fact_771_hd__Nil__eq__last,axiom,
( ( hd_set_a @ nil_set_a )
= ( last_set_a @ nil_set_a ) ) ).
% hd_Nil_eq_last
thf(fact_772_Pow__mono,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ord_le5722252365846178494_set_a @ ( pow_set_a @ A2 ) @ ( pow_set_a @ B2 ) ) ) ).
% Pow_mono
thf(fact_773_Pow__mono,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( ord_le8049040685576063006_set_a @ ( pow_set_set_a @ A2 ) @ ( pow_set_set_a @ B2 ) ) ) ).
% Pow_mono
thf(fact_774_Pow__mono,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( pow_a @ A2 ) @ ( pow_a @ B2 ) ) ) ).
% Pow_mono
thf(fact_775_comp__sgraph_Ohas__loop__def,axiom,
! [S: set_a,V: a] :
( ( undire3617971648856834880loop_a @ ( undire2918257014606996450dges_a @ S ) @ V )
= ( member_set_a2 @ ( insert_a2 @ V @ bot_bot_set_a ) @ ( undire2918257014606996450dges_a @ S ) ) ) ).
% comp_sgraph.has_loop_def
thf(fact_776_list_Oexhaust__sel,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( List
= ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_777_list_Oexhaust__sel,axiom,
! [List: list_set_a] :
( ( List != nil_set_a )
=> ( List
= ( cons_set_a @ ( hd_set_a @ List ) @ ( tl_set_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_778_rotate1__hd__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( rotate1_a @ Xs )
= ( append_a @ ( tl_a @ Xs ) @ ( cons_a @ ( hd_a @ Xs ) @ nil_a ) ) ) ) ).
% rotate1_hd_tl
thf(fact_779_rotate1__hd__tl,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
=> ( ( rotate1_set_a @ Xs )
= ( append_set_a @ ( tl_set_a @ Xs ) @ ( cons_set_a @ ( hd_set_a @ Xs ) @ nil_set_a ) ) ) ) ).
% rotate1_hd_tl
thf(fact_780_all__subset__image,axiom,
! [F: set_a > set_a,A2: set_set_a,P: set_set_a > $o] :
( ( ! [B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ ( image_set_a_set_a @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A2 )
=> ( P @ ( image_set_a_set_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_781_all__subset__image,axiom,
! [F: set_set_a > set_a,A2: set_set_set_a,P: set_set_a > $o] :
( ( ! [B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ ( image_6061375613820669477_set_a @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B4 @ A2 )
=> ( P @ ( image_6061375613820669477_set_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_782_all__subset__image,axiom,
! [F: a > set_a,A2: set_a,P: set_set_a > $o] :
( ( ! [B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ ( image_a_set_a @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A2 )
=> ( P @ ( image_a_set_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_783_all__subset__image,axiom,
! [F: set_a > set_set_a,A2: set_set_a,P: set_set_set_a > $o] :
( ( ! [B4: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B4 @ ( image_4955109552351689957_set_a @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A2 )
=> ( P @ ( image_4955109552351689957_set_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_784_all__subset__image,axiom,
! [F: set_set_a > set_set_a,A2: set_set_set_a,P: set_set_set_a > $o] :
( ( ! [B4: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B4 @ ( image_1042221919965026181_set_a @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B4 @ A2 )
=> ( P @ ( image_1042221919965026181_set_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_785_all__subset__image,axiom,
! [F: a > set_set_a,A2: set_a,P: set_set_set_a > $o] :
( ( ! [B4: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B4 @ ( image_a_set_set_a @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A2 )
=> ( P @ ( image_a_set_set_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_786_all__subset__image,axiom,
! [F: set_a > a,A2: set_set_a,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_set_a_a @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A2 )
=> ( P @ ( image_set_a_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_787_all__subset__image,axiom,
! [F: set_set_a > a,A2: set_set_set_a,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_set_set_a_a @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B4 @ A2 )
=> ( P @ ( image_set_set_a_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_788_all__subset__image,axiom,
! [F: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A2 )
=> ( P @ ( image_a_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_789_distinct__adj__append__iff,axiom,
! [Xs: list_a,Ys: list_a] :
( ( distinct_adj_a @ ( append_a @ Xs @ Ys ) )
= ( ( distinct_adj_a @ Xs )
& ( distinct_adj_a @ Ys )
& ( ( Xs = nil_a )
| ( Ys = nil_a )
| ( ( last_a @ Xs )
!= ( hd_a @ Ys ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_790_distinct__adj__append__iff,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( distinct_adj_set_a @ ( append_set_a @ Xs @ Ys ) )
= ( ( distinct_adj_set_a @ Xs )
& ( distinct_adj_set_a @ Ys )
& ( ( Xs = nil_set_a )
| ( Ys = nil_set_a )
| ( ( last_set_a @ Xs )
!= ( hd_set_a @ Ys ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_791_all__edges__loops__ss_I1_J,axiom,
! [S: set_a] : ( ord_le3724670747650509150_set_a @ ( undire2918257014606996450dges_a @ S ) @ ( undire9065700607645037417oops_a @ S ) ) ).
% all_edges_loops_ss(1)
thf(fact_792_all__edges__loops__ss_I1_J,axiom,
! [S: set_set_a] : ( ord_le5722252365846178494_set_a @ ( undire8247866692393712962_set_a @ S ) @ ( undire3533089111843156169_set_a @ S ) ) ).
% all_edges_loops_ss(1)
thf(fact_793_distinct__adj__Cons__Cons,axiom,
! [X2: a,Y: a,Xs: list_a] :
( ( distinct_adj_a @ ( cons_a @ X2 @ ( cons_a @ Y @ Xs ) ) )
= ( ( X2 != Y )
& ( distinct_adj_a @ ( cons_a @ Y @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_794_distinct__adj__Cons__Cons,axiom,
! [X2: set_a,Y: set_a,Xs: list_set_a] :
( ( distinct_adj_set_a @ ( cons_set_a @ X2 @ ( cons_set_a @ Y @ Xs ) ) )
= ( ( X2 != Y )
& ( distinct_adj_set_a @ ( cons_set_a @ Y @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_795_rotate1__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rotate1_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_796_rotate1__is__Nil__conv,axiom,
! [Xs: list_set_a] :
( ( ( rotate1_set_a @ Xs )
= nil_set_a )
= ( Xs = nil_set_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_797_set__rotate1,axiom,
! [Xs: list_set_a] :
( ( set_set_a2 @ ( rotate1_set_a @ Xs ) )
= ( set_set_a2 @ Xs ) ) ).
% set_rotate1
thf(fact_798_set__rotate1,axiom,
! [Xs: list_set_set_a] :
( ( set_set_set_a2 @ ( rotate1_set_set_a @ Xs ) )
= ( set_set_set_a2 @ Xs ) ) ).
% set_rotate1
thf(fact_799_distinct__adj__ConsD,axiom,
! [X2: a,Xs: list_a] :
( ( distinct_adj_a @ ( cons_a @ X2 @ Xs ) )
=> ( distinct_adj_a @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_800_distinct__adj__ConsD,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( distinct_adj_set_a @ ( cons_set_a @ X2 @ Xs ) )
=> ( distinct_adj_set_a @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_801_distinct__adj__Nil,axiom,
distinct_adj_a @ nil_a ).
% distinct_adj_Nil
thf(fact_802_distinct__adj__Nil,axiom,
distinct_adj_set_a @ nil_set_a ).
% distinct_adj_Nil
thf(fact_803_distinct__adj__appendD2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( distinct_adj_a @ ( append_a @ Xs @ Ys ) )
=> ( distinct_adj_a @ Ys ) ) ).
% distinct_adj_appendD2
thf(fact_804_distinct__adj__appendD2,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( distinct_adj_set_a @ ( append_set_a @ Xs @ Ys ) )
=> ( distinct_adj_set_a @ Ys ) ) ).
% distinct_adj_appendD2
thf(fact_805_distinct__adj__appendD1,axiom,
! [Xs: list_a,Ys: list_a] :
( ( distinct_adj_a @ ( append_a @ Xs @ Ys ) )
=> ( distinct_adj_a @ Xs ) ) ).
% distinct_adj_appendD1
thf(fact_806_distinct__adj__appendD1,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( distinct_adj_set_a @ ( append_set_a @ Xs @ Ys ) )
=> ( distinct_adj_set_a @ Xs ) ) ).
% distinct_adj_appendD1
thf(fact_807_rotate1_Osimps_I1_J,axiom,
( ( rotate1_a @ nil_a )
= nil_a ) ).
% rotate1.simps(1)
thf(fact_808_rotate1_Osimps_I1_J,axiom,
( ( rotate1_set_a @ nil_set_a )
= nil_set_a ) ).
% rotate1.simps(1)
thf(fact_809_distinct__adj__singleton,axiom,
! [X2: a] : ( distinct_adj_a @ ( cons_a @ X2 @ nil_a ) ) ).
% distinct_adj_singleton
thf(fact_810_distinct__adj__singleton,axiom,
! [X2: set_a] : ( distinct_adj_set_a @ ( cons_set_a @ X2 @ nil_set_a ) ) ).
% distinct_adj_singleton
thf(fact_811_distinct__adj__Cons,axiom,
! [X2: a,Xs: list_a] :
( ( distinct_adj_a @ ( cons_a @ X2 @ Xs ) )
= ( ( Xs = nil_a )
| ( ( X2
!= ( hd_a @ Xs ) )
& ( distinct_adj_a @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_812_distinct__adj__Cons,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( distinct_adj_set_a @ ( cons_set_a @ X2 @ Xs ) )
= ( ( Xs = nil_set_a )
| ( ( X2
!= ( hd_set_a @ Xs ) )
& ( distinct_adj_set_a @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_813_rotate1_Osimps_I2_J,axiom,
! [X2: a,Xs: list_a] :
( ( rotate1_a @ ( cons_a @ X2 @ Xs ) )
= ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) ) ).
% rotate1.simps(2)
thf(fact_814_rotate1_Osimps_I2_J,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( rotate1_set_a @ ( cons_set_a @ X2 @ Xs ) )
= ( append_set_a @ Xs @ ( cons_set_a @ X2 @ nil_set_a ) ) ) ).
% rotate1.simps(2)
thf(fact_815_successively__append__iff,axiom,
! [P: a > a > $o,Xs: list_a,Ys: list_a] :
( ( successively_a @ P @ ( append_a @ Xs @ Ys ) )
= ( ( successively_a @ P @ Xs )
& ( successively_a @ P @ Ys )
& ( ( Xs = nil_a )
| ( Ys = nil_a )
| ( P @ ( last_a @ Xs ) @ ( hd_a @ Ys ) ) ) ) ) ).
% successively_append_iff
thf(fact_816_successively__append__iff,axiom,
! [P: set_a > set_a > $o,Xs: list_set_a,Ys: list_set_a] :
( ( successively_set_a @ P @ ( append_set_a @ Xs @ Ys ) )
= ( ( successively_set_a @ P @ Xs )
& ( successively_set_a @ P @ Ys )
& ( ( Xs = nil_set_a )
| ( Ys = nil_set_a )
| ( P @ ( last_set_a @ Xs ) @ ( hd_set_a @ Ys ) ) ) ) ) ).
% successively_append_iff
thf(fact_817_comp__sgraph_Oincident__sedges__empty,axiom,
! [V: a,S: set_a] :
( ~ ( member_a2 @ V @ S )
=> ( ( undire1270416042309875431dges_a @ ( undire2918257014606996450dges_a @ S ) @ V )
= bot_bot_set_set_a ) ) ).
% comp_sgraph.incident_sedges_empty
thf(fact_818_comp__sgraph_Oincident__sedges__empty,axiom,
! [V: set_a,S: set_set_a] :
( ~ ( member_set_a2 @ V @ S )
=> ( ( undire5844230293943614535_set_a @ ( undire8247866692393712962_set_a @ S ) @ V )
= bot_bo3380559777022489994_set_a ) ) ).
% comp_sgraph.incident_sedges_empty
thf(fact_819_Cons__in__shuffles__iff,axiom,
! [Z2: a,Zs: list_a,Xs: list_a,Ys: list_a] :
( ( member_list_a @ ( cons_a @ Z2 @ Zs ) @ ( shuffles_a @ Xs @ Ys ) )
= ( ( ( Xs != nil_a )
& ( ( hd_a @ Xs )
= Z2 )
& ( member_list_a @ Zs @ ( shuffles_a @ ( tl_a @ Xs ) @ Ys ) ) )
| ( ( Ys != nil_a )
& ( ( hd_a @ Ys )
= Z2 )
& ( member_list_a @ Zs @ ( shuffles_a @ Xs @ ( tl_a @ Ys ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_820_Cons__in__shuffles__iff,axiom,
! [Z2: set_a,Zs: list_set_a,Xs: list_set_a,Ys: list_set_a] :
( ( member_list_set_a @ ( cons_set_a @ Z2 @ Zs ) @ ( shuffles_set_a @ Xs @ Ys ) )
= ( ( ( Xs != nil_set_a )
& ( ( hd_set_a @ Xs )
= Z2 )
& ( member_list_set_a @ Zs @ ( shuffles_set_a @ ( tl_set_a @ Xs ) @ Ys ) ) )
| ( ( Ys != nil_set_a )
& ( ( hd_set_a @ Ys )
= Z2 )
& ( member_list_set_a @ Zs @ ( shuffles_set_a @ Xs @ ( tl_set_a @ Ys ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_821_Nil__in__shuffles,axiom,
! [Xs: list_a,Ys: list_a] :
( ( member_list_a @ nil_a @ ( shuffles_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_in_shuffles
thf(fact_822_Nil__in__shuffles,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( member_list_set_a @ nil_set_a @ ( shuffles_set_a @ Xs @ Ys ) )
= ( ( Xs = nil_set_a )
& ( Ys = nil_set_a ) ) ) ).
% Nil_in_shuffles
thf(fact_823_Cons__in__shuffles__leftI,axiom,
! [Zs: list_a,Xs: list_a,Ys: list_a,Z2: a] :
( ( member_list_a @ Zs @ ( shuffles_a @ Xs @ Ys ) )
=> ( member_list_a @ ( cons_a @ Z2 @ Zs ) @ ( shuffles_a @ ( cons_a @ Z2 @ Xs ) @ Ys ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_824_Cons__in__shuffles__leftI,axiom,
! [Zs: list_set_a,Xs: list_set_a,Ys: list_set_a,Z2: set_a] :
( ( member_list_set_a @ Zs @ ( shuffles_set_a @ Xs @ Ys ) )
=> ( member_list_set_a @ ( cons_set_a @ Z2 @ Zs ) @ ( shuffles_set_a @ ( cons_set_a @ Z2 @ Xs ) @ Ys ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_825_Cons__in__shuffles__rightI,axiom,
! [Zs: list_a,Xs: list_a,Ys: list_a,Z2: a] :
( ( member_list_a @ Zs @ ( shuffles_a @ Xs @ Ys ) )
=> ( member_list_a @ ( cons_a @ Z2 @ Zs ) @ ( shuffles_a @ Xs @ ( cons_a @ Z2 @ Ys ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_826_Cons__in__shuffles__rightI,axiom,
! [Zs: list_set_a,Xs: list_set_a,Ys: list_set_a,Z2: set_a] :
( ( member_list_set_a @ Zs @ ( shuffles_set_a @ Xs @ Ys ) )
=> ( member_list_set_a @ ( cons_set_a @ Z2 @ Zs ) @ ( shuffles_set_a @ Xs @ ( cons_set_a @ Z2 @ Ys ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_827_Nil__in__shufflesI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = nil_a )
=> ( ( Ys = nil_a )
=> ( member_list_a @ nil_a @ ( shuffles_a @ Xs @ Ys ) ) ) ) ).
% Nil_in_shufflesI
thf(fact_828_Nil__in__shufflesI,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( Xs = nil_set_a )
=> ( ( Ys = nil_set_a )
=> ( member_list_set_a @ nil_set_a @ ( shuffles_set_a @ Xs @ Ys ) ) ) ) ).
% Nil_in_shufflesI
thf(fact_829_successively_Oelims_I3_J,axiom,
! [X2: a > a > $o,Xa2: list_a] :
( ~ ( successively_a @ X2 @ Xa2 )
=> ~ ! [X: a,Y2: a,Xs3: list_a] :
( ( Xa2
= ( cons_a @ X @ ( cons_a @ Y2 @ Xs3 ) ) )
=> ( ( X2 @ X @ Y2 )
& ( successively_a @ X2 @ ( cons_a @ Y2 @ Xs3 ) ) ) ) ) ).
% successively.elims(3)
thf(fact_830_successively_Oelims_I3_J,axiom,
! [X2: set_a > set_a > $o,Xa2: list_set_a] :
( ~ ( successively_set_a @ X2 @ Xa2 )
=> ~ ! [X: set_a,Y2: set_a,Xs3: list_set_a] :
( ( Xa2
= ( cons_set_a @ X @ ( cons_set_a @ Y2 @ Xs3 ) ) )
=> ( ( X2 @ X @ Y2 )
& ( successively_set_a @ X2 @ ( cons_set_a @ Y2 @ Xs3 ) ) ) ) ) ).
% successively.elims(3)
thf(fact_831_successively_Osimps_I3_J,axiom,
! [P: a > a > $o,X2: a,Y: a,Xs: list_a] :
( ( successively_a @ P @ ( cons_a @ X2 @ ( cons_a @ Y @ Xs ) ) )
= ( ( P @ X2 @ Y )
& ( successively_a @ P @ ( cons_a @ Y @ Xs ) ) ) ) ).
% successively.simps(3)
thf(fact_832_successively_Osimps_I3_J,axiom,
! [P: set_a > set_a > $o,X2: set_a,Y: set_a,Xs: list_set_a] :
( ( successively_set_a @ P @ ( cons_set_a @ X2 @ ( cons_set_a @ Y @ Xs ) ) )
= ( ( P @ X2 @ Y )
& ( successively_set_a @ P @ ( cons_set_a @ Y @ Xs ) ) ) ) ).
% successively.simps(3)
thf(fact_833_successively_Osimps_I1_J,axiom,
! [P: a > a > $o] : ( successively_a @ P @ nil_a ) ).
% successively.simps(1)
thf(fact_834_successively_Osimps_I1_J,axiom,
! [P: set_a > set_a > $o] : ( successively_set_a @ P @ nil_set_a ) ).
% successively.simps(1)
thf(fact_835_successively__cong,axiom,
! [Xs: list_a,P: a > a > $o,Q: a > a > $o,Ys: list_a] :
( ! [X: a,Y2: a] :
( ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ( ( member_a2 @ Y2 @ ( set_a2 @ Xs ) )
=> ( ( P @ X @ Y2 )
= ( Q @ X @ Y2 ) ) ) )
=> ( ( Xs = Ys )
=> ( ( successively_a @ P @ Xs )
= ( successively_a @ Q @ Ys ) ) ) ) ).
% successively_cong
thf(fact_836_successively__cong,axiom,
! [Xs: list_set_a,P: set_a > set_a > $o,Q: set_a > set_a > $o,Ys: list_set_a] :
( ! [X: set_a,Y2: set_a] :
( ( member_set_a2 @ X @ ( set_set_a2 @ Xs ) )
=> ( ( member_set_a2 @ Y2 @ ( set_set_a2 @ Xs ) )
=> ( ( P @ X @ Y2 )
= ( Q @ X @ Y2 ) ) ) )
=> ( ( Xs = Ys )
=> ( ( successively_set_a @ P @ Xs )
= ( successively_set_a @ Q @ Ys ) ) ) ) ).
% successively_cong
thf(fact_837_successively__cong,axiom,
! [Xs: list_set_set_a,P: set_set_a > set_set_a > $o,Q: set_set_a > set_set_a > $o,Ys: list_set_set_a] :
( ! [X: set_set_a,Y2: set_set_a] :
( ( member_set_set_a2 @ X @ ( set_set_set_a2 @ Xs ) )
=> ( ( member_set_set_a2 @ Y2 @ ( set_set_set_a2 @ Xs ) )
=> ( ( P @ X @ Y2 )
= ( Q @ X @ Y2 ) ) ) )
=> ( ( Xs = Ys )
=> ( ( succes33185699143464721_set_a @ P @ Xs )
= ( succes33185699143464721_set_a @ Q @ Ys ) ) ) ) ).
% successively_cong
thf(fact_838_successively__mono,axiom,
! [P: a > a > $o,Xs: list_a,Q: a > a > $o] :
( ( successively_a @ P @ Xs )
=> ( ! [X: a,Y2: a] :
( ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ( ( member_a2 @ Y2 @ ( set_a2 @ Xs ) )
=> ( ( P @ X @ Y2 )
=> ( Q @ X @ Y2 ) ) ) )
=> ( successively_a @ Q @ Xs ) ) ) ).
% successively_mono
thf(fact_839_successively__mono,axiom,
! [P: set_a > set_a > $o,Xs: list_set_a,Q: set_a > set_a > $o] :
( ( successively_set_a @ P @ Xs )
=> ( ! [X: set_a,Y2: set_a] :
( ( member_set_a2 @ X @ ( set_set_a2 @ Xs ) )
=> ( ( member_set_a2 @ Y2 @ ( set_set_a2 @ Xs ) )
=> ( ( P @ X @ Y2 )
=> ( Q @ X @ Y2 ) ) ) )
=> ( successively_set_a @ Q @ Xs ) ) ) ).
% successively_mono
thf(fact_840_successively__mono,axiom,
! [P: set_set_a > set_set_a > $o,Xs: list_set_set_a,Q: set_set_a > set_set_a > $o] :
( ( succes33185699143464721_set_a @ P @ Xs )
=> ( ! [X: set_set_a,Y2: set_set_a] :
( ( member_set_set_a2 @ X @ ( set_set_set_a2 @ Xs ) )
=> ( ( member_set_set_a2 @ Y2 @ ( set_set_set_a2 @ Xs ) )
=> ( ( P @ X @ Y2 )
=> ( Q @ X @ Y2 ) ) ) )
=> ( succes33185699143464721_set_a @ Q @ Xs ) ) ) ).
% successively_mono
thf(fact_841_empty__in__Fpow,axiom,
! [A2: set_a] : ( member_set_a2 @ bot_bot_set_a @ ( finite_Fpow_a @ A2 ) ) ).
% empty_in_Fpow
thf(fact_842_shufflesE,axiom,
! [Zs: list_a,Xs: list_a,Ys: list_a] :
( ( member_list_a @ Zs @ ( shuffles_a @ Xs @ Ys ) )
=> ( ( ( Zs = Xs )
=> ( Ys != nil_a ) )
=> ( ( ( Zs = Ys )
=> ( Xs != nil_a ) )
=> ( ! [X: a,Xs6: list_a] :
( ( Xs
= ( cons_a @ X @ Xs6 ) )
=> ! [Z3: a,Zs4: list_a] :
( ( Zs
= ( cons_a @ Z3 @ Zs4 ) )
=> ( ( X = Z3 )
=> ~ ( member_list_a @ Zs4 @ ( shuffles_a @ Xs6 @ Ys ) ) ) ) )
=> ~ ! [Y2: a,Ys6: list_a] :
( ( Ys
= ( cons_a @ Y2 @ Ys6 ) )
=> ! [Z3: a,Zs4: list_a] :
( ( Zs
= ( cons_a @ Z3 @ Zs4 ) )
=> ( ( Y2 = Z3 )
=> ~ ( member_list_a @ Zs4 @ ( shuffles_a @ Xs @ Ys6 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_843_shufflesE,axiom,
! [Zs: list_set_a,Xs: list_set_a,Ys: list_set_a] :
( ( member_list_set_a @ Zs @ ( shuffles_set_a @ Xs @ Ys ) )
=> ( ( ( Zs = Xs )
=> ( Ys != nil_set_a ) )
=> ( ( ( Zs = Ys )
=> ( Xs != nil_set_a ) )
=> ( ! [X: set_a,Xs6: list_set_a] :
( ( Xs
= ( cons_set_a @ X @ Xs6 ) )
=> ! [Z3: set_a,Zs4: list_set_a] :
( ( Zs
= ( cons_set_a @ Z3 @ Zs4 ) )
=> ( ( X = Z3 )
=> ~ ( member_list_set_a @ Zs4 @ ( shuffles_set_a @ Xs6 @ Ys ) ) ) ) )
=> ~ ! [Y2: set_a,Ys6: list_set_a] :
( ( Ys
= ( cons_set_a @ Y2 @ Ys6 ) )
=> ! [Z3: set_a,Zs4: list_set_a] :
( ( Zs
= ( cons_set_a @ Z3 @ Zs4 ) )
=> ( ( Y2 = Z3 )
=> ~ ( member_list_set_a @ Zs4 @ ( shuffles_set_a @ Xs @ Ys6 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_844_successively_Oelims_I2_J,axiom,
! [X2: a > a > $o,Xa2: list_a] :
( ( successively_a @ X2 @ Xa2 )
=> ( ( Xa2 != nil_a )
=> ( ! [X: a] :
( Xa2
!= ( cons_a @ X @ nil_a ) )
=> ~ ! [X: a,Y2: a,Xs3: list_a] :
( ( Xa2
= ( cons_a @ X @ ( cons_a @ Y2 @ Xs3 ) ) )
=> ~ ( ( X2 @ X @ Y2 )
& ( successively_a @ X2 @ ( cons_a @ Y2 @ Xs3 ) ) ) ) ) ) ) ).
% successively.elims(2)
thf(fact_845_successively_Oelims_I2_J,axiom,
! [X2: set_a > set_a > $o,Xa2: list_set_a] :
( ( successively_set_a @ X2 @ Xa2 )
=> ( ( Xa2 != nil_set_a )
=> ( ! [X: set_a] :
( Xa2
!= ( cons_set_a @ X @ nil_set_a ) )
=> ~ ! [X: set_a,Y2: set_a,Xs3: list_set_a] :
( ( Xa2
= ( cons_set_a @ X @ ( cons_set_a @ Y2 @ Xs3 ) ) )
=> ~ ( ( X2 @ X @ Y2 )
& ( successively_set_a @ X2 @ ( cons_set_a @ Y2 @ Xs3 ) ) ) ) ) ) ) ).
% successively.elims(2)
thf(fact_846_successively_Oelims_I1_J,axiom,
! [X2: a > a > $o,Xa2: list_a,Y: $o] :
( ( ( successively_a @ X2 @ Xa2 )
= Y )
=> ( ( ( Xa2 = nil_a )
=> ~ Y )
=> ( ( ? [X: a] :
( Xa2
= ( cons_a @ X @ nil_a ) )
=> ~ Y )
=> ~ ! [X: a,Y2: a,Xs3: list_a] :
( ( Xa2
= ( cons_a @ X @ ( cons_a @ Y2 @ Xs3 ) ) )
=> ( Y
= ( ~ ( ( X2 @ X @ Y2 )
& ( successively_a @ X2 @ ( cons_a @ Y2 @ Xs3 ) ) ) ) ) ) ) ) ) ).
% successively.elims(1)
thf(fact_847_successively_Oelims_I1_J,axiom,
! [X2: set_a > set_a > $o,Xa2: list_set_a,Y: $o] :
( ( ( successively_set_a @ X2 @ Xa2 )
= Y )
=> ( ( ( Xa2 = nil_set_a )
=> ~ Y )
=> ( ( ? [X: set_a] :
( Xa2
= ( cons_set_a @ X @ nil_set_a ) )
=> ~ Y )
=> ~ ! [X: set_a,Y2: set_a,Xs3: list_set_a] :
( ( Xa2
= ( cons_set_a @ X @ ( cons_set_a @ Y2 @ Xs3 ) ) )
=> ( Y
= ( ~ ( ( X2 @ X @ Y2 )
& ( successively_set_a @ X2 @ ( cons_set_a @ Y2 @ Xs3 ) ) ) ) ) ) ) ) ) ).
% successively.elims(1)
thf(fact_848_successively_Osimps_I2_J,axiom,
! [P: a > a > $o,X2: a] : ( successively_a @ P @ ( cons_a @ X2 @ nil_a ) ) ).
% successively.simps(2)
thf(fact_849_successively_Osimps_I2_J,axiom,
! [P: set_a > set_a > $o,X2: set_a] : ( successively_set_a @ P @ ( cons_set_a @ X2 @ nil_set_a ) ) ).
% successively.simps(2)
thf(fact_850_successively__remdups__adj__iff,axiom,
! [Xs: list_a,P: a > a > $o] :
( ! [X: a] :
( ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ( P @ X @ X ) )
=> ( ( successively_a @ P @ ( remdups_adj_a @ Xs ) )
= ( successively_a @ P @ Xs ) ) ) ).
% successively_remdups_adj_iff
thf(fact_851_successively__remdups__adj__iff,axiom,
! [Xs: list_set_a,P: set_a > set_a > $o] :
( ! [X: set_a] :
( ( member_set_a2 @ X @ ( set_set_a2 @ Xs ) )
=> ( P @ X @ X ) )
=> ( ( successively_set_a @ P @ ( remdups_adj_set_a @ Xs ) )
= ( successively_set_a @ P @ Xs ) ) ) ).
% successively_remdups_adj_iff
thf(fact_852_successively__remdups__adj__iff,axiom,
! [Xs: list_set_set_a,P: set_set_a > set_set_a > $o] :
( ! [X: set_set_a] :
( ( member_set_set_a2 @ X @ ( set_set_set_a2 @ Xs ) )
=> ( P @ X @ X ) )
=> ( ( succes33185699143464721_set_a @ P @ ( remdup1882599702278213626_set_a @ Xs ) )
= ( succes33185699143464721_set_a @ P @ Xs ) ) ) ).
% successively_remdups_adj_iff
thf(fact_853_Cons__shuffles__subset2,axiom,
! [Y: a,Xs: list_a,Ys: list_a] : ( ord_le8861187494160871172list_a @ ( image_list_a_list_a @ ( cons_a @ Y ) @ ( shuffles_a @ Xs @ Ys ) ) @ ( shuffles_a @ Xs @ ( cons_a @ Y @ Ys ) ) ) ).
% Cons_shuffles_subset2
thf(fact_854_Cons__shuffles__subset2,axiom,
! [Y: set_a,Xs: list_set_a,Ys: list_set_a] : ( ord_le864617614081865828_set_a @ ( image_6999107473886120325_set_a @ ( cons_set_a @ Y ) @ ( shuffles_set_a @ Xs @ Ys ) ) @ ( shuffles_set_a @ Xs @ ( cons_set_a @ Y @ Ys ) ) ) ).
% Cons_shuffles_subset2
thf(fact_855_Cons__shuffles__subset1,axiom,
! [X2: a,Xs: list_a,Ys: list_a] : ( ord_le8861187494160871172list_a @ ( image_list_a_list_a @ ( cons_a @ X2 ) @ ( shuffles_a @ Xs @ Ys ) ) @ ( shuffles_a @ ( cons_a @ X2 @ Xs ) @ Ys ) ) ).
% Cons_shuffles_subset1
thf(fact_856_Cons__shuffles__subset1,axiom,
! [X2: set_a,Xs: list_set_a,Ys: list_set_a] : ( ord_le864617614081865828_set_a @ ( image_6999107473886120325_set_a @ ( cons_set_a @ X2 ) @ ( shuffles_set_a @ Xs @ Ys ) ) @ ( shuffles_set_a @ ( cons_set_a @ X2 @ Xs ) @ Ys ) ) ).
% Cons_shuffles_subset1
thf(fact_857_successively__Cons,axiom,
! [P: a > a > $o,X2: a,Xs: list_a] :
( ( successively_a @ P @ ( cons_a @ X2 @ Xs ) )
= ( ( Xs = nil_a )
| ( ( P @ X2 @ ( hd_a @ Xs ) )
& ( successively_a @ P @ Xs ) ) ) ) ).
% successively_Cons
thf(fact_858_successively__Cons,axiom,
! [P: set_a > set_a > $o,X2: set_a,Xs: list_set_a] :
( ( successively_set_a @ P @ ( cons_set_a @ X2 @ Xs ) )
= ( ( Xs = nil_set_a )
| ( ( P @ X2 @ ( hd_set_a @ Xs ) )
& ( successively_set_a @ P @ Xs ) ) ) ) ).
% successively_Cons
thf(fact_859_ulgraph_Oincident__sedges__empty,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ~ ( member_a2 @ V @ Vertices )
=> ( ( undire1270416042309875431dges_a @ Edges @ V )
= bot_bot_set_set_a ) ) ) ).
% ulgraph.incident_sedges_empty
thf(fact_860_ulgraph_Oincident__sedges__empty,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ~ ( member_set_a2 @ V @ Vertices )
=> ( ( undire5844230293943614535_set_a @ Edges @ V )
= bot_bo3380559777022489994_set_a ) ) ) ).
% ulgraph.incident_sedges_empty
thf(fact_861_Fpow__mono,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ord_le5722252365846178494_set_a @ ( finite_Fpow_set_a @ A2 ) @ ( finite_Fpow_set_a @ B2 ) ) ) ).
% Fpow_mono
thf(fact_862_Fpow__mono,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( ord_le8049040685576063006_set_a @ ( finite3131931315176300580_set_a @ A2 ) @ ( finite3131931315176300580_set_a @ B2 ) ) ) ).
% Fpow_mono
thf(fact_863_Fpow__mono,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( finite_Fpow_a @ A2 ) @ ( finite_Fpow_a @ B2 ) ) ) ).
% Fpow_mono
thf(fact_864_shuffles_Osimps_I2_J,axiom,
! [Xs: list_a] :
( ( shuffles_a @ Xs @ nil_a )
= ( insert_list_a @ Xs @ bot_bot_set_list_a ) ) ).
% shuffles.simps(2)
thf(fact_865_shuffles_Osimps_I2_J,axiom,
! [Xs: list_set_a] :
( ( shuffles_set_a @ Xs @ nil_set_a )
= ( insert_list_set_a @ Xs @ bot_bo4397488018069675312_set_a ) ) ).
% shuffles.simps(2)
thf(fact_866_shuffles_Osimps_I1_J,axiom,
! [Ys: list_a] :
( ( shuffles_a @ nil_a @ Ys )
= ( insert_list_a @ Ys @ bot_bot_set_list_a ) ) ).
% shuffles.simps(1)
thf(fact_867_shuffles_Osimps_I1_J,axiom,
! [Ys: list_set_a] :
( ( shuffles_set_a @ nil_set_a @ Ys )
= ( insert_list_set_a @ Ys @ bot_bo4397488018069675312_set_a ) ) ).
% shuffles.simps(1)
thf(fact_868_Fpow__subset__Pow,axiom,
! [A2: set_a] : ( ord_le3724670747650509150_set_a @ ( finite_Fpow_a @ A2 ) @ ( pow_a @ A2 ) ) ).
% Fpow_subset_Pow
thf(fact_869_Fpow__subset__Pow,axiom,
! [A2: set_set_a] : ( ord_le5722252365846178494_set_a @ ( finite_Fpow_set_a @ A2 ) @ ( pow_set_a @ A2 ) ) ).
% Fpow_subset_Pow
thf(fact_870_shuffles_Oelims,axiom,
! [X2: list_a,Xa2: list_a,Y: set_list_a] :
( ( ( shuffles_a @ X2 @ Xa2 )
= Y )
=> ( ( ( X2 = nil_a )
=> ( Y
!= ( insert_list_a @ Xa2 @ bot_bot_set_list_a ) ) )
=> ( ( ( Xa2 = nil_a )
=> ( Y
!= ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) )
=> ~ ! [X: a,Xs3: list_a] :
( ( X2
= ( cons_a @ X @ Xs3 ) )
=> ! [Y2: a,Ys2: list_a] :
( ( Xa2
= ( cons_a @ Y2 @ Ys2 ) )
=> ( Y
!= ( sup_sup_set_list_a @ ( image_list_a_list_a @ ( cons_a @ X ) @ ( shuffles_a @ Xs3 @ ( cons_a @ Y2 @ Ys2 ) ) ) @ ( image_list_a_list_a @ ( cons_a @ Y2 ) @ ( shuffles_a @ ( cons_a @ X @ Xs3 ) @ Ys2 ) ) ) ) ) ) ) ) ) ).
% shuffles.elims
thf(fact_871_shuffles_Oelims,axiom,
! [X2: list_set_a,Xa2: list_set_a,Y: set_list_set_a] :
( ( ( shuffles_set_a @ X2 @ Xa2 )
= Y )
=> ( ( ( X2 = nil_set_a )
=> ( Y
!= ( insert_list_set_a @ Xa2 @ bot_bo4397488018069675312_set_a ) ) )
=> ( ( ( Xa2 = nil_set_a )
=> ( Y
!= ( insert_list_set_a @ X2 @ bot_bo4397488018069675312_set_a ) ) )
=> ~ ! [X: set_a,Xs3: list_set_a] :
( ( X2
= ( cons_set_a @ X @ Xs3 ) )
=> ! [Y2: set_a,Ys2: list_set_a] :
( ( Xa2
= ( cons_set_a @ Y2 @ Ys2 ) )
=> ( Y
!= ( sup_su5748565005391983768_set_a @ ( image_6999107473886120325_set_a @ ( cons_set_a @ X ) @ ( shuffles_set_a @ Xs3 @ ( cons_set_a @ Y2 @ Ys2 ) ) ) @ ( image_6999107473886120325_set_a @ ( cons_set_a @ Y2 ) @ ( shuffles_set_a @ ( cons_set_a @ X @ Xs3 ) @ Ys2 ) ) ) ) ) ) ) ) ) ).
% shuffles.elims
thf(fact_872_Un__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a2 @ C @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( ( member_a2 @ C @ A2 )
| ( member_a2 @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_873_Un__iff,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a2 @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) )
= ( ( member_set_a2 @ C @ A2 )
| ( member_set_a2 @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_874_UnCI,axiom,
! [C: a,B2: set_a,A2: set_a] :
( ( ~ ( member_a2 @ C @ B2 )
=> ( member_a2 @ C @ A2 ) )
=> ( member_a2 @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_875_UnCI,axiom,
! [C: set_a,B2: set_set_a,A2: set_set_a] :
( ( ~ ( member_set_a2 @ C @ B2 )
=> ( member_set_a2 @ C @ A2 ) )
=> ( member_set_a2 @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_876_Un__empty,axiom,
! [A2: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_877_Un__subset__iff,axiom,
! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ C2 )
= ( ( ord_le3724670747650509150_set_a @ A2 @ C2 )
& ( ord_le3724670747650509150_set_a @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_878_Un__subset__iff,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,C2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( sup_su2076012971530813682_set_a @ A2 @ B2 ) @ C2 )
= ( ( ord_le5722252365846178494_set_a @ A2 @ C2 )
& ( ord_le5722252365846178494_set_a @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_879_Un__subset__iff,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C2 )
= ( ( ord_less_eq_set_a @ A2 @ C2 )
& ( ord_less_eq_set_a @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_880_Un__insert__right,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( sup_sup_set_a @ A2 @ ( insert_a2 @ A @ B2 ) )
= ( insert_a2 @ A @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_881_Un__insert__left,axiom,
! [A: a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( insert_a2 @ A @ B2 ) @ C2 )
= ( insert_a2 @ A @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_882_set__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( set_a2 @ ( append_a @ Xs @ Ys ) )
= ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) ) ).
% set_append
thf(fact_883_set__append,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( set_set_a2 @ ( append_set_a @ Xs @ Ys ) )
= ( sup_sup_set_set_a @ ( set_set_a2 @ Xs ) @ ( set_set_a2 @ Ys ) ) ) ).
% set_append
thf(fact_884_set__append,axiom,
! [Xs: list_set_set_a,Ys: list_set_set_a] :
( ( set_set_set_a2 @ ( append_set_set_a @ Xs @ Ys ) )
= ( sup_su2076012971530813682_set_a @ ( set_set_set_a2 @ Xs ) @ ( set_set_set_a2 @ Ys ) ) ) ).
% set_append
thf(fact_885_Un__empty__left,axiom,
! [B2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_886_Un__empty__right,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Un_empty_right
thf(fact_887_Un__mono,axiom,
! [A2: set_set_a,C2: set_set_a,B2: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ D )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ ( sup_sup_set_set_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_888_Un__mono,axiom,
! [A2: set_set_set_a,C2: set_set_set_a,B2: set_set_set_a,D: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ C2 )
=> ( ( ord_le5722252365846178494_set_a @ B2 @ D )
=> ( ord_le5722252365846178494_set_a @ ( sup_su2076012971530813682_set_a @ A2 @ B2 ) @ ( sup_su2076012971530813682_set_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_889_Un__mono,axiom,
! [A2: set_a,C2: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_890_Un__least,axiom,
! [A2: set_set_a,C2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_891_Un__least,axiom,
! [A2: set_set_set_a,C2: set_set_set_a,B2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ C2 )
=> ( ( ord_le5722252365846178494_set_a @ B2 @ C2 )
=> ( ord_le5722252365846178494_set_a @ ( sup_su2076012971530813682_set_a @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_892_Un__least,axiom,
! [A2: set_a,C2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_893_Un__upper1,axiom,
! [A2: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_894_Un__upper1,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] : ( ord_le5722252365846178494_set_a @ A2 @ ( sup_su2076012971530813682_set_a @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_895_Un__upper1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_896_Un__upper2,axiom,
! [B2: set_set_a,A2: set_set_a] : ( ord_le3724670747650509150_set_a @ B2 @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_897_Un__upper2,axiom,
! [B2: set_set_set_a,A2: set_set_set_a] : ( ord_le5722252365846178494_set_a @ B2 @ ( sup_su2076012971530813682_set_a @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_898_Un__upper2,axiom,
! [B2: set_a,A2: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_899_Un__absorb1,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_set_a @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_900_Un__absorb1,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( ( sup_su2076012971530813682_set_a @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_901_Un__absorb1,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_902_Un__absorb2,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_set_a @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_903_Un__absorb2,axiom,
! [B2: set_set_set_a,A2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B2 @ A2 )
=> ( ( sup_su2076012971530813682_set_a @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_904_Un__absorb2,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_905_subset__UnE,axiom,
! [C2: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ ( sup_sup_set_set_a @ A2 @ B2 ) )
=> ~ ! [A6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A6 @ A2 )
=> ! [B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B6 @ B2 )
=> ( C2
!= ( sup_sup_set_set_a @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_906_subset__UnE,axiom,
! [C2: set_set_set_a,A2: set_set_set_a,B2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ C2 @ ( sup_su2076012971530813682_set_a @ A2 @ B2 ) )
=> ~ ! [A6: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A6 @ A2 )
=> ! [B6: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B6 @ B2 )
=> ( C2
!= ( sup_su2076012971530813682_set_a @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_907_subset__UnE,axiom,
! [C2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A2 @ B2 ) )
=> ~ ! [A6: set_a] :
( ( ord_less_eq_set_a @ A6 @ A2 )
=> ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ B2 )
=> ( C2
!= ( sup_sup_set_a @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_908_subset__Un__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( sup_sup_set_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_909_subset__Un__eq,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [A4: set_set_set_a,B4: set_set_set_a] :
( ( sup_su2076012971530813682_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_910_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( sup_sup_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_911_UnI2,axiom,
! [C: a,B2: set_a,A2: set_a] :
( ( member_a2 @ C @ B2 )
=> ( member_a2 @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_912_UnI2,axiom,
! [C: set_a,B2: set_set_a,A2: set_set_a] :
( ( member_set_a2 @ C @ B2 )
=> ( member_set_a2 @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_913_UnI1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a2 @ C @ A2 )
=> ( member_a2 @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_914_UnI1,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a2 @ C @ A2 )
=> ( member_set_a2 @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_915_UnE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a2 @ C @ ( sup_sup_set_a @ A2 @ B2 ) )
=> ( ~ ( member_a2 @ C @ A2 )
=> ( member_a2 @ C @ B2 ) ) ) ).
% UnE
thf(fact_916_UnE,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a2 @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) )
=> ( ~ ( member_set_a2 @ C @ A2 )
=> ( member_set_a2 @ C @ B2 ) ) ) ).
% UnE
thf(fact_917_Un__Pow__subset,axiom,
! [A2: set_a,B2: set_a] : ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ ( pow_a @ A2 ) @ ( pow_a @ B2 ) ) @ ( pow_a @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% Un_Pow_subset
thf(fact_918_Un__Pow__subset,axiom,
! [A2: set_set_a,B2: set_set_a] : ( ord_le5722252365846178494_set_a @ ( sup_su2076012971530813682_set_a @ ( pow_set_a @ A2 ) @ ( pow_set_a @ B2 ) ) @ ( pow_set_a @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% Un_Pow_subset
thf(fact_919_insert__is__Un,axiom,
( insert_a2
= ( ^ [A3: a] : ( sup_sup_set_a @ ( insert_a2 @ A3 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_920_Un__singleton__iff,axiom,
! [A2: set_a,B2: set_a,X2: a] :
( ( ( sup_sup_set_a @ A2 @ B2 )
= ( insert_a2 @ X2 @ bot_bot_set_a ) )
= ( ( ( A2 = bot_bot_set_a )
& ( B2
= ( insert_a2 @ X2 @ bot_bot_set_a ) ) )
| ( ( A2
= ( insert_a2 @ X2 @ bot_bot_set_a ) )
& ( B2 = bot_bot_set_a ) )
| ( ( A2
= ( insert_a2 @ X2 @ bot_bot_set_a ) )
& ( B2
= ( insert_a2 @ X2 @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_921_singleton__Un__iff,axiom,
! [X2: a,A2: set_a,B2: set_a] :
( ( ( insert_a2 @ X2 @ bot_bot_set_a )
= ( sup_sup_set_a @ A2 @ B2 ) )
= ( ( ( A2 = bot_bot_set_a )
& ( B2
= ( insert_a2 @ X2 @ bot_bot_set_a ) ) )
| ( ( A2
= ( insert_a2 @ X2 @ bot_bot_set_a ) )
& ( B2 = bot_bot_set_a ) )
| ( ( A2
= ( insert_a2 @ X2 @ bot_bot_set_a ) )
& ( B2
= ( insert_a2 @ X2 @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_922_shuffles_Osimps_I3_J,axiom,
! [X2: a,Xs: list_a,Y: a,Ys: list_a] :
( ( shuffles_a @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ Y @ Ys ) )
= ( sup_sup_set_list_a @ ( image_list_a_list_a @ ( cons_a @ X2 ) @ ( shuffles_a @ Xs @ ( cons_a @ Y @ Ys ) ) ) @ ( image_list_a_list_a @ ( cons_a @ Y ) @ ( shuffles_a @ ( cons_a @ X2 @ Xs ) @ Ys ) ) ) ) ).
% shuffles.simps(3)
thf(fact_923_shuffles_Osimps_I3_J,axiom,
! [X2: set_a,Xs: list_set_a,Y: set_a,Ys: list_set_a] :
( ( shuffles_set_a @ ( cons_set_a @ X2 @ Xs ) @ ( cons_set_a @ Y @ Ys ) )
= ( sup_su5748565005391983768_set_a @ ( image_6999107473886120325_set_a @ ( cons_set_a @ X2 ) @ ( shuffles_set_a @ Xs @ ( cons_set_a @ Y @ Ys ) ) ) @ ( image_6999107473886120325_set_a @ ( cons_set_a @ Y ) @ ( shuffles_set_a @ ( cons_set_a @ X2 @ Xs ) @ Ys ) ) ) ) ).
% shuffles.simps(3)
thf(fact_924_set__shuffles,axiom,
! [Zs: list_set_a,Xs: list_set_a,Ys: list_set_a] :
( ( member_list_set_a @ Zs @ ( shuffles_set_a @ Xs @ Ys ) )
=> ( ( set_set_a2 @ Zs )
= ( sup_sup_set_set_a @ ( set_set_a2 @ Xs ) @ ( set_set_a2 @ Ys ) ) ) ) ).
% set_shuffles
thf(fact_925_set__shuffles,axiom,
! [Zs: list_set_set_a,Xs: list_set_set_a,Ys: list_set_set_a] :
( ( member6684481465865166061_set_a @ Zs @ ( shuffles_set_set_a @ Xs @ Ys ) )
=> ( ( set_set_set_a2 @ Zs )
= ( sup_su2076012971530813682_set_a @ ( set_set_set_a2 @ Xs ) @ ( set_set_set_a2 @ Ys ) ) ) ) ).
% set_shuffles
thf(fact_926_comp__sgraph_Oincident__edges__empty,axiom,
! [V: a,S: set_a] :
( ~ ( member_a2 @ V @ S )
=> ( ( undire3231912044278729248dges_a @ ( undire2918257014606996450dges_a @ S ) @ V )
= bot_bot_set_set_a ) ) ).
% comp_sgraph.incident_edges_empty
thf(fact_927_comp__sgraph_Oincident__edges__empty,axiom,
! [V: set_a,S: set_set_a] :
( ~ ( member_set_a2 @ V @ S )
=> ( ( undire4631953023069350784_set_a @ ( undire8247866692393712962_set_a @ S ) @ V )
= bot_bo3380559777022489994_set_a ) ) ).
% comp_sgraph.incident_edges_empty
thf(fact_928_sup__bot__left,axiom,
! [X2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_929_sup__bot__right,axiom,
! [X2: set_a] :
( ( sup_sup_set_a @ X2 @ bot_bot_set_a )
= X2 ) ).
% sup_bot_right
thf(fact_930_bot__eq__sup__iff,axiom,
! [X2: set_a,Y: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X2 @ Y ) )
= ( ( X2 = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_931_sup__eq__bot__iff,axiom,
! [X2: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X2 @ Y )
= bot_bot_set_a )
= ( ( X2 = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_932_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A @ B )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_933_le__sup__iff,axiom,
! [X2: set_set_a,Y: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ X2 @ Y ) @ Z2 )
= ( ( ord_le3724670747650509150_set_a @ X2 @ Z2 )
& ( ord_le3724670747650509150_set_a @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_934_le__sup__iff,axiom,
! [X2: set_set_set_a,Y: set_set_set_a,Z2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( sup_su2076012971530813682_set_a @ X2 @ Y ) @ Z2 )
= ( ( ord_le5722252365846178494_set_a @ X2 @ Z2 )
& ( ord_le5722252365846178494_set_a @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_935_le__sup__iff,axiom,
! [X2: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X2 @ Y ) @ Z2 )
= ( ( ord_less_eq_set_a @ X2 @ Z2 )
& ( ord_less_eq_set_a @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_936_sup_Obounded__iff,axiom,
! [B: set_set_a,C: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B @ C ) @ A )
= ( ( ord_le3724670747650509150_set_a @ B @ A )
& ( ord_le3724670747650509150_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_937_sup_Obounded__iff,axiom,
! [B: set_set_set_a,C: set_set_set_a,A: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( sup_su2076012971530813682_set_a @ B @ C ) @ A )
= ( ( ord_le5722252365846178494_set_a @ B @ A )
& ( ord_le5722252365846178494_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_938_sup_Obounded__iff,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
= ( ( ord_less_eq_set_a @ B @ A )
& ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_939_sup__bot_Oright__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_940_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A @ B ) )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_941_sup__bot_Oleft__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_942_Pow__insert,axiom,
! [A: a,A2: set_a] :
( ( pow_a @ ( insert_a2 @ A @ A2 ) )
= ( sup_sup_set_set_a @ ( pow_a @ A2 ) @ ( image_set_a_set_a @ ( insert_a2 @ A ) @ ( pow_a @ A2 ) ) ) ) ).
% Pow_insert
thf(fact_943_inf__sup__ord_I4_J,axiom,
! [Y: set_set_a,X2: set_set_a] : ( ord_le3724670747650509150_set_a @ Y @ ( sup_sup_set_set_a @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_944_inf__sup__ord_I4_J,axiom,
! [Y: set_set_set_a,X2: set_set_set_a] : ( ord_le5722252365846178494_set_a @ Y @ ( sup_su2076012971530813682_set_a @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_945_inf__sup__ord_I4_J,axiom,
! [Y: set_a,X2: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_946_inf__sup__ord_I3_J,axiom,
! [X2: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ X2 @ ( sup_sup_set_set_a @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_947_inf__sup__ord_I3_J,axiom,
! [X2: set_set_set_a,Y: set_set_set_a] : ( ord_le5722252365846178494_set_a @ X2 @ ( sup_su2076012971530813682_set_a @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_948_inf__sup__ord_I3_J,axiom,
! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_949_le__supE,axiom,
! [A: set_set_a,B: set_set_a,X2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A @ B ) @ X2 )
=> ~ ( ( ord_le3724670747650509150_set_a @ A @ X2 )
=> ~ ( ord_le3724670747650509150_set_a @ B @ X2 ) ) ) ).
% le_supE
thf(fact_950_le__supE,axiom,
! [A: set_set_set_a,B: set_set_set_a,X2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( sup_su2076012971530813682_set_a @ A @ B ) @ X2 )
=> ~ ( ( ord_le5722252365846178494_set_a @ A @ X2 )
=> ~ ( ord_le5722252365846178494_set_a @ B @ X2 ) ) ) ).
% le_supE
thf(fact_951_le__supE,axiom,
! [A: set_a,B: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ X2 )
=> ~ ( ( ord_less_eq_set_a @ A @ X2 )
=> ~ ( ord_less_eq_set_a @ B @ X2 ) ) ) ).
% le_supE
thf(fact_952_le__supI,axiom,
! [A: set_set_a,X2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ X2 )
=> ( ( ord_le3724670747650509150_set_a @ B @ X2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_953_le__supI,axiom,
! [A: set_set_set_a,X2: set_set_set_a,B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ X2 )
=> ( ( ord_le5722252365846178494_set_a @ B @ X2 )
=> ( ord_le5722252365846178494_set_a @ ( sup_su2076012971530813682_set_a @ A @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_954_le__supI,axiom,
! [A: set_a,X2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X2 )
=> ( ( ord_less_eq_set_a @ B @ X2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_955_sup__ge1,axiom,
! [X2: set_set_a,Y: set_set_a] : ( ord_le3724670747650509150_set_a @ X2 @ ( sup_sup_set_set_a @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_956_sup__ge1,axiom,
! [X2: set_set_set_a,Y: set_set_set_a] : ( ord_le5722252365846178494_set_a @ X2 @ ( sup_su2076012971530813682_set_a @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_957_sup__ge1,axiom,
! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_958_sup__ge2,axiom,
! [Y: set_set_a,X2: set_set_a] : ( ord_le3724670747650509150_set_a @ Y @ ( sup_sup_set_set_a @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_959_sup__ge2,axiom,
! [Y: set_set_set_a,X2: set_set_set_a] : ( ord_le5722252365846178494_set_a @ Y @ ( sup_su2076012971530813682_set_a @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_960_sup__ge2,axiom,
! [Y: set_a,X2: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_961_le__supI1,axiom,
! [X2: set_set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ A )
=> ( ord_le3724670747650509150_set_a @ X2 @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% le_supI1
thf(fact_962_le__supI1,axiom,
! [X2: set_set_set_a,A: set_set_set_a,B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X2 @ A )
=> ( ord_le5722252365846178494_set_a @ X2 @ ( sup_su2076012971530813682_set_a @ A @ B ) ) ) ).
% le_supI1
thf(fact_963_le__supI1,axiom,
! [X2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% le_supI1
thf(fact_964_le__supI2,axiom,
! [X2: set_set_a,B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ B )
=> ( ord_le3724670747650509150_set_a @ X2 @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% le_supI2
thf(fact_965_le__supI2,axiom,
! [X2: set_set_set_a,B: set_set_set_a,A: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X2 @ B )
=> ( ord_le5722252365846178494_set_a @ X2 @ ( sup_su2076012971530813682_set_a @ A @ B ) ) ) ).
% le_supI2
thf(fact_966_le__supI2,axiom,
! [X2: set_a,B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ X2 @ B )
=> ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% le_supI2
thf(fact_967_sup_Omono,axiom,
! [C: set_set_a,A: set_set_a,D2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ A )
=> ( ( ord_le3724670747650509150_set_a @ D2 @ B )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ C @ D2 ) @ ( sup_sup_set_set_a @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_968_sup_Omono,axiom,
! [C: set_set_set_a,A: set_set_set_a,D2: set_set_set_a,B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ C @ A )
=> ( ( ord_le5722252365846178494_set_a @ D2 @ B )
=> ( ord_le5722252365846178494_set_a @ ( sup_su2076012971530813682_set_a @ C @ D2 ) @ ( sup_su2076012971530813682_set_a @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_969_sup_Omono,axiom,
! [C: set_a,A: set_a,D2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ( ord_less_eq_set_a @ D2 @ B )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D2 ) @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_970_sup__mono,axiom,
! [A: set_set_a,C: set_set_a,B: set_set_a,D2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C )
=> ( ( ord_le3724670747650509150_set_a @ B @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ A @ B ) @ ( sup_sup_set_set_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_971_sup__mono,axiom,
! [A: set_set_set_a,C: set_set_set_a,B: set_set_set_a,D2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ C )
=> ( ( ord_le5722252365846178494_set_a @ B @ D2 )
=> ( ord_le5722252365846178494_set_a @ ( sup_su2076012971530813682_set_a @ A @ B ) @ ( sup_su2076012971530813682_set_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_972_sup__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_973_sup__least,axiom,
! [Y: set_set_a,X2: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X2 )
=> ( ( ord_le3724670747650509150_set_a @ Z2 @ X2 )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ Y @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_974_sup__least,axiom,
! [Y: set_set_set_a,X2: set_set_set_a,Z2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ Y @ X2 )
=> ( ( ord_le5722252365846178494_set_a @ Z2 @ X2 )
=> ( ord_le5722252365846178494_set_a @ ( sup_su2076012971530813682_set_a @ Y @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_975_sup__least,axiom,
! [Y: set_a,X2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ( ( ord_less_eq_set_a @ Z2 @ X2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_976_le__iff__sup,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [X3: set_set_a,Y4: set_set_a] :
( ( sup_sup_set_set_a @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_977_le__iff__sup,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [X3: set_set_set_a,Y4: set_set_set_a] :
( ( sup_su2076012971530813682_set_a @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_978_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( sup_sup_set_a @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_979_sup_OorderE,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( A
= ( sup_sup_set_set_a @ A @ B ) ) ) ).
% sup.orderE
thf(fact_980_sup_OorderE,axiom,
! [B: set_set_set_a,A: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B @ A )
=> ( A
= ( sup_su2076012971530813682_set_a @ A @ B ) ) ) ).
% sup.orderE
thf(fact_981_sup_OorderE,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( A
= ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.orderE
thf(fact_982_sup_OorderI,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A
= ( sup_sup_set_set_a @ A @ B ) )
=> ( ord_le3724670747650509150_set_a @ B @ A ) ) ).
% sup.orderI
thf(fact_983_sup_OorderI,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ( A
= ( sup_su2076012971530813682_set_a @ A @ B ) )
=> ( ord_le5722252365846178494_set_a @ B @ A ) ) ).
% sup.orderI
thf(fact_984_sup_OorderI,axiom,
! [A: set_a,B: set_a] :
( ( A
= ( sup_sup_set_a @ A @ B ) )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% sup.orderI
thf(fact_985_sup__unique,axiom,
! [F: set_set_a > set_set_a > set_set_a,X2: set_set_a,Y: set_set_a] :
( ! [X: set_set_a,Y2: set_set_a] : ( ord_le3724670747650509150_set_a @ X @ ( F @ X @ Y2 ) )
=> ( ! [X: set_set_a,Y2: set_set_a] : ( ord_le3724670747650509150_set_a @ Y2 @ ( F @ X @ Y2 ) )
=> ( ! [X: set_set_a,Y2: set_set_a,Z3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y2 @ X )
=> ( ( ord_le3724670747650509150_set_a @ Z3 @ X )
=> ( ord_le3724670747650509150_set_a @ ( F @ Y2 @ Z3 ) @ X ) ) )
=> ( ( sup_sup_set_set_a @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_986_sup__unique,axiom,
! [F: set_set_set_a > set_set_set_a > set_set_set_a,X2: set_set_set_a,Y: set_set_set_a] :
( ! [X: set_set_set_a,Y2: set_set_set_a] : ( ord_le5722252365846178494_set_a @ X @ ( F @ X @ Y2 ) )
=> ( ! [X: set_set_set_a,Y2: set_set_set_a] : ( ord_le5722252365846178494_set_a @ Y2 @ ( F @ X @ Y2 ) )
=> ( ! [X: set_set_set_a,Y2: set_set_set_a,Z3: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ Y2 @ X )
=> ( ( ord_le5722252365846178494_set_a @ Z3 @ X )
=> ( ord_le5722252365846178494_set_a @ ( F @ Y2 @ Z3 ) @ X ) ) )
=> ( ( sup_su2076012971530813682_set_a @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_987_sup__unique,axiom,
! [F: set_a > set_a > set_a,X2: set_a,Y: set_a] :
( ! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X @ ( F @ X @ Y2 ) )
=> ( ! [X: set_a,Y2: set_a] : ( ord_less_eq_set_a @ Y2 @ ( F @ X @ Y2 ) )
=> ( ! [X: set_a,Y2: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X )
=> ( ( ord_less_eq_set_a @ Z3 @ X )
=> ( ord_less_eq_set_a @ ( F @ Y2 @ Z3 ) @ X ) ) )
=> ( ( sup_sup_set_a @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_988_sup_Oabsorb1,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( sup_sup_set_set_a @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_989_sup_Oabsorb1,axiom,
! [B: set_set_set_a,A: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B @ A )
=> ( ( sup_su2076012971530813682_set_a @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_990_sup_Oabsorb1,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_991_sup_Oabsorb2,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( sup_sup_set_set_a @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_992_sup_Oabsorb2,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( sup_su2076012971530813682_set_a @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_993_sup_Oabsorb2,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_994_sup__absorb1,axiom,
! [Y: set_set_a,X2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X2 )
=> ( ( sup_sup_set_set_a @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_995_sup__absorb1,axiom,
! [Y: set_set_set_a,X2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ Y @ X2 )
=> ( ( sup_su2076012971530813682_set_a @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_996_sup__absorb1,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ( ( sup_sup_set_a @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_997_sup__absorb2,axiom,
! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ( sup_sup_set_set_a @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_998_sup__absorb2,axiom,
! [X2: set_set_set_a,Y: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X2 @ Y )
=> ( ( sup_su2076012971530813682_set_a @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_999_sup__absorb2,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( sup_sup_set_a @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1000_sup_OboundedE,axiom,
! [B: set_set_a,C: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B @ C ) @ A )
=> ~ ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ~ ( ord_le3724670747650509150_set_a @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_1001_sup_OboundedE,axiom,
! [B: set_set_set_a,C: set_set_set_a,A: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( sup_su2076012971530813682_set_a @ B @ C ) @ A )
=> ~ ( ( ord_le5722252365846178494_set_a @ B @ A )
=> ~ ( ord_le5722252365846178494_set_a @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_1002_sup_OboundedE,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_set_a @ B @ A )
=> ~ ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_1003_sup_OboundedI,axiom,
! [B: set_set_a,A: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( ord_le3724670747650509150_set_a @ C @ A )
=> ( ord_le3724670747650509150_set_a @ ( sup_sup_set_set_a @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_1004_sup_OboundedI,axiom,
! [B: set_set_set_a,A: set_set_set_a,C: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ B @ A )
=> ( ( ord_le5722252365846178494_set_a @ C @ A )
=> ( ord_le5722252365846178494_set_a @ ( sup_su2076012971530813682_set_a @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_1005_sup_OboundedI,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_1006_sup_Oorder__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B3: set_set_a,A3: set_set_a] :
( A3
= ( sup_sup_set_set_a @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_1007_sup_Oorder__iff,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [B3: set_set_set_a,A3: set_set_set_a] :
( A3
= ( sup_su2076012971530813682_set_a @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_1008_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A3: set_a] :
( A3
= ( sup_sup_set_a @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_1009_sup_Ocobounded1,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ ( sup_sup_set_set_a @ A @ B ) ) ).
% sup.cobounded1
thf(fact_1010_sup_Ocobounded1,axiom,
! [A: set_set_set_a,B: set_set_set_a] : ( ord_le5722252365846178494_set_a @ A @ ( sup_su2076012971530813682_set_a @ A @ B ) ) ).
% sup.cobounded1
thf(fact_1011_sup_Ocobounded1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).
% sup.cobounded1
thf(fact_1012_sup_Ocobounded2,axiom,
! [B: set_set_a,A: set_set_a] : ( ord_le3724670747650509150_set_a @ B @ ( sup_sup_set_set_a @ A @ B ) ) ).
% sup.cobounded2
thf(fact_1013_sup_Ocobounded2,axiom,
! [B: set_set_set_a,A: set_set_set_a] : ( ord_le5722252365846178494_set_a @ B @ ( sup_su2076012971530813682_set_a @ A @ B ) ) ).
% sup.cobounded2
thf(fact_1014_sup_Ocobounded2,axiom,
! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).
% sup.cobounded2
thf(fact_1015_sup_Oabsorb__iff1,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [B3: set_set_a,A3: set_set_a] :
( ( sup_sup_set_set_a @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_1016_sup_Oabsorb__iff1,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [B3: set_set_set_a,A3: set_set_set_a] :
( ( sup_su2076012971530813682_set_a @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_1017_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( sup_sup_set_a @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_1018_sup_Oabsorb__iff2,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( sup_sup_set_set_a @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_1019_sup_Oabsorb__iff2,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [A3: set_set_set_a,B3: set_set_set_a] :
( ( sup_su2076012971530813682_set_a @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_1020_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( sup_sup_set_a @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_1021_sup_OcoboundedI1,axiom,
! [C: set_set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ A )
=> ( ord_le3724670747650509150_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_1022_sup_OcoboundedI1,axiom,
! [C: set_set_set_a,A: set_set_set_a,B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ C @ A )
=> ( ord_le5722252365846178494_set_a @ C @ ( sup_su2076012971530813682_set_a @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_1023_sup_OcoboundedI1,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_1024_sup_OcoboundedI2,axiom,
! [C: set_set_a,B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C @ B )
=> ( ord_le3724670747650509150_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_1025_sup_OcoboundedI2,axiom,
! [C: set_set_set_a,B: set_set_set_a,A: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ C @ B )
=> ( ord_le5722252365846178494_set_a @ C @ ( sup_su2076012971530813682_set_a @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_1026_sup_OcoboundedI2,axiom,
! [C: set_a,B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_1027_sup__bot_Omonoid__axioms,axiom,
monoid_set_a @ sup_sup_set_a @ bot_bot_set_a ).
% sup_bot.monoid_axioms
thf(fact_1028_set__union,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( set_set_a2 @ ( union_set_a @ Xs @ Ys ) )
= ( sup_sup_set_set_a @ ( set_set_a2 @ Xs ) @ ( set_set_a2 @ Ys ) ) ) ).
% set_union
thf(fact_1029_set__union,axiom,
! [Xs: list_set_set_a,Ys: list_set_set_a] :
( ( set_set_set_a2 @ ( union_set_set_a @ Xs @ Ys ) )
= ( sup_su2076012971530813682_set_a @ ( set_set_set_a2 @ Xs ) @ ( set_set_set_a2 @ Ys ) ) ) ).
% set_union
thf(fact_1030_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_a] :
( ( sup_sup_set_a @ X2 @ bot_bot_set_a )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_1031_ulgraph_Oneighborhood__incident,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,U: set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( member_set_a2 @ U @ ( undire2074812191327625774_set_a @ Vertices @ Edges @ V ) )
= ( member_set_set_a2 @ ( insert_set_a2 @ U @ ( insert_set_a2 @ V @ bot_bot_set_set_a ) ) @ ( undire4631953023069350784_set_a @ Edges @ V ) ) ) ) ).
% ulgraph.neighborhood_incident
thf(fact_1032_ulgraph_Oneighborhood__incident,axiom,
! [Vertices: set_a,Edges: set_set_a,U: a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( member_a2 @ U @ ( undire8504279938402040014hood_a @ Vertices @ Edges @ V ) )
= ( member_set_a2 @ ( insert_a2 @ U @ ( insert_a2 @ V @ bot_bot_set_a ) ) @ ( undire3231912044278729248dges_a @ Edges @ V ) ) ) ) ).
% ulgraph.neighborhood_incident
thf(fact_1033_comp__sgraph_Oneighborhood__incident,axiom,
! [U: set_a,S: set_set_a,V: set_a] :
( ( member_set_a2 @ U @ ( undire2074812191327625774_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ V ) )
= ( member_set_set_a2 @ ( insert_set_a2 @ U @ ( insert_set_a2 @ V @ bot_bot_set_set_a ) ) @ ( undire4631953023069350784_set_a @ ( undire8247866692393712962_set_a @ S ) @ V ) ) ) ).
% comp_sgraph.neighborhood_incident
thf(fact_1034_comp__sgraph_Oneighborhood__incident,axiom,
! [U: a,S: set_a,V: a] :
( ( member_a2 @ U @ ( undire8504279938402040014hood_a @ S @ ( undire2918257014606996450dges_a @ S ) @ V ) )
= ( member_set_a2 @ ( insert_a2 @ U @ ( insert_a2 @ V @ bot_bot_set_a ) ) @ ( undire3231912044278729248dges_a @ ( undire2918257014606996450dges_a @ S ) @ V ) ) ) ).
% comp_sgraph.neighborhood_incident
thf(fact_1035_ulgraph_Oiso__vertex__empty__neighborhood,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V )
=> ( ( undire8504279938402040014hood_a @ Vertices @ Edges @ V )
= bot_bot_set_a ) ) ) ).
% ulgraph.iso_vertex_empty_neighborhood
thf(fact_1036_comp__sgraph_Oiso__vertex__empty__neighborhood,axiom,
! [S: set_a,V: a] :
( ( undire8931668460104145173rtex_a @ S @ ( undire2918257014606996450dges_a @ S ) @ V )
=> ( ( undire8504279938402040014hood_a @ S @ ( undire2918257014606996450dges_a @ S ) @ V )
= bot_bot_set_a ) ) ).
% comp_sgraph.iso_vertex_empty_neighborhood
thf(fact_1037_chains__extend,axiom,
! [C: set_set_set_a,S: set_set_set_a,Z2: set_set_a] :
( ( member_set_set_set_a @ C @ ( chains_set_a @ S ) )
=> ( ( member_set_set_a2 @ Z2 @ S )
=> ( ! [X: set_set_a] :
( ( member_set_set_a2 @ X @ C )
=> ( ord_le3724670747650509150_set_a @ X @ Z2 ) )
=> ( member_set_set_set_a @ ( sup_su2076012971530813682_set_a @ ( insert_set_set_a2 @ Z2 @ bot_bo3380559777022489994_set_a ) @ C ) @ ( chains_set_a @ S ) ) ) ) ) ).
% chains_extend
thf(fact_1038_chains__extend,axiom,
! [C: set_set_set_set_a,S: set_set_set_set_a,Z2: set_set_set_a] :
( ( member7634106644413650855_set_a @ C @ ( chains_set_set_a @ S ) )
=> ( ( member_set_set_set_a @ Z2 @ S )
=> ( ! [X: set_set_set_a] :
( ( member_set_set_set_a @ X @ C )
=> ( ord_le5722252365846178494_set_a @ X @ Z2 ) )
=> ( member7634106644413650855_set_a @ ( sup_su6872963709084814930_set_a @ ( insert_set_set_set_a @ Z2 @ bot_bo4178452617224790762_set_a ) @ C ) @ ( chains_set_set_a @ S ) ) ) ) ) ).
% chains_extend
thf(fact_1039_chains__extend,axiom,
! [C: set_set_a,S: set_set_a,Z2: set_a] :
( ( member_set_set_a2 @ C @ ( chains_a @ S ) )
=> ( ( member_set_a2 @ Z2 @ S )
=> ( ! [X: set_a] :
( ( member_set_a2 @ X @ C )
=> ( ord_less_eq_set_a @ X @ Z2 ) )
=> ( member_set_set_a2 @ ( sup_sup_set_set_a @ ( insert_set_a2 @ Z2 @ bot_bot_set_set_a ) @ C ) @ ( chains_a @ S ) ) ) ) ) ).
% chains_extend
thf(fact_1040_chainsD2,axiom,
! [C: set_set_a,S: set_set_a] :
( ( member_set_set_a2 @ C @ ( chains_a @ S ) )
=> ( ord_le3724670747650509150_set_a @ C @ S ) ) ).
% chainsD2
thf(fact_1041_chainsD2,axiom,
! [C: set_set_set_a,S: set_set_set_a] :
( ( member_set_set_set_a @ C @ ( chains_set_a @ S ) )
=> ( ord_le5722252365846178494_set_a @ C @ S ) ) ).
% chainsD2
thf(fact_1042_Zorn__Lemma2,axiom,
! [A2: set_set_set_a] :
( ! [X: set_set_set_a] :
( ( member_set_set_set_a @ X @ ( chains_set_a @ A2 ) )
=> ? [Xa: set_set_a] :
( ( member_set_set_a2 @ Xa @ A2 )
& ! [Xb: set_set_a] :
( ( member_set_set_a2 @ Xb @ X )
=> ( ord_le3724670747650509150_set_a @ Xb @ Xa ) ) ) )
=> ? [X: set_set_a] :
( ( member_set_set_a2 @ X @ A2 )
& ! [Xa: set_set_a] :
( ( member_set_set_a2 @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ X @ Xa )
=> ( Xa = X ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_1043_Zorn__Lemma2,axiom,
! [A2: set_set_set_set_a] :
( ! [X: set_set_set_set_a] :
( ( member7634106644413650855_set_a @ X @ ( chains_set_set_a @ A2 ) )
=> ? [Xa: set_set_set_a] :
( ( member_set_set_set_a @ Xa @ A2 )
& ! [Xb: set_set_set_a] :
( ( member_set_set_set_a @ Xb @ X )
=> ( ord_le5722252365846178494_set_a @ Xb @ Xa ) ) ) )
=> ? [X: set_set_set_a] :
( ( member_set_set_set_a @ X @ A2 )
& ! [Xa: set_set_set_a] :
( ( member_set_set_set_a @ Xa @ A2 )
=> ( ( ord_le5722252365846178494_set_a @ X @ Xa )
=> ( Xa = X ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_1044_Zorn__Lemma2,axiom,
! [A2: set_set_a] :
( ! [X: set_set_a] :
( ( member_set_set_a2 @ X @ ( chains_a @ A2 ) )
=> ? [Xa: set_a] :
( ( member_set_a2 @ Xa @ A2 )
& ! [Xb: set_a] :
( ( member_set_a2 @ Xb @ X )
=> ( ord_less_eq_set_a @ Xb @ Xa ) ) ) )
=> ? [X: set_a] :
( ( member_set_a2 @ X @ A2 )
& ! [Xa: set_a] :
( ( member_set_a2 @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X @ Xa )
=> ( Xa = X ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_1045_chainsD,axiom,
! [C: set_set_set_a,S: set_set_set_a,X2: set_set_a,Y: set_set_a] :
( ( member_set_set_set_a @ C @ ( chains_set_a @ S ) )
=> ( ( member_set_set_a2 @ X2 @ C )
=> ( ( member_set_set_a2 @ Y @ C )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Y )
| ( ord_le3724670747650509150_set_a @ Y @ X2 ) ) ) ) ) ).
% chainsD
thf(fact_1046_chainsD,axiom,
! [C: set_set_set_set_a,S: set_set_set_set_a,X2: set_set_set_a,Y: set_set_set_a] :
( ( member7634106644413650855_set_a @ C @ ( chains_set_set_a @ S ) )
=> ( ( member_set_set_set_a @ X2 @ C )
=> ( ( member_set_set_set_a @ Y @ C )
=> ( ( ord_le5722252365846178494_set_a @ X2 @ Y )
| ( ord_le5722252365846178494_set_a @ Y @ X2 ) ) ) ) ) ).
% chainsD
thf(fact_1047_chainsD,axiom,
! [C: set_set_a,S: set_set_a,X2: set_a,Y: set_a] :
( ( member_set_set_a2 @ C @ ( chains_a @ S ) )
=> ( ( member_set_a2 @ X2 @ C )
=> ( ( member_set_a2 @ Y @ C )
=> ( ( ord_less_eq_set_a @ X2 @ Y )
| ( ord_less_eq_set_a @ Y @ X2 ) ) ) ) ) ).
% chainsD
thf(fact_1048_transpose__empty,axiom,
! [Xs: list_list_a] :
( ( ( transpose_a @ Xs )
= nil_list_a )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( X3 = nil_a ) ) ) ) ).
% transpose_empty
thf(fact_1049_transpose__empty,axiom,
! [Xs: list_list_set_a] :
( ( ( transpose_set_a @ Xs )
= nil_list_set_a )
= ( ! [X3: list_set_a] :
( ( member_list_set_a @ X3 @ ( set_list_set_a2 @ Xs ) )
=> ( X3 = nil_set_a ) ) ) ) ).
% transpose_empty
thf(fact_1050_ulgraph_Overt__adj__def,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V22: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V22 )
= ( member_set_a2 @ ( insert_a2 @ V1 @ ( insert_a2 @ V22 @ bot_bot_set_a ) ) @ Edges ) ) ) ).
% ulgraph.vert_adj_def
thf(fact_1051_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a2 @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_1052_List_Ofinite__set,axiom,
! [Xs: list_set_a] : ( finite_finite_set_a @ ( set_set_a2 @ Xs ) ) ).
% List.finite_set
thf(fact_1053_List_Ofinite__set,axiom,
! [Xs: list_set_set_a] : ( finite7209287970140883943_set_a @ ( set_set_set_a2 @ Xs ) ) ).
% List.finite_set
thf(fact_1054_finite__has__maximal2,axiom,
! [A2: set_set_set_a,A: set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( member_set_set_a2 @ A @ A2 )
=> ? [X: set_set_a] :
( ( member_set_set_a2 @ X @ A2 )
& ( ord_le3724670747650509150_set_a @ A @ X )
& ! [Xa: set_set_a] :
( ( member_set_set_a2 @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1055_finite__has__maximal2,axiom,
! [A2: set_set_set_set_a,A: set_set_set_a] :
( ( finite5318320746233006407_set_a @ A2 )
=> ( ( member_set_set_set_a @ A @ A2 )
=> ? [X: set_set_set_a] :
( ( member_set_set_set_a @ X @ A2 )
& ( ord_le5722252365846178494_set_a @ A @ X )
& ! [Xa: set_set_set_a] :
( ( member_set_set_set_a @ Xa @ A2 )
=> ( ( ord_le5722252365846178494_set_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1056_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a2 @ A @ A2 )
=> ? [X: set_a] :
( ( member_set_a2 @ X @ A2 )
& ( ord_less_eq_set_a @ A @ X )
& ! [Xa: set_a] :
( ( member_set_a2 @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1057_finite__has__minimal2,axiom,
! [A2: set_set_set_a,A: set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( member_set_set_a2 @ A @ A2 )
=> ? [X: set_set_a] :
( ( member_set_set_a2 @ X @ A2 )
& ( ord_le3724670747650509150_set_a @ X @ A )
& ! [Xa: set_set_a] :
( ( member_set_set_a2 @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1058_finite__has__minimal2,axiom,
! [A2: set_set_set_set_a,A: set_set_set_a] :
( ( finite5318320746233006407_set_a @ A2 )
=> ( ( member_set_set_set_a @ A @ A2 )
=> ? [X: set_set_set_a] :
( ( member_set_set_set_a @ X @ A2 )
& ( ord_le5722252365846178494_set_a @ X @ A )
& ! [Xa: set_set_set_a] :
( ( member_set_set_set_a @ Xa @ A2 )
=> ( ( ord_le5722252365846178494_set_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1059_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a2 @ A @ A2 )
=> ? [X: set_a] :
( ( member_set_a2 @ X @ A2 )
& ( ord_less_eq_set_a @ X @ A )
& ! [Xa: set_a] :
( ( member_set_a2 @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1060_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_1061_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_1062_finite__list,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ? [Xs3: list_set_a] :
( ( set_set_a2 @ Xs3 )
= A2 ) ) ).
% finite_list
thf(fact_1063_finite__list,axiom,
! [A2: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ? [Xs3: list_set_set_a] :
( ( set_set_set_a2 @ Xs3 )
= A2 ) ) ).
% finite_list
thf(fact_1064_finite_OinsertI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a2 @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_1065_finite__subset,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( finite_finite_set_a @ B2 )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% finite_subset
thf(fact_1066_finite__subset,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( ( finite7209287970140883943_set_a @ B2 )
=> ( finite7209287970140883943_set_a @ A2 ) ) ) ).
% finite_subset
thf(fact_1067_finite__subset,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_1068_infinite__super,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ S @ T2 )
=> ( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ T2 ) ) ) ).
% infinite_super
thf(fact_1069_infinite__super,axiom,
! [S: set_set_set_a,T2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ S @ T2 )
=> ( ~ ( finite7209287970140883943_set_a @ S )
=> ~ ( finite7209287970140883943_set_a @ T2 ) ) ) ).
% infinite_super
thf(fact_1070_infinite__super,axiom,
! [S: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_1071_rev__finite__subset,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_1072_rev__finite__subset,axiom,
! [B2: set_set_set_a,A2: set_set_set_a] :
( ( finite7209287970140883943_set_a @ B2 )
=> ( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( finite7209287970140883943_set_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_1073_rev__finite__subset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_1074_comp__sgraph_Overt__adj__imp__inV,axiom,
! [S: set_a,V1: a,V22: a] :
( ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ V1 @ V22 )
=> ( ( member_a2 @ V1 @ S )
& ( member_a2 @ V22 @ S ) ) ) ).
% comp_sgraph.vert_adj_imp_inV
thf(fact_1075_comp__sgraph_Overt__adj__imp__inV,axiom,
! [S: set_set_a,V1: set_a,V22: set_a] :
( ( undire3510646817838285160_set_a @ ( undire8247866692393712962_set_a @ S ) @ V1 @ V22 )
=> ( ( member_set_a2 @ V1 @ S )
& ( member_set_a2 @ V22 @ S ) ) ) ).
% comp_sgraph.vert_adj_imp_inV
thf(fact_1076_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V22: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V22 )
=> ( ( member_a2 @ V1 @ Vertices )
& ( member_a2 @ V22 @ Vertices ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_1077_ulgraph_Overt__adj__imp__inV,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V1: set_a,V22: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire3510646817838285160_set_a @ Edges @ V1 @ V22 )
=> ( ( member_set_a2 @ V1 @ Vertices )
& ( member_set_a2 @ V22 @ Vertices ) ) ) ) ).
% ulgraph.vert_adj_imp_inV
thf(fact_1078_finite__has__maximal,axiom,
! [A2: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( A2 != bot_bo3380559777022489994_set_a )
=> ? [X: set_set_a] :
( ( member_set_set_a2 @ X @ A2 )
& ! [Xa: set_set_a] :
( ( member_set_set_a2 @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1079_finite__has__maximal,axiom,
! [A2: set_set_set_set_a] :
( ( finite5318320746233006407_set_a @ A2 )
=> ( ( A2 != bot_bo4178452617224790762_set_a )
=> ? [X: set_set_set_a] :
( ( member_set_set_set_a @ X @ A2 )
& ! [Xa: set_set_set_a] :
( ( member_set_set_set_a @ Xa @ A2 )
=> ( ( ord_le5722252365846178494_set_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1080_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X: set_a] :
( ( member_set_a2 @ X @ A2 )
& ! [Xa: set_a] :
( ( member_set_a2 @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1081_finite__has__minimal,axiom,
! [A2: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( A2 != bot_bo3380559777022489994_set_a )
=> ? [X: set_set_a] :
( ( member_set_set_a2 @ X @ A2 )
& ! [Xa: set_set_a] :
( ( member_set_set_a2 @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1082_finite__has__minimal,axiom,
! [A2: set_set_set_set_a] :
( ( finite5318320746233006407_set_a @ A2 )
=> ( ( A2 != bot_bo4178452617224790762_set_a )
=> ? [X: set_set_set_a] :
( ( member_set_set_set_a @ X @ A2 )
& ! [Xa: set_set_set_a] :
( ( member_set_set_set_a @ Xa @ A2 )
=> ( ( ord_le5722252365846178494_set_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1083_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X: set_a] :
( ( member_set_a2 @ X @ A2 )
& ! [Xa: set_a] :
( ( member_set_a2 @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1084_all__finite__subset__image,axiom,
! [F: set_a > set_a,A2: set_set_a,P: set_set_a > $o] :
( ( ! [B4: set_set_a] :
( ( ( finite_finite_set_a @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ ( image_set_a_set_a @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_a] :
( ( ( finite_finite_set_a @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ A2 ) )
=> ( P @ ( image_set_a_set_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1085_all__finite__subset__image,axiom,
! [F: set_set_a > set_a,A2: set_set_set_a,P: set_set_a > $o] :
( ( ! [B4: set_set_a] :
( ( ( finite_finite_set_a @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ ( image_6061375613820669477_set_a @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_set_a] :
( ( ( finite7209287970140883943_set_a @ B4 )
& ( ord_le5722252365846178494_set_a @ B4 @ A2 ) )
=> ( P @ ( image_6061375613820669477_set_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1086_all__finite__subset__image,axiom,
! [F: a > set_a,A2: set_a,P: set_set_a > $o] :
( ( ! [B4: set_set_a] :
( ( ( finite_finite_set_a @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ ( image_a_set_a @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 ) )
=> ( P @ ( image_a_set_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1087_all__finite__subset__image,axiom,
! [F: set_a > set_set_a,A2: set_set_a,P: set_set_set_a > $o] :
( ( ! [B4: set_set_set_a] :
( ( ( finite7209287970140883943_set_a @ B4 )
& ( ord_le5722252365846178494_set_a @ B4 @ ( image_4955109552351689957_set_a @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_a] :
( ( ( finite_finite_set_a @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ A2 ) )
=> ( P @ ( image_4955109552351689957_set_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1088_all__finite__subset__image,axiom,
! [F: set_set_a > set_set_a,A2: set_set_set_a,P: set_set_set_a > $o] :
( ( ! [B4: set_set_set_a] :
( ( ( finite7209287970140883943_set_a @ B4 )
& ( ord_le5722252365846178494_set_a @ B4 @ ( image_1042221919965026181_set_a @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_set_a] :
( ( ( finite7209287970140883943_set_a @ B4 )
& ( ord_le5722252365846178494_set_a @ B4 @ A2 ) )
=> ( P @ ( image_1042221919965026181_set_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1089_all__finite__subset__image,axiom,
! [F: a > set_set_a,A2: set_a,P: set_set_set_a > $o] :
( ( ! [B4: set_set_set_a] :
( ( ( finite7209287970140883943_set_a @ B4 )
& ( ord_le5722252365846178494_set_a @ B4 @ ( image_a_set_set_a @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 ) )
=> ( P @ ( image_a_set_set_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1090_all__finite__subset__image,axiom,
! [F: set_a > a,A2: set_set_a,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_set_a_a @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_a] :
( ( ( finite_finite_set_a @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ A2 ) )
=> ( P @ ( image_set_a_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1091_all__finite__subset__image,axiom,
! [F: set_set_a > a,A2: set_set_set_a,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_set_set_a_a @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_set_a] :
( ( ( finite7209287970140883943_set_a @ B4 )
& ( ord_le5722252365846178494_set_a @ B4 @ A2 ) )
=> ( P @ ( image_set_set_a_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1092_all__finite__subset__image,axiom,
! [F: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 ) )
=> ( P @ ( image_a_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1093_ex__finite__subset__image,axiom,
! [F: set_a > set_a,A2: set_set_a,P: set_set_a > $o] :
( ( ? [B4: set_set_a] :
( ( finite_finite_set_a @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ ( image_set_a_set_a @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_set_a] :
( ( finite_finite_set_a @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ A2 )
& ( P @ ( image_set_a_set_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1094_ex__finite__subset__image,axiom,
! [F: set_set_a > set_a,A2: set_set_set_a,P: set_set_a > $o] :
( ( ? [B4: set_set_a] :
( ( finite_finite_set_a @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ ( image_6061375613820669477_set_a @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_set_set_a] :
( ( finite7209287970140883943_set_a @ B4 )
& ( ord_le5722252365846178494_set_a @ B4 @ A2 )
& ( P @ ( image_6061375613820669477_set_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1095_ex__finite__subset__image,axiom,
! [F: a > set_a,A2: set_a,P: set_set_a > $o] :
( ( ? [B4: set_set_a] :
( ( finite_finite_set_a @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ ( image_a_set_a @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 )
& ( P @ ( image_a_set_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1096_ex__finite__subset__image,axiom,
! [F: set_a > set_set_a,A2: set_set_a,P: set_set_set_a > $o] :
( ( ? [B4: set_set_set_a] :
( ( finite7209287970140883943_set_a @ B4 )
& ( ord_le5722252365846178494_set_a @ B4 @ ( image_4955109552351689957_set_a @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_set_a] :
( ( finite_finite_set_a @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ A2 )
& ( P @ ( image_4955109552351689957_set_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1097_ex__finite__subset__image,axiom,
! [F: set_set_a > set_set_a,A2: set_set_set_a,P: set_set_set_a > $o] :
( ( ? [B4: set_set_set_a] :
( ( finite7209287970140883943_set_a @ B4 )
& ( ord_le5722252365846178494_set_a @ B4 @ ( image_1042221919965026181_set_a @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_set_set_a] :
( ( finite7209287970140883943_set_a @ B4 )
& ( ord_le5722252365846178494_set_a @ B4 @ A2 )
& ( P @ ( image_1042221919965026181_set_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1098_ex__finite__subset__image,axiom,
! [F: a > set_set_a,A2: set_a,P: set_set_set_a > $o] :
( ( ? [B4: set_set_set_a] :
( ( finite7209287970140883943_set_a @ B4 )
& ( ord_le5722252365846178494_set_a @ B4 @ ( image_a_set_set_a @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 )
& ( P @ ( image_a_set_set_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1099_ex__finite__subset__image,axiom,
! [F: set_a > a,A2: set_set_a,P: set_a > $o] :
( ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_set_a_a @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_set_a] :
( ( finite_finite_set_a @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ A2 )
& ( P @ ( image_set_a_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1100_ex__finite__subset__image,axiom,
! [F: set_set_a > a,A2: set_set_set_a,P: set_a > $o] :
( ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_set_set_a_a @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_set_set_a] :
( ( finite7209287970140883943_set_a @ B4 )
& ( ord_le5722252365846178494_set_a @ B4 @ A2 )
& ( P @ ( image_set_set_a_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1101_ex__finite__subset__image,axiom,
! [F: a > a,A2: set_a,P: set_a > $o] :
( ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 )
& ( P @ ( image_a_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1102_finite__subset__image,axiom,
! [B2: set_set_a,F: set_a > set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ ( image_set_a_set_a @ F @ A2 ) )
=> ? [C4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C4 @ A2 )
& ( finite_finite_set_a @ C4 )
& ( B2
= ( image_set_a_set_a @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1103_finite__subset__image,axiom,
! [B2: set_set_a,F: set_set_a > set_a,A2: set_set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ ( image_6061375613820669477_set_a @ F @ A2 ) )
=> ? [C4: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ C4 @ A2 )
& ( finite7209287970140883943_set_a @ C4 )
& ( B2
= ( image_6061375613820669477_set_a @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1104_finite__subset__image,axiom,
! [B2: set_set_a,F: a > set_a,A2: set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ ( image_a_set_a @ F @ A2 ) )
=> ? [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A2 )
& ( finite_finite_a @ C4 )
& ( B2
= ( image_a_set_a @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1105_finite__subset__image,axiom,
! [B2: set_set_set_a,F: set_a > set_set_a,A2: set_set_a] :
( ( finite7209287970140883943_set_a @ B2 )
=> ( ( ord_le5722252365846178494_set_a @ B2 @ ( image_4955109552351689957_set_a @ F @ A2 ) )
=> ? [C4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C4 @ A2 )
& ( finite_finite_set_a @ C4 )
& ( B2
= ( image_4955109552351689957_set_a @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1106_finite__subset__image,axiom,
! [B2: set_set_set_a,F: set_set_a > set_set_a,A2: set_set_set_a] :
( ( finite7209287970140883943_set_a @ B2 )
=> ( ( ord_le5722252365846178494_set_a @ B2 @ ( image_1042221919965026181_set_a @ F @ A2 ) )
=> ? [C4: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ C4 @ A2 )
& ( finite7209287970140883943_set_a @ C4 )
& ( B2
= ( image_1042221919965026181_set_a @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1107_finite__subset__image,axiom,
! [B2: set_set_set_a,F: a > set_set_a,A2: set_a] :
( ( finite7209287970140883943_set_a @ B2 )
=> ( ( ord_le5722252365846178494_set_a @ B2 @ ( image_a_set_set_a @ F @ A2 ) )
=> ? [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A2 )
& ( finite_finite_a @ C4 )
& ( B2
= ( image_a_set_set_a @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1108_finite__subset__image,axiom,
! [B2: set_a,F: set_a > a,A2: set_set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_set_a_a @ F @ A2 ) )
=> ? [C4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C4 @ A2 )
& ( finite_finite_set_a @ C4 )
& ( B2
= ( image_set_a_a @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1109_finite__subset__image,axiom,
! [B2: set_a,F: set_set_a > a,A2: set_set_set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_set_set_a_a @ F @ A2 ) )
=> ? [C4: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ C4 @ A2 )
& ( finite7209287970140883943_set_a @ C4 )
& ( B2
= ( image_set_set_a_a @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1110_finite__subset__image,axiom,
! [B2: set_a,F: a > a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) )
=> ? [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A2 )
& ( finite_finite_a @ C4 )
& ( B2
= ( image_a_a @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1111_infinite__finite__induct,axiom,
! [P: set_set_a > $o,A2: set_set_a] :
( ! [A7: set_set_a] :
( ~ ( finite_finite_set_a @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ~ ( member_set_a2 @ X @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_a2 @ X @ F2 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1112_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A7: set_a] :
( ~ ( finite_finite_a @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ~ ( member_a2 @ X @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a2 @ X @ F2 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1113_finite__ne__induct,axiom,
! [F3: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F3 )
=> ( ( F3 != bot_bot_set_set_a )
=> ( ! [X: set_a] : ( P @ ( insert_set_a2 @ X @ bot_bot_set_set_a ) )
=> ( ! [X: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ( F2 != bot_bot_set_set_a )
=> ( ~ ( member_set_a2 @ X @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_a2 @ X @ F2 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1114_finite__ne__induct,axiom,
! [F3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ! [X: a] : ( P @ ( insert_a2 @ X @ bot_bot_set_a ) )
=> ( ! [X: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ~ ( member_a2 @ X @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a2 @ X @ F2 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1115_finite__induct,axiom,
! [F3: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F3 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ~ ( member_set_a2 @ X @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_a2 @ X @ F2 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_1116_finite__induct,axiom,
! [F3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ~ ( member_a2 @ X @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a2 @ X @ F2 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_1117_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A3: set_a] :
( ( A3 = bot_bot_set_a )
| ? [A4: set_a,B3: a] :
( ( A3
= ( insert_a2 @ B3 @ A4 ) )
& ( finite_finite_a @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_1118_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A7: set_a] :
( ? [A5: a] :
( A
= ( insert_a2 @ A5 @ A7 ) )
=> ~ ( finite_finite_a @ A7 ) ) ) ) ).
% finite.cases
thf(fact_1119_transpose_Osimps_I2_J,axiom,
! [Xss2: list_list_a] :
( ( transpose_a @ ( cons_list_a @ nil_a @ Xss2 ) )
= ( transpose_a @ Xss2 ) ) ).
% transpose.simps(2)
thf(fact_1120_transpose_Osimps_I2_J,axiom,
! [Xss2: list_list_set_a] :
( ( transpose_set_a @ ( cons_list_set_a @ nil_set_a @ Xss2 ) )
= ( transpose_set_a @ Xss2 ) ) ).
% transpose.simps(2)
thf(fact_1121_finite__subset__induct,axiom,
! [F3: set_set_a,A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F3 )
=> ( ( ord_le3724670747650509150_set_a @ F3 @ A2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A5: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ( member_set_a2 @ A5 @ A2 )
=> ( ~ ( member_set_a2 @ A5 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_a2 @ A5 @ F2 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1122_finite__subset__induct,axiom,
! [F3: set_set_set_a,A2: set_set_set_a,P: set_set_set_a > $o] :
( ( finite7209287970140883943_set_a @ F3 )
=> ( ( ord_le5722252365846178494_set_a @ F3 @ A2 )
=> ( ( P @ bot_bo3380559777022489994_set_a )
=> ( ! [A5: set_set_a,F2: set_set_set_a] :
( ( finite7209287970140883943_set_a @ F2 )
=> ( ( member_set_set_a2 @ A5 @ A2 )
=> ( ~ ( member_set_set_a2 @ A5 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_set_a2 @ A5 @ F2 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1123_finite__subset__induct,axiom,
! [F3: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( member_a2 @ A5 @ A2 )
=> ( ~ ( member_a2 @ A5 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a2 @ A5 @ F2 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1124_finite__subset__induct_H,axiom,
! [F3: set_set_a,A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F3 )
=> ( ( ord_le3724670747650509150_set_a @ F3 @ A2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A5: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ( member_set_a2 @ A5 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ F2 @ A2 )
=> ( ~ ( member_set_a2 @ A5 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_a2 @ A5 @ F2 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1125_finite__subset__induct_H,axiom,
! [F3: set_set_set_a,A2: set_set_set_a,P: set_set_set_a > $o] :
( ( finite7209287970140883943_set_a @ F3 )
=> ( ( ord_le5722252365846178494_set_a @ F3 @ A2 )
=> ( ( P @ bot_bo3380559777022489994_set_a )
=> ( ! [A5: set_set_a,F2: set_set_set_a] :
( ( finite7209287970140883943_set_a @ F2 )
=> ( ( member_set_set_a2 @ A5 @ A2 )
=> ( ( ord_le5722252365846178494_set_a @ F2 @ A2 )
=> ( ~ ( member_set_set_a2 @ A5 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_set_set_a2 @ A5 @ F2 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1126_finite__subset__induct_H,axiom,
! [F3: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( member_a2 @ A5 @ A2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ~ ( member_a2 @ A5 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a2 @ A5 @ F2 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1127_comp__sgraph_Ois__isolated__vertex__def,axiom,
! [S: set_a,V: a] :
( ( undire8931668460104145173rtex_a @ S @ ( undire2918257014606996450dges_a @ S ) @ V )
= ( ( member_a2 @ V @ S )
& ! [X3: a] :
( ( member_a2 @ X3 @ S )
=> ~ ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ X3 @ V ) ) ) ) ).
% comp_sgraph.is_isolated_vertex_def
thf(fact_1128_comp__sgraph_Ois__isolated__vertex__def,axiom,
! [S: set_set_a,V: set_a] :
( ( undire6879241558604981877_set_a @ S @ ( undire8247866692393712962_set_a @ S ) @ V )
= ( ( member_set_a2 @ V @ S )
& ! [X3: set_a] :
( ( member_set_a2 @ X3 @ S )
=> ~ ( undire3510646817838285160_set_a @ ( undire8247866692393712962_set_a @ S ) @ X3 @ V ) ) ) ) ).
% comp_sgraph.is_isolated_vertex_def
thf(fact_1129_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V )
= ( ( member_a2 @ V @ Vertices )
& ! [X3: a] :
( ( member_a2 @ X3 @ Vertices )
=> ~ ( undire397441198561214472_adj_a @ Edges @ X3 @ V ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_1130_ulgraph_Ois__isolated__vertex__def,axiom,
! [Vertices: set_set_a,Edges: set_set_set_a,V: set_a] :
( ( undire6886684016831807756_set_a @ Vertices @ Edges )
=> ( ( undire6879241558604981877_set_a @ Vertices @ Edges @ V )
= ( ( member_set_a2 @ V @ Vertices )
& ! [X3: set_a] :
( ( member_set_a2 @ X3 @ Vertices )
=> ~ ( undire3510646817838285160_set_a @ Edges @ X3 @ V ) ) ) ) ) ).
% ulgraph.is_isolated_vertex_def
thf(fact_1131_comp__sgraph_Onot__vert__adj,axiom,
! [S: set_a,V: a,U: a] :
( ~ ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ V @ U )
=> ~ ( member_set_a2 @ ( insert_a2 @ V @ ( insert_a2 @ U @ bot_bot_set_a ) ) @ ( undire2918257014606996450dges_a @ S ) ) ) ).
% comp_sgraph.not_vert_adj
thf(fact_1132_comp__sgraph_Overt__adj__def,axiom,
! [S: set_a,V1: a,V22: a] :
( ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ V1 @ V22 )
= ( member_set_a2 @ ( insert_a2 @ V1 @ ( insert_a2 @ V22 @ bot_bot_set_a ) ) @ ( undire2918257014606996450dges_a @ S ) ) ) ).
% comp_sgraph.vert_adj_def
thf(fact_1133_ulgraph_Onot__vert__adj,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a,U: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ~ ( undire397441198561214472_adj_a @ Edges @ V @ U )
=> ~ ( member_set_a2 @ ( insert_a2 @ V @ ( insert_a2 @ U @ bot_bot_set_a ) ) @ Edges ) ) ) ).
% ulgraph.not_vert_adj
thf(fact_1134_finite__transitivity__chain,axiom,
! [A2: set_set_a,R: set_a > set_a > $o] :
( ( finite_finite_set_a @ A2 )
=> ( ! [X: set_a] :
~ ( R @ X @ X )
=> ( ! [X: set_a,Y2: set_a,Z3: set_a] :
( ( R @ X @ Y2 )
=> ( ( R @ Y2 @ Z3 )
=> ( R @ X @ Z3 ) ) )
=> ( ! [X: set_a] :
( ( member_set_a2 @ X @ A2 )
=> ? [Y5: set_a] :
( ( member_set_a2 @ Y5 @ A2 )
& ( R @ X @ Y5 ) ) )
=> ( A2 = bot_bot_set_set_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1135_finite__transitivity__chain,axiom,
! [A2: set_a,R: a > a > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [X: a] :
~ ( R @ X @ X )
=> ( ! [X: a,Y2: a,Z3: a] :
( ( R @ X @ Y2 )
=> ( ( R @ Y2 @ Z3 )
=> ( R @ X @ Z3 ) ) )
=> ( ! [X: a] :
( ( member_a2 @ X @ A2 )
=> ? [Y5: a] :
( ( member_a2 @ Y5 @ A2 )
& ( R @ X @ Y5 ) ) )
=> ( A2 = bot_bot_set_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1136_transpose_Opsimps_I2_J,axiom,
! [Xss2: list_list_a] :
( ( accp_list_list_a @ transpose_rel_a @ ( cons_list_a @ nil_a @ Xss2 ) )
=> ( ( transpose_a @ ( cons_list_a @ nil_a @ Xss2 ) )
= ( transpose_a @ Xss2 ) ) ) ).
% transpose.psimps(2)
thf(fact_1137_transpose_Opsimps_I2_J,axiom,
! [Xss2: list_list_set_a] :
( ( accp_list_list_set_a @ transpose_rel_set_a @ ( cons_list_set_a @ nil_set_a @ Xss2 ) )
=> ( ( transpose_set_a @ ( cons_list_set_a @ nil_set_a @ Xss2 ) )
= ( transpose_set_a @ Xss2 ) ) ) ).
% transpose.psimps(2)
thf(fact_1138_ulgraph_Overt__adj__inc__edge__iff,axiom,
! [Vertices: set_a,Edges: set_set_a,V1: a,V22: a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire397441198561214472_adj_a @ Edges @ V1 @ V22 )
= ( ( undire1521409233611534436dent_a @ V1 @ ( insert_a2 @ V1 @ ( insert_a2 @ V22 @ bot_bot_set_a ) ) )
& ( undire1521409233611534436dent_a @ V22 @ ( insert_a2 @ V1 @ ( insert_a2 @ V22 @ bot_bot_set_a ) ) )
& ( member_set_a2 @ ( insert_a2 @ V1 @ ( insert_a2 @ V22 @ bot_bot_set_a ) ) @ Edges ) ) ) ) ).
% ulgraph.vert_adj_inc_edge_iff
thf(fact_1139_comp__sgraph_Oincident__def,axiom,
undire1521409233611534436dent_a = member_a2 ).
% comp_sgraph.incident_def
thf(fact_1140_comp__sgraph_Oincident__def,axiom,
undire2320338297334612420_set_a = member_set_a2 ).
% comp_sgraph.incident_def
thf(fact_1141_comp__sgraph_Oincident__edge__in__wf,axiom,
! [E: set_set_a,S: set_set_a,V: set_a] :
( ( member_set_set_a2 @ E @ ( undire8247866692393712962_set_a @ S ) )
=> ( ( undire2320338297334612420_set_a @ V @ E )
=> ( member_set_a2 @ V @ S ) ) ) ).
% comp_sgraph.incident_edge_in_wf
thf(fact_1142_comp__sgraph_Oincident__edge__in__wf,axiom,
! [E: set_a,S: set_a,V: a] :
( ( member_set_a2 @ E @ ( undire2918257014606996450dges_a @ S ) )
=> ( ( undire1521409233611534436dent_a @ V @ E )
=> ( member_a2 @ V @ S ) ) ) ).
% comp_sgraph.incident_edge_in_wf
thf(fact_1143_comp__sgraph_Ois__isolated__vertex__edge,axiom,
! [S: set_a,V: a,E: set_a] :
( ( undire8931668460104145173rtex_a @ S @ ( undire2918257014606996450dges_a @ S ) @ V )
=> ( ( member_set_a2 @ E @ ( undire2918257014606996450dges_a @ S ) )
=> ~ ( undire1521409233611534436dent_a @ V @ E ) ) ) ).
% comp_sgraph.is_isolated_vertex_edge
thf(fact_1144_ulgraph_Ois__isolated__vertex__edge,axiom,
! [Vertices: set_a,Edges: set_set_a,V: a,E: set_a] :
( ( undire7251896706689453996raph_a @ Vertices @ Edges )
=> ( ( undire8931668460104145173rtex_a @ Vertices @ Edges @ V )
=> ( ( member_set_a2 @ E @ Edges )
=> ~ ( undire1521409233611534436dent_a @ V @ E ) ) ) ) ).
% ulgraph.is_isolated_vertex_edge
thf(fact_1145_comp__sgraph_Overt__adj__inc__edge__iff,axiom,
! [S: set_a,V1: a,V22: a] :
( ( undire397441198561214472_adj_a @ ( undire2918257014606996450dges_a @ S ) @ V1 @ V22 )
= ( ( undire1521409233611534436dent_a @ V1 @ ( insert_a2 @ V1 @ ( insert_a2 @ V22 @ bot_bot_set_a ) ) )
& ( undire1521409233611534436dent_a @ V22 @ ( insert_a2 @ V1 @ ( insert_a2 @ V22 @ bot_bot_set_a ) ) )
& ( member_set_a2 @ ( insert_a2 @ V1 @ ( insert_a2 @ V22 @ bot_bot_set_a ) ) @ ( undire2918257014606996450dges_a @ S ) ) ) ) ).
% comp_sgraph.vert_adj_inc_edge_iff
thf(fact_1146_Sup__fin_OcoboundedI,axiom,
! [A2: set_set_set_a,A: set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( member_set_set_a2 @ A @ A2 )
=> ( ord_le3724670747650509150_set_a @ A @ ( lattic338143333561554293_set_a @ A2 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_1147_Sup__fin_OcoboundedI,axiom,
! [A2: set_set_set_set_a,A: set_set_set_a] :
( ( finite5318320746233006407_set_a @ A2 )
=> ( ( member_set_set_set_a @ A @ A2 )
=> ( ord_le5722252365846178494_set_a @ A @ ( lattic5824591902637136597_set_a @ A2 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_1148_Sup__fin_OcoboundedI,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a2 @ A @ A2 )
=> ( ord_less_eq_set_a @ A @ ( lattic2918178356826803221_set_a @ A2 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_1149_Sup__fin_Obounded__iff,axiom,
! [A2: set_set_set_a,X2: set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( A2 != bot_bo3380559777022489994_set_a )
=> ( ( ord_le3724670747650509150_set_a @ ( lattic338143333561554293_set_a @ A2 ) @ X2 )
= ( ! [X3: set_set_a] :
( ( member_set_set_a2 @ X3 @ A2 )
=> ( ord_le3724670747650509150_set_a @ X3 @ X2 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1150_Sup__fin_Obounded__iff,axiom,
! [A2: set_set_set_set_a,X2: set_set_set_a] :
( ( finite5318320746233006407_set_a @ A2 )
=> ( ( A2 != bot_bo4178452617224790762_set_a )
=> ( ( ord_le5722252365846178494_set_a @ ( lattic5824591902637136597_set_a @ A2 ) @ X2 )
= ( ! [X3: set_set_set_a] :
( ( member_set_set_set_a @ X3 @ A2 )
=> ( ord_le5722252365846178494_set_a @ X3 @ X2 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1151_Sup__fin_Obounded__iff,axiom,
! [A2: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A2 ) @ X2 )
= ( ! [X3: set_a] :
( ( member_set_a2 @ X3 @ A2 )
=> ( ord_less_eq_set_a @ X3 @ X2 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1152_Sup__fin_OboundedI,axiom,
! [A2: set_set_set_a,X2: set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( A2 != bot_bo3380559777022489994_set_a )
=> ( ! [A5: set_set_a] :
( ( member_set_set_a2 @ A5 @ A2 )
=> ( ord_le3724670747650509150_set_a @ A5 @ X2 ) )
=> ( ord_le3724670747650509150_set_a @ ( lattic338143333561554293_set_a @ A2 ) @ X2 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1153_Sup__fin_OboundedI,axiom,
! [A2: set_set_set_set_a,X2: set_set_set_a] :
( ( finite5318320746233006407_set_a @ A2 )
=> ( ( A2 != bot_bo4178452617224790762_set_a )
=> ( ! [A5: set_set_set_a] :
( ( member_set_set_set_a @ A5 @ A2 )
=> ( ord_le5722252365846178494_set_a @ A5 @ X2 ) )
=> ( ord_le5722252365846178494_set_a @ ( lattic5824591902637136597_set_a @ A2 ) @ X2 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1154_Sup__fin_OboundedI,axiom,
! [A2: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ! [A5: set_a] :
( ( member_set_a2 @ A5 @ A2 )
=> ( ord_less_eq_set_a @ A5 @ X2 ) )
=> ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A2 ) @ X2 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1155_Sup__fin_OboundedE,axiom,
! [A2: set_set_set_a,X2: set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( A2 != bot_bo3380559777022489994_set_a )
=> ( ( ord_le3724670747650509150_set_a @ ( lattic338143333561554293_set_a @ A2 ) @ X2 )
=> ! [A8: set_set_a] :
( ( member_set_set_a2 @ A8 @ A2 )
=> ( ord_le3724670747650509150_set_a @ A8 @ X2 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1156_Sup__fin_OboundedE,axiom,
! [A2: set_set_set_set_a,X2: set_set_set_a] :
( ( finite5318320746233006407_set_a @ A2 )
=> ( ( A2 != bot_bo4178452617224790762_set_a )
=> ( ( ord_le5722252365846178494_set_a @ ( lattic5824591902637136597_set_a @ A2 ) @ X2 )
=> ! [A8: set_set_set_a] :
( ( member_set_set_set_a @ A8 @ A2 )
=> ( ord_le5722252365846178494_set_a @ A8 @ X2 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1157_Sup__fin_OboundedE,axiom,
! [A2: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A2 ) @ X2 )
=> ! [A8: set_a] :
( ( member_set_a2 @ A8 @ A2 )
=> ( ord_less_eq_set_a @ A8 @ X2 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1158_Sup__fin_Osubset__imp,axiom,
! [A2: set_set_set_set_a,B2: set_set_set_set_a] :
( ( ord_le8049040685576063006_set_a @ A2 @ B2 )
=> ( ( A2 != bot_bo4178452617224790762_set_a )
=> ( ( finite5318320746233006407_set_a @ B2 )
=> ( ord_le5722252365846178494_set_a @ ( lattic5824591902637136597_set_a @ A2 ) @ ( lattic5824591902637136597_set_a @ B2 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1159_Sup__fin_Osubset__imp,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B2 )
=> ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A2 ) @ ( lattic2918178356826803221_set_a @ B2 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1160_Sup__fin_Osubset__imp,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( ( A2 != bot_bo3380559777022489994_set_a )
=> ( ( finite7209287970140883943_set_a @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( lattic338143333561554293_set_a @ A2 ) @ ( lattic338143333561554293_set_a @ B2 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1161_Sup__fin_Osubset,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( B2 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_a @ ( lattic2918178356826803221_set_a @ B2 ) @ ( lattic2918178356826803221_set_a @ A2 ) )
= ( lattic2918178356826803221_set_a @ A2 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1162_Sup__fin_Osubset,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( B2 != bot_bo3380559777022489994_set_a )
=> ( ( ord_le5722252365846178494_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_set_a @ ( lattic338143333561554293_set_a @ B2 ) @ ( lattic338143333561554293_set_a @ A2 ) )
= ( lattic338143333561554293_set_a @ A2 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1163_Sup__fin_Oinsert__not__elem,axiom,
! [A2: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ~ ( member_set_a2 @ X2 @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ ( insert_set_a2 @ X2 @ A2 ) )
= ( sup_sup_set_a @ X2 @ ( lattic2918178356826803221_set_a @ A2 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1164_Sup__fin_Oclosed,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ! [X: set_a,Y2: set_a] : ( member_set_a2 @ ( sup_sup_set_a @ X @ Y2 ) @ ( insert_set_a2 @ X @ ( insert_set_a2 @ Y2 @ bot_bot_set_set_a ) ) )
=> ( member_set_a2 @ ( lattic2918178356826803221_set_a @ A2 ) @ A2 ) ) ) ) ).
% Sup_fin.closed
thf(fact_1165_Sup__fin_Oremove,axiom,
! [A2: set_set_a,X2: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a2 @ X2 @ A2 )
=> ( ( ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a2 @ X2 @ bot_bot_set_set_a ) )
= bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ A2 )
= X2 ) )
& ( ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a2 @ X2 @ bot_bot_set_set_a ) )
!= bot_bot_set_set_a )
=> ( ( lattic2918178356826803221_set_a @ A2 )
= ( sup_sup_set_a @ X2 @ ( lattic2918178356826803221_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a2 @ X2 @ bot_bot_set_set_a ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1166_DiffI,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a2 @ C @ A2 )
=> ( ~ ( member_a2 @ C @ B2 )
=> ( member_a2 @ C @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_1167_DiffI,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a2 @ C @ A2 )
=> ( ~ ( member_set_a2 @ C @ B2 )
=> ( member_set_a2 @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_1168_Diff__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a2 @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( ( member_a2 @ C @ A2 )
& ~ ( member_a2 @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_1169_Diff__iff,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a2 @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
= ( ( member_set_a2 @ C @ A2 )
& ~ ( member_set_a2 @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_1170_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_1171_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_1172_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_1173_insert__Diff1,axiom,
! [X2: a,B2: set_a,A2: set_a] :
( ( member_a2 @ X2 @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a2 @ X2 @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1174_insert__Diff1,axiom,
! [X2: set_a,B2: set_set_a,A2: set_set_a] :
( ( member_set_a2 @ X2 @ B2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a2 @ X2 @ A2 ) @ B2 )
= ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1175_Diff__insert0,axiom,
! [X2: a,A2: set_a,B2: set_a] :
( ~ ( member_a2 @ X2 @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a2 @ X2 @ B2 ) )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1176_Diff__insert0,axiom,
! [X2: set_a,A2: set_set_a,B2: set_set_a] :
( ~ ( member_set_a2 @ X2 @ A2 )
=> ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a2 @ X2 @ B2 ) )
= ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1177_Diff__eq__empty__iff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ( minus_5736297505244876581_set_a @ A2 @ B2 )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_1178_Diff__eq__empty__iff,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( ( minus_3359197881701045381_set_a @ A2 @ B2 )
= bot_bo3380559777022489994_set_a )
= ( ord_le5722252365846178494_set_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_1179_Diff__eq__empty__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( minus_minus_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_1180_insert__Diff__single,axiom,
! [A: a,A2: set_a] :
( ( insert_a2 @ A @ ( minus_minus_set_a @ A2 @ ( insert_a2 @ A @ bot_bot_set_a ) ) )
= ( insert_a2 @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_1181_finite__Diff__insert,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ ( insert_a2 @ A @ B2 ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_1182_insert__Diff__if,axiom,
! [X2: a,B2: set_a,A2: set_a] :
( ( ( member_a2 @ X2 @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a2 @ X2 @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) )
& ( ~ ( member_a2 @ X2 @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a2 @ X2 @ A2 ) @ B2 )
= ( insert_a2 @ X2 @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1183_insert__Diff__if,axiom,
! [X2: set_a,B2: set_set_a,A2: set_set_a] :
( ( ( member_set_a2 @ X2 @ B2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a2 @ X2 @ A2 ) @ B2 )
= ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) )
& ( ~ ( member_set_a2 @ X2 @ B2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a2 @ X2 @ A2 ) @ B2 )
= ( insert_set_a2 @ X2 @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1184_DiffE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a2 @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a2 @ C @ A2 )
=> ( member_a2 @ C @ B2 ) ) ) ).
% DiffE
thf(fact_1185_DiffE,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a2 @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
=> ~ ( ( member_set_a2 @ C @ A2 )
=> ( member_set_a2 @ C @ B2 ) ) ) ).
% DiffE
thf(fact_1186_DiffD1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a2 @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ( member_a2 @ C @ A2 ) ) ).
% DiffD1
thf(fact_1187_DiffD1,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a2 @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
=> ( member_set_a2 @ C @ A2 ) ) ).
% DiffD1
thf(fact_1188_DiffD2,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a2 @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( member_a2 @ C @ B2 ) ) ).
% DiffD2
thf(fact_1189_DiffD2,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a2 @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
=> ~ ( member_set_a2 @ C @ B2 ) ) ).
% DiffD2
thf(fact_1190_double__diff,axiom,
! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C2 )
=> ( ( minus_5736297505244876581_set_a @ B2 @ ( minus_5736297505244876581_set_a @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_1191_double__diff,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,C2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( ( ord_le5722252365846178494_set_a @ B2 @ C2 )
=> ( ( minus_3359197881701045381_set_a @ B2 @ ( minus_3359197881701045381_set_a @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_1192_double__diff,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ( minus_minus_set_a @ B2 @ ( minus_minus_set_a @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_1193_Diff__subset,axiom,
! [A2: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_1194_Diff__subset,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] : ( ord_le5722252365846178494_set_a @ ( minus_3359197881701045381_set_a @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_1195_Diff__subset,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_1196_Diff__mono,axiom,
! [A2: set_set_a,C2: set_set_a,D: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ D @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) @ ( minus_5736297505244876581_set_a @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_1197_Diff__mono,axiom,
! [A2: set_set_set_a,C2: set_set_set_a,D: set_set_set_a,B2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ C2 )
=> ( ( ord_le5722252365846178494_set_a @ D @ B2 )
=> ( ord_le5722252365846178494_set_a @ ( minus_3359197881701045381_set_a @ A2 @ B2 ) @ ( minus_3359197881701045381_set_a @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_1198_Diff__mono,axiom,
! [A2: set_a,C2: set_a,D: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ D @ B2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( minus_minus_set_a @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_1199_diff__shunt__var,axiom,
! [X2: set_set_a,Y: set_set_a] :
( ( ( minus_5736297505244876581_set_a @ X2 @ Y )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_1200_diff__shunt__var,axiom,
! [X2: set_set_set_a,Y: set_set_set_a] :
( ( ( minus_3359197881701045381_set_a @ X2 @ Y )
= bot_bo3380559777022489994_set_a )
= ( ord_le5722252365846178494_set_a @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_1201_diff__shunt__var,axiom,
! [X2: set_a,Y: set_a] :
( ( ( minus_minus_set_a @ X2 @ Y )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_1202_Diff__insert__absorb,axiom,
! [X2: set_a,A2: set_set_a] :
( ~ ( member_set_a2 @ X2 @ A2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a2 @ X2 @ A2 ) @ ( insert_set_a2 @ X2 @ bot_bot_set_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_1203_Diff__insert__absorb,axiom,
! [X2: a,A2: set_a] :
( ~ ( member_a2 @ X2 @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a2 @ X2 @ A2 ) @ ( insert_a2 @ X2 @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_1204_Diff__insert2,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a2 @ A @ B2 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a2 @ A @ bot_bot_set_a ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_1205_insert__Diff,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a2 @ A @ A2 )
=> ( ( insert_set_a2 @ A @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a2 @ A @ bot_bot_set_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_1206_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a2 @ A @ A2 )
=> ( ( insert_a2 @ A @ ( minus_minus_set_a @ A2 @ ( insert_a2 @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_1207_Diff__insert,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a2 @ A @ B2 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( insert_a2 @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_1208_subset__Diff__insert,axiom,
! [A2: set_set_a,B2: set_set_a,X2: set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B2 @ ( insert_set_a2 @ X2 @ C2 ) ) )
= ( ( ord_le3724670747650509150_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B2 @ C2 ) )
& ~ ( member_set_a2 @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1209_subset__Diff__insert,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,X2: set_set_a,C2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ ( minus_3359197881701045381_set_a @ B2 @ ( insert_set_set_a2 @ X2 @ C2 ) ) )
= ( ( ord_le5722252365846178494_set_a @ A2 @ ( minus_3359197881701045381_set_a @ B2 @ C2 ) )
& ~ ( member_set_set_a2 @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1210_subset__Diff__insert,axiom,
! [A2: set_a,B2: set_a,X2: a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ ( insert_a2 @ X2 @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ C2 ) )
& ~ ( member_a2 @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1211_Diff__subset__conv,axiom,
! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) @ C2 )
= ( ord_le3724670747650509150_set_a @ A2 @ ( sup_sup_set_set_a @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1212_Diff__subset__conv,axiom,
! [A2: set_set_set_a,B2: set_set_set_a,C2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( minus_3359197881701045381_set_a @ A2 @ B2 ) @ C2 )
= ( ord_le5722252365846178494_set_a @ A2 @ ( sup_su2076012971530813682_set_a @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1213_Diff__subset__conv,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ C2 )
= ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1214_Diff__partition,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_1215_Diff__partition,axiom,
! [A2: set_set_set_a,B2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B2 )
=> ( ( sup_su2076012971530813682_set_a @ A2 @ ( minus_3359197881701045381_set_a @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_1216_Diff__partition,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_1217_finite__empty__induct,axiom,
! [A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: set_a,A7: set_set_a] :
( ( finite_finite_set_a @ A7 )
=> ( ( member_set_a2 @ A5 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_5736297505244876581_set_a @ A7 @ ( insert_set_a2 @ A5 @ bot_bot_set_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1218_finite__empty__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: a,A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( ( member_a2 @ A5 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_minus_set_a @ A7 @ ( insert_a2 @ A5 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1219_infinite__coinduct,axiom,
! [X6: set_a > $o,A2: set_a] :
( ( X6 @ A2 )
=> ( ! [A7: set_a] :
( ( X6 @ A7 )
=> ? [X5: a] :
( ( member_a2 @ X5 @ A7 )
& ( ( X6 @ ( minus_minus_set_a @ A7 @ ( insert_a2 @ X5 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A7 @ ( insert_a2 @ X5 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_1220_infinite__remove,axiom,
! [S: set_a,A: a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ ( insert_a2 @ A @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_1221_Diff__single__insert,axiom,
! [A2: set_set_a,X2: set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a2 @ X2 @ bot_bot_set_set_a ) ) @ B2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a2 @ X2 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_1222_Diff__single__insert,axiom,
! [A2: set_set_set_a,X2: set_set_a,B2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( minus_3359197881701045381_set_a @ A2 @ ( insert_set_set_a2 @ X2 @ bot_bo3380559777022489994_set_a ) ) @ B2 )
=> ( ord_le5722252365846178494_set_a @ A2 @ ( insert_set_set_a2 @ X2 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_1223_Diff__single__insert,axiom,
! [A2: set_a,X2: a,B2: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a2 @ X2 @ bot_bot_set_a ) ) @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a2 @ X2 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_1224_subset__insert__iff,axiom,
! [A2: set_set_a,X2: set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a2 @ X2 @ B2 ) )
= ( ( ( member_set_a2 @ X2 @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a2 @ X2 @ bot_bot_set_set_a ) ) @ B2 ) )
& ( ~ ( member_set_a2 @ X2 @ A2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1225_subset__insert__iff,axiom,
! [A2: set_set_set_a,X2: set_set_a,B2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ ( insert_set_set_a2 @ X2 @ B2 ) )
= ( ( ( member_set_set_a2 @ X2 @ A2 )
=> ( ord_le5722252365846178494_set_a @ ( minus_3359197881701045381_set_a @ A2 @ ( insert_set_set_a2 @ X2 @ bot_bo3380559777022489994_set_a ) ) @ B2 ) )
& ( ~ ( member_set_set_a2 @ X2 @ A2 )
=> ( ord_le5722252365846178494_set_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1226_subset__insert__iff,axiom,
! [A2: set_a,X2: a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a2 @ X2 @ B2 ) )
= ( ( ( member_a2 @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a2 @ X2 @ bot_bot_set_a ) ) @ B2 ) )
& ( ~ ( member_a2 @ X2 @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1227_in__image__insert__iff,axiom,
! [B2: set_set_set_a,X2: set_a,A2: set_set_a] :
( ! [C4: set_set_a] :
( ( member_set_set_a2 @ C4 @ B2 )
=> ~ ( member_set_a2 @ X2 @ C4 ) )
=> ( ( member_set_set_a2 @ A2 @ ( image_1042221919965026181_set_a @ ( insert_set_a2 @ X2 ) @ B2 ) )
= ( ( member_set_a2 @ X2 @ A2 )
& ( member_set_set_a2 @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a2 @ X2 @ bot_bot_set_set_a ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1228_in__image__insert__iff,axiom,
! [B2: set_set_a,X2: a,A2: set_a] :
( ! [C4: set_a] :
( ( member_set_a2 @ C4 @ B2 )
=> ~ ( member_a2 @ X2 @ C4 ) )
=> ( ( member_set_a2 @ A2 @ ( image_set_a_set_a @ ( insert_a2 @ X2 ) @ B2 ) )
= ( ( member_a2 @ X2 @ A2 )
& ( member_set_a2 @ ( minus_minus_set_a @ A2 @ ( insert_a2 @ X2 @ bot_bot_set_a ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1229_remove__def,axiom,
( remove_a
= ( ^ [X3: a,A4: set_a] : ( minus_minus_set_a @ A4 @ ( insert_a2 @ X3 @ bot_bot_set_a ) ) ) ) ).
% remove_def
thf(fact_1230_finite__remove__induct,axiom,
! [B2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ B2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A7: set_set_a] :
( ( finite_finite_set_a @ A7 )
=> ( ( A7 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A7 @ B2 )
=> ( ! [X5: set_a] :
( ( member_set_a2 @ X5 @ A7 )
=> ( P @ ( minus_5736297505244876581_set_a @ A7 @ ( insert_set_a2 @ X5 @ bot_bot_set_set_a ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1231_finite__remove__induct,axiom,
! [B2: set_set_set_a,P: set_set_set_a > $o] :
( ( finite7209287970140883943_set_a @ B2 )
=> ( ( P @ bot_bo3380559777022489994_set_a )
=> ( ! [A7: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A7 )
=> ( ( A7 != bot_bo3380559777022489994_set_a )
=> ( ( ord_le5722252365846178494_set_a @ A7 @ B2 )
=> ( ! [X5: set_set_a] :
( ( member_set_set_a2 @ X5 @ A7 )
=> ( P @ ( minus_3359197881701045381_set_a @ A7 @ ( insert_set_set_a2 @ X5 @ bot_bo3380559777022489994_set_a ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1232_finite__remove__induct,axiom,
! [B2: set_a,P: set_a > $o] :
( ( finite_finite_a @ B2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( ( A7 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A7 @ B2 )
=> ( ! [X5: a] :
( ( member_a2 @ X5 @ A7 )
=> ( P @ ( minus_minus_set_a @ A7 @ ( insert_a2 @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1233_remove__induct,axiom,
! [P: set_set_a > $o,B2: set_set_a] :
( ( P @ bot_bot_set_set_a )
=> ( ( ~ ( finite_finite_set_a @ B2 )
=> ( P @ B2 ) )
=> ( ! [A7: set_set_a] :
( ( finite_finite_set_a @ A7 )
=> ( ( A7 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A7 @ B2 )
=> ( ! [X5: set_a] :
( ( member_set_a2 @ X5 @ A7 )
=> ( P @ ( minus_5736297505244876581_set_a @ A7 @ ( insert_set_a2 @ X5 @ bot_bot_set_set_a ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1234_remove__induct,axiom,
! [P: set_set_set_a > $o,B2: set_set_set_a] :
( ( P @ bot_bo3380559777022489994_set_a )
=> ( ( ~ ( finite7209287970140883943_set_a @ B2 )
=> ( P @ B2 ) )
=> ( ! [A7: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A7 )
=> ( ( A7 != bot_bo3380559777022489994_set_a )
=> ( ( ord_le5722252365846178494_set_a @ A7 @ B2 )
=> ( ! [X5: set_set_a] :
( ( member_set_set_a2 @ X5 @ A7 )
=> ( P @ ( minus_3359197881701045381_set_a @ A7 @ ( insert_set_set_a2 @ X5 @ bot_bo3380559777022489994_set_a ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1235_remove__induct,axiom,
! [P: set_a > $o,B2: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B2 )
=> ( P @ B2 ) )
=> ( ! [A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( ( A7 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A7 @ B2 )
=> ( ! [X5: a] :
( ( member_a2 @ X5 @ A7 )
=> ( P @ ( minus_minus_set_a @ A7 @ ( insert_a2 @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1236_set__removeAll,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( set_set_a2 @ ( removeAll_set_a @ X2 @ Xs ) )
= ( minus_5736297505244876581_set_a @ ( set_set_a2 @ Xs ) @ ( insert_set_a2 @ X2 @ bot_bot_set_set_a ) ) ) ).
% set_removeAll
thf(fact_1237_set__removeAll,axiom,
! [X2: set_set_a,Xs: list_set_set_a] :
( ( set_set_set_a2 @ ( removeAll_set_set_a @ X2 @ Xs ) )
= ( minus_3359197881701045381_set_a @ ( set_set_set_a2 @ Xs ) @ ( insert_set_set_a2 @ X2 @ bot_bo3380559777022489994_set_a ) ) ) ).
% set_removeAll
thf(fact_1238_set__removeAll,axiom,
! [X2: a,Xs: list_a] :
( ( set_a2 @ ( removeAll_a @ X2 @ Xs ) )
= ( minus_minus_set_a @ ( set_a2 @ Xs ) @ ( insert_a2 @ X2 @ bot_bot_set_a ) ) ) ).
% set_removeAll
thf(fact_1239_min__bot,axiom,
! [X2: set_a] :
( ( ord_min_set_a @ bot_bot_set_a @ X2 )
= bot_bot_set_a ) ).
% min_bot
thf(fact_1240_min__bot2,axiom,
! [X2: set_a] :
( ( ord_min_set_a @ X2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% min_bot2
thf(fact_1241_in__set__remove1,axiom,
! [A: a,B: a,Xs: list_a] :
( ( A != B )
=> ( ( member_a2 @ A @ ( set_a2 @ ( remove1_a @ B @ Xs ) ) )
= ( member_a2 @ A @ ( set_a2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_1242_in__set__remove1,axiom,
! [A: set_a,B: set_a,Xs: list_set_a] :
( ( A != B )
=> ( ( member_set_a2 @ A @ ( set_set_a2 @ ( remove1_set_a @ B @ Xs ) ) )
= ( member_set_a2 @ A @ ( set_set_a2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_1243_in__set__remove1,axiom,
! [A: set_set_a,B: set_set_a,Xs: list_set_set_a] :
( ( A != B )
=> ( ( member_set_set_a2 @ A @ ( set_set_set_a2 @ ( remove1_set_set_a @ B @ Xs ) ) )
= ( member_set_set_a2 @ A @ ( set_set_set_a2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_1244_removeAll__id,axiom,
! [X2: a,Xs: list_a] :
( ~ ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ( removeAll_a @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_1245_removeAll__id,axiom,
! [X2: set_a,Xs: list_set_a] :
( ~ ( member_set_a2 @ X2 @ ( set_set_a2 @ Xs ) )
=> ( ( removeAll_set_a @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_1246_removeAll__id,axiom,
! [X2: set_set_a,Xs: list_set_set_a] :
( ~ ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ Xs ) )
=> ( ( removeAll_set_set_a @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_1247_removeAll__append,axiom,
! [X2: a,Xs: list_a,Ys: list_a] :
( ( removeAll_a @ X2 @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( removeAll_a @ X2 @ Xs ) @ ( removeAll_a @ X2 @ Ys ) ) ) ).
% removeAll_append
thf(fact_1248_removeAll__append,axiom,
! [X2: set_a,Xs: list_set_a,Ys: list_set_a] :
( ( removeAll_set_a @ X2 @ ( append_set_a @ Xs @ Ys ) )
= ( append_set_a @ ( removeAll_set_a @ X2 @ Xs ) @ ( removeAll_set_a @ X2 @ Ys ) ) ) ).
% removeAll_append
thf(fact_1249_removeAll_Osimps_I2_J,axiom,
! [X2: a,Y: a,Xs: list_a] :
( ( ( X2 = Y )
=> ( ( removeAll_a @ X2 @ ( cons_a @ Y @ Xs ) )
= ( removeAll_a @ X2 @ Xs ) ) )
& ( ( X2 != Y )
=> ( ( removeAll_a @ X2 @ ( cons_a @ Y @ Xs ) )
= ( cons_a @ Y @ ( removeAll_a @ X2 @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_1250_removeAll_Osimps_I2_J,axiom,
! [X2: set_a,Y: set_a,Xs: list_set_a] :
( ( ( X2 = Y )
=> ( ( removeAll_set_a @ X2 @ ( cons_set_a @ Y @ Xs ) )
= ( removeAll_set_a @ X2 @ Xs ) ) )
& ( ( X2 != Y )
=> ( ( removeAll_set_a @ X2 @ ( cons_set_a @ Y @ Xs ) )
= ( cons_set_a @ Y @ ( removeAll_set_a @ X2 @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_1251_removeAll_Osimps_I1_J,axiom,
! [X2: a] :
( ( removeAll_a @ X2 @ nil_a )
= nil_a ) ).
% removeAll.simps(1)
thf(fact_1252_removeAll_Osimps_I1_J,axiom,
! [X2: set_a] :
( ( removeAll_set_a @ X2 @ nil_set_a )
= nil_set_a ) ).
% removeAll.simps(1)
thf(fact_1253_min__def,axiom,
( ord_min_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] : ( if_set_set_a @ ( ord_le3724670747650509150_set_a @ A3 @ B3 ) @ A3 @ B3 ) ) ) ).
% min_def
thf(fact_1254_min__def,axiom,
( ord_mi3815083042109949829_set_a
= ( ^ [A3: set_set_set_a,B3: set_set_set_a] : ( if_set_set_set_a @ ( ord_le5722252365846178494_set_a @ A3 @ B3 ) @ A3 @ B3 ) ) ) ).
% min_def
thf(fact_1255_min__def,axiom,
( ord_min_set_a
= ( ^ [A3: set_a,B3: set_a] : ( if_set_a @ ( ord_less_eq_set_a @ A3 @ B3 ) @ A3 @ B3 ) ) ) ).
% min_def
thf(fact_1256_min__absorb1,axiom,
! [X2: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y )
=> ( ( ord_min_set_set_a @ X2 @ Y )
= X2 ) ) ).
% min_absorb1
thf(fact_1257_min__absorb1,axiom,
! [X2: set_set_set_a,Y: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ X2 @ Y )
=> ( ( ord_mi3815083042109949829_set_a @ X2 @ Y )
= X2 ) ) ).
% min_absorb1
thf(fact_1258_min__absorb1,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_min_set_a @ X2 @ Y )
= X2 ) ) ).
% min_absorb1
thf(fact_1259_min__absorb2,axiom,
! [Y: set_set_a,X2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ X2 )
=> ( ( ord_min_set_set_a @ X2 @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_1260_min__absorb2,axiom,
! [Y: set_set_set_a,X2: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ Y @ X2 )
=> ( ( ord_mi3815083042109949829_set_a @ X2 @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_1261_min__absorb2,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ( ( ord_min_set_a @ X2 @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_1262_remove1__idem,axiom,
! [X2: a,Xs: list_a] :
( ~ ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ( remove1_a @ X2 @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_1263_remove1__idem,axiom,
! [X2: set_a,Xs: list_set_a] :
( ~ ( member_set_a2 @ X2 @ ( set_set_a2 @ Xs ) )
=> ( ( remove1_set_a @ X2 @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_1264_remove1__idem,axiom,
! [X2: set_set_a,Xs: list_set_set_a] :
( ~ ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ Xs ) )
=> ( ( remove1_set_set_a @ X2 @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_1265_notin__set__remove1,axiom,
! [X2: a,Xs: list_a,Y: a] :
( ~ ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ~ ( member_a2 @ X2 @ ( set_a2 @ ( remove1_a @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_1266_notin__set__remove1,axiom,
! [X2: set_a,Xs: list_set_a,Y: set_a] :
( ~ ( member_set_a2 @ X2 @ ( set_set_a2 @ Xs ) )
=> ~ ( member_set_a2 @ X2 @ ( set_set_a2 @ ( remove1_set_a @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_1267_notin__set__remove1,axiom,
! [X2: set_set_a,Xs: list_set_set_a,Y: set_set_a] :
( ~ ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ Xs ) )
=> ~ ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ ( remove1_set_set_a @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_1268_remove1_Osimps_I1_J,axiom,
! [X2: a] :
( ( remove1_a @ X2 @ nil_a )
= nil_a ) ).
% remove1.simps(1)
thf(fact_1269_remove1_Osimps_I1_J,axiom,
! [X2: set_a] :
( ( remove1_set_a @ X2 @ nil_set_a )
= nil_set_a ) ).
% remove1.simps(1)
thf(fact_1270_remove1_Osimps_I2_J,axiom,
! [X2: a,Y: a,Xs: list_a] :
( ( ( X2 = Y )
=> ( ( remove1_a @ X2 @ ( cons_a @ Y @ Xs ) )
= Xs ) )
& ( ( X2 != Y )
=> ( ( remove1_a @ X2 @ ( cons_a @ Y @ Xs ) )
= ( cons_a @ Y @ ( remove1_a @ X2 @ Xs ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_1271_remove1_Osimps_I2_J,axiom,
! [X2: set_a,Y: set_a,Xs: list_set_a] :
( ( ( X2 = Y )
=> ( ( remove1_set_a @ X2 @ ( cons_set_a @ Y @ Xs ) )
= Xs ) )
& ( ( X2 != Y )
=> ( ( remove1_set_a @ X2 @ ( cons_set_a @ Y @ Xs ) )
= ( cons_set_a @ Y @ ( remove1_set_a @ X2 @ Xs ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_1272_set__remove1__subset,axiom,
! [X2: set_a,Xs: list_set_a] : ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( remove1_set_a @ X2 @ Xs ) ) @ ( set_set_a2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_1273_set__remove1__subset,axiom,
! [X2: set_set_a,Xs: list_set_set_a] : ( ord_le5722252365846178494_set_a @ ( set_set_set_a2 @ ( remove1_set_set_a @ X2 @ Xs ) ) @ ( set_set_set_a2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_1274_set__remove1__subset,axiom,
! [X2: a,Xs: list_a] : ( ord_less_eq_set_a @ ( set_a2 @ ( remove1_a @ X2 @ Xs ) ) @ ( set_a2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_1275_remove1__append,axiom,
! [X2: set_a,Xs: list_set_a,Ys: list_set_a] :
( ( ( member_set_a2 @ X2 @ ( set_set_a2 @ Xs ) )
=> ( ( remove1_set_a @ X2 @ ( append_set_a @ Xs @ Ys ) )
= ( append_set_a @ ( remove1_set_a @ X2 @ Xs ) @ Ys ) ) )
& ( ~ ( member_set_a2 @ X2 @ ( set_set_a2 @ Xs ) )
=> ( ( remove1_set_a @ X2 @ ( append_set_a @ Xs @ Ys ) )
= ( append_set_a @ Xs @ ( remove1_set_a @ X2 @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_1276_remove1__append,axiom,
! [X2: set_set_a,Xs: list_set_set_a,Ys: list_set_set_a] :
( ( ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ Xs ) )
=> ( ( remove1_set_set_a @ X2 @ ( append_set_set_a @ Xs @ Ys ) )
= ( append_set_set_a @ ( remove1_set_set_a @ X2 @ Xs ) @ Ys ) ) )
& ( ~ ( member_set_set_a2 @ X2 @ ( set_set_set_a2 @ Xs ) )
=> ( ( remove1_set_set_a @ X2 @ ( append_set_set_a @ Xs @ Ys ) )
= ( append_set_set_a @ Xs @ ( remove1_set_set_a @ X2 @ Ys ) ) ) ) ) ).
% remove1_append
% Helper facts (13)
thf(help_If_2_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X2: set_a,Y: set_a] :
( ( if_set_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X2: set_a,Y: set_a] :
( ( if_set_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X2: list_a,Y: list_a] :
( ( if_list_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X2: list_a,Y: list_a] :
( ( if_list_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_T,axiom,
! [X2: set_set_a,Y: set_set_a] :
( ( if_set_set_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_T,axiom,
! [X2: set_set_a,Y: set_set_a] :
( ( if_set_set_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Set__Oset_Itf__a_J_J_T,axiom,
! [X2: list_set_a,Y: list_set_a] :
( ( if_list_set_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Set__Oset_Itf__a_J_J_T,axiom,
! [X2: list_set_a,Y: list_set_a] :
( ( if_list_set_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_T,axiom,
! [X2: set_set_set_a,Y: set_set_set_a] :
( ( if_set_set_set_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_T,axiom,
! [X2: set_set_set_a,Y: set_set_set_a] :
( ( if_set_set_set_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_T,axiom,
! [X2: list_set_set_a,Y: list_set_set_a] :
( ( if_list_set_set_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_T,axiom,
! [X2: list_set_set_a,Y: list_set_set_a] :
( ( if_list_set_set_a @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_le3724670747650509150_set_a @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ xs ) ) @ ( set_set_a2 @ ( undire7337870655677353998dges_a @ ( append_a @ xs @ ys ) ) ) ).
%------------------------------------------------------------------------------