TPTP Problem File: SLH0679^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% 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 : VYDRA_MDL/0007_MDL/prob_00336_013793__16215774_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1449 ( 494 unt; 170 typ; 0 def)
% Number of atoms : 3795 (1261 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 11314 ( 536 ~; 53 |; 250 &;8567 @)
% ( 0 <=>;1908 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 459 ( 459 >; 0 *; 0 +; 0 <<)
% Number of symbols : 158 ( 155 usr; 19 con; 0-3 aty)
% Number of variables : 3383 ( 184 ^;3076 !; 123 ?;3383 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:52:36.456
%------------------------------------------------------------------------------
% Could-be-implicit typings (15)
thf(ty_n_t__List__Olist_It__List__Olist_It__MDL__Oformula_Itf__b_Mtf__a_J_J_J,type,
list_l1896549967257458927la_b_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__MDL__Oformula_Itf__b_Mtf__a_J_J_J,type,
set_list_formula_b_a: $tType ).
thf(ty_n_t__List__Olist_It__MDL__Oformula_Itf__b_Mtf__a_J_J,type,
list_formula_b_a: $tType ).
thf(ty_n_t__Set__Oset_It__MDL__Oformula_Itf__b_Mtf__a_J_J,type,
set_formula_b_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__Interval__O__092__060I__062_Itf__a_J,type,
i_a: $tType ).
thf(ty_n_t__MDL__Oformula_Itf__b_Mtf__a_J,type,
formula_b_a: $tType ).
thf(ty_n_t__MDL__Oregex_Itf__b_Mtf__a_J,type,
regex_b_a: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $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_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (155)
thf(sy_c_Finite__Set_Ocard_001t__MDL__Oformula_Itf__b_Mtf__a_J,type,
finite7932102720334033959la_b_a: set_formula_b_a > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__MDL__Oformula_Itf__b_Mtf__a_J_J,type,
finite4601825613486469110la_b_a: set_list_formula_b_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_Itf__a_J,type,
finite_finite_list_a: set_list_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__MDL__Oformula_Itf__b_Mtf__a_J,type,
finite4096952451150804198la_b_a: set_formula_b_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__MDL__Oformula_Itf__b_Mtf__a_J_J,type,
minus_2577195155700852062la_b_a: set_formula_b_a > set_formula_b_a > set_formula_b_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
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_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__a,type,
zero_zero_a: a ).
thf(sy_c_If_001t__List__Olist_It__MDL__Oformula_Itf__b_Mtf__a_J_J,type,
if_list_formula_b_a: $o > list_formula_b_a > list_formula_b_a > list_formula_b_a ).
thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
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__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
infini8530281810654367211te_nat: set_nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__MDL__Oformula_Itf__b_Mtf__a_J_J,type,
inf_in5034913211621613591la_b_a: set_formula_b_a > set_formula_b_a > set_formula_b_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__MDL__Oformula_Itf__b_Mtf__a_J_J,type,
sup_su8125743748909105329la_b_a: set_formula_b_a > set_formula_b_a > set_formula_b_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
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_Osup__class_Osup_001tf__a,type,
sup_sup_a: a > a > a ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
lattic8265883725875713057ax_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Nat__Onat,type,
lattic8721135487736765967in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__MDL__Oformula_Itf__b_Mtf__a_J_001t__Nat__Onat,type,
lattic257815086748031546_a_nat: ( formula_b_a > nat ) > set_formula_b_a > formula_b_a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001tf__a_001t__Nat__Onat,type,
lattic6340287419671400565_a_nat: ( a > nat ) > set_a > a ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Nat__Onat,type,
lattic5238388535129920115in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Nat__Onat,type,
lattic1093996805478795353in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001tf__a,type,
lattic6792493950031347381_fin_a: set_a > a ).
thf(sy_c_List_Ocan__select_001t__MDL__Oformula_Itf__b_Mtf__a_J,type,
can_se6587019059267173266la_b_a: ( formula_b_a > $o ) > set_formula_b_a > $o ).
thf(sy_c_List_Ocan__select_001t__Nat__Onat,type,
can_select_nat: ( nat > $o ) > set_nat > $o ).
thf(sy_c_List_Ocan__select_001tf__a,type,
can_select_a: ( a > $o ) > set_a > $o ).
thf(sy_c_List_Oconcat_001t__MDL__Oformula_Itf__b_Mtf__a_J,type,
concat_formula_b_a: list_l1896549967257458927la_b_a > list_formula_b_a ).
thf(sy_c_List_Oconcat_001tf__a,type,
concat_a: list_list_a > list_a ).
thf(sy_c_List_Ocoset_001t__MDL__Oformula_Itf__b_Mtf__a_J,type,
coset_formula_b_a: list_formula_b_a > set_formula_b_a ).
thf(sy_c_List_Ocoset_001t__Nat__Onat,type,
coset_nat: list_nat > set_nat ).
thf(sy_c_List_Ocoset_001tf__a,type,
coset_a: list_a > set_a ).
thf(sy_c_List_Odistinct_001t__List__Olist_It__MDL__Oformula_Itf__b_Mtf__a_J_J,type,
distin8544179423821613478la_b_a: list_l1896549967257458927la_b_a > $o ).
thf(sy_c_List_Odistinct_001t__List__Olist_Itf__a_J,type,
distinct_list_a: list_list_a > $o ).
thf(sy_c_List_Odistinct_001t__MDL__Oformula_Itf__b_Mtf__a_J,type,
distinct_formula_b_a: list_formula_b_a > $o ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Odistinct_001tf__a,type,
distinct_a: list_a > $o ).
thf(sy_c_List_Oinsert_001t__MDL__Oformula_Itf__b_Mtf__a_J,type,
insert_formula_b_a: formula_b_a > list_formula_b_a > list_formula_b_a ).
thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oinsert_001tf__a,type,
insert_a: a > list_a > list_a ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
linord2614967742042102400et_nat: set_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__MDL__Oformula_Itf__b_Mtf__a_J,type,
cons_formula_b_a: formula_b_a > list_formula_b_a > list_formula_b_a ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001t__MDL__Oformula_Itf__b_Mtf__a_J,type,
nil_formula_b_a: list_formula_b_a ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__MDL__Oformula_Itf__b_Mtf__a_J_J,type,
set_list_formula_b_a2: list_l1896549967257458927la_b_a > set_list_formula_b_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__MDL__Oformula_Itf__b_Mtf__a_J,type,
set_formula_b_a2: list_formula_b_a > set_formula_b_a ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist__ex1_001t__MDL__Oformula_Itf__b_Mtf__a_J,type,
list_ex1_formula_b_a: ( formula_b_a > $o ) > list_formula_b_a > $o ).
thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex1_001tf__a,type,
list_ex1_a: ( a > $o ) > list_a > $o ).
thf(sy_c_List_Olist__update_001t__MDL__Oformula_Itf__b_Mtf__a_J,type,
list_u3614645878711143009la_b_a: list_formula_b_a > nat > formula_b_a > list_formula_b_a ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_Olist__update_001tf__a,type,
list_update_a: list_a > nat > a > list_a ).
thf(sy_c_List_Omember_001t__MDL__Oformula_Itf__b_Mtf__a_J,type,
member_formula_b_a: list_formula_b_a > formula_b_a > $o ).
thf(sy_c_List_Omember_001t__Nat__Onat,type,
member_nat: list_nat > nat > $o ).
thf(sy_c_List_Omember_001tf__a,type,
member_a: list_a > a > $o ).
thf(sy_c_List_On__lists_001tf__a,type,
n_lists_a: nat > list_a > list_list_a ).
thf(sy_c_List_Onth_001t__MDL__Oformula_Itf__b_Mtf__a_J,type,
nth_formula_b_a: list_formula_b_a > nat > formula_b_a ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_Onull_001tf__a,type,
null_a: list_a > $o ).
thf(sy_c_List_Oproduct__lists_001tf__a,type,
product_lists_a: list_list_a > list_list_a ).
thf(sy_c_List_Oremove1_001t__MDL__Oformula_Itf__b_Mtf__a_J,type,
remove1_formula_b_a: formula_b_a > list_formula_b_a > list_formula_b_a ).
thf(sy_c_List_Oremove1_001t__Nat__Onat,type,
remove1_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oremove1_001tf__a,type,
remove1_a: a > list_a > list_a ).
thf(sy_c_List_OremoveAll_001t__List__Olist_It__MDL__Oformula_Itf__b_Mtf__a_J_J,type,
remove8334650765243156947la_b_a: list_formula_b_a > list_l1896549967257458927la_b_a > list_l1896549967257458927la_b_a ).
thf(sy_c_List_OremoveAll_001t__List__Olist_Itf__a_J,type,
removeAll_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_OremoveAll_001t__MDL__Oformula_Itf__b_Mtf__a_J,type,
remove5015878800009109827la_b_a: formula_b_a > list_formula_b_a > list_formula_b_a ).
thf(sy_c_List_OremoveAll_001t__Nat__Onat,type,
removeAll_nat: nat > list_nat > list_nat ).
thf(sy_c_List_OremoveAll_001tf__a,type,
removeAll_a: a > list_a > list_a ).
thf(sy_c_List_Ounion_001tf__a,type,
union_a: list_a > list_a > list_a ).
thf(sy_c_MDL_Oatms_001tf__b_001tf__a,type,
atms_b_a: regex_b_a > set_formula_b_a ).
thf(sy_c_MDL_Oformula_OMatchP_001tf__a_001tf__b,type,
matchP_a_b: i_a > regex_b_a > formula_b_a ).
thf(sy_c_MDL_Oprogress_001tf__b_001tf__a,type,
progress_b_a: formula_b_a > list_a > nat ).
thf(sy_c_MDL_Orderive_001tf__b_001tf__a,type,
rderive_b_a: regex_b_a > regex_b_a ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
size_s349497388124573686list_a: list_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__MDL__Oformula_Itf__b_Mtf__a_J_J,type,
size_s6861460340215666547la_b_a: list_formula_b_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__MDL__Oformula_Itf__b_Mtf__a_J,type,
size_s1229512387538370275la_b_a: formula_b_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__MDL__Oregex_Itf__b_Mtf__a_J,type,
size_size_regex_b_a: regex_b_a > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__MDL__Oformula_Itf__b_Mtf__a_J_M_Eo_J,type,
bot_bo6442232419959819692_b_a_o: formula_b_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__MDL__Oformula_Itf__b_Mtf__a_J_J,type,
bot_bo7861856631361375769la_b_a: set_formula_b_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__MDL__Oformula_Itf__b_Mtf__a_J_J,type,
ord_le976137276181116377la_b_a: set_formula_b_a > set_formula_b_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__MDL__Oformula_Itf__b_Mtf__a_J_M_Eo_J,type,
ord_le2585840208699550432_b_a_o: ( formula_b_a > $o ) > ( formula_b_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__MDL__Oformula_Itf__b_Mtf__a_J_J_J,type,
ord_le7364111620167700725la_b_a: set_list_formula_b_a > set_list_formula_b_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_le8861187494160871172list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__MDL__Oformula_Itf__b_Mtf__a_J_J,type,
ord_le5472159299058833381la_b_a: set_formula_b_a > set_formula_b_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
ord_less_eq_a: a > a > $o ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__MDL__Oformula_Itf__b_Mtf__a_J_J,type,
ord_ma3619480027648345712la_b_a: set_formula_b_a > set_formula_b_a > set_formula_b_a ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
ord_min_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Set__Oset_It__MDL__Oformula_Itf__b_Mtf__a_J_J,type,
ord_mi284285648586469982la_b_a: set_formula_b_a > set_formula_b_a > set_formula_b_a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__MDL__Oformula_Itf__b_Mtf__a_J_J,type,
collec167716771123398964la_b_a: ( list_formula_b_a > $o ) > set_list_formula_b_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001t__MDL__Oformula_Itf__b_Mtf__a_J,type,
collect_formula_b_a: ( formula_b_a > $o ) > set_formula_b_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001t__MDL__Oformula_Itf__b_Mtf__a_J,type,
insert_formula_b_a2: formula_b_a > set_formula_b_a > set_formula_b_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat2: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a2: a > set_a > set_a ).
thf(sy_c_Set_Ois__empty_001t__MDL__Oformula_Itf__b_Mtf__a_J,type,
is_empty_formula_b_a: set_formula_b_a > $o ).
thf(sy_c_Set_Ois__empty_001tf__a,type,
is_empty_a: set_a > $o ).
thf(sy_c_Set_Ois__singleton_001t__MDL__Oformula_Itf__b_Mtf__a_J,type,
is_sin2179658930933931611la_b_a: set_formula_b_a > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Oremove_001t__MDL__Oformula_Itf__b_Mtf__a_J,type,
remove_formula_b_a: formula_b_a > set_formula_b_a > set_formula_b_a ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oremove_001tf__a,type,
remove_a: a > set_a > set_a ).
thf(sy_c_Set_Othe__elem_001t__MDL__Oformula_Itf__b_Mtf__a_J,type,
the_elem_formula_b_a: set_formula_b_a > formula_b_a ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_Timestamp_Otfin__class_Otfin_001t__Nat__Onat,type,
tfin_tfin_nat: set_nat ).
thf(sy_c_Timestamp_Otfin__class_Otfin_001tf__a,type,
tfin_tfin_a: set_a ).
thf(sy_c_member_001t__List__Olist_It__MDL__Oformula_Itf__b_Mtf__a_J_J,type,
member6997303959060339446la_b_a: list_formula_b_a > set_list_formula_b_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__MDL__Oformula_Itf__b_Mtf__a_J,type,
member_formula_b_a2: formula_b_a > set_formula_b_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat2: nat > set_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a2: a > set_a > $o ).
thf(sy_v_I____,type,
i: i_a ).
thf(sy_v_r____,type,
r: regex_b_a ).
thf(sy_v_ts,type,
ts: list_a ).
thf(sy_v_tsa____,type,
tsa: list_a ).
% Relevant facts (1269)
thf(fact_0__C9_Oprems_C,axiom,
! [T: a] :
( ( member_a2 @ T @ ( set_a2 @ tsa ) )
=> ( member_a2 @ T @ tfin_tfin_a ) ) ).
% "9.prems"
thf(fact_1__092_060open_062atms_Ar_A_092_060noteq_062_A_123_125_092_060close_062,axiom,
( ( atms_b_a @ r )
!= bot_bo7861856631361375769la_b_a ) ).
% \<open>atms r \<noteq> {}\<close>
thf(fact_2__092_060open_062finite_A_Iatms_Ar_J_092_060close_062,axiom,
finite4096952451150804198la_b_a @ ( atms_b_a @ r ) ).
% \<open>finite (atms r)\<close>
thf(fact_3_formula_Oinject_I9_J,axiom,
! [X91: i_a,X92: regex_b_a,Y91: i_a,Y92: regex_b_a] :
( ( ( matchP_a_b @ X91 @ X92 )
= ( matchP_a_b @ Y91 @ Y92 ) )
= ( ( X91 = Y91 )
& ( X92 = Y92 ) ) ) ).
% formula.inject(9)
thf(fact_4__C9_OIH_C,axiom,
! [X: formula_b_a] :
( ( member_formula_b_a2 @ X @ ( atms_b_a @ r ) )
=> ( ! [T2: a] :
( ( member_a2 @ T2 @ ( set_a2 @ tsa ) )
=> ( member_a2 @ T2 @ tfin_tfin_a ) )
=> ( ord_less_eq_nat @ ( progress_b_a @ X @ tsa ) @ ( size_size_list_a @ tsa ) ) ) ) ).
% "9.IH"
thf(fact_5_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_6_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_7_order__refl,axiom,
! [X: set_formula_b_a] : ( ord_le5472159299058833381la_b_a @ X @ X ) ).
% order_refl
thf(fact_8_order__refl,axiom,
! [X: a] : ( ord_less_eq_a @ X @ X ) ).
% order_refl
thf(fact_9_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_10_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_11_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_12_dual__order_Orefl,axiom,
! [A: set_formula_b_a] : ( ord_le5472159299058833381la_b_a @ A @ A ) ).
% dual_order.refl
thf(fact_13_dual__order_Orefl,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% dual_order.refl
thf(fact_14_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_15_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_nat] :
( ( size_size_list_nat @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_16_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_formula_b_a] :
( ( size_s6861460340215666547la_b_a @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_17_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_list_a] :
( ( size_s349497388124573686list_a @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_18_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_a] :
( ( size_size_list_a @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_19_neq__if__length__neq,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_20_neq__if__length__neq,axiom,
! [Xs2: list_formula_b_a,Ys: list_formula_b_a] :
( ( ( size_s6861460340215666547la_b_a @ Xs2 )
!= ( size_s6861460340215666547la_b_a @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_21_neq__if__length__neq,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
!= ( size_s349497388124573686list_a @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_22_neq__if__length__neq,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs2 )
!= ( size_size_list_a @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_23_size__neq__size__imp__neq,axiom,
! [X: formula_b_a,Y: formula_b_a] :
( ( ( size_s1229512387538370275la_b_a @ X )
!= ( size_s1229512387538370275la_b_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_24_size__neq__size__imp__neq,axiom,
! [X: regex_b_a,Y: regex_b_a] :
( ( ( size_size_regex_b_a @ X )
!= ( size_size_regex_b_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_25_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_26_size__neq__size__imp__neq,axiom,
! [X: list_formula_b_a,Y: list_formula_b_a] :
( ( ( size_s6861460340215666547la_b_a @ X )
!= ( size_s6861460340215666547la_b_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_27_size__neq__size__imp__neq,axiom,
! [X: list_list_a,Y: list_list_a] :
( ( ( size_s349497388124573686list_a @ X )
!= ( size_s349497388124573686list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_28_size__neq__size__imp__neq,axiom,
! [X: list_a,Y: list_a] :
( ( ( size_size_list_a @ X )
!= ( size_size_list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_29_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_30_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_31_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_32_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_33_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_34_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_35_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M2: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M2 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_36_bot__apply,axiom,
( bot_bo6442232419959819692_b_a_o
= ( ^ [X4: formula_b_a] : bot_bot_o ) ) ).
% bot_apply
thf(fact_37_bot__apply,axiom,
( bot_bot_nat_o
= ( ^ [X4: nat] : bot_bot_o ) ) ).
% bot_apply
thf(fact_38_bot__apply,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : bot_bot_o ) ) ).
% bot_apply
thf(fact_39_assms,axiom,
! [T: a] :
( ( member_a2 @ T @ ( set_a2 @ ts ) )
=> ( member_a2 @ T @ tfin_tfin_a ) ) ).
% assms
thf(fact_40_List_Ofinite__set,axiom,
! [Xs2: list_l1896549967257458927la_b_a] : ( finite4601825613486469110la_b_a @ ( set_list_formula_b_a2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_41_List_Ofinite__set,axiom,
! [Xs2: list_list_a] : ( finite_finite_list_a @ ( set_list_a2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_42_List_Ofinite__set,axiom,
! [Xs2: list_a] : ( finite_finite_a @ ( set_a2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_43_List_Ofinite__set,axiom,
! [Xs2: list_formula_b_a] : ( finite4096952451150804198la_b_a @ ( set_formula_b_a2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_44_List_Ofinite__set,axiom,
! [Xs2: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_45_subset__code_I1_J,axiom,
! [Xs2: list_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs2 ) @ B2 )
= ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Xs2 ) )
=> ( member_list_a @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_46_subset__code_I1_J,axiom,
! [Xs2: list_l1896549967257458927la_b_a,B2: set_list_formula_b_a] :
( ( ord_le7364111620167700725la_b_a @ ( set_list_formula_b_a2 @ Xs2 ) @ B2 )
= ( ! [X4: list_formula_b_a] :
( ( member6997303959060339446la_b_a @ X4 @ ( set_list_formula_b_a2 @ Xs2 ) )
=> ( member6997303959060339446la_b_a @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_47_subset__code_I1_J,axiom,
! [Xs2: list_formula_b_a,B2: set_formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ ( set_formula_b_a2 @ Xs2 ) @ B2 )
= ( ! [X4: formula_b_a] :
( ( member_formula_b_a2 @ X4 @ ( set_formula_b_a2 @ Xs2 ) )
=> ( member_formula_b_a2 @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_48_subset__code_I1_J,axiom,
! [Xs2: list_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B2 )
= ( ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs2 ) )
=> ( member_nat2 @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_49_subset__code_I1_J,axiom,
! [Xs2: list_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ B2 )
= ( ! [X4: a] :
( ( member_a2 @ X4 @ ( set_a2 @ Xs2 ) )
=> ( member_a2 @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_50_finite__list,axiom,
! [A2: set_list_formula_b_a] :
( ( finite4601825613486469110la_b_a @ A2 )
=> ? [Xs: list_l1896549967257458927la_b_a] :
( ( set_list_formula_b_a2 @ Xs )
= A2 ) ) ).
% finite_list
thf(fact_51_finite__list,axiom,
! [A2: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ? [Xs: list_list_a] :
( ( set_list_a2 @ Xs )
= A2 ) ) ).
% finite_list
thf(fact_52_finite__list,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ? [Xs: list_a] :
( ( set_a2 @ Xs )
= A2 ) ) ).
% finite_list
thf(fact_53_finite__list,axiom,
! [A2: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ A2 )
=> ? [Xs: list_formula_b_a] :
( ( set_formula_b_a2 @ Xs )
= A2 ) ) ).
% finite_list
thf(fact_54_finite__list,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ? [Xs: list_nat] :
( ( set_nat2 @ Xs )
= A2 ) ) ).
% finite_list
thf(fact_55_bot__fun__def,axiom,
( bot_bo6442232419959819692_b_a_o
= ( ^ [X4: formula_b_a] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_56_bot__fun__def,axiom,
( bot_bot_nat_o
= ( ^ [X4: nat] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_57_bot__fun__def,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_58_atms__finite,axiom,
! [R: regex_b_a] : ( finite4096952451150804198la_b_a @ ( atms_b_a @ R ) ) ).
% atms_finite
thf(fact_59_atms__nonempty,axiom,
! [R: regex_b_a] :
( ( atms_b_a @ R )
!= bot_bo7861856631361375769la_b_a ) ).
% atms_nonempty
thf(fact_60_bot_Oextremum__uniqueI,axiom,
! [A: formula_b_a > $o] :
( ( ord_le2585840208699550432_b_a_o @ A @ bot_bo6442232419959819692_b_a_o )
=> ( A = bot_bo6442232419959819692_b_a_o ) ) ).
% bot.extremum_uniqueI
thf(fact_61_bot_Oextremum__uniqueI,axiom,
! [A: nat > $o] :
( ( ord_less_eq_nat_o @ A @ bot_bot_nat_o )
=> ( A = bot_bot_nat_o ) ) ).
% bot.extremum_uniqueI
thf(fact_62_bot_Oextremum__uniqueI,axiom,
! [A: a > $o] :
( ( ord_less_eq_a_o @ A @ bot_bot_a_o )
=> ( A = bot_bot_a_o ) ) ).
% bot.extremum_uniqueI
thf(fact_63_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_64_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_65_bot_Oextremum__uniqueI,axiom,
! [A: set_formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ A @ bot_bo7861856631361375769la_b_a )
=> ( A = bot_bo7861856631361375769la_b_a ) ) ).
% bot.extremum_uniqueI
thf(fact_66_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_67_bot_Oextremum__unique,axiom,
! [A: formula_b_a > $o] :
( ( ord_le2585840208699550432_b_a_o @ A @ bot_bo6442232419959819692_b_a_o )
= ( A = bot_bo6442232419959819692_b_a_o ) ) ).
% bot.extremum_unique
thf(fact_68_bot_Oextremum__unique,axiom,
! [A: nat > $o] :
( ( ord_less_eq_nat_o @ A @ bot_bot_nat_o )
= ( A = bot_bot_nat_o ) ) ).
% bot.extremum_unique
thf(fact_69_bot_Oextremum__unique,axiom,
! [A: a > $o] :
( ( ord_less_eq_a_o @ A @ bot_bot_a_o )
= ( A = bot_bot_a_o ) ) ).
% bot.extremum_unique
thf(fact_70_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_71_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_72_bot_Oextremum__unique,axiom,
! [A: set_formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ A @ bot_bo7861856631361375769la_b_a )
= ( A = bot_bo7861856631361375769la_b_a ) ) ).
% bot.extremum_unique
thf(fact_73_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_74_bot_Oextremum,axiom,
! [A: formula_b_a > $o] : ( ord_le2585840208699550432_b_a_o @ bot_bo6442232419959819692_b_a_o @ A ) ).
% bot.extremum
thf(fact_75_bot_Oextremum,axiom,
! [A: nat > $o] : ( ord_less_eq_nat_o @ bot_bot_nat_o @ A ) ).
% bot.extremum
thf(fact_76_bot_Oextremum,axiom,
! [A: a > $o] : ( ord_less_eq_a_o @ bot_bot_a_o @ A ) ).
% bot.extremum
thf(fact_77_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_78_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_79_bot_Oextremum,axiom,
! [A: set_formula_b_a] : ( ord_le5472159299058833381la_b_a @ bot_bo7861856631361375769la_b_a @ A ) ).
% bot.extremum
thf(fact_80_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_81_order__antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_82_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_83_order__antisym__conv,axiom,
! [Y: set_formula_b_a,X: set_formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ Y @ X )
=> ( ( ord_le5472159299058833381la_b_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_84_order__antisym__conv,axiom,
! [Y: a,X: a] :
( ( ord_less_eq_a @ Y @ X )
=> ( ( ord_less_eq_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_85_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_86_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_87_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_88_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_89_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_90_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_91_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_92_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_93_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_94_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_95_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_96_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > a,C: a] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_97_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_98_ord__eq__le__subst,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_99_ord__eq__le__subst,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_100_ord__eq__le__subst,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_101_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_102_ord__eq__le__subst,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_103_ord__eq__le__subst,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_104_ord__eq__le__subst,axiom,
! [A: a,F: set_a > a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_105_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_106_ord__eq__le__subst,axiom,
! [A: a,F: set_nat > a,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_107_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_108_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_109_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_110_order__eq__refl,axiom,
! [X: set_formula_b_a,Y: set_formula_b_a] :
( ( X = Y )
=> ( ord_le5472159299058833381la_b_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_111_order__eq__refl,axiom,
! [X: a,Y: a] :
( ( X = Y )
=> ( ord_less_eq_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_112_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_113_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_114_order__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_115_order__subst2,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_116_order__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_117_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_118_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_119_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_120_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_121_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_122_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > a,C: a] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_123_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_124_order__subst1,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_125_order__subst1,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_126_order__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_127_order__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_128_order__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_129_order__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_130_order__subst1,axiom,
! [A: set_a,F: a > set_a,B: a,C: a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_131_order__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_132_order__subst1,axiom,
! [A: set_nat,F: a > set_nat,B: a,C: a] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X2: a,Y2: a] :
( ( ord_less_eq_a @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_133_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = 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_134_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_135_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_formula_b_a,Z: set_formula_b_a] : ( Y4 = Z ) )
= ( ^ [A3: set_formula_b_a,B3: set_formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ A3 @ B3 )
& ( ord_le5472159299058833381la_b_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_136_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: a,Z: a] : ( Y4 = Z ) )
= ( ^ [A3: a,B3: a] :
( ( ord_less_eq_a @ A3 @ B3 )
& ( ord_less_eq_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_137_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_138_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_139_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_140_antisym,axiom,
! [A: set_formula_b_a,B: set_formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ A @ B )
=> ( ( ord_le5472159299058833381la_b_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_141_antisym,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_142_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_143_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_144_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_145_dual__order_Otrans,axiom,
! [B: set_formula_b_a,A: set_formula_b_a,C: set_formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ B @ A )
=> ( ( ord_le5472159299058833381la_b_a @ C @ B )
=> ( ord_le5472159299058833381la_b_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_146_dual__order_Otrans,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_eq_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_147_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_148_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_149_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_150_dual__order_Oantisym,axiom,
! [B: set_formula_b_a,A: set_formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ B @ A )
=> ( ( ord_le5472159299058833381la_b_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_151_dual__order_Oantisym,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_152_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_153_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = 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_154_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
& ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_155_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_formula_b_a,Z: set_formula_b_a] : ( Y4 = Z ) )
= ( ^ [A3: set_formula_b_a,B3: set_formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ B3 @ A3 )
& ( ord_le5472159299058833381la_b_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_156_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: a,Z: a] : ( Y4 = Z ) )
= ( ^ [A3: a,B3: a] :
( ( ord_less_eq_a @ B3 @ A3 )
& ( ord_less_eq_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_157_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_158_mem__Collect__eq,axiom,
! [A: list_a,P: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_159_mem__Collect__eq,axiom,
! [A: list_formula_b_a,P: list_formula_b_a > $o] :
( ( member6997303959060339446la_b_a @ A @ ( collec167716771123398964la_b_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_160_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a2 @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_161_mem__Collect__eq,axiom,
! [A: formula_b_a,P: formula_b_a > $o] :
( ( member_formula_b_a2 @ A @ ( collect_formula_b_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_162_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat2 @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_163_Collect__mem__eq,axiom,
! [A2: set_list_a] :
( ( collect_list_a
@ ^ [X4: list_a] : ( member_list_a @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_164_Collect__mem__eq,axiom,
! [A2: set_list_formula_b_a] :
( ( collec167716771123398964la_b_a
@ ^ [X4: list_formula_b_a] : ( member6997303959060339446la_b_a @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_165_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a2 @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_166_Collect__mem__eq,axiom,
! [A2: set_formula_b_a] :
( ( collect_formula_b_a
@ ^ [X4: formula_b_a] : ( member_formula_b_a2 @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_167_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat2 @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_168_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_169_Collect__cong,axiom,
! [P: formula_b_a > $o,Q: formula_b_a > $o] :
( ! [X2: formula_b_a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_formula_b_a @ P )
= ( collect_formula_b_a @ Q ) ) ) ).
% Collect_cong
thf(fact_170_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_171_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_172_order__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_173_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z2 )
=> ( ord_less_eq_set_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_174_order__trans,axiom,
! [X: set_formula_b_a,Y: set_formula_b_a,Z2: set_formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ X @ Y )
=> ( ( ord_le5472159299058833381la_b_a @ Y @ Z2 )
=> ( ord_le5472159299058833381la_b_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_175_order__trans,axiom,
! [X: a,Y: a,Z2: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ Z2 )
=> ( ord_less_eq_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_176_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_177_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_178_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_179_order_Otrans,axiom,
! [A: set_formula_b_a,B: set_formula_b_a,C: set_formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ A @ B )
=> ( ( ord_le5472159299058833381la_b_a @ B @ C )
=> ( ord_le5472159299058833381la_b_a @ A @ C ) ) ) ).
% order.trans
thf(fact_180_order_Otrans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% order.trans
thf(fact_181_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_182_order__antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_183_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_184_order__antisym,axiom,
! [X: set_formula_b_a,Y: set_formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ X @ Y )
=> ( ( ord_le5472159299058833381la_b_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_185_order__antisym,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_186_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_187_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_188_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_189_ord__le__eq__trans,axiom,
! [A: set_formula_b_a,B: set_formula_b_a,C: set_formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ A @ B )
=> ( ( B = C )
=> ( ord_le5472159299058833381la_b_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_190_ord__le__eq__trans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_191_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_192_ord__eq__le__trans,axiom,
! [A: set_formula_b_a,B: set_formula_b_a,C: set_formula_b_a] :
( ( A = B )
=> ( ( ord_le5472159299058833381la_b_a @ B @ C )
=> ( ord_le5472159299058833381la_b_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_193_ord__eq__le__trans,axiom,
! [A: a,B: a,C: a] :
( ( A = B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_194_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_195_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_196_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_197_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_198_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat2 @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_199_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat2 @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_200_empty__iff,axiom,
! [C: a] :
~ ( member_a2 @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_201_empty__iff,axiom,
! [C: nat] :
~ ( member_nat2 @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_202_empty__iff,axiom,
! [C: formula_b_a] :
~ ( member_formula_b_a2 @ C @ bot_bo7861856631361375769la_b_a ) ).
% empty_iff
thf(fact_203_subset__empty,axiom,
! [A2: set_formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ A2 @ bot_bo7861856631361375769la_b_a )
= ( A2 = bot_bo7861856631361375769la_b_a ) ) ).
% subset_empty
thf(fact_204_empty__subsetI,axiom,
! [A2: set_formula_b_a] : ( ord_le5472159299058833381la_b_a @ bot_bo7861856631361375769la_b_a @ A2 ) ).
% empty_subsetI
thf(fact_205_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X4: a] :
~ ( member_a2 @ X4 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_206_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X4: nat] :
~ ( member_nat2 @ X4 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_207_all__not__in__conv,axiom,
! [A2: set_formula_b_a] :
( ( ! [X4: formula_b_a] :
~ ( member_formula_b_a2 @ X4 @ A2 ) )
= ( A2 = bot_bo7861856631361375769la_b_a ) ) ).
% all_not_in_conv
thf(fact_208_Collect__empty__eq,axiom,
! [P: formula_b_a > $o] :
( ( ( collect_formula_b_a @ P )
= bot_bo7861856631361375769la_b_a )
= ( ! [X4: formula_b_a] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_209_empty__Collect__eq,axiom,
! [P: formula_b_a > $o] :
( ( bot_bo7861856631361375769la_b_a
= ( collect_formula_b_a @ P ) )
= ( ! [X4: formula_b_a] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_210_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_211_finite_OemptyI,axiom,
finite4096952451150804198la_b_a @ bot_bo7861856631361375769la_b_a ).
% finite.emptyI
thf(fact_212_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_213_infinite__imp__nonempty,axiom,
! [S: set_formula_b_a] :
( ~ ( finite4096952451150804198la_b_a @ S )
=> ( S != bot_bo7861856631361375769la_b_a ) ) ).
% infinite_imp_nonempty
thf(fact_214_finite__transitivity__chain,axiom,
! [A2: set_a,R2: a > a > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [X2: a] :
~ ( R2 @ X2 @ X2 )
=> ( ! [X2: a,Y2: a,Z3: a] :
( ( R2 @ X2 @ Y2 )
=> ( ( R2 @ Y2 @ Z3 )
=> ( R2 @ X2 @ Z3 ) ) )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ A2 )
=> ? [Y3: a] :
( ( member_a2 @ Y3 @ A2 )
& ( R2 @ X2 @ Y3 ) ) )
=> ( A2 = bot_bot_set_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_215_finite__transitivity__chain,axiom,
! [A2: set_nat,R2: nat > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [X2: nat] :
~ ( R2 @ X2 @ X2 )
=> ( ! [X2: nat,Y2: nat,Z3: nat] :
( ( R2 @ X2 @ Y2 )
=> ( ( R2 @ Y2 @ Z3 )
=> ( R2 @ X2 @ Z3 ) ) )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ? [Y3: nat] :
( ( member_nat2 @ Y3 @ A2 )
& ( R2 @ X2 @ Y3 ) ) )
=> ( A2 = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_216_finite__transitivity__chain,axiom,
! [A2: set_formula_b_a,R2: formula_b_a > formula_b_a > $o] :
( ( finite4096952451150804198la_b_a @ A2 )
=> ( ! [X2: formula_b_a] :
~ ( R2 @ X2 @ X2 )
=> ( ! [X2: formula_b_a,Y2: formula_b_a,Z3: formula_b_a] :
( ( R2 @ X2 @ Y2 )
=> ( ( R2 @ Y2 @ Z3 )
=> ( R2 @ X2 @ Z3 ) ) )
=> ( ! [X2: formula_b_a] :
( ( member_formula_b_a2 @ X2 @ A2 )
=> ? [Y3: formula_b_a] :
( ( member_formula_b_a2 @ Y3 @ A2 )
& ( R2 @ X2 @ Y3 ) ) )
=> ( A2 = bot_bo7861856631361375769la_b_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_217_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X2: a] :
( ( member_a2 @ X2 @ A2 )
=> ( member_a2 @ X2 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_218_subsetI,axiom,
! [A2: set_formula_b_a,B2: set_formula_b_a] :
( ! [X2: formula_b_a] :
( ( member_formula_b_a2 @ X2 @ A2 )
=> ( member_formula_b_a2 @ X2 @ B2 ) )
=> ( ord_le5472159299058833381la_b_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_219_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( member_nat2 @ X2 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_220_in__mono,axiom,
! [A2: set_a,B2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a2 @ X @ A2 )
=> ( member_a2 @ X @ B2 ) ) ) ).
% in_mono
thf(fact_221_in__mono,axiom,
! [A2: set_formula_b_a,B2: set_formula_b_a,X: formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ A2 @ B2 )
=> ( ( member_formula_b_a2 @ X @ A2 )
=> ( member_formula_b_a2 @ X @ B2 ) ) ) ).
% in_mono
thf(fact_222_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( member_nat2 @ X @ B2 ) ) ) ).
% in_mono
thf(fact_223_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_224_subsetD,axiom,
! [A2: set_formula_b_a,B2: set_formula_b_a,C: formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ A2 @ B2 )
=> ( ( member_formula_b_a2 @ C @ A2 )
=> ( member_formula_b_a2 @ C @ B2 ) ) ) ).
% subsetD
thf(fact_225_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat2 @ C @ A2 )
=> ( member_nat2 @ C @ B2 ) ) ) ).
% subsetD
thf(fact_226_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X4: a] :
( ( member_a2 @ X4 @ A5 )
=> ( member_a2 @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_227_subset__eq,axiom,
( ord_le5472159299058833381la_b_a
= ( ^ [A5: set_formula_b_a,B5: set_formula_b_a] :
! [X4: formula_b_a] :
( ( member_formula_b_a2 @ X4 @ A5 )
=> ( member_formula_b_a2 @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_228_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X4: nat] :
( ( member_nat2 @ X4 @ A5 )
=> ( member_nat2 @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_229_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T3: a] :
( ( member_a2 @ T3 @ A5 )
=> ( member_a2 @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_230_subset__iff,axiom,
( ord_le5472159299058833381la_b_a
= ( ^ [A5: set_formula_b_a,B5: set_formula_b_a] :
! [T3: formula_b_a] :
( ( member_formula_b_a2 @ T3 @ A5 )
=> ( member_formula_b_a2 @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_231_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T3: nat] :
( ( member_nat2 @ T3 @ A5 )
=> ( member_nat2 @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_232_bot__set__def,axiom,
( bot_bo7861856631361375769la_b_a
= ( collect_formula_b_a @ bot_bo6442232419959819692_b_a_o ) ) ).
% bot_set_def
thf(fact_233_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M4: nat] :
? [N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
& ( member_nat2 @ N2 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_234_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M4: nat] :
! [X4: nat] :
( ( member_nat2 @ X4 @ N3 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_235_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X4: a] : ( member_a2 @ X4 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_236_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X4: nat] : ( member_nat2 @ X4 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_237_ex__in__conv,axiom,
! [A2: set_formula_b_a] :
( ( ? [X4: formula_b_a] : ( member_formula_b_a2 @ X4 @ A2 ) )
= ( A2 != bot_bo7861856631361375769la_b_a ) ) ).
% ex_in_conv
thf(fact_238_equals0I,axiom,
! [A2: set_a] :
( ! [Y2: a] :
~ ( member_a2 @ Y2 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_239_equals0I,axiom,
! [A2: set_nat] :
( ! [Y2: nat] :
~ ( member_nat2 @ Y2 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_240_equals0I,axiom,
! [A2: set_formula_b_a] :
( ! [Y2: formula_b_a] :
~ ( member_formula_b_a2 @ Y2 @ A2 )
=> ( A2 = bot_bo7861856631361375769la_b_a ) ) ).
% equals0I
thf(fact_241_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a2 @ A @ A2 ) ) ).
% equals0D
thf(fact_242_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat2 @ A @ A2 ) ) ).
% equals0D
thf(fact_243_equals0D,axiom,
! [A2: set_formula_b_a,A: formula_b_a] :
( ( A2 = bot_bo7861856631361375769la_b_a )
=> ~ ( member_formula_b_a2 @ A @ A2 ) ) ).
% equals0D
thf(fact_244_emptyE,axiom,
! [A: a] :
~ ( member_a2 @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_245_emptyE,axiom,
! [A: nat] :
~ ( member_nat2 @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_246_emptyE,axiom,
! [A: formula_b_a] :
~ ( member_formula_b_a2 @ A @ bot_bo7861856631361375769la_b_a ) ).
% emptyE
thf(fact_247_rev__finite__subset,axiom,
! [B2: set_formula_b_a,A2: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ B2 )
=> ( ( ord_le5472159299058833381la_b_a @ A2 @ B2 )
=> ( finite4096952451150804198la_b_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_248_rev__finite__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_249_infinite__super,axiom,
! [S: set_formula_b_a,T4: set_formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ S @ T4 )
=> ( ~ ( finite4096952451150804198la_b_a @ S )
=> ~ ( finite4096952451150804198la_b_a @ T4 ) ) ) ).
% infinite_super
thf(fact_250_infinite__super,axiom,
! [S: set_nat,T4: set_nat] :
( ( ord_less_eq_set_nat @ S @ T4 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T4 ) ) ) ).
% infinite_super
thf(fact_251_finite__subset,axiom,
! [A2: set_formula_b_a,B2: set_formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ A2 @ B2 )
=> ( ( finite4096952451150804198la_b_a @ B2 )
=> ( finite4096952451150804198la_b_a @ A2 ) ) ) ).
% finite_subset
thf(fact_252_finite__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_253_finite__has__maximal2,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a2 @ A @ A2 )
=> ? [X2: a] :
( ( member_a2 @ X2 @ A2 )
& ( ord_less_eq_a @ A @ X2 )
& ! [Xa: a] :
( ( member_a2 @ Xa @ A2 )
=> ( ( ord_less_eq_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_254_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
& ( ord_less_eq_nat @ A @ X2 )
& ! [Xa: nat] :
( ( member_nat2 @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_255_finite__has__minimal2,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a2 @ A @ A2 )
=> ? [X2: a] :
( ( member_a2 @ X2 @ A2 )
& ( ord_less_eq_a @ X2 @ A )
& ! [Xa: a] :
( ( member_a2 @ Xa @ A2 )
=> ( ( ord_less_eq_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_256_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
& ( ord_less_eq_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat2 @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_257_subset__emptyI,axiom,
! [A2: set_a] :
( ! [X2: a] :
~ ( member_a2 @ X2 @ A2 )
=> ( ord_less_eq_set_a @ A2 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_258_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X2: nat] :
~ ( member_nat2 @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_259_subset__emptyI,axiom,
! [A2: set_formula_b_a] :
( ! [X2: formula_b_a] :
~ ( member_formula_b_a2 @ X2 @ A2 )
=> ( ord_le5472159299058833381la_b_a @ A2 @ bot_bo7861856631361375769la_b_a ) ) ).
% subset_emptyI
thf(fact_260_Set_Ois__empty__def,axiom,
( is_empty_formula_b_a
= ( ^ [A5: set_formula_b_a] : ( A5 = bot_bo7861856631361375769la_b_a ) ) ) ).
% Set.is_empty_def
thf(fact_261_enumerate__mono__le__iff,axiom,
! [S: set_nat,M: nat,N: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( ord_less_eq_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% enumerate_mono_le_iff
thf(fact_262_arg__min__least,axiom,
! [S: set_a,Y: a,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ( ( member_a2 @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic6340287419671400565_a_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_263_arg__min__least,axiom,
! [S: set_nat,Y: nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat2 @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_264_arg__min__least,axiom,
! [S: set_formula_b_a,Y: formula_b_a,F: formula_b_a > nat] :
( ( finite4096952451150804198la_b_a @ S )
=> ( ( S != bot_bo7861856631361375769la_b_a )
=> ( ( member_formula_b_a2 @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic257815086748031546_a_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_265_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( set_nat2 @ ( linord2614967742042102400et_nat @ A2 ) )
= A2 ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_266_GreatestI2__order,axiom,
! [P: nat > $o,X: nat,Q: nat > $o] :
( ( P @ X )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_267_Greatest__equality,axiom,
! [P: nat > $o,X: nat] :
( ( P @ X )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( order_Greatest_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_268_subset__code_I2_J,axiom,
! [A2: set_formula_b_a,Ys: list_formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ A2 @ ( coset_formula_b_a @ Ys ) )
= ( ! [X4: formula_b_a] :
( ( member_formula_b_a2 @ X4 @ ( set_formula_b_a2 @ Ys ) )
=> ~ ( member_formula_b_a2 @ X4 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_269_subset__code_I2_J,axiom,
! [A2: set_nat,Ys: list_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( coset_nat @ Ys ) )
= ( ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Ys ) )
=> ~ ( member_nat2 @ X4 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_270_subset__code_I2_J,axiom,
! [A2: set_a,Ys: list_a] :
( ( ord_less_eq_set_a @ A2 @ ( coset_a @ Ys ) )
= ( ! [X4: a] :
( ( member_a2 @ X4 @ ( set_a2 @ Ys ) )
=> ~ ( member_a2 @ X4 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_271_in__set__member,axiom,
! [X: formula_b_a,Xs2: list_formula_b_a] :
( ( member_formula_b_a2 @ X @ ( set_formula_b_a2 @ Xs2 ) )
= ( member_formula_b_a @ Xs2 @ X ) ) ).
% in_set_member
thf(fact_272_in__set__member,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs2 ) )
= ( member_nat @ Xs2 @ X ) ) ).
% in_set_member
thf(fact_273_in__set__member,axiom,
! [X: a,Xs2: list_a] :
( ( member_a2 @ X @ ( set_a2 @ Xs2 ) )
= ( member_a @ Xs2 @ X ) ) ).
% in_set_member
thf(fact_274_enumerate__Ex,axiom,
! [S: set_nat,S2: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( member_nat2 @ S2 @ S )
=> ? [N4: nat] :
( ( infini8530281810654367211te_nat @ S @ N4 )
= S2 ) ) ) ).
% enumerate_Ex
thf(fact_275_le__enumerate,axiom,
! [S: set_nat,N: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ).
% le_enumerate
thf(fact_276_enumerate__in__set,axiom,
! [S: set_nat,N: nat] :
( ~ ( finite_finite_nat @ S )
=> ( member_nat2 @ ( infini8530281810654367211te_nat @ S @ N ) @ S ) ) ).
% enumerate_in_set
thf(fact_277_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( linord2614967742042102400et_nat @ A2 )
= ( linord2614967742042102400et_nat @ B2 ) )
=> ( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_278_GreatestI__ex__nat,axiom,
! [P: nat > $o,B: nat] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_279_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_280_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_281_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : ( member_a2 @ X4 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_282_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X4: nat] : ( member_nat2 @ X4 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_283_bot__empty__eq,axiom,
( bot_bo6442232419959819692_b_a_o
= ( ^ [X4: formula_b_a] : ( member_formula_b_a2 @ X4 @ bot_bo7861856631361375769la_b_a ) ) ) ).
% bot_empty_eq
thf(fact_284_Collect__empty__eq__bot,axiom,
! [P: formula_b_a > $o] :
( ( ( collect_formula_b_a @ P )
= bot_bo7861856631361375769la_b_a )
= ( P = bot_bo6442232419959819692_b_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_285_is__empty__set,axiom,
! [Xs2: list_a] :
( ( is_empty_a @ ( set_a2 @ Xs2 ) )
= ( null_a @ Xs2 ) ) ).
% is_empty_set
thf(fact_286_Sup__fin_Osubset__imp,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ ( lattic1093996805478795353in_nat @ B2 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_287_Inf__fin_Osubset__imp,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B2 ) @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_288_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( linord2614967742042102400et_nat @ A2 )
= nil_nat )
= ( A2 = bot_bot_set_nat ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_289_Max__mono,axiom,
! [M2: set_nat,N5: set_nat] :
( ( ord_less_eq_set_nat @ M2 @ N5 )
=> ( ( M2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N5 )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ M2 ) @ ( lattic8265883725875713057ax_nat @ N5 ) ) ) ) ) ).
% Max_mono
thf(fact_290_Max_Osubset__imp,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A2 ) @ ( lattic8265883725875713057ax_nat @ B2 ) ) ) ) ) ).
% Max.subset_imp
thf(fact_291_Min__antimono,axiom,
! [M2: set_nat,N5: set_nat] :
( ( ord_less_eq_set_nat @ M2 @ N5 )
=> ( ( M2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N5 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ N5 ) @ ( lattic8721135487736765967in_nat @ M2 ) ) ) ) ) ).
% Min_antimono
thf(fact_292_Min_Osubset__imp,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ B2 ) @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min.subset_imp
thf(fact_293_set__empty2,axiom,
! [Xs2: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs2 ) )
= ( Xs2 = nil_a ) ) ).
% set_empty2
thf(fact_294_set__empty2,axiom,
! [Xs2: list_formula_b_a] :
( ( bot_bo7861856631361375769la_b_a
= ( set_formula_b_a2 @ Xs2 ) )
= ( Xs2 = nil_formula_b_a ) ) ).
% set_empty2
thf(fact_295_set__empty,axiom,
! [Xs2: list_a] :
( ( ( set_a2 @ Xs2 )
= bot_bot_set_a )
= ( Xs2 = nil_a ) ) ).
% set_empty
thf(fact_296_set__empty,axiom,
! [Xs2: list_formula_b_a] :
( ( ( set_formula_b_a2 @ Xs2 )
= bot_bo7861856631361375769la_b_a )
= ( Xs2 = nil_formula_b_a ) ) ).
% set_empty
thf(fact_297_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( linord2614967742042102400et_nat @ A2 )
= nil_nat ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
thf(fact_298_Min_Obounded__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) )
= ( ! [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
=> ( ord_less_eq_nat @ X @ X4 ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_299_Max_Obounded__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A2 ) @ X )
= ( ! [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
=> ( ord_less_eq_nat @ X4 @ X ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_300_Inf__fin__le__Sup__fin,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_301_Min__le,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ X ) ) ) ).
% Min_le
thf(fact_302_Min__eqI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [Y2: nat] :
( ( member_nat2 @ Y2 @ A2 )
=> ( ord_less_eq_nat @ X @ Y2 ) )
=> ( ( member_nat2 @ X @ A2 )
=> ( ( lattic8721135487736765967in_nat @ A2 )
= X ) ) ) ) ).
% Min_eqI
thf(fact_303_Min_OcoboundedI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ A @ A2 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ A ) ) ) ).
% Min.coboundedI
thf(fact_304_Min__in,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( member_nat2 @ ( lattic8721135487736765967in_nat @ A2 ) @ A2 ) ) ) ).
% Min_in
thf(fact_305_Max__ge,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( ord_less_eq_nat @ X @ ( lattic8265883725875713057ax_nat @ A2 ) ) ) ) ).
% Max_ge
thf(fact_306_Max__eqI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [Y2: nat] :
( ( member_nat2 @ Y2 @ A2 )
=> ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( member_nat2 @ X @ A2 )
=> ( ( lattic8265883725875713057ax_nat @ A2 )
= X ) ) ) ) ).
% Max_eqI
thf(fact_307_Max__eq__if,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ? [Xa: nat] :
( ( member_nat2 @ Xa @ B2 )
& ( ord_less_eq_nat @ X2 @ Xa ) ) )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ B2 )
=> ? [Xa: nat] :
( ( member_nat2 @ Xa @ A2 )
& ( ord_less_eq_nat @ X2 @ Xa ) ) )
=> ( ( lattic8265883725875713057ax_nat @ A2 )
= ( lattic8265883725875713057ax_nat @ B2 ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_308_Max_OcoboundedI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ A @ A2 )
=> ( ord_less_eq_nat @ A @ ( lattic8265883725875713057ax_nat @ A2 ) ) ) ) ).
% Max.coboundedI
thf(fact_309_Max__in,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( member_nat2 @ ( lattic8265883725875713057ax_nat @ A2 ) @ A2 ) ) ) ).
% Max_in
thf(fact_310_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_311_empty__set,axiom,
( bot_bo7861856631361375769la_b_a
= ( set_formula_b_a2 @ nil_formula_b_a ) ) ).
% empty_set
thf(fact_312_Inf__fin_OcoboundedI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ A @ A2 )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_313_Sup__fin_OcoboundedI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a2 @ A @ A2 )
=> ( ord_less_eq_a @ A @ ( lattic6792493950031347381_fin_a @ A2 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_314_Sup__fin_OcoboundedI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ A @ A2 )
=> ( ord_less_eq_nat @ A @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_315_Min_OboundedI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat2 @ A4 @ A2 )
=> ( ord_less_eq_nat @ X @ A4 ) )
=> ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min.boundedI
thf(fact_316_Min_OboundedE,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) )
=> ! [A6: nat] :
( ( member_nat2 @ A6 @ A2 )
=> ( ord_less_eq_nat @ X @ A6 ) ) ) ) ) ).
% Min.boundedE
thf(fact_317_eq__Min__iff,axiom,
! [A2: set_nat,M: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( M
= ( lattic8721135487736765967in_nat @ A2 ) )
= ( ( member_nat2 @ M @ A2 )
& ! [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
=> ( ord_less_eq_nat @ M @ X4 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_318_Min__le__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ X )
= ( ? [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
& ( ord_less_eq_nat @ X4 @ X ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_319_Min__eq__iff,axiom,
! [A2: set_nat,M: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ( lattic8721135487736765967in_nat @ A2 )
= M )
= ( ( member_nat2 @ M @ A2 )
& ! [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
=> ( ord_less_eq_nat @ M @ X4 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_320_Max_OboundedI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat2 @ A4 @ A2 )
=> ( ord_less_eq_nat @ A4 @ X ) )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A2 ) @ X ) ) ) ) ).
% Max.boundedI
thf(fact_321_Max_OboundedE,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A2 ) @ X )
=> ! [A6: nat] :
( ( member_nat2 @ A6 @ A2 )
=> ( ord_less_eq_nat @ A6 @ X ) ) ) ) ) ).
% Max.boundedE
thf(fact_322_eq__Max__iff,axiom,
! [A2: set_nat,M: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( M
= ( lattic8265883725875713057ax_nat @ A2 ) )
= ( ( member_nat2 @ M @ A2 )
& ! [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
=> ( ord_less_eq_nat @ X4 @ M ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_323_Max__ge__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8265883725875713057ax_nat @ A2 ) )
= ( ? [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
& ( ord_less_eq_nat @ X @ X4 ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_324_Max__eq__iff,axiom,
! [A2: set_nat,M: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ( lattic8265883725875713057ax_nat @ A2 )
= M )
= ( ( member_nat2 @ M @ A2 )
& ! [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
=> ( ord_less_eq_nat @ X4 @ M ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_325_Inf__fin_OboundedE,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) )
=> ! [A6: nat] :
( ( member_nat2 @ A6 @ A2 )
=> ( ord_less_eq_nat @ X @ A6 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_326_Inf__fin_OboundedI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat2 @ A4 @ A2 )
=> ( ord_less_eq_nat @ X @ A4 ) )
=> ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_327_Sup__fin_OboundedE,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( A2 != bot_bot_set_a )
=> ( ( ord_less_eq_a @ ( lattic6792493950031347381_fin_a @ A2 ) @ X )
=> ! [A6: a] :
( ( member_a2 @ A6 @ A2 )
=> ( ord_less_eq_a @ A6 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_328_Sup__fin_OboundedE,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X )
=> ! [A6: nat] :
( ( member_nat2 @ A6 @ A2 )
=> ( ord_less_eq_nat @ A6 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_329_Sup__fin_OboundedI,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( A2 != bot_bot_set_a )
=> ( ! [A4: a] :
( ( member_a2 @ A4 @ A2 )
=> ( ord_less_eq_a @ A4 @ X ) )
=> ( ord_less_eq_a @ ( lattic6792493950031347381_fin_a @ A2 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_330_Sup__fin_OboundedI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat2 @ A4 @ A2 )
=> ( ord_less_eq_nat @ A4 @ X ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_331_Inf__fin_Obounded__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) )
= ( ! [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
=> ( ord_less_eq_nat @ X @ X4 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_332_Sup__fin_Obounded__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X )
= ( ! [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
=> ( ord_less_eq_nat @ X4 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_333_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_a @ ( coset_a @ nil_a ) @ ( set_a2 @ nil_a ) ) ).
% subset_code(3)
thf(fact_334_Inf__fin_Osubset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B2 ) @ ( lattic5238388535129920115in_nat @ A2 ) )
= ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_335_Sup__fin_Osubset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B2 ) @ ( lattic1093996805478795353in_nat @ A2 ) )
= ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_336_Max__less__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( lattic8265883725875713057ax_nat @ A2 ) @ X )
= ( ! [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
=> ( ord_less_nat @ X4 @ X ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_337_Min__gr__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) )
= ( ! [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
=> ( ord_less_nat @ X @ X4 ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_338_Max_Osubset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_max_nat @ ( lattic8265883725875713057ax_nat @ B2 ) @ ( lattic8265883725875713057ax_nat @ A2 ) )
= ( lattic8265883725875713057ax_nat @ A2 ) ) ) ) ) ).
% Max.subset
thf(fact_339_Min_Osubset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_min_nat @ ( lattic8721135487736765967in_nat @ B2 ) @ ( lattic8721135487736765967in_nat @ A2 ) )
= ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min.subset
thf(fact_340_Un__empty,axiom,
! [A2: set_formula_b_a,B2: set_formula_b_a] :
( ( ( sup_su8125743748909105329la_b_a @ A2 @ B2 )
= bot_bo7861856631361375769la_b_a )
= ( ( A2 = bot_bo7861856631361375769la_b_a )
& ( B2 = bot_bo7861856631361375769la_b_a ) ) ) ).
% Un_empty
thf(fact_341_finite__Un,axiom,
! [F2: set_formula_b_a,G: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ ( sup_su8125743748909105329la_b_a @ F2 @ G ) )
= ( ( finite4096952451150804198la_b_a @ F2 )
& ( finite4096952451150804198la_b_a @ G ) ) ) ).
% finite_Un
thf(fact_342_finite__Un,axiom,
! [F2: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) )
= ( ( finite_finite_nat @ F2 )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_343_finite__Int,axiom,
! [F2: set_formula_b_a,G: set_formula_b_a] :
( ( ( finite4096952451150804198la_b_a @ F2 )
| ( finite4096952451150804198la_b_a @ G ) )
=> ( finite4096952451150804198la_b_a @ ( inf_in5034913211621613591la_b_a @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_344_finite__Int,axiom,
! [F2: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_345_min__bot,axiom,
! [X: set_formula_b_a] :
( ( ord_mi284285648586469982la_b_a @ bot_bo7861856631361375769la_b_a @ X )
= bot_bo7861856631361375769la_b_a ) ).
% min_bot
thf(fact_346_min__bot,axiom,
! [X: nat] :
( ( ord_min_nat @ bot_bot_nat @ X )
= bot_bot_nat ) ).
% min_bot
thf(fact_347_min__bot2,axiom,
! [X: set_formula_b_a] :
( ( ord_mi284285648586469982la_b_a @ X @ bot_bo7861856631361375769la_b_a )
= bot_bo7861856631361375769la_b_a ) ).
% min_bot2
thf(fact_348_min__bot2,axiom,
! [X: nat] :
( ( ord_min_nat @ X @ bot_bot_nat )
= bot_bot_nat ) ).
% min_bot2
thf(fact_349_max__bot,axiom,
! [X: set_formula_b_a] :
( ( ord_ma3619480027648345712la_b_a @ bot_bo7861856631361375769la_b_a @ X )
= X ) ).
% max_bot
thf(fact_350_max__bot,axiom,
! [X: nat] :
( ( ord_max_nat @ bot_bot_nat @ X )
= X ) ).
% max_bot
thf(fact_351_max__bot2,axiom,
! [X: set_formula_b_a] :
( ( ord_ma3619480027648345712la_b_a @ X @ bot_bo7861856631361375769la_b_a )
= X ) ).
% max_bot2
thf(fact_352_max__bot2,axiom,
! [X: nat] :
( ( ord_max_nat @ X @ bot_bot_nat )
= X ) ).
% max_bot2
thf(fact_353_max__min__same_I1_J,axiom,
! [X: nat,Y: nat] :
( ( ord_max_nat @ X @ ( ord_min_nat @ X @ Y ) )
= X ) ).
% max_min_same(1)
thf(fact_354_max__min__same_I2_J,axiom,
! [X: nat,Y: nat] :
( ( ord_max_nat @ ( ord_min_nat @ X @ Y ) @ X )
= X ) ).
% max_min_same(2)
thf(fact_355_max__min__same_I3_J,axiom,
! [X: nat,Y: nat] :
( ( ord_max_nat @ ( ord_min_nat @ X @ Y ) @ Y )
= Y ) ).
% max_min_same(3)
thf(fact_356_max__min__same_I4_J,axiom,
! [Y: nat,X: nat] :
( ( ord_max_nat @ Y @ ( ord_min_nat @ X @ Y ) )
= Y ) ).
% max_min_same(4)
thf(fact_357_enumerate__mono__iff,axiom,
! [S: set_nat,M: nat,N: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% enumerate_mono_iff
thf(fact_358_inf__Sup__absorb,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ A @ A2 )
=> ( ( inf_inf_nat @ A @ ( lattic1093996805478795353in_nat @ A2 ) )
= A ) ) ) ).
% inf_Sup_absorb
thf(fact_359_sup__Inf__absorb,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ A @ A2 )
=> ( ( sup_sup_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ A )
= A ) ) ) ).
% sup_Inf_absorb
thf(fact_360_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_361_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_362_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_363_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_364_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_365_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_366_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_367_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_368_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_369_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_370_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_371_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_372_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_373_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_374_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_375_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_376_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_377_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_378_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_379_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_380_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_381_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_382_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ~ ( P3 @ M4 ) ) ) ) ) ).
% exists_least_iff
thf(fact_383_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_384_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_385_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_386_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_387_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X2: nat] :
( ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ X2 )
=> ( P @ Y3 ) )
=> ( P @ X2 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_388_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_389_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_390_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_391_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_392_gt__ex,axiom,
! [X: nat] :
? [X_12: nat] : ( ord_less_nat @ X @ X_12 ) ).
% gt_ex
thf(fact_393_max__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_max_nat @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_394_max__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_max_nat @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_395_max__def,axiom,
( ord_max_nat
= ( ^ [A3: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).
% max_def
thf(fact_396_min__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_min_nat @ X @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_397_min__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_min_nat @ X @ Y )
= X ) ) ).
% min_absorb1
thf(fact_398_min__def,axiom,
( ord_min_nat
= ( ^ [A3: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B3 ) @ A3 @ B3 ) ) ) ).
% min_def
thf(fact_399_enumerate__mono,axiom,
! [M: nat,N: nat,S: set_nat] :
( ( ord_less_nat @ M @ N )
=> ( ~ ( finite_finite_nat @ S )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ).
% enumerate_mono
thf(fact_400_Un__empty__left,axiom,
! [B2: set_formula_b_a] :
( ( sup_su8125743748909105329la_b_a @ bot_bo7861856631361375769la_b_a @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_401_Un__empty__right,axiom,
! [A2: set_formula_b_a] :
( ( sup_su8125743748909105329la_b_a @ A2 @ bot_bo7861856631361375769la_b_a )
= A2 ) ).
% Un_empty_right
thf(fact_402_finite__UnI,axiom,
! [F2: set_formula_b_a,G: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ F2 )
=> ( ( finite4096952451150804198la_b_a @ G )
=> ( finite4096952451150804198la_b_a @ ( sup_su8125743748909105329la_b_a @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_403_finite__UnI,axiom,
! [F2: set_nat,G: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( finite_finite_nat @ G )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_404_Un__infinite,axiom,
! [S: set_formula_b_a,T4: set_formula_b_a] :
( ~ ( finite4096952451150804198la_b_a @ S )
=> ~ ( finite4096952451150804198la_b_a @ ( sup_su8125743748909105329la_b_a @ S @ T4 ) ) ) ).
% Un_infinite
thf(fact_405_Un__infinite,axiom,
! [S: set_nat,T4: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T4 ) ) ) ).
% Un_infinite
thf(fact_406_infinite__Un,axiom,
! [S: set_formula_b_a,T4: set_formula_b_a] :
( ( ~ ( finite4096952451150804198la_b_a @ ( sup_su8125743748909105329la_b_a @ S @ T4 ) ) )
= ( ~ ( finite4096952451150804198la_b_a @ S )
| ~ ( finite4096952451150804198la_b_a @ T4 ) ) ) ).
% infinite_Un
thf(fact_407_infinite__Un,axiom,
! [S: set_nat,T4: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T4 ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T4 ) ) ) ).
% infinite_Un
thf(fact_408_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_409_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_410_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_411_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_412_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_413_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_414_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_415_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_416_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_417_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_418_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_419_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_420_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_421_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_422_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_423_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_424_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_425_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_426_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_427_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_428_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_429_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_430_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_431_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_432_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_433_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_434_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_435_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_436_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_437_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_438_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_439_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_440_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_441_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_442_Int__emptyI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X2: a] :
( ( member_a2 @ X2 @ A2 )
=> ~ ( member_a2 @ X2 @ B2 ) )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_443_Int__emptyI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ~ ( member_nat2 @ X2 @ B2 ) )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_444_Int__emptyI,axiom,
! [A2: set_formula_b_a,B2: set_formula_b_a] :
( ! [X2: formula_b_a] :
( ( member_formula_b_a2 @ X2 @ A2 )
=> ~ ( member_formula_b_a2 @ X2 @ B2 ) )
=> ( ( inf_in5034913211621613591la_b_a @ A2 @ B2 )
= bot_bo7861856631361375769la_b_a ) ) ).
% Int_emptyI
thf(fact_445_disjoint__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ! [X4: a] :
( ( member_a2 @ X4 @ A2 )
=> ~ ( member_a2 @ X4 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_446_disjoint__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ! [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
=> ~ ( member_nat2 @ X4 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_447_disjoint__iff,axiom,
! [A2: set_formula_b_a,B2: set_formula_b_a] :
( ( ( inf_in5034913211621613591la_b_a @ A2 @ B2 )
= bot_bo7861856631361375769la_b_a )
= ( ! [X4: formula_b_a] :
( ( member_formula_b_a2 @ X4 @ A2 )
=> ~ ( member_formula_b_a2 @ X4 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_448_Int__empty__left,axiom,
! [B2: set_formula_b_a] :
( ( inf_in5034913211621613591la_b_a @ bot_bo7861856631361375769la_b_a @ B2 )
= bot_bo7861856631361375769la_b_a ) ).
% Int_empty_left
thf(fact_449_Int__empty__right,axiom,
! [A2: set_formula_b_a] :
( ( inf_in5034913211621613591la_b_a @ A2 @ bot_bo7861856631361375769la_b_a )
= bot_bo7861856631361375769la_b_a ) ).
% Int_empty_right
thf(fact_450_disjoint__iff__not__equal,axiom,
! [A2: set_formula_b_a,B2: set_formula_b_a] :
( ( ( inf_in5034913211621613591la_b_a @ A2 @ B2 )
= bot_bo7861856631361375769la_b_a )
= ( ! [X4: formula_b_a] :
( ( member_formula_b_a2 @ X4 @ A2 )
=> ! [Y5: formula_b_a] :
( ( member_formula_b_a2 @ Y5 @ B2 )
=> ( X4 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_451_Int__Collect__mono,axiom,
! [A2: set_a,B2: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B2 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_452_Int__Collect__mono,axiom,
! [A2: set_formula_b_a,B2: set_formula_b_a,P: formula_b_a > $o,Q: formula_b_a > $o] :
( ( ord_le5472159299058833381la_b_a @ A2 @ B2 )
=> ( ! [X2: formula_b_a] :
( ( member_formula_b_a2 @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le5472159299058833381la_b_a @ ( inf_in5034913211621613591la_b_a @ A2 @ ( collect_formula_b_a @ P ) ) @ ( inf_in5034913211621613591la_b_a @ B2 @ ( collect_formula_b_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_453_Int__Collect__mono,axiom,
! [A2: set_nat,B2: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B2 @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_454_bot_Oextremum__strict,axiom,
! [A: set_formula_b_a] :
~ ( ord_le976137276181116377la_b_a @ A @ bot_bo7861856631361375769la_b_a ) ).
% bot.extremum_strict
thf(fact_455_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_456_bot_Onot__eq__extremum,axiom,
! [A: set_formula_b_a] :
( ( A != bot_bo7861856631361375769la_b_a )
= ( ord_le976137276181116377la_b_a @ bot_bo7861856631361375769la_b_a @ A ) ) ).
% bot.not_eq_extremum
thf(fact_457_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_458_not__psubset__empty,axiom,
! [A2: set_formula_b_a] :
~ ( ord_le976137276181116377la_b_a @ A2 @ bot_bo7861856631361375769la_b_a ) ).
% not_psubset_empty
thf(fact_459_finite__psubset__induct,axiom,
! [A2: set_formula_b_a,P: set_formula_b_a > $o] :
( ( finite4096952451150804198la_b_a @ A2 )
=> ( ! [A7: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ A7 )
=> ( ! [B6: set_formula_b_a] :
( ( ord_le976137276181116377la_b_a @ B6 @ A7 )
=> ( P @ B6 ) )
=> ( P @ A7 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_460_finite__psubset__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [B6: set_nat] :
( ( ord_less_set_nat @ B6 @ A7 )
=> ( P @ B6 ) )
=> ( P @ A7 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_461_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_462_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_463_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_464_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
| ( M4 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_465_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_466_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
& ( M4 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_467_length__induct,axiom,
! [P: list_a > $o,Xs2: list_a] :
( ! [Xs: list_a] :
( ! [Ys2: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys2 ) @ ( size_size_list_a @ Xs ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_468_finite__maxlen,axiom,
! [M2: set_list_a] :
( ( finite_finite_list_a @ M2 )
=> ? [N4: nat] :
! [X3: list_a] :
( ( member_list_a @ X3 @ M2 )
=> ( ord_less_nat @ ( size_size_list_a @ X3 ) @ N4 ) ) ) ).
% finite_maxlen
thf(fact_469_unbounded__k__infinite,axiom,
! [K: nat,S: set_nat] :
( ! [M3: nat] :
( ( ord_less_nat @ K @ M3 )
=> ? [N6: nat] :
( ( ord_less_nat @ M3 @ N6 )
& ( member_nat2 @ N6 @ S ) ) )
=> ~ ( finite_finite_nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_470_bounded__nat__set__is__finite,axiom,
! [N5: set_nat,N: nat] :
( ! [X2: nat] :
( ( member_nat2 @ X2 @ N5 )
=> ( ord_less_nat @ X2 @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_471_infinite__nat__iff__unbounded,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M4: nat] :
? [N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ( member_nat2 @ N2 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_472_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M4: nat] :
! [X4: nat] :
( ( member_nat2 @ X4 @ N3 )
=> ( ord_less_nat @ X4 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_473_size__atms,axiom,
! [Phi: formula_b_a,R: regex_b_a] :
( ( member_formula_b_a2 @ Phi @ ( atms_b_a @ R ) )
=> ( ord_less_nat @ ( size_s1229512387538370275la_b_a @ Phi ) @ ( size_size_regex_b_a @ R ) ) ) ).
% size_atms
thf(fact_474_Min_Ounion,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B2 )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ord_min_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ ( lattic8721135487736765967in_nat @ B2 ) ) ) ) ) ) ) ).
% Min.union
thf(fact_475_Max_Ounion,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B2 )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ord_max_nat @ ( lattic8265883725875713057ax_nat @ A2 ) @ ( lattic8265883725875713057ax_nat @ B2 ) ) ) ) ) ) ) ).
% Max.union
thf(fact_476_Sup__fin_Ounion,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B2 )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ ( lattic1093996805478795353in_nat @ B2 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_477_Inf__fin_Ounion,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B2 )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ ( lattic5238388535129920115in_nat @ B2 ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_478_Min_Oin__idem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( ( ord_min_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) )
= ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ).
% Min.in_idem
thf(fact_479_Max_Oin__idem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ A2 ) )
= ( lattic8265883725875713057ax_nat @ A2 ) ) ) ) ).
% Max.in_idem
thf(fact_480_infinite__growing,axiom,
! [X6: set_nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat2 @ X2 @ X6 )
=> ? [Xa: nat] :
( ( member_nat2 @ Xa @ X6 )
& ( ord_less_nat @ X2 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X6 ) ) ) ).
% infinite_growing
thf(fact_481_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat2 @ X2 @ S )
& ~ ? [Xa: nat] :
( ( member_nat2 @ Xa @ S )
& ( ord_less_nat @ Xa @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_482_Sup__fin_Oin__idem,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a2 @ X @ A2 )
=> ( ( sup_sup_a @ X @ ( lattic6792493950031347381_fin_a @ A2 ) )
= ( lattic6792493950031347381_fin_a @ A2 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_483_Sup__fin_Oin__idem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A2 ) )
= ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_484_Inf__fin_Oin__idem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) )
= ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_485_list__ex1__iff,axiom,
( list_ex1_formula_b_a
= ( ^ [P3: formula_b_a > $o,Xs3: list_formula_b_a] :
? [X4: formula_b_a] :
( ( member_formula_b_a2 @ X4 @ ( set_formula_b_a2 @ Xs3 ) )
& ( P3 @ X4 )
& ! [Y5: formula_b_a] :
( ( ( member_formula_b_a2 @ Y5 @ ( set_formula_b_a2 @ Xs3 ) )
& ( P3 @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_486_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P3: nat > $o,Xs3: list_nat] :
? [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs3 ) )
& ( P3 @ X4 )
& ! [Y5: nat] :
( ( ( member_nat2 @ Y5 @ ( set_nat2 @ Xs3 ) )
& ( P3 @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_487_list__ex1__iff,axiom,
( list_ex1_a
= ( ^ [P3: a > $o,Xs3: list_a] :
? [X4: a] :
( ( member_a2 @ X4 @ ( set_a2 @ Xs3 ) )
& ( P3 @ X4 )
& ! [Y5: a] :
( ( ( member_a2 @ Y5 @ ( set_a2 @ Xs3 ) )
& ( P3 @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_488_Min__less__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ X )
= ( ? [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
& ( ord_less_nat @ X4 @ X ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_489_Max__gr__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_nat @ X @ ( lattic8265883725875713057ax_nat @ A2 ) )
= ( ? [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
& ( ord_less_nat @ X @ X4 ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_490_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X3: nat] :
( ( member_nat2 @ X3 @ S )
& ( ord_less_nat @ ( F @ X3 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_491_arg__min__if__finite_I2_J,axiom,
! [S: set_formula_b_a,F: formula_b_a > nat] :
( ( finite4096952451150804198la_b_a @ S )
=> ( ( S != bot_bo7861856631361375769la_b_a )
=> ~ ? [X3: formula_b_a] :
( ( member_formula_b_a2 @ X3 @ S )
& ( ord_less_nat @ ( F @ X3 ) @ ( F @ ( lattic257815086748031546_a_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_492_max_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_493_max_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_494_max__less__iff__conj,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z2 )
= ( ( ord_less_nat @ X @ Z2 )
& ( ord_less_nat @ Y @ Z2 ) ) ) ).
% max_less_iff_conj
thf(fact_495_min_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_min_nat @ A @ B )
= A ) ) ).
% min.absorb3
thf(fact_496_min_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_min_nat @ A @ B )
= B ) ) ).
% min.absorb4
thf(fact_497_min__less__iff__conj,axiom,
! [Z2: nat,X: nat,Y: nat] :
( ( ord_less_nat @ Z2 @ ( ord_min_nat @ X @ Y ) )
= ( ( ord_less_nat @ Z2 @ X )
& ( ord_less_nat @ Z2 @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_498_max_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% max.bounded_iff
thf(fact_499_max_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_500_inf__right__idem,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Y )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_right_idem
thf(fact_501_inf_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ B )
= ( inf_inf_nat @ A @ B ) ) ).
% inf.right_idem
thf(fact_502_inf__left__idem,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_left_idem
thf(fact_503_inf_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( inf_inf_nat @ A @ ( inf_inf_nat @ A @ B ) )
= ( inf_inf_nat @ A @ B ) ) ).
% inf.left_idem
thf(fact_504_inf__idem,axiom,
! [X: nat] :
( ( inf_inf_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_505_inf_Oidem,axiom,
! [A: nat] :
( ( inf_inf_nat @ A @ A )
= A ) ).
% inf.idem
thf(fact_506_sup_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A @ B ) @ B )
= ( sup_sup_nat @ A @ B ) ) ).
% sup.right_idem
thf(fact_507_sup__left__idem,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= ( sup_sup_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_508_sup_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( sup_sup_nat @ A @ ( sup_sup_nat @ A @ B ) )
= ( sup_sup_nat @ A @ B ) ) ).
% sup.left_idem
thf(fact_509_sup__idem,axiom,
! [X: nat] :
( ( sup_sup_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_510_sup_Oidem,axiom,
! [A: nat] :
( ( sup_sup_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_511_Int__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a2 @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( ( member_a2 @ C @ A2 )
& ( member_a2 @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_512_Int__iff,axiom,
! [C: formula_b_a,A2: set_formula_b_a,B2: set_formula_b_a] :
( ( member_formula_b_a2 @ C @ ( inf_in5034913211621613591la_b_a @ A2 @ B2 ) )
= ( ( member_formula_b_a2 @ C @ A2 )
& ( member_formula_b_a2 @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_513_Int__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( ( member_nat2 @ C @ A2 )
& ( member_nat2 @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_514_IntI,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a2 @ C @ A2 )
=> ( ( member_a2 @ C @ B2 )
=> ( member_a2 @ C @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_515_IntI,axiom,
! [C: formula_b_a,A2: set_formula_b_a,B2: set_formula_b_a] :
( ( member_formula_b_a2 @ C @ A2 )
=> ( ( member_formula_b_a2 @ C @ B2 )
=> ( member_formula_b_a2 @ C @ ( inf_in5034913211621613591la_b_a @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_516_IntI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ A2 )
=> ( ( member_nat2 @ C @ B2 )
=> ( member_nat2 @ C @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_517_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_518_Un__iff,axiom,
! [C: formula_b_a,A2: set_formula_b_a,B2: set_formula_b_a] :
( ( member_formula_b_a2 @ C @ ( sup_su8125743748909105329la_b_a @ A2 @ B2 ) )
= ( ( member_formula_b_a2 @ C @ A2 )
| ( member_formula_b_a2 @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_519_Un__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( member_nat2 @ C @ A2 )
| ( member_nat2 @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_520_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_521_UnCI,axiom,
! [C: formula_b_a,B2: set_formula_b_a,A2: set_formula_b_a] :
( ( ~ ( member_formula_b_a2 @ C @ B2 )
=> ( member_formula_b_a2 @ C @ A2 ) )
=> ( member_formula_b_a2 @ C @ ( sup_su8125743748909105329la_b_a @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_522_UnCI,axiom,
! [C: nat,B2: set_nat,A2: set_nat] :
( ( ~ ( member_nat2 @ C @ B2 )
=> ( member_nat2 @ C @ A2 ) )
=> ( member_nat2 @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_523_min_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( ord_min_nat @ ( ord_min_nat @ A @ B ) @ B )
= ( ord_min_nat @ A @ B ) ) ).
% min.right_idem
thf(fact_524_min_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( ord_min_nat @ A @ ( ord_min_nat @ A @ B ) )
= ( ord_min_nat @ A @ B ) ) ).
% min.left_idem
thf(fact_525_min_Oidem,axiom,
! [A: nat] :
( ( ord_min_nat @ A @ A )
= A ) ).
% min.idem
thf(fact_526_max_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ B )
= ( ord_max_nat @ A @ B ) ) ).
% max.right_idem
thf(fact_527_max_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( ord_max_nat @ A @ ( ord_max_nat @ A @ B ) )
= ( ord_max_nat @ A @ B ) ) ).
% max.left_idem
thf(fact_528_max_Oidem,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ A )
= A ) ).
% max.idem
thf(fact_529_le__inf__iff,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_530_inf_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_531_le__sup__iff,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X @ Z2 )
& ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_532_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_533_inf__bot__right,axiom,
! [X: set_formula_b_a] :
( ( inf_in5034913211621613591la_b_a @ X @ bot_bo7861856631361375769la_b_a )
= bot_bo7861856631361375769la_b_a ) ).
% inf_bot_right
thf(fact_534_inf__bot__left,axiom,
! [X: set_formula_b_a] :
( ( inf_in5034913211621613591la_b_a @ bot_bo7861856631361375769la_b_a @ X )
= bot_bo7861856631361375769la_b_a ) ).
% inf_bot_left
thf(fact_535_sup__bot_Oright__neutral,axiom,
! [A: set_formula_b_a] :
( ( sup_su8125743748909105329la_b_a @ A @ bot_bo7861856631361375769la_b_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_536_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_formula_b_a,B: set_formula_b_a] :
( ( bot_bo7861856631361375769la_b_a
= ( sup_su8125743748909105329la_b_a @ A @ B ) )
= ( ( A = bot_bo7861856631361375769la_b_a )
& ( B = bot_bo7861856631361375769la_b_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_537_sup__bot_Oleft__neutral,axiom,
! [A: set_formula_b_a] :
( ( sup_su8125743748909105329la_b_a @ bot_bo7861856631361375769la_b_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_538_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_formula_b_a,B: set_formula_b_a] :
( ( ( sup_su8125743748909105329la_b_a @ A @ B )
= bot_bo7861856631361375769la_b_a )
= ( ( A = bot_bo7861856631361375769la_b_a )
& ( B = bot_bo7861856631361375769la_b_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_539_sup__eq__bot__iff,axiom,
! [X: set_formula_b_a,Y: set_formula_b_a] :
( ( ( sup_su8125743748909105329la_b_a @ X @ Y )
= bot_bo7861856631361375769la_b_a )
= ( ( X = bot_bo7861856631361375769la_b_a )
& ( Y = bot_bo7861856631361375769la_b_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_540_bot__eq__sup__iff,axiom,
! [X: set_formula_b_a,Y: set_formula_b_a] :
( ( bot_bo7861856631361375769la_b_a
= ( sup_su8125743748909105329la_b_a @ X @ Y ) )
= ( ( X = bot_bo7861856631361375769la_b_a )
& ( Y = bot_bo7861856631361375769la_b_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_541_sup__bot__right,axiom,
! [X: set_formula_b_a] :
( ( sup_su8125743748909105329la_b_a @ X @ bot_bo7861856631361375769la_b_a )
= X ) ).
% sup_bot_right
thf(fact_542_sup__bot__left,axiom,
! [X: set_formula_b_a] :
( ( sup_su8125743748909105329la_b_a @ bot_bo7861856631361375769la_b_a @ X )
= X ) ).
% sup_bot_left
thf(fact_543_sup__inf__absorb,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( inf_inf_nat @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_544_inf__sup__absorb,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_545_min_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_min_nat @ A @ B )
= A ) ) ).
% min.absorb1
thf(fact_546_min_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_min_nat @ A @ B )
= B ) ) ).
% min.absorb2
thf(fact_547_min_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% min.bounded_iff
thf(fact_548_max_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_549_psubsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( member_a2 @ C @ A2 )
=> ( member_a2 @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_550_psubsetD,axiom,
! [A2: set_formula_b_a,B2: set_formula_b_a,C: formula_b_a] :
( ( ord_le976137276181116377la_b_a @ A2 @ B2 )
=> ( ( member_formula_b_a2 @ C @ A2 )
=> ( member_formula_b_a2 @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_551_psubsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( member_nat2 @ C @ A2 )
=> ( member_nat2 @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_552_IntD2,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a2 @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ( member_a2 @ C @ B2 ) ) ).
% IntD2
thf(fact_553_IntD2,axiom,
! [C: formula_b_a,A2: set_formula_b_a,B2: set_formula_b_a] :
( ( member_formula_b_a2 @ C @ ( inf_in5034913211621613591la_b_a @ A2 @ B2 ) )
=> ( member_formula_b_a2 @ C @ B2 ) ) ).
% IntD2
thf(fact_554_IntD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( member_nat2 @ C @ B2 ) ) ).
% IntD2
thf(fact_555_IntD1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a2 @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ( member_a2 @ C @ A2 ) ) ).
% IntD1
thf(fact_556_IntD1,axiom,
! [C: formula_b_a,A2: set_formula_b_a,B2: set_formula_b_a] :
( ( member_formula_b_a2 @ C @ ( inf_in5034913211621613591la_b_a @ A2 @ B2 ) )
=> ( member_formula_b_a2 @ C @ A2 ) ) ).
% IntD1
thf(fact_557_IntD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( member_nat2 @ C @ A2 ) ) ).
% IntD1
thf(fact_558_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_559_UnI2,axiom,
! [C: formula_b_a,B2: set_formula_b_a,A2: set_formula_b_a] :
( ( member_formula_b_a2 @ C @ B2 )
=> ( member_formula_b_a2 @ C @ ( sup_su8125743748909105329la_b_a @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_560_UnI2,axiom,
! [C: nat,B2: set_nat,A2: set_nat] :
( ( member_nat2 @ C @ B2 )
=> ( member_nat2 @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_561_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_562_UnI1,axiom,
! [C: formula_b_a,A2: set_formula_b_a,B2: set_formula_b_a] :
( ( member_formula_b_a2 @ C @ A2 )
=> ( member_formula_b_a2 @ C @ ( sup_su8125743748909105329la_b_a @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_563_UnI1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ A2 )
=> ( member_nat2 @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_564_IntE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a2 @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a2 @ C @ A2 )
=> ~ ( member_a2 @ C @ B2 ) ) ) ).
% IntE
thf(fact_565_IntE,axiom,
! [C: formula_b_a,A2: set_formula_b_a,B2: set_formula_b_a] :
( ( member_formula_b_a2 @ C @ ( inf_in5034913211621613591la_b_a @ A2 @ B2 ) )
=> ~ ( ( member_formula_b_a2 @ C @ A2 )
=> ~ ( member_formula_b_a2 @ C @ B2 ) ) ) ).
% IntE
thf(fact_566_IntE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat2 @ C @ A2 )
=> ~ ( member_nat2 @ C @ B2 ) ) ) ).
% IntE
thf(fact_567_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_568_UnE,axiom,
! [C: formula_b_a,A2: set_formula_b_a,B2: set_formula_b_a] :
( ( member_formula_b_a2 @ C @ ( sup_su8125743748909105329la_b_a @ A2 @ B2 ) )
=> ( ~ ( member_formula_b_a2 @ C @ A2 )
=> ( member_formula_b_a2 @ C @ B2 ) ) ) ).
% UnE
thf(fact_569_UnE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( ~ ( member_nat2 @ C @ A2 )
=> ( member_nat2 @ C @ B2 ) ) ) ).
% UnE
thf(fact_570_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_571_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_572_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ( P @ M5 ) )
=> ( P @ N4 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_573_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_574_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_575_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_576_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_577_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_578_inf__left__commute,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) )
= ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_579_inf_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( inf_inf_nat @ B @ ( inf_inf_nat @ A @ C ) )
= ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_580_inf__commute,axiom,
( inf_inf_nat
= ( ^ [X4: nat,Y5: nat] : ( inf_inf_nat @ Y5 @ X4 ) ) ) ).
% inf_commute
thf(fact_581_inf_Ocommute,axiom,
( inf_inf_nat
= ( ^ [A3: nat,B3: nat] : ( inf_inf_nat @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_582_inf__assoc,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z2 )
= ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) ) ) ).
% inf_assoc
thf(fact_583_inf_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ C )
= ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).
% inf.assoc
thf(fact_584_inf__sup__aci_I1_J,axiom,
( inf_inf_nat
= ( ^ [X4: nat,Y5: nat] : ( inf_inf_nat @ Y5 @ X4 ) ) ) ).
% inf_sup_aci(1)
thf(fact_585_inf__sup__aci_I2_J,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z2 )
= ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_586_inf__sup__aci_I3_J,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) )
= ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_587_inf__sup__aci_I4_J,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_588_sup__left__commute,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_589_sup_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( sup_sup_nat @ B @ ( sup_sup_nat @ A @ C ) )
= ( sup_sup_nat @ A @ ( sup_sup_nat @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_590_sup__commute,axiom,
( sup_sup_nat
= ( ^ [X4: nat,Y5: nat] : ( sup_sup_nat @ Y5 @ X4 ) ) ) ).
% sup_commute
thf(fact_591_sup_Ocommute,axiom,
( sup_sup_nat
= ( ^ [A3: nat,B3: nat] : ( sup_sup_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_592_sup__assoc,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z2 )
= ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_593_sup_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A @ B ) @ C )
= ( sup_sup_nat @ A @ ( sup_sup_nat @ B @ C ) ) ) ).
% sup.assoc
thf(fact_594_inf__sup__aci_I5_J,axiom,
( sup_sup_nat
= ( ^ [X4: nat,Y5: nat] : ( sup_sup_nat @ Y5 @ X4 ) ) ) ).
% inf_sup_aci(5)
thf(fact_595_inf__sup__aci_I6_J,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z2 )
= ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_596_inf__sup__aci_I7_J,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_597_inf__sup__aci_I8_J,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_598_min_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_min_nat @ B @ ( ord_min_nat @ A @ C ) )
= ( ord_min_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ).
% min.left_commute
thf(fact_599_min_Ocommute,axiom,
( ord_min_nat
= ( ^ [A3: nat,B3: nat] : ( ord_min_nat @ B3 @ A3 ) ) ) ).
% min.commute
thf(fact_600_min_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_min_nat @ ( ord_min_nat @ A @ B ) @ C )
= ( ord_min_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ).
% min.assoc
thf(fact_601_max_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_max_nat @ B @ ( ord_max_nat @ A @ C ) )
= ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).
% max.left_commute
thf(fact_602_max_Ocommute,axiom,
( ord_max_nat
= ( ^ [A3: nat,B3: nat] : ( ord_max_nat @ B3 @ A3 ) ) ) ).
% max.commute
thf(fact_603_max_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ C )
= ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).
% max.assoc
thf(fact_604_inf_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_605_inf_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_606_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( inf_inf_nat @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_607_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( inf_inf_nat @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_608_inf_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_609_inf_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_610_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( A3
= ( inf_inf_nat @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_611_inf__greatest,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z2 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_612_inf_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_613_inf_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_614_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_615_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_616_inf_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_617_inf_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_618_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y5: nat] :
( ( inf_inf_nat @ X4 @ Y5 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_619_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ( ord_less_eq_nat @ X2 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_620_inf_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( inf_inf_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_621_inf_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( inf_inf_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_622_le__infI2,axiom,
! [B: nat,X: nat,A: nat] :
( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_623_le__infI1,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_624_inf__mono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_625_le__infI,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_626_le__infE,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_nat @ X @ A )
=> ~ ( ord_less_eq_nat @ X @ B ) ) ) ).
% le_infE
thf(fact_627_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_628_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_629_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_630_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_631_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_632_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_633_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( sup_sup_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_634_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( sup_sup_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_635_sup_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_636_sup_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_637_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( A3
= ( sup_sup_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_638_sup_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_639_sup_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_640_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_641_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_642_sup_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_643_sup_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_644_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ( ord_less_eq_nat @ ( F @ Y2 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_645_sup_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( sup_sup_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_646_sup_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( sup_sup_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_647_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y5: nat] :
( ( sup_sup_nat @ X4 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_648_sup__least,axiom,
! [Y: nat,X: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_649_sup__mono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_650_sup_Omono,axiom,
! [C: nat,A: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_651_le__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_652_le__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_653_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_654_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_655_le__supI,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_656_le__supE,axiom,
! [A: nat,B: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_nat @ A @ X )
=> ~ ( ord_less_eq_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_657_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_658_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_659_inf_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_660_inf_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_661_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( A3
= ( inf_inf_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_662_inf_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_663_inf_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_664_inf_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_665_less__infI2,axiom,
! [B: nat,X: nat,A: nat] :
( ( ord_less_nat @ B @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% less_infI2
thf(fact_666_less__infI1,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_nat @ A @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% less_infI1
thf(fact_667_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_668_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_669_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( A3
= ( sup_sup_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_670_sup_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_671_sup_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_672_sup_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_673_less__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( ( ord_less_nat @ X @ B )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_674_less__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_nat @ X @ A )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_675_sup__inf__distrib2,axiom,
! [Y: nat,Z2: nat,X: nat] :
( ( sup_sup_nat @ ( inf_inf_nat @ Y @ Z2 ) @ X )
= ( inf_inf_nat @ ( sup_sup_nat @ Y @ X ) @ ( sup_sup_nat @ Z2 @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_676_sup__inf__distrib1,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) )
= ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_677_inf__sup__distrib2,axiom,
! [Y: nat,Z2: nat,X: nat] :
( ( inf_inf_nat @ ( sup_sup_nat @ Y @ Z2 ) @ X )
= ( sup_sup_nat @ ( inf_inf_nat @ Y @ X ) @ ( inf_inf_nat @ Z2 @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_678_inf__sup__distrib1,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) )
= ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_679_distrib__imp2,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ! [X2: nat,Y2: nat,Z3: nat] :
( ( sup_sup_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z3 ) )
= ( inf_inf_nat @ ( sup_sup_nat @ X2 @ Y2 ) @ ( sup_sup_nat @ X2 @ Z3 ) ) )
=> ( ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) )
= ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_680_distrib__imp1,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ! [X2: nat,Y2: nat,Z3: nat] :
( ( inf_inf_nat @ X2 @ ( sup_sup_nat @ Y2 @ Z3 ) )
= ( sup_sup_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ ( inf_inf_nat @ X2 @ Z3 ) ) )
=> ( ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) )
= ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_681_min__le__iff__disj,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( ord_min_nat @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X @ Z2 )
| ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% min_le_iff_disj
thf(fact_682_min_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.coboundedI2
thf(fact_683_min_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.coboundedI1
thf(fact_684_min_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_min_nat @ A3 @ B3 )
= B3 ) ) ) ).
% min.absorb_iff2
thf(fact_685_min_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_min_nat @ A3 @ B3 )
= A3 ) ) ) ).
% min.absorb_iff1
thf(fact_686_min_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ B ) ).
% min.cobounded2
thf(fact_687_min_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ A ) ).
% min.cobounded1
thf(fact_688_min_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( A3
= ( ord_min_nat @ A3 @ B3 ) ) ) ) ).
% min.order_iff
thf(fact_689_min_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ) ).
% min.boundedI
thf(fact_690_min_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% min.boundedE
thf(fact_691_min_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( ord_min_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% min.orderI
thf(fact_692_min_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( ord_min_nat @ A @ B ) ) ) ).
% min.orderE
thf(fact_693_min_Omono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ ( ord_min_nat @ C @ D ) ) ) ) ).
% min.mono
thf(fact_694_max_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_695_max_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_696_max_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_max_nat @ A3 @ B3 )
= B3 ) ) ) ).
% max.absorb_iff2
thf(fact_697_max_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_max_nat @ A3 @ B3 )
= A3 ) ) ) ).
% max.absorb_iff1
thf(fact_698_le__max__iff__disj,axiom,
! [Z2: nat,X: nat,Y: nat] :
( ( ord_less_eq_nat @ Z2 @ ( ord_max_nat @ X @ Y ) )
= ( ( ord_less_eq_nat @ Z2 @ X )
| ( ord_less_eq_nat @ Z2 @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_699_max_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded2
thf(fact_700_max_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded1
thf(fact_701_max_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( A3
= ( ord_max_nat @ A3 @ B3 ) ) ) ) ).
% max.order_iff
thf(fact_702_max_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_703_max_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% max.boundedE
thf(fact_704_max_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( ord_max_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% max.orderI
thf(fact_705_max_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( ord_max_nat @ A @ B ) ) ) ).
% max.orderE
thf(fact_706_max_Omono,axiom,
! [C: nat,A: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).
% max.mono
thf(fact_707_min_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.strict_coboundedI2
thf(fact_708_min_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.strict_coboundedI1
thf(fact_709_min_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( A3
= ( ord_min_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% min.strict_order_iff
thf(fact_710_min_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( ord_min_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C ) ) ) ).
% min.strict_boundedE
thf(fact_711_min__less__iff__disj,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ ( ord_min_nat @ X @ Y ) @ Z2 )
= ( ( ord_less_nat @ X @ Z2 )
| ( ord_less_nat @ Y @ Z2 ) ) ) ).
% min_less_iff_disj
thf(fact_712_max_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_713_max_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_714_max_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( A3
= ( ord_max_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% max.strict_order_iff
thf(fact_715_max_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% max.strict_boundedE
thf(fact_716_less__max__iff__disj,axiom,
! [Z2: nat,X: nat,Y: nat] :
( ( ord_less_nat @ Z2 @ ( ord_max_nat @ X @ Y ) )
= ( ( ord_less_nat @ Z2 @ X )
| ( ord_less_nat @ Z2 @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_717_inf__min,axiom,
inf_inf_nat = ord_min_nat ).
% inf_min
thf(fact_718_sup__max,axiom,
sup_sup_nat = ord_max_nat ).
% sup_max
thf(fact_719_min__max__distrib2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_min_nat @ A @ ( ord_max_nat @ B @ C ) )
= ( ord_max_nat @ ( ord_min_nat @ A @ B ) @ ( ord_min_nat @ A @ C ) ) ) ).
% min_max_distrib2
thf(fact_720_min__max__distrib1,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_min_nat @ ( ord_max_nat @ B @ C ) @ A )
= ( ord_max_nat @ ( ord_min_nat @ B @ A ) @ ( ord_min_nat @ C @ A ) ) ) ).
% min_max_distrib1
thf(fact_721_max__min__distrib2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_max_nat @ A @ ( ord_min_nat @ B @ C ) )
= ( ord_min_nat @ ( ord_max_nat @ A @ B ) @ ( ord_max_nat @ A @ C ) ) ) ).
% max_min_distrib2
thf(fact_722_max__min__distrib1,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_max_nat @ ( ord_min_nat @ B @ C ) @ A )
= ( ord_min_nat @ ( ord_max_nat @ B @ A ) @ ( ord_max_nat @ C @ A ) ) ) ).
% max_min_distrib1
thf(fact_723_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_724_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z2 ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_725_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_formula_b_a] :
( ( inf_in5034913211621613591la_b_a @ X @ bot_bo7861856631361375769la_b_a )
= bot_bo7861856631361375769la_b_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_726_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_formula_b_a] :
( ( inf_in5034913211621613591la_b_a @ bot_bo7861856631361375769la_b_a @ X )
= bot_bo7861856631361375769la_b_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_727_set__union,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( set_a2 @ ( union_a @ Xs2 @ Ys ) )
= ( sup_sup_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ Ys ) ) ) ).
% set_union
thf(fact_728_can__select__set__list__ex1,axiom,
! [P: a > $o,A2: list_a] :
( ( can_select_a @ P @ ( set_a2 @ A2 ) )
= ( list_ex1_a @ P @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_729_can__select__def,axiom,
( can_select_a
= ( ^ [P3: a > $o,A5: set_a] :
? [X4: a] :
( ( member_a2 @ X4 @ A5 )
& ( P3 @ X4 )
& ! [Y5: a] :
( ( ( member_a2 @ Y5 @ A5 )
& ( P3 @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% can_select_def
thf(fact_730_can__select__def,axiom,
( can_se6587019059267173266la_b_a
= ( ^ [P3: formula_b_a > $o,A5: set_formula_b_a] :
? [X4: formula_b_a] :
( ( member_formula_b_a2 @ X4 @ A5 )
& ( P3 @ X4 )
& ! [Y5: formula_b_a] :
( ( ( member_formula_b_a2 @ Y5 @ A5 )
& ( P3 @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% can_select_def
thf(fact_731_can__select__def,axiom,
( can_select_nat
= ( ^ [P3: nat > $o,A5: set_nat] :
? [X4: nat] :
( ( member_nat2 @ X4 @ A5 )
& ( P3 @ X4 )
& ! [Y5: nat] :
( ( ( member_nat2 @ Y5 @ A5 )
& ( P3 @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% can_select_def
thf(fact_732_boolean__algebra__cancel_Oinf1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( inf_inf_nat @ K @ A ) )
=> ( ( inf_inf_nat @ A2 @ B )
= ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_733_boolean__algebra__cancel_Oinf2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( inf_inf_nat @ K @ B ) )
=> ( ( inf_inf_nat @ A @ B2 )
= ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_734_boolean__algebra__cancel_Osup1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( sup_sup_nat @ K @ A ) )
=> ( ( sup_sup_nat @ A2 @ B )
= ( sup_sup_nat @ K @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_735_boolean__algebra__cancel_Osup2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( sup_sup_nat @ K @ B ) )
=> ( ( sup_sup_nat @ A @ B2 )
= ( sup_sup_nat @ K @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_736_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_formula_b_a] :
( ( sup_su8125743748909105329la_b_a @ X @ bot_bo7861856631361375769la_b_a )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_737_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_738_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X3 ) ) ).
% minf(8)
thf(fact_739_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ord_less_eq_nat @ X3 @ T ) ) ).
% minf(6)
thf(fact_740_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_741_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_742_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_743_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_744_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ~ ( ord_less_nat @ X3 @ T ) ) ).
% pinf(5)
thf(fact_745_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ord_less_nat @ T @ X3 ) ) ).
% pinf(7)
thf(fact_746_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_747_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_748_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_749_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_750_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ord_less_nat @ X3 @ T ) ) ).
% minf(5)
thf(fact_751_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ~ ( ord_less_nat @ T @ X3 ) ) ).
% minf(7)
thf(fact_752_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ T ) ) ).
% pinf(6)
thf(fact_753_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ord_less_eq_nat @ T @ X3 ) ) ).
% pinf(8)
thf(fact_754_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ C2 ) )
=> ( P @ X3 ) )
& ! [D2: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ D2 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D2 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_755_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A8: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A8 ) )
= ( ord_less_nat @ A8 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_756_size__rderive,axiom,
! [Phi: formula_b_a,R: regex_b_a] :
( ( member_formula_b_a2 @ Phi @ ( atms_b_a @ ( rderive_b_a @ R ) ) )
=> ( ord_less_nat @ ( size_s1229512387538370275la_b_a @ Phi ) @ ( size_size_regex_b_a @ R ) ) ) ).
% size_rderive
thf(fact_757_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_758_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_759_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_760_Max__insert,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat2 @ X @ A2 ) )
= ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ A2 ) ) ) ) ) ).
% Max_insert
thf(fact_761_Min__insert,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( ord_min_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min_insert
thf(fact_762_Sup__fin_Oinsert,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_763_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_764_insertCI,axiom,
! [A: formula_b_a,B2: set_formula_b_a,B: formula_b_a] :
( ( ~ ( member_formula_b_a2 @ A @ B2 )
=> ( A = B ) )
=> ( member_formula_b_a2 @ A @ ( insert_formula_b_a2 @ B @ B2 ) ) ) ).
% insertCI
thf(fact_765_insertCI,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat2 @ A @ B2 )
=> ( A = B ) )
=> ( member_nat2 @ A @ ( insert_nat2 @ B @ B2 ) ) ) ).
% insertCI
thf(fact_766_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_767_insert__iff,axiom,
! [A: formula_b_a,B: formula_b_a,A2: set_formula_b_a] :
( ( member_formula_b_a2 @ A @ ( insert_formula_b_a2 @ B @ A2 ) )
= ( ( A = B )
| ( member_formula_b_a2 @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_768_insert__iff,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat2 @ A @ ( insert_nat2 @ B @ A2 ) )
= ( ( A = B )
| ( member_nat2 @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_769_singletonI,axiom,
! [A: a] : ( member_a2 @ A @ ( insert_a2 @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_770_singletonI,axiom,
! [A: nat] : ( member_nat2 @ A @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_771_singletonI,axiom,
! [A: formula_b_a] : ( member_formula_b_a2 @ A @ ( insert_formula_b_a2 @ A @ bot_bo7861856631361375769la_b_a ) ) ).
% singletonI
thf(fact_772_finite__insert,axiom,
! [A: formula_b_a,A2: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ ( insert_formula_b_a2 @ A @ A2 ) )
= ( finite4096952451150804198la_b_a @ A2 ) ) ).
% finite_insert
thf(fact_773_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat2 @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_774_insert__subset,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( insert_a2 @ X @ A2 ) @ B2 )
= ( ( member_a2 @ X @ B2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_775_insert__subset,axiom,
! [X: formula_b_a,A2: set_formula_b_a,B2: set_formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ ( insert_formula_b_a2 @ X @ A2 ) @ B2 )
= ( ( member_formula_b_a2 @ X @ B2 )
& ( ord_le5472159299058833381la_b_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_776_insert__subset,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat2 @ X @ A2 ) @ B2 )
= ( ( member_nat2 @ X @ B2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_777_Int__insert__left__if0,axiom,
! [A: a,C3: set_a,B2: set_a] :
( ~ ( member_a2 @ A @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a2 @ A @ B2 ) @ C3 )
= ( inf_inf_set_a @ B2 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_778_Int__insert__left__if0,axiom,
! [A: formula_b_a,C3: set_formula_b_a,B2: set_formula_b_a] :
( ~ ( member_formula_b_a2 @ A @ C3 )
=> ( ( inf_in5034913211621613591la_b_a @ ( insert_formula_b_a2 @ A @ B2 ) @ C3 )
= ( inf_in5034913211621613591la_b_a @ B2 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_779_Int__insert__left__if0,axiom,
! [A: nat,C3: set_nat,B2: set_nat] :
( ~ ( member_nat2 @ A @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ B2 ) @ C3 )
= ( inf_inf_set_nat @ B2 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_780_Int__insert__left__if1,axiom,
! [A: a,C3: set_a,B2: set_a] :
( ( member_a2 @ A @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a2 @ A @ B2 ) @ C3 )
= ( insert_a2 @ A @ ( inf_inf_set_a @ B2 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_781_Int__insert__left__if1,axiom,
! [A: formula_b_a,C3: set_formula_b_a,B2: set_formula_b_a] :
( ( member_formula_b_a2 @ A @ C3 )
=> ( ( inf_in5034913211621613591la_b_a @ ( insert_formula_b_a2 @ A @ B2 ) @ C3 )
= ( insert_formula_b_a2 @ A @ ( inf_in5034913211621613591la_b_a @ B2 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_782_Int__insert__left__if1,axiom,
! [A: nat,C3: set_nat,B2: set_nat] :
( ( member_nat2 @ A @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ B2 ) @ C3 )
= ( insert_nat2 @ A @ ( inf_inf_set_nat @ B2 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_783_Int__insert__right__if0,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ~ ( member_a2 @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a2 @ A @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_784_Int__insert__right__if0,axiom,
! [A: formula_b_a,A2: set_formula_b_a,B2: set_formula_b_a] :
( ~ ( member_formula_b_a2 @ A @ A2 )
=> ( ( inf_in5034913211621613591la_b_a @ A2 @ ( insert_formula_b_a2 @ A @ B2 ) )
= ( inf_in5034913211621613591la_b_a @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_785_Int__insert__right__if0,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat2 @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat2 @ A @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_786_Int__insert__right__if1,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( member_a2 @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a2 @ A @ B2 ) )
= ( insert_a2 @ A @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_787_Int__insert__right__if1,axiom,
! [A: formula_b_a,A2: set_formula_b_a,B2: set_formula_b_a] :
( ( member_formula_b_a2 @ A @ A2 )
=> ( ( inf_in5034913211621613591la_b_a @ A2 @ ( insert_formula_b_a2 @ A @ B2 ) )
= ( insert_formula_b_a2 @ A @ ( inf_in5034913211621613591la_b_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_788_Int__insert__right__if1,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( member_nat2 @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat2 @ A @ B2 ) )
= ( insert_nat2 @ A @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_789_singleton__insert__inj__eq_H,axiom,
! [A: formula_b_a,A2: set_formula_b_a,B: formula_b_a] :
( ( ( insert_formula_b_a2 @ A @ A2 )
= ( insert_formula_b_a2 @ B @ bot_bo7861856631361375769la_b_a ) )
= ( ( A = B )
& ( ord_le5472159299058833381la_b_a @ A2 @ ( insert_formula_b_a2 @ B @ bot_bo7861856631361375769la_b_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_790_singleton__insert__inj__eq,axiom,
! [B: formula_b_a,A: formula_b_a,A2: set_formula_b_a] :
( ( ( insert_formula_b_a2 @ B @ bot_bo7861856631361375769la_b_a )
= ( insert_formula_b_a2 @ A @ A2 ) )
= ( ( A = B )
& ( ord_le5472159299058833381la_b_a @ A2 @ ( insert_formula_b_a2 @ B @ bot_bo7861856631361375769la_b_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_791_insert__disjoint_I1_J,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ ( insert_a2 @ A @ A2 ) @ B2 )
= bot_bot_set_a )
= ( ~ ( member_a2 @ A @ B2 )
& ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_792_insert__disjoint_I1_J,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ A2 ) @ B2 )
= bot_bot_set_nat )
= ( ~ ( member_nat2 @ A @ B2 )
& ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_793_insert__disjoint_I1_J,axiom,
! [A: formula_b_a,A2: set_formula_b_a,B2: set_formula_b_a] :
( ( ( inf_in5034913211621613591la_b_a @ ( insert_formula_b_a2 @ A @ A2 ) @ B2 )
= bot_bo7861856631361375769la_b_a )
= ( ~ ( member_formula_b_a2 @ A @ B2 )
& ( ( inf_in5034913211621613591la_b_a @ A2 @ B2 )
= bot_bo7861856631361375769la_b_a ) ) ) ).
% insert_disjoint(1)
thf(fact_794_insert__disjoint_I2_J,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a2 @ A @ A2 ) @ B2 ) )
= ( ~ ( member_a2 @ A @ B2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_795_insert__disjoint_I2_J,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat2 @ A @ A2 ) @ B2 ) )
= ( ~ ( member_nat2 @ A @ B2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_796_insert__disjoint_I2_J,axiom,
! [A: formula_b_a,A2: set_formula_b_a,B2: set_formula_b_a] :
( ( bot_bo7861856631361375769la_b_a
= ( inf_in5034913211621613591la_b_a @ ( insert_formula_b_a2 @ A @ A2 ) @ B2 ) )
= ( ~ ( member_formula_b_a2 @ A @ B2 )
& ( bot_bo7861856631361375769la_b_a
= ( inf_in5034913211621613591la_b_a @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_797_disjoint__insert_I1_J,axiom,
! [B2: set_a,A: a,A2: set_a] :
( ( ( inf_inf_set_a @ B2 @ ( insert_a2 @ A @ A2 ) )
= bot_bot_set_a )
= ( ~ ( member_a2 @ A @ B2 )
& ( ( inf_inf_set_a @ B2 @ A2 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_798_disjoint__insert_I1_J,axiom,
! [B2: set_nat,A: nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ B2 @ ( insert_nat2 @ A @ A2 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat2 @ A @ B2 )
& ( ( inf_inf_set_nat @ B2 @ A2 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_799_disjoint__insert_I1_J,axiom,
! [B2: set_formula_b_a,A: formula_b_a,A2: set_formula_b_a] :
( ( ( inf_in5034913211621613591la_b_a @ B2 @ ( insert_formula_b_a2 @ A @ A2 ) )
= bot_bo7861856631361375769la_b_a )
= ( ~ ( member_formula_b_a2 @ A @ B2 )
& ( ( inf_in5034913211621613591la_b_a @ B2 @ A2 )
= bot_bo7861856631361375769la_b_a ) ) ) ).
% disjoint_insert(1)
thf(fact_800_disjoint__insert_I2_J,axiom,
! [A2: set_a,B: a,B2: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ ( insert_a2 @ B @ B2 ) ) )
= ( ~ ( member_a2 @ B @ A2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_801_disjoint__insert_I2_J,axiom,
! [A2: set_nat,B: nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ ( insert_nat2 @ B @ B2 ) ) )
= ( ~ ( member_nat2 @ B @ A2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_802_disjoint__insert_I2_J,axiom,
! [A2: set_formula_b_a,B: formula_b_a,B2: set_formula_b_a] :
( ( bot_bo7861856631361375769la_b_a
= ( inf_in5034913211621613591la_b_a @ A2 @ ( insert_formula_b_a2 @ B @ B2 ) ) )
= ( ~ ( member_formula_b_a2 @ B @ A2 )
& ( bot_bo7861856631361375769la_b_a
= ( inf_in5034913211621613591la_b_a @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_803_Inf__fin_Oinsert,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_804_singletonD,axiom,
! [B: a,A: a] :
( ( member_a2 @ B @ ( insert_a2 @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_805_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat2 @ B @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_806_singletonD,axiom,
! [B: formula_b_a,A: formula_b_a] :
( ( member_formula_b_a2 @ B @ ( insert_formula_b_a2 @ A @ bot_bo7861856631361375769la_b_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_807_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a2 @ B @ ( insert_a2 @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_808_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat2 @ B @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_809_singleton__iff,axiom,
! [B: formula_b_a,A: formula_b_a] :
( ( member_formula_b_a2 @ B @ ( insert_formula_b_a2 @ A @ bot_bo7861856631361375769la_b_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_810_doubleton__eq__iff,axiom,
! [A: formula_b_a,B: formula_b_a,C: formula_b_a,D: formula_b_a] :
( ( ( insert_formula_b_a2 @ A @ ( insert_formula_b_a2 @ B @ bot_bo7861856631361375769la_b_a ) )
= ( insert_formula_b_a2 @ C @ ( insert_formula_b_a2 @ D @ bot_bo7861856631361375769la_b_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_811_insert__not__empty,axiom,
! [A: formula_b_a,A2: set_formula_b_a] :
( ( insert_formula_b_a2 @ A @ A2 )
!= bot_bo7861856631361375769la_b_a ) ).
% insert_not_empty
thf(fact_812_singleton__inject,axiom,
! [A: formula_b_a,B: formula_b_a] :
( ( ( insert_formula_b_a2 @ A @ bot_bo7861856631361375769la_b_a )
= ( insert_formula_b_a2 @ B @ bot_bo7861856631361375769la_b_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_813_finite_OinsertI,axiom,
! [A2: set_formula_b_a,A: formula_b_a] :
( ( finite4096952451150804198la_b_a @ A2 )
=> ( finite4096952451150804198la_b_a @ ( insert_formula_b_a2 @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_814_finite_OinsertI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat2 @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_815_insert__subsetI,axiom,
! [X: a,A2: set_a,X6: set_a] :
( ( member_a2 @ X @ A2 )
=> ( ( ord_less_eq_set_a @ X6 @ A2 )
=> ( ord_less_eq_set_a @ ( insert_a2 @ X @ X6 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_816_insert__subsetI,axiom,
! [X: formula_b_a,A2: set_formula_b_a,X6: set_formula_b_a] :
( ( member_formula_b_a2 @ X @ A2 )
=> ( ( ord_le5472159299058833381la_b_a @ X6 @ A2 )
=> ( ord_le5472159299058833381la_b_a @ ( insert_formula_b_a2 @ X @ X6 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_817_insert__subsetI,axiom,
! [X: nat,A2: set_nat,X6: set_nat] :
( ( member_nat2 @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ X6 @ A2 )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ X @ X6 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_818_subset__insert,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a2 @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a2 @ X @ B2 ) )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_819_subset__insert,axiom,
! [X: formula_b_a,A2: set_formula_b_a,B2: set_formula_b_a] :
( ~ ( member_formula_b_a2 @ X @ A2 )
=> ( ( ord_le5472159299058833381la_b_a @ A2 @ ( insert_formula_b_a2 @ X @ B2 ) )
= ( ord_le5472159299058833381la_b_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_820_subset__insert,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat2 @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ X @ B2 ) )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_821_Int__insert__left,axiom,
! [A: a,C3: set_a,B2: set_a] :
( ( ( member_a2 @ A @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a2 @ A @ B2 ) @ C3 )
= ( insert_a2 @ A @ ( inf_inf_set_a @ B2 @ C3 ) ) ) )
& ( ~ ( member_a2 @ A @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a2 @ A @ B2 ) @ C3 )
= ( inf_inf_set_a @ B2 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_822_Int__insert__left,axiom,
! [A: formula_b_a,C3: set_formula_b_a,B2: set_formula_b_a] :
( ( ( member_formula_b_a2 @ A @ C3 )
=> ( ( inf_in5034913211621613591la_b_a @ ( insert_formula_b_a2 @ A @ B2 ) @ C3 )
= ( insert_formula_b_a2 @ A @ ( inf_in5034913211621613591la_b_a @ B2 @ C3 ) ) ) )
& ( ~ ( member_formula_b_a2 @ A @ C3 )
=> ( ( inf_in5034913211621613591la_b_a @ ( insert_formula_b_a2 @ A @ B2 ) @ C3 )
= ( inf_in5034913211621613591la_b_a @ B2 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_823_Int__insert__left,axiom,
! [A: nat,C3: set_nat,B2: set_nat] :
( ( ( member_nat2 @ A @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ B2 ) @ C3 )
= ( insert_nat2 @ A @ ( inf_inf_set_nat @ B2 @ C3 ) ) ) )
& ( ~ ( member_nat2 @ A @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ B2 ) @ C3 )
= ( inf_inf_set_nat @ B2 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_824_Int__insert__right,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( ( member_a2 @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a2 @ A @ B2 ) )
= ( insert_a2 @ A @ ( inf_inf_set_a @ A2 @ B2 ) ) ) )
& ( ~ ( member_a2 @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a2 @ A @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_825_Int__insert__right,axiom,
! [A: formula_b_a,A2: set_formula_b_a,B2: set_formula_b_a] :
( ( ( member_formula_b_a2 @ A @ A2 )
=> ( ( inf_in5034913211621613591la_b_a @ A2 @ ( insert_formula_b_a2 @ A @ B2 ) )
= ( insert_formula_b_a2 @ A @ ( inf_in5034913211621613591la_b_a @ A2 @ B2 ) ) ) )
& ( ~ ( member_formula_b_a2 @ A @ A2 )
=> ( ( inf_in5034913211621613591la_b_a @ A2 @ ( insert_formula_b_a2 @ A @ B2 ) )
= ( inf_in5034913211621613591la_b_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_826_Int__insert__right,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( ( member_nat2 @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat2 @ A @ B2 ) )
= ( insert_nat2 @ A @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) )
& ( ~ ( member_nat2 @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat2 @ A @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_827_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_828_insertE,axiom,
! [A: formula_b_a,B: formula_b_a,A2: set_formula_b_a] :
( ( member_formula_b_a2 @ A @ ( insert_formula_b_a2 @ B @ A2 ) )
=> ( ( A != B )
=> ( member_formula_b_a2 @ A @ A2 ) ) ) ).
% insertE
thf(fact_829_insertE,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat2 @ A @ ( insert_nat2 @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat2 @ A @ A2 ) ) ) ).
% insertE
thf(fact_830_insertI1,axiom,
! [A: a,B2: set_a] : ( member_a2 @ A @ ( insert_a2 @ A @ B2 ) ) ).
% insertI1
thf(fact_831_insertI1,axiom,
! [A: formula_b_a,B2: set_formula_b_a] : ( member_formula_b_a2 @ A @ ( insert_formula_b_a2 @ A @ B2 ) ) ).
% insertI1
thf(fact_832_insertI1,axiom,
! [A: nat,B2: set_nat] : ( member_nat2 @ A @ ( insert_nat2 @ A @ B2 ) ) ).
% insertI1
thf(fact_833_insertI2,axiom,
! [A: a,B2: set_a,B: a] :
( ( member_a2 @ A @ B2 )
=> ( member_a2 @ A @ ( insert_a2 @ B @ B2 ) ) ) ).
% insertI2
thf(fact_834_insertI2,axiom,
! [A: formula_b_a,B2: set_formula_b_a,B: formula_b_a] :
( ( member_formula_b_a2 @ A @ B2 )
=> ( member_formula_b_a2 @ A @ ( insert_formula_b_a2 @ B @ B2 ) ) ) ).
% insertI2
thf(fact_835_insertI2,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( member_nat2 @ A @ B2 )
=> ( member_nat2 @ A @ ( insert_nat2 @ B @ B2 ) ) ) ).
% insertI2
thf(fact_836_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a2 @ X @ A2 )
=> ~ ! [B8: set_a] :
( ( A2
= ( insert_a2 @ X @ B8 ) )
=> ( member_a2 @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_837_Set_Oset__insert,axiom,
! [X: formula_b_a,A2: set_formula_b_a] :
( ( member_formula_b_a2 @ X @ A2 )
=> ~ ! [B8: set_formula_b_a] :
( ( A2
= ( insert_formula_b_a2 @ X @ B8 ) )
=> ( member_formula_b_a2 @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_838_Set_Oset__insert,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat2 @ X @ A2 )
=> ~ ! [B8: set_nat] :
( ( A2
= ( insert_nat2 @ X @ B8 ) )
=> ( member_nat2 @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_839_insert__ident,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a2 @ X @ A2 )
=> ( ~ ( member_a2 @ X @ B2 )
=> ( ( ( insert_a2 @ X @ A2 )
= ( insert_a2 @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_840_insert__ident,axiom,
! [X: formula_b_a,A2: set_formula_b_a,B2: set_formula_b_a] :
( ~ ( member_formula_b_a2 @ X @ A2 )
=> ( ~ ( member_formula_b_a2 @ X @ B2 )
=> ( ( ( insert_formula_b_a2 @ X @ A2 )
= ( insert_formula_b_a2 @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_841_insert__ident,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat2 @ X @ A2 )
=> ( ~ ( member_nat2 @ X @ B2 )
=> ( ( ( insert_nat2 @ X @ A2 )
= ( insert_nat2 @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_842_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a2 @ A @ A2 )
=> ( ( insert_a2 @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_843_insert__absorb,axiom,
! [A: formula_b_a,A2: set_formula_b_a] :
( ( member_formula_b_a2 @ A @ A2 )
=> ( ( insert_formula_b_a2 @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_844_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat2 @ A @ A2 )
=> ( ( insert_nat2 @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_845_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 )
=> ? [C4: set_a] :
( ( A2
= ( insert_a2 @ B @ C4 ) )
& ~ ( member_a2 @ B @ C4 )
& ( B2
= ( insert_a2 @ A @ C4 ) )
& ~ ( member_a2 @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_846_insert__eq__iff,axiom,
! [A: formula_b_a,A2: set_formula_b_a,B: formula_b_a,B2: set_formula_b_a] :
( ~ ( member_formula_b_a2 @ A @ A2 )
=> ( ~ ( member_formula_b_a2 @ B @ B2 )
=> ( ( ( insert_formula_b_a2 @ A @ A2 )
= ( insert_formula_b_a2 @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C4: set_formula_b_a] :
( ( A2
= ( insert_formula_b_a2 @ B @ C4 ) )
& ~ ( member_formula_b_a2 @ B @ C4 )
& ( B2
= ( insert_formula_b_a2 @ A @ C4 ) )
& ~ ( member_formula_b_a2 @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_847_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
( ~ ( member_nat2 @ A @ A2 )
=> ( ~ ( member_nat2 @ B @ B2 )
=> ( ( ( insert_nat2 @ A @ A2 )
= ( insert_nat2 @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C4: set_nat] :
( ( A2
= ( insert_nat2 @ B @ C4 ) )
& ~ ( member_nat2 @ B @ C4 )
& ( B2
= ( insert_nat2 @ A @ C4 ) )
& ~ ( member_nat2 @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_848_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a2 @ A @ A2 )
=> ? [B8: set_a] :
( ( A2
= ( insert_a2 @ A @ B8 ) )
& ~ ( member_a2 @ A @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_849_mk__disjoint__insert,axiom,
! [A: formula_b_a,A2: set_formula_b_a] :
( ( member_formula_b_a2 @ A @ A2 )
=> ? [B8: set_formula_b_a] :
( ( A2
= ( insert_formula_b_a2 @ A @ B8 ) )
& ~ ( member_formula_b_a2 @ A @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_850_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat2 @ A @ A2 )
=> ? [B8: set_nat] :
( ( A2
= ( insert_nat2 @ A @ B8 ) )
& ~ ( member_nat2 @ A @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_851_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 )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a2 @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a2 @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_852_infinite__finite__induct,axiom,
! [P: set_nat > $o,A2: set_nat] :
( ! [A7: set_nat] :
( ~ ( finite_finite_nat @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat2 @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_853_infinite__finite__induct,axiom,
! [P: set_formula_b_a > $o,A2: set_formula_b_a] :
( ! [A7: set_formula_b_a] :
( ~ ( finite4096952451150804198la_b_a @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bo7861856631361375769la_b_a )
=> ( ! [X2: formula_b_a,F3: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ F3 )
=> ( ~ ( member_formula_b_a2 @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_formula_b_a2 @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_854_finite__ne__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ! [X2: a] : ( P @ ( insert_a2 @ X2 @ bot_bot_set_a ) )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a2 @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a2 @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_855_finite__ne__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X2: nat] : ( P @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat2 @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_856_finite__ne__induct,axiom,
! [F2: set_formula_b_a,P: set_formula_b_a > $o] :
( ( finite4096952451150804198la_b_a @ F2 )
=> ( ( F2 != bot_bo7861856631361375769la_b_a )
=> ( ! [X2: formula_b_a] : ( P @ ( insert_formula_b_a2 @ X2 @ bot_bo7861856631361375769la_b_a ) )
=> ( ! [X2: formula_b_a,F3: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ F3 )
=> ( ( F3 != bot_bo7861856631361375769la_b_a )
=> ( ~ ( member_formula_b_a2 @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_formula_b_a2 @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_857_finite__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a2 @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a2 @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_858_finite__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat2 @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_859_finite__induct,axiom,
! [F2: set_formula_b_a,P: set_formula_b_a > $o] :
( ( finite4096952451150804198la_b_a @ F2 )
=> ( ( P @ bot_bo7861856631361375769la_b_a )
=> ( ! [X2: formula_b_a,F3: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ F3 )
=> ( ~ ( member_formula_b_a2 @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_formula_b_a2 @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_860_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A3: set_nat] :
( ( A3 = bot_bot_set_nat )
| ? [A5: set_nat,B3: nat] :
( ( A3
= ( insert_nat2 @ B3 @ A5 ) )
& ( finite_finite_nat @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_861_finite_Osimps,axiom,
( finite4096952451150804198la_b_a
= ( ^ [A3: set_formula_b_a] :
( ( A3 = bot_bo7861856631361375769la_b_a )
| ? [A5: set_formula_b_a,B3: formula_b_a] :
( ( A3
= ( insert_formula_b_a2 @ B3 @ A5 ) )
& ( finite4096952451150804198la_b_a @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_862_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A7: set_nat] :
( ? [A4: nat] :
( A
= ( insert_nat2 @ A4 @ A7 ) )
=> ~ ( finite_finite_nat @ A7 ) ) ) ) ).
% finite.cases
thf(fact_863_finite_Ocases,axiom,
! [A: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ A )
=> ( ( A != bot_bo7861856631361375769la_b_a )
=> ~ ! [A7: set_formula_b_a] :
( ? [A4: formula_b_a] :
( A
= ( insert_formula_b_a2 @ A4 @ A7 ) )
=> ~ ( finite4096952451150804198la_b_a @ A7 ) ) ) ) ).
% finite.cases
thf(fact_864_subset__singleton__iff,axiom,
! [X6: set_formula_b_a,A: formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ X6 @ ( insert_formula_b_a2 @ A @ bot_bo7861856631361375769la_b_a ) )
= ( ( X6 = bot_bo7861856631361375769la_b_a )
| ( X6
= ( insert_formula_b_a2 @ A @ bot_bo7861856631361375769la_b_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_865_subset__singletonD,axiom,
! [A2: set_formula_b_a,X: formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ A2 @ ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a ) )
=> ( ( A2 = bot_bo7861856631361375769la_b_a )
| ( A2
= ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a ) ) ) ) ).
% subset_singletonD
thf(fact_866_insert__is__Un,axiom,
( insert_formula_b_a2
= ( ^ [A3: formula_b_a] : ( sup_su8125743748909105329la_b_a @ ( insert_formula_b_a2 @ A3 @ bot_bo7861856631361375769la_b_a ) ) ) ) ).
% insert_is_Un
thf(fact_867_Un__singleton__iff,axiom,
! [A2: set_formula_b_a,B2: set_formula_b_a,X: formula_b_a] :
( ( ( sup_su8125743748909105329la_b_a @ A2 @ B2 )
= ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a ) )
= ( ( ( A2 = bot_bo7861856631361375769la_b_a )
& ( B2
= ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a ) ) )
| ( ( A2
= ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a ) )
& ( B2 = bot_bo7861856631361375769la_b_a ) )
| ( ( A2
= ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a ) )
& ( B2
= ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_868_singleton__Un__iff,axiom,
! [X: formula_b_a,A2: set_formula_b_a,B2: set_formula_b_a] :
( ( ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a )
= ( sup_su8125743748909105329la_b_a @ A2 @ B2 ) )
= ( ( ( A2 = bot_bo7861856631361375769la_b_a )
& ( B2
= ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a ) ) )
| ( ( A2
= ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a ) )
& ( B2 = bot_bo7861856631361375769la_b_a ) )
| ( ( A2
= ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a ) )
& ( B2
= ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_869_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,S3: set_a] :
( ( finite_finite_a @ S3 )
=> ( ! [Y3: a] :
( ( member_a2 @ Y3 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X2 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_a2 @ X2 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_870_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [Y3: nat] :
( ( member_nat2 @ Y3 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X2 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_nat2 @ X2 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_871_finite__ranking__induct,axiom,
! [S: set_formula_b_a,P: set_formula_b_a > $o,F: formula_b_a > nat] :
( ( finite4096952451150804198la_b_a @ S )
=> ( ( P @ bot_bo7861856631361375769la_b_a )
=> ( ! [X2: formula_b_a,S3: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ S3 )
=> ( ! [Y3: formula_b_a] :
( ( member_formula_b_a2 @ Y3 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X2 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_formula_b_a2 @ X2 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_872_finite__linorder__min__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B4: nat,A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ A7 )
=> ( ord_less_nat @ B4 @ X3 ) )
=> ( ( P @ A7 )
=> ( P @ ( insert_nat2 @ B4 @ A7 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_873_finite__linorder__max__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B4: nat,A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ A7 )
=> ( ord_less_nat @ X3 @ B4 ) )
=> ( ( P @ A7 )
=> ( P @ ( insert_nat2 @ B4 @ A7 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_874_finite__subset__induct_H,axiom,
! [F2: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A4: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a2 @ A4 @ A2 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ~ ( member_a2 @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a2 @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_875_finite__subset__induct_H,axiom,
! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat2 @ A4 @ A2 )
=> ( ( ord_less_eq_set_nat @ F3 @ A2 )
=> ( ~ ( member_nat2 @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_876_finite__subset__induct_H,axiom,
! [F2: set_formula_b_a,A2: set_formula_b_a,P: set_formula_b_a > $o] :
( ( finite4096952451150804198la_b_a @ F2 )
=> ( ( ord_le5472159299058833381la_b_a @ F2 @ A2 )
=> ( ( P @ bot_bo7861856631361375769la_b_a )
=> ( ! [A4: formula_b_a,F3: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ F3 )
=> ( ( member_formula_b_a2 @ A4 @ A2 )
=> ( ( ord_le5472159299058833381la_b_a @ F3 @ A2 )
=> ( ~ ( member_formula_b_a2 @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_formula_b_a2 @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_877_finite__subset__induct,axiom,
! [F2: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A4: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a2 @ A4 @ A2 )
=> ( ~ ( member_a2 @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a2 @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_878_finite__subset__induct,axiom,
! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat2 @ A4 @ A2 )
=> ( ~ ( member_nat2 @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_879_finite__subset__induct,axiom,
! [F2: set_formula_b_a,A2: set_formula_b_a,P: set_formula_b_a > $o] :
( ( finite4096952451150804198la_b_a @ F2 )
=> ( ( ord_le5472159299058833381la_b_a @ F2 @ A2 )
=> ( ( P @ bot_bo7861856631361375769la_b_a )
=> ( ! [A4: formula_b_a,F3: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ F3 )
=> ( ( member_formula_b_a2 @ A4 @ A2 )
=> ( ~ ( member_formula_b_a2 @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_formula_b_a2 @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_880_Min__insert2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [B4: nat] :
( ( member_nat2 @ B4 @ A2 )
=> ( ord_less_eq_nat @ A @ B4 ) )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ A @ A2 ) )
= A ) ) ) ).
% Min_insert2
thf(fact_881_Max__insert2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [B4: nat] :
( ( member_nat2 @ B4 @ A2 )
=> ( ord_less_eq_nat @ B4 @ A ) )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat2 @ A @ A2 ) )
= A ) ) ) ).
% Max_insert2
thf(fact_882_Inf__fin_Oclosed,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat,Y2: nat] : ( member_nat2 @ ( inf_inf_nat @ X2 @ Y2 ) @ ( insert_nat2 @ X2 @ ( insert_nat2 @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat2 @ ( lattic5238388535129920115in_nat @ A2 ) @ A2 ) ) ) ) ).
% Inf_fin.closed
thf(fact_883_Inf__fin_Oinsert__not__elem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat2 @ X @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_884_Sup__fin_Oclosed,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( A2 != bot_bot_set_a )
=> ( ! [X2: a,Y2: a] : ( member_a2 @ ( sup_sup_a @ X2 @ Y2 ) @ ( insert_a2 @ X2 @ ( insert_a2 @ Y2 @ bot_bot_set_a ) ) )
=> ( member_a2 @ ( lattic6792493950031347381_fin_a @ A2 ) @ A2 ) ) ) ) ).
% Sup_fin.closed
thf(fact_885_Sup__fin_Oclosed,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat,Y2: nat] : ( member_nat2 @ ( sup_sup_nat @ X2 @ Y2 ) @ ( insert_nat2 @ X2 @ ( insert_nat2 @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat2 @ ( lattic1093996805478795353in_nat @ A2 ) @ A2 ) ) ) ) ).
% Sup_fin.closed
thf(fact_886_Sup__fin_Oinsert__not__elem,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ~ ( member_a2 @ X @ A2 )
=> ( ( A2 != bot_bot_set_a )
=> ( ( lattic6792493950031347381_fin_a @ ( insert_a2 @ X @ A2 ) )
= ( sup_sup_a @ X @ ( lattic6792493950031347381_fin_a @ A2 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_887_Sup__fin_Oinsert__not__elem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat2 @ X @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_888_Min_Oinsert__not__elem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat2 @ X @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( ord_min_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ) ).
% Min.insert_not_elem
thf(fact_889_Min_Oclosed,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat,Y2: nat] : ( member_nat2 @ ( ord_min_nat @ X2 @ Y2 ) @ ( insert_nat2 @ X2 @ ( insert_nat2 @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat2 @ ( lattic8721135487736765967in_nat @ A2 ) @ A2 ) ) ) ) ).
% Min.closed
thf(fact_890_Max_Oinsert__not__elem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat2 @ X @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat2 @ X @ A2 ) )
= ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ A2 ) ) ) ) ) ) ).
% Max.insert_not_elem
thf(fact_891_Max_Oclosed,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat,Y2: nat] : ( member_nat2 @ ( ord_max_nat @ X2 @ Y2 ) @ ( insert_nat2 @ X2 @ ( insert_nat2 @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat2 @ ( lattic8265883725875713057ax_nat @ A2 ) @ A2 ) ) ) ) ).
% Max.closed
thf(fact_892_the__elem__eq,axiom,
! [X: formula_b_a] :
( ( the_elem_formula_b_a @ ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a ) )
= X ) ).
% the_elem_eq
thf(fact_893_is__singletonI,axiom,
! [X: formula_b_a] : ( is_sin2179658930933931611la_b_a @ ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a ) ) ).
% is_singletonI
thf(fact_894_Max_Oremove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ A2 )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ A2 )
= ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_895_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_896_Diff__iff,axiom,
! [C: formula_b_a,A2: set_formula_b_a,B2: set_formula_b_a] :
( ( member_formula_b_a2 @ C @ ( minus_2577195155700852062la_b_a @ A2 @ B2 ) )
= ( ( member_formula_b_a2 @ C @ A2 )
& ~ ( member_formula_b_a2 @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_897_Diff__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( ( member_nat2 @ C @ A2 )
& ~ ( member_nat2 @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_898_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_899_DiffI,axiom,
! [C: formula_b_a,A2: set_formula_b_a,B2: set_formula_b_a] :
( ( member_formula_b_a2 @ C @ A2 )
=> ( ~ ( member_formula_b_a2 @ C @ B2 )
=> ( member_formula_b_a2 @ C @ ( minus_2577195155700852062la_b_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_900_DiffI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ A2 )
=> ( ~ ( member_nat2 @ C @ B2 )
=> ( member_nat2 @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_901_Diff__cancel,axiom,
! [A2: set_formula_b_a] :
( ( minus_2577195155700852062la_b_a @ A2 @ A2 )
= bot_bo7861856631361375769la_b_a ) ).
% Diff_cancel
thf(fact_902_empty__Diff,axiom,
! [A2: set_formula_b_a] :
( ( minus_2577195155700852062la_b_a @ bot_bo7861856631361375769la_b_a @ A2 )
= bot_bo7861856631361375769la_b_a ) ).
% empty_Diff
thf(fact_903_Diff__empty,axiom,
! [A2: set_formula_b_a] :
( ( minus_2577195155700852062la_b_a @ A2 @ bot_bo7861856631361375769la_b_a )
= A2 ) ).
% Diff_empty
thf(fact_904_finite__Diff2,axiom,
! [B2: set_formula_b_a,A2: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ B2 )
=> ( ( finite4096952451150804198la_b_a @ ( minus_2577195155700852062la_b_a @ A2 @ B2 ) )
= ( finite4096952451150804198la_b_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_905_finite__Diff2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_906_finite__Diff,axiom,
! [A2: set_formula_b_a,B2: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ A2 )
=> ( finite4096952451150804198la_b_a @ ( minus_2577195155700852062la_b_a @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_907_finite__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_908_Diff__insert0,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a2 @ X @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a2 @ X @ B2 ) )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_909_Diff__insert0,axiom,
! [X: formula_b_a,A2: set_formula_b_a,B2: set_formula_b_a] :
( ~ ( member_formula_b_a2 @ X @ A2 )
=> ( ( minus_2577195155700852062la_b_a @ A2 @ ( insert_formula_b_a2 @ X @ B2 ) )
= ( minus_2577195155700852062la_b_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_910_Diff__insert0,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat2 @ X @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ B2 ) )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_911_insert__Diff1,axiom,
! [X: a,B2: set_a,A2: set_a] :
( ( member_a2 @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a2 @ X @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_912_insert__Diff1,axiom,
! [X: formula_b_a,B2: set_formula_b_a,A2: set_formula_b_a] :
( ( member_formula_b_a2 @ X @ B2 )
=> ( ( minus_2577195155700852062la_b_a @ ( insert_formula_b_a2 @ X @ A2 ) @ B2 )
= ( minus_2577195155700852062la_b_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_913_insert__Diff1,axiom,
! [X: nat,B2: set_nat,A2: set_nat] :
( ( member_nat2 @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_914_Diff__eq__empty__iff,axiom,
! [A2: set_formula_b_a,B2: set_formula_b_a] :
( ( ( minus_2577195155700852062la_b_a @ A2 @ B2 )
= bot_bo7861856631361375769la_b_a )
= ( ord_le5472159299058833381la_b_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_915_insert__Diff__single,axiom,
! [A: formula_b_a,A2: set_formula_b_a] :
( ( insert_formula_b_a2 @ A @ ( minus_2577195155700852062la_b_a @ A2 @ ( insert_formula_b_a2 @ A @ bot_bo7861856631361375769la_b_a ) ) )
= ( insert_formula_b_a2 @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_916_finite__Diff__insert,axiom,
! [A2: set_formula_b_a,A: formula_b_a,B2: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ ( minus_2577195155700852062la_b_a @ A2 @ ( insert_formula_b_a2 @ A @ B2 ) ) )
= ( finite4096952451150804198la_b_a @ ( minus_2577195155700852062la_b_a @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_917_finite__Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ B2 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_918_Diff__disjoint,axiom,
! [A2: set_formula_b_a,B2: set_formula_b_a] :
( ( inf_in5034913211621613591la_b_a @ A2 @ ( minus_2577195155700852062la_b_a @ B2 @ A2 ) )
= bot_bo7861856631361375769la_b_a ) ).
% Diff_disjoint
thf(fact_919_insert__Diff__if,axiom,
! [X: a,B2: set_a,A2: set_a] :
( ( ( member_a2 @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a2 @ X @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) )
& ( ~ ( member_a2 @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a2 @ X @ A2 ) @ B2 )
= ( insert_a2 @ X @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_920_insert__Diff__if,axiom,
! [X: formula_b_a,B2: set_formula_b_a,A2: set_formula_b_a] :
( ( ( member_formula_b_a2 @ X @ B2 )
=> ( ( minus_2577195155700852062la_b_a @ ( insert_formula_b_a2 @ X @ A2 ) @ B2 )
= ( minus_2577195155700852062la_b_a @ A2 @ B2 ) ) )
& ( ~ ( member_formula_b_a2 @ X @ B2 )
=> ( ( minus_2577195155700852062la_b_a @ ( insert_formula_b_a2 @ X @ A2 ) @ B2 )
= ( insert_formula_b_a2 @ X @ ( minus_2577195155700852062la_b_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_921_insert__Diff__if,axiom,
! [X: nat,B2: set_nat,A2: set_nat] :
( ( ( member_nat2 @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) )
& ( ~ ( member_nat2 @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A2 ) @ B2 )
= ( insert_nat2 @ X @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_922_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_923_DiffD2,axiom,
! [C: formula_b_a,A2: set_formula_b_a,B2: set_formula_b_a] :
( ( member_formula_b_a2 @ C @ ( minus_2577195155700852062la_b_a @ A2 @ B2 ) )
=> ~ ( member_formula_b_a2 @ C @ B2 ) ) ).
% DiffD2
thf(fact_924_DiffD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( member_nat2 @ C @ B2 ) ) ).
% DiffD2
thf(fact_925_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_926_DiffD1,axiom,
! [C: formula_b_a,A2: set_formula_b_a,B2: set_formula_b_a] :
( ( member_formula_b_a2 @ C @ ( minus_2577195155700852062la_b_a @ A2 @ B2 ) )
=> ( member_formula_b_a2 @ C @ A2 ) ) ).
% DiffD1
thf(fact_927_DiffD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ( member_nat2 @ C @ A2 ) ) ).
% DiffD1
thf(fact_928_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_929_DiffE,axiom,
! [C: formula_b_a,A2: set_formula_b_a,B2: set_formula_b_a] :
( ( member_formula_b_a2 @ C @ ( minus_2577195155700852062la_b_a @ A2 @ B2 ) )
=> ~ ( ( member_formula_b_a2 @ C @ A2 )
=> ( member_formula_b_a2 @ C @ B2 ) ) ) ).
% DiffE
thf(fact_930_DiffE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat2 @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat2 @ C @ A2 )
=> ( member_nat2 @ C @ B2 ) ) ) ).
% DiffE
thf(fact_931_psubset__imp__ex__mem,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ? [B4: a] : ( member_a2 @ B4 @ ( minus_minus_set_a @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_932_psubset__imp__ex__mem,axiom,
! [A2: set_formula_b_a,B2: set_formula_b_a] :
( ( ord_le976137276181116377la_b_a @ A2 @ B2 )
=> ? [B4: formula_b_a] : ( member_formula_b_a2 @ B4 @ ( minus_2577195155700852062la_b_a @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_933_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ? [B4: nat] : ( member_nat2 @ B4 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_934_Diff__infinite__finite,axiom,
! [T4: set_formula_b_a,S: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ T4 )
=> ( ~ ( finite4096952451150804198la_b_a @ S )
=> ~ ( finite4096952451150804198la_b_a @ ( minus_2577195155700852062la_b_a @ S @ T4 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_935_Diff__infinite__finite,axiom,
! [T4: set_nat,S: set_nat] :
( ( finite_finite_nat @ T4 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T4 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_936_diff__shunt__var,axiom,
! [X: set_formula_b_a,Y: set_formula_b_a] :
( ( ( minus_2577195155700852062la_b_a @ X @ Y )
= bot_bo7861856631361375769la_b_a )
= ( ord_le5472159299058833381la_b_a @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_937_Diff__insert,axiom,
! [A2: set_formula_b_a,A: formula_b_a,B2: set_formula_b_a] :
( ( minus_2577195155700852062la_b_a @ A2 @ ( insert_formula_b_a2 @ A @ B2 ) )
= ( minus_2577195155700852062la_b_a @ ( minus_2577195155700852062la_b_a @ A2 @ B2 ) @ ( insert_formula_b_a2 @ A @ bot_bo7861856631361375769la_b_a ) ) ) ).
% Diff_insert
thf(fact_938_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_939_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat2 @ A @ A2 )
=> ( ( insert_nat2 @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_940_insert__Diff,axiom,
! [A: formula_b_a,A2: set_formula_b_a] :
( ( member_formula_b_a2 @ A @ A2 )
=> ( ( insert_formula_b_a2 @ A @ ( minus_2577195155700852062la_b_a @ A2 @ ( insert_formula_b_a2 @ A @ bot_bo7861856631361375769la_b_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_941_Diff__insert2,axiom,
! [A2: set_formula_b_a,A: formula_b_a,B2: set_formula_b_a] :
( ( minus_2577195155700852062la_b_a @ A2 @ ( insert_formula_b_a2 @ A @ B2 ) )
= ( minus_2577195155700852062la_b_a @ ( minus_2577195155700852062la_b_a @ A2 @ ( insert_formula_b_a2 @ A @ bot_bo7861856631361375769la_b_a ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_942_Diff__insert__absorb,axiom,
! [X: a,A2: set_a] :
( ~ ( member_a2 @ X @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a2 @ X @ A2 ) @ ( insert_a2 @ X @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_943_Diff__insert__absorb,axiom,
! [X: nat,A2: set_nat] :
( ~ ( member_nat2 @ X @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A2 ) @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_944_Diff__insert__absorb,axiom,
! [X: formula_b_a,A2: set_formula_b_a] :
( ~ ( member_formula_b_a2 @ X @ A2 )
=> ( ( minus_2577195155700852062la_b_a @ ( insert_formula_b_a2 @ X @ A2 ) @ ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_945_subset__Diff__insert,axiom,
! [A2: set_a,B2: set_a,X: a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ ( insert_a2 @ X @ C3 ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ C3 ) )
& ~ ( member_a2 @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_946_subset__Diff__insert,axiom,
! [A2: set_formula_b_a,B2: set_formula_b_a,X: formula_b_a,C3: set_formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ A2 @ ( minus_2577195155700852062la_b_a @ B2 @ ( insert_formula_b_a2 @ X @ C3 ) ) )
= ( ( ord_le5472159299058833381la_b_a @ A2 @ ( minus_2577195155700852062la_b_a @ B2 @ C3 ) )
& ~ ( member_formula_b_a2 @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_947_subset__Diff__insert,axiom,
! [A2: set_nat,B2: set_nat,X: nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ ( insert_nat2 @ X @ C3 ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C3 ) )
& ~ ( member_nat2 @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_948_Diff__triv,axiom,
! [A2: set_formula_b_a,B2: set_formula_b_a] :
( ( ( inf_in5034913211621613591la_b_a @ A2 @ B2 )
= bot_bo7861856631361375769la_b_a )
=> ( ( minus_2577195155700852062la_b_a @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_949_Int__Diff__disjoint,axiom,
! [A2: set_formula_b_a,B2: set_formula_b_a] :
( ( inf_in5034913211621613591la_b_a @ ( inf_in5034913211621613591la_b_a @ A2 @ B2 ) @ ( minus_2577195155700852062la_b_a @ A2 @ B2 ) )
= bot_bo7861856631361375769la_b_a ) ).
% Int_Diff_disjoint
thf(fact_950_is__singleton__the__elem,axiom,
( is_sin2179658930933931611la_b_a
= ( ^ [A5: set_formula_b_a] :
( A5
= ( insert_formula_b_a2 @ ( the_elem_formula_b_a @ A5 ) @ bot_bo7861856631361375769la_b_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_951_is__singletonI_H,axiom,
! [A2: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ! [X2: a,Y2: a] :
( ( member_a2 @ X2 @ A2 )
=> ( ( member_a2 @ Y2 @ A2 )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_952_is__singletonI_H,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat,Y2: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( ( member_nat2 @ Y2 @ A2 )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_953_is__singletonI_H,axiom,
! [A2: set_formula_b_a] :
( ( A2 != bot_bo7861856631361375769la_b_a )
=> ( ! [X2: formula_b_a,Y2: formula_b_a] :
( ( member_formula_b_a2 @ X2 @ A2 )
=> ( ( member_formula_b_a2 @ Y2 @ A2 )
=> ( X2 = Y2 ) ) )
=> ( is_sin2179658930933931611la_b_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_954_finite__empty__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A4: a,A7: set_a] :
( ( finite_finite_a @ A7 )
=> ( ( member_a2 @ A4 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_minus_set_a @ A7 @ ( insert_a2 @ A4 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_955_finite__empty__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A4: nat,A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( member_nat2 @ A4 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat2 @ A4 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_956_finite__empty__induct,axiom,
! [A2: set_formula_b_a,P: set_formula_b_a > $o] :
( ( finite4096952451150804198la_b_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A4: formula_b_a,A7: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ A7 )
=> ( ( member_formula_b_a2 @ A4 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_2577195155700852062la_b_a @ A7 @ ( insert_formula_b_a2 @ A4 @ bot_bo7861856631361375769la_b_a ) ) ) ) ) )
=> ( P @ bot_bo7861856631361375769la_b_a ) ) ) ) ).
% finite_empty_induct
thf(fact_957_infinite__coinduct,axiom,
! [X6: set_nat > $o,A2: set_nat] :
( ( X6 @ A2 )
=> ( ! [A7: set_nat] :
( ( X6 @ A7 )
=> ? [X3: nat] :
( ( member_nat2 @ X3 @ A7 )
& ( ( X6 @ ( minus_minus_set_nat @ A7 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A7 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_958_infinite__coinduct,axiom,
! [X6: set_formula_b_a > $o,A2: set_formula_b_a] :
( ( X6 @ A2 )
=> ( ! [A7: set_formula_b_a] :
( ( X6 @ A7 )
=> ? [X3: formula_b_a] :
( ( member_formula_b_a2 @ X3 @ A7 )
& ( ( X6 @ ( minus_2577195155700852062la_b_a @ A7 @ ( insert_formula_b_a2 @ X3 @ bot_bo7861856631361375769la_b_a ) ) )
| ~ ( finite4096952451150804198la_b_a @ ( minus_2577195155700852062la_b_a @ A7 @ ( insert_formula_b_a2 @ X3 @ bot_bo7861856631361375769la_b_a ) ) ) ) ) )
=> ~ ( finite4096952451150804198la_b_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_959_infinite__remove,axiom,
! [S: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_960_infinite__remove,axiom,
! [S: set_formula_b_a,A: formula_b_a] :
( ~ ( finite4096952451150804198la_b_a @ S )
=> ~ ( finite4096952451150804198la_b_a @ ( minus_2577195155700852062la_b_a @ S @ ( insert_formula_b_a2 @ A @ bot_bo7861856631361375769la_b_a ) ) ) ) ).
% infinite_remove
thf(fact_961_subset__insert__iff,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a2 @ X @ B2 ) )
= ( ( ( member_a2 @ X @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a2 @ X @ bot_bot_set_a ) ) @ B2 ) )
& ( ~ ( member_a2 @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_962_subset__insert__iff,axiom,
! [A2: set_nat,X: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ X @ B2 ) )
= ( ( ( member_nat2 @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat2 @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_963_subset__insert__iff,axiom,
! [A2: set_formula_b_a,X: formula_b_a,B2: set_formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ A2 @ ( insert_formula_b_a2 @ X @ B2 ) )
= ( ( ( member_formula_b_a2 @ X @ A2 )
=> ( ord_le5472159299058833381la_b_a @ ( minus_2577195155700852062la_b_a @ A2 @ ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a ) ) @ B2 ) )
& ( ~ ( member_formula_b_a2 @ X @ A2 )
=> ( ord_le5472159299058833381la_b_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_964_Diff__single__insert,axiom,
! [A2: set_formula_b_a,X: formula_b_a,B2: set_formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ ( minus_2577195155700852062la_b_a @ A2 @ ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a ) ) @ B2 )
=> ( ord_le5472159299058833381la_b_a @ A2 @ ( insert_formula_b_a2 @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_965_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 )
=> ( ! [X3: a] :
( ( member_a2 @ X3 @ A7 )
=> ( P @ ( minus_minus_set_a @ A7 @ ( insert_a2 @ X3 @ bot_bot_set_a ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_966_finite__remove__induct,axiom,
! [B2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( A7 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A7 @ B2 )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ A7 )
=> ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_967_finite__remove__induct,axiom,
! [B2: set_formula_b_a,P: set_formula_b_a > $o] :
( ( finite4096952451150804198la_b_a @ B2 )
=> ( ( P @ bot_bo7861856631361375769la_b_a )
=> ( ! [A7: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ A7 )
=> ( ( A7 != bot_bo7861856631361375769la_b_a )
=> ( ( ord_le5472159299058833381la_b_a @ A7 @ B2 )
=> ( ! [X3: formula_b_a] :
( ( member_formula_b_a2 @ X3 @ A7 )
=> ( P @ ( minus_2577195155700852062la_b_a @ A7 @ ( insert_formula_b_a2 @ X3 @ bot_bo7861856631361375769la_b_a ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_968_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 )
=> ( ! [X3: a] :
( ( member_a2 @ X3 @ A7 )
=> ( P @ ( minus_minus_set_a @ A7 @ ( insert_a2 @ X3 @ bot_bot_set_a ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_969_remove__induct,axiom,
! [P: set_nat > $o,B2: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B2 )
=> ( P @ B2 ) )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( A7 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A7 @ B2 )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ A7 )
=> ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_970_remove__induct,axiom,
! [P: set_formula_b_a > $o,B2: set_formula_b_a] :
( ( P @ bot_bo7861856631361375769la_b_a )
=> ( ( ~ ( finite4096952451150804198la_b_a @ B2 )
=> ( P @ B2 ) )
=> ( ! [A7: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ A7 )
=> ( ( A7 != bot_bo7861856631361375769la_b_a )
=> ( ( ord_le5472159299058833381la_b_a @ A7 @ B2 )
=> ( ! [X3: formula_b_a] :
( ( member_formula_b_a2 @ X3 @ A7 )
=> ( P @ ( minus_2577195155700852062la_b_a @ A7 @ ( insert_formula_b_a2 @ X3 @ bot_bo7861856631361375769la_b_a ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_971_finite__induct__select,axiom,
! [S: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [T5: set_nat] :
( ( ord_less_set_nat @ T5 @ S )
=> ( ( P @ T5 )
=> ? [X3: nat] :
( ( member_nat2 @ X3 @ ( minus_minus_set_nat @ S @ T5 ) )
& ( P @ ( insert_nat2 @ X3 @ T5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_972_finite__induct__select,axiom,
! [S: set_formula_b_a,P: set_formula_b_a > $o] :
( ( finite4096952451150804198la_b_a @ S )
=> ( ( P @ bot_bo7861856631361375769la_b_a )
=> ( ! [T5: set_formula_b_a] :
( ( ord_le976137276181116377la_b_a @ T5 @ S )
=> ( ( P @ T5 )
=> ? [X3: formula_b_a] :
( ( member_formula_b_a2 @ X3 @ ( minus_2577195155700852062la_b_a @ S @ T5 ) )
& ( P @ ( insert_formula_b_a2 @ X3 @ T5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_973_psubset__insert__iff,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( ord_less_set_a @ A2 @ ( insert_a2 @ X @ B2 ) )
= ( ( ( member_a2 @ X @ B2 )
=> ( ord_less_set_a @ A2 @ B2 ) )
& ( ~ ( member_a2 @ X @ B2 )
=> ( ( ( member_a2 @ X @ A2 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a2 @ X @ bot_bot_set_a ) ) @ B2 ) )
& ( ~ ( member_a2 @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_974_psubset__insert__iff,axiom,
! [A2: set_nat,X: nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ ( insert_nat2 @ X @ B2 ) )
= ( ( ( member_nat2 @ X @ B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) )
& ( ~ ( member_nat2 @ X @ B2 )
=> ( ( ( member_nat2 @ X @ A2 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat2 @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_975_psubset__insert__iff,axiom,
! [A2: set_formula_b_a,X: formula_b_a,B2: set_formula_b_a] :
( ( ord_le976137276181116377la_b_a @ A2 @ ( insert_formula_b_a2 @ X @ B2 ) )
= ( ( ( member_formula_b_a2 @ X @ B2 )
=> ( ord_le976137276181116377la_b_a @ A2 @ B2 ) )
& ( ~ ( member_formula_b_a2 @ X @ B2 )
=> ( ( ( member_formula_b_a2 @ X @ A2 )
=> ( ord_le976137276181116377la_b_a @ ( minus_2577195155700852062la_b_a @ A2 @ ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a ) ) @ B2 ) )
& ( ~ ( member_formula_b_a2 @ X @ A2 )
=> ( ord_le5472159299058833381la_b_a @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_976_is__singletonE,axiom,
! [A2: set_formula_b_a] :
( ( is_sin2179658930933931611la_b_a @ A2 )
=> ~ ! [X2: formula_b_a] :
( A2
!= ( insert_formula_b_a2 @ X2 @ bot_bo7861856631361375769la_b_a ) ) ) ).
% is_singletonE
thf(fact_977_is__singleton__def,axiom,
( is_sin2179658930933931611la_b_a
= ( ^ [A5: set_formula_b_a] :
? [X4: formula_b_a] :
( A5
= ( insert_formula_b_a2 @ X4 @ bot_bo7861856631361375769la_b_a ) ) ) ) ).
% is_singleton_def
thf(fact_978_Inf__fin_Oinsert__remove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X @ A2 ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_979_Inf__fin_Oremove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A2 )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A2 )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_980_Sup__fin_Oinsert__remove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X @ A2 ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_981_Sup__fin_Oremove,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a2 @ X @ A2 )
=> ( ( ( ( minus_minus_set_a @ A2 @ ( insert_a2 @ X @ bot_bot_set_a ) )
= bot_bot_set_a )
=> ( ( lattic6792493950031347381_fin_a @ A2 )
= X ) )
& ( ( ( minus_minus_set_a @ A2 @ ( insert_a2 @ X @ bot_bot_set_a ) )
!= bot_bot_set_a )
=> ( ( lattic6792493950031347381_fin_a @ A2 )
= ( sup_sup_a @ X @ ( lattic6792493950031347381_fin_a @ ( minus_minus_set_a @ A2 @ ( insert_a2 @ X @ bot_bot_set_a ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_982_Sup__fin_Oremove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A2 )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A2 )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_983_Min_Oinsert__remove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ X @ A2 ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( ord_min_nat @ X @ ( lattic8721135487736765967in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Min.insert_remove
thf(fact_984_Min_Oremove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ A2 )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ A2 )
= ( ord_min_nat @ X @ ( lattic8721135487736765967in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Min.remove
thf(fact_985_Max_Oinsert__remove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat2 @ X @ A2 ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat2 @ X @ A2 ) )
= ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_986_set__removeAll,axiom,
! [X: a,Xs2: list_a] :
( ( set_a2 @ ( removeAll_a @ X @ Xs2 ) )
= ( minus_minus_set_a @ ( set_a2 @ Xs2 ) @ ( insert_a2 @ X @ bot_bot_set_a ) ) ) ).
% set_removeAll
thf(fact_987_set__removeAll,axiom,
! [X: formula_b_a,Xs2: list_formula_b_a] :
( ( set_formula_b_a2 @ ( remove5015878800009109827la_b_a @ X @ Xs2 ) )
= ( minus_2577195155700852062la_b_a @ ( set_formula_b_a2 @ Xs2 ) @ ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a ) ) ) ).
% set_removeAll
thf(fact_988_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_989_removeAll__id,axiom,
! [X: formula_b_a,Xs2: list_formula_b_a] :
( ~ ( member_formula_b_a2 @ X @ ( set_formula_b_a2 @ Xs2 ) )
=> ( ( remove5015878800009109827la_b_a @ X @ Xs2 )
= Xs2 ) ) ).
% removeAll_id
thf(fact_990_removeAll__id,axiom,
! [X: nat,Xs2: list_nat] :
( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( removeAll_nat @ X @ Xs2 )
= Xs2 ) ) ).
% removeAll_id
thf(fact_991_removeAll__id,axiom,
! [X: a,Xs2: list_a] :
( ~ ( member_a2 @ X @ ( set_a2 @ Xs2 ) )
=> ( ( removeAll_a @ X @ Xs2 )
= Xs2 ) ) ).
% removeAll_id
thf(fact_992_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_993_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_994_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_995_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_996_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_997_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_998_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_999_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1000_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1001_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1002_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1003_length__removeAll__less__eq,axiom,
! [X: a,Xs2: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( removeAll_a @ X @ Xs2 ) ) @ ( size_size_list_a @ Xs2 ) ) ).
% length_removeAll_less_eq
thf(fact_1004_length__removeAll__less,axiom,
! [X: formula_b_a,Xs2: list_formula_b_a] :
( ( member_formula_b_a2 @ X @ ( set_formula_b_a2 @ Xs2 ) )
=> ( ord_less_nat @ ( size_s6861460340215666547la_b_a @ ( remove5015878800009109827la_b_a @ X @ Xs2 ) ) @ ( size_s6861460340215666547la_b_a @ Xs2 ) ) ) ).
% length_removeAll_less
thf(fact_1005_length__removeAll__less,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs2 ) )
=> ( ord_less_nat @ ( size_size_list_nat @ ( removeAll_nat @ X @ Xs2 ) ) @ ( size_size_list_nat @ Xs2 ) ) ) ).
% length_removeAll_less
thf(fact_1006_length__removeAll__less,axiom,
! [X: a,Xs2: list_a] :
( ( member_a2 @ X @ ( set_a2 @ Xs2 ) )
=> ( ord_less_nat @ ( size_size_list_a @ ( removeAll_a @ X @ Xs2 ) ) @ ( size_size_list_a @ Xs2 ) ) ) ).
% length_removeAll_less
thf(fact_1007_sorted__list__of__set__nonempty,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( linord2614967742042102400et_nat @ A2 )
= ( cons_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ ( lattic8721135487736765967in_nat @ A2 ) @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% sorted_list_of_set_nonempty
thf(fact_1008_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
= ( remove1_nat @ X @ ( linord2614967742042102400et_nat @ A2 ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_1009_in__set__remove1,axiom,
! [A: formula_b_a,B: formula_b_a,Xs2: list_formula_b_a] :
( ( A != B )
=> ( ( member_formula_b_a2 @ A @ ( set_formula_b_a2 @ ( remove1_formula_b_a @ B @ Xs2 ) ) )
= ( member_formula_b_a2 @ A @ ( set_formula_b_a2 @ Xs2 ) ) ) ) ).
% in_set_remove1
thf(fact_1010_in__set__remove1,axiom,
! [A: nat,B: nat,Xs2: list_nat] :
( ( A != B )
=> ( ( member_nat2 @ A @ ( set_nat2 @ ( remove1_nat @ B @ Xs2 ) ) )
= ( member_nat2 @ A @ ( set_nat2 @ Xs2 ) ) ) ) ).
% in_set_remove1
thf(fact_1011_in__set__remove1,axiom,
! [A: a,B: a,Xs2: list_a] :
( ( A != B )
=> ( ( member_a2 @ A @ ( set_a2 @ ( remove1_a @ B @ Xs2 ) ) )
= ( member_a2 @ A @ ( set_a2 @ Xs2 ) ) ) ) ).
% in_set_remove1
thf(fact_1012_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_1013_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1014_min__diff,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
= ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).
% min_diff
thf(fact_1015_set__subset__Cons,axiom,
! [Xs2: list_a,X: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ ( cons_a @ X @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_1016_impossible__Cons,axiom,
! [Xs2: list_a,Ys: list_a,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ ( size_size_list_a @ Ys ) )
=> ( Xs2
!= ( cons_a @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_1017_list__induct2,axiom,
! [Xs2: list_a,Ys: list_a,P: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_a @ nil_a )
=> ( ! [X2: a,Xs: list_a,Y2: a,Ys3: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys3 ) )
=> ( ( P @ Xs @ Ys3 )
=> ( P @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ Y2 @ Ys3 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1018_list__induct3,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_a,P: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_a )
=> ( ! [X2: a,Xs: list_a,Y2: a,Ys3: list_a,Z3: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs @ Ys3 @ Zs2 )
=> ( P @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ Y2 @ Ys3 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_1019_list__induct4,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_a,Ws: list_a,P: list_a > list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X2: a,Xs: list_a,Y2: a,Ys3: list_a,Z3: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ Y2 @ Ys3 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1020_notin__set__remove1,axiom,
! [X: formula_b_a,Xs2: list_formula_b_a,Y: formula_b_a] :
( ~ ( member_formula_b_a2 @ X @ ( set_formula_b_a2 @ Xs2 ) )
=> ~ ( member_formula_b_a2 @ X @ ( set_formula_b_a2 @ ( remove1_formula_b_a @ Y @ Xs2 ) ) ) ) ).
% notin_set_remove1
thf(fact_1021_notin__set__remove1,axiom,
! [X: nat,Xs2: list_nat,Y: nat] :
( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs2 ) )
=> ~ ( member_nat2 @ X @ ( set_nat2 @ ( remove1_nat @ Y @ Xs2 ) ) ) ) ).
% notin_set_remove1
thf(fact_1022_notin__set__remove1,axiom,
! [X: a,Xs2: list_a,Y: a] :
( ~ ( member_a2 @ X @ ( set_a2 @ Xs2 ) )
=> ~ ( member_a2 @ X @ ( set_a2 @ ( remove1_a @ Y @ Xs2 ) ) ) ) ).
% notin_set_remove1
thf(fact_1023_remove1__idem,axiom,
! [X: formula_b_a,Xs2: list_formula_b_a] :
( ~ ( member_formula_b_a2 @ X @ ( set_formula_b_a2 @ Xs2 ) )
=> ( ( remove1_formula_b_a @ X @ Xs2 )
= Xs2 ) ) ).
% remove1_idem
thf(fact_1024_remove1__idem,axiom,
! [X: nat,Xs2: list_nat] :
( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( remove1_nat @ X @ Xs2 )
= Xs2 ) ) ).
% remove1_idem
thf(fact_1025_remove1__idem,axiom,
! [X: a,Xs2: list_a] :
( ~ ( member_a2 @ X @ ( set_a2 @ Xs2 ) )
=> ( ( remove1_a @ X @ Xs2 )
= Xs2 ) ) ).
% remove1_idem
thf(fact_1026_set__ConsD,axiom,
! [Y: formula_b_a,X: formula_b_a,Xs2: list_formula_b_a] :
( ( member_formula_b_a2 @ Y @ ( set_formula_b_a2 @ ( cons_formula_b_a @ X @ Xs2 ) ) )
=> ( ( Y = X )
| ( member_formula_b_a2 @ Y @ ( set_formula_b_a2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_1027_set__ConsD,axiom,
! [Y: nat,X: nat,Xs2: list_nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs2 ) ) )
=> ( ( Y = X )
| ( member_nat2 @ Y @ ( set_nat2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_1028_set__ConsD,axiom,
! [Y: a,X: a,Xs2: list_a] :
( ( member_a2 @ Y @ ( set_a2 @ ( cons_a @ X @ Xs2 ) ) )
=> ( ( Y = X )
| ( member_a2 @ Y @ ( set_a2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_1029_list_Oset__cases,axiom,
! [E: formula_b_a,A: list_formula_b_a] :
( ( member_formula_b_a2 @ E @ ( set_formula_b_a2 @ A ) )
=> ( ! [Z22: list_formula_b_a] :
( A
!= ( cons_formula_b_a @ E @ Z22 ) )
=> ~ ! [Z1: formula_b_a,Z22: list_formula_b_a] :
( ( A
= ( cons_formula_b_a @ Z1 @ Z22 ) )
=> ~ ( member_formula_b_a2 @ E @ ( set_formula_b_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1030_list_Oset__cases,axiom,
! [E: nat,A: list_nat] :
( ( member_nat2 @ E @ ( set_nat2 @ A ) )
=> ( ! [Z22: list_nat] :
( A
!= ( cons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat2 @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1031_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_1032_list_Oset__intros_I1_J,axiom,
! [X21: formula_b_a,X22: list_formula_b_a] : ( member_formula_b_a2 @ X21 @ ( set_formula_b_a2 @ ( cons_formula_b_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_1033_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat2 @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_1034_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_1035_list_Oset__intros_I2_J,axiom,
! [Y: formula_b_a,X22: list_formula_b_a,X21: formula_b_a] :
( ( member_formula_b_a2 @ Y @ ( set_formula_b_a2 @ X22 ) )
=> ( member_formula_b_a2 @ Y @ ( set_formula_b_a2 @ ( cons_formula_b_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1036_list_Oset__intros_I2_J,axiom,
! [Y: nat,X22: list_nat,X21: nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ X22 ) )
=> ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1037_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_1038_set__remove1__subset,axiom,
! [X: a,Xs2: list_a] : ( ord_less_eq_set_a @ ( set_a2 @ ( remove1_a @ X @ Xs2 ) ) @ ( set_a2 @ Xs2 ) ) ).
% set_remove1_subset
thf(fact_1039_the__elem__set,axiom,
! [X: a] :
( ( the_elem_a @ ( set_a2 @ ( cons_a @ X @ nil_a ) ) )
= X ) ).
% the_elem_set
thf(fact_1040_remove__def,axiom,
( remove_formula_b_a
= ( ^ [X4: formula_b_a,A5: set_formula_b_a] : ( minus_2577195155700852062la_b_a @ A5 @ ( insert_formula_b_a2 @ X4 @ bot_bo7861856631361375769la_b_a ) ) ) ) ).
% remove_def
thf(fact_1041_set__remove1__eq,axiom,
! [Xs2: list_a,X: a] :
( ( distinct_a @ Xs2 )
=> ( ( set_a2 @ ( remove1_a @ X @ Xs2 ) )
= ( minus_minus_set_a @ ( set_a2 @ Xs2 ) @ ( insert_a2 @ X @ bot_bot_set_a ) ) ) ) ).
% set_remove1_eq
thf(fact_1042_set__remove1__eq,axiom,
! [Xs2: list_formula_b_a,X: formula_b_a] :
( ( distinct_formula_b_a @ Xs2 )
=> ( ( set_formula_b_a2 @ ( remove1_formula_b_a @ X @ Xs2 ) )
= ( minus_2577195155700852062la_b_a @ ( set_formula_b_a2 @ Xs2 ) @ ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a ) ) ) ) ).
% set_remove1_eq
thf(fact_1043_member__remove,axiom,
! [X: a,Y: a,A2: set_a] :
( ( member_a2 @ X @ ( remove_a @ Y @ A2 ) )
= ( ( member_a2 @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_1044_member__remove,axiom,
! [X: formula_b_a,Y: formula_b_a,A2: set_formula_b_a] :
( ( member_formula_b_a2 @ X @ ( remove_formula_b_a @ Y @ A2 ) )
= ( ( member_formula_b_a2 @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_1045_member__remove,axiom,
! [X: nat,Y: nat,A2: set_nat] :
( ( member_nat2 @ X @ ( remove_nat @ Y @ A2 ) )
= ( ( member_nat2 @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_1046_distinct_Osimps_I2_J,axiom,
! [X: formula_b_a,Xs2: list_formula_b_a] :
( ( distinct_formula_b_a @ ( cons_formula_b_a @ X @ Xs2 ) )
= ( ~ ( member_formula_b_a2 @ X @ ( set_formula_b_a2 @ Xs2 ) )
& ( distinct_formula_b_a @ Xs2 ) ) ) ).
% distinct.simps(2)
thf(fact_1047_distinct_Osimps_I2_J,axiom,
! [X: nat,Xs2: list_nat] :
( ( distinct_nat @ ( cons_nat @ X @ Xs2 ) )
= ( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs2 ) )
& ( distinct_nat @ Xs2 ) ) ) ).
% distinct.simps(2)
thf(fact_1048_distinct_Osimps_I2_J,axiom,
! [X: a,Xs2: list_a] :
( ( distinct_a @ ( cons_a @ X @ Xs2 ) )
= ( ~ ( member_a2 @ X @ ( set_a2 @ Xs2 ) )
& ( distinct_a @ Xs2 ) ) ) ).
% distinct.simps(2)
thf(fact_1049_finite__distinct__list,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ? [Xs: list_a] :
( ( ( set_a2 @ Xs )
= A2 )
& ( distinct_a @ Xs ) ) ) ).
% finite_distinct_list
thf(fact_1050_finite__distinct__list,axiom,
! [A2: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ A2 )
=> ? [Xs: list_formula_b_a] :
( ( ( set_formula_b_a2 @ Xs )
= A2 )
& ( distinct_formula_b_a @ Xs ) ) ) ).
% finite_distinct_list
thf(fact_1051_finite__distinct__list,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ? [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= A2 )
& ( distinct_nat @ Xs ) ) ) ).
% finite_distinct_list
thf(fact_1052_inf__nat__def,axiom,
inf_inf_nat = ord_min_nat ).
% inf_nat_def
thf(fact_1053_remove__code_I1_J,axiom,
! [X: a,Xs2: list_a] :
( ( remove_a @ X @ ( set_a2 @ Xs2 ) )
= ( set_a2 @ ( removeAll_a @ X @ Xs2 ) ) ) ).
% remove_code(1)
thf(fact_1054_in__set__product__lists__length,axiom,
! [Xs2: list_a,Xss: list_list_a] :
( ( member_list_a @ Xs2 @ ( set_list_a2 @ ( product_lists_a @ Xss ) ) )
=> ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_1055_distinct__concat__iff,axiom,
! [Xs2: list_list_a] :
( ( distinct_a @ ( concat_a @ Xs2 ) )
= ( ( distinct_list_a @ ( removeAll_list_a @ nil_a @ Xs2 ) )
& ! [Ys4: list_a] :
( ( member_list_a @ Ys4 @ ( set_list_a2 @ Xs2 ) )
=> ( distinct_a @ Ys4 ) )
& ! [Ys4: list_a,Zs3: list_a] :
( ( ( member_list_a @ Ys4 @ ( set_list_a2 @ Xs2 ) )
& ( member_list_a @ Zs3 @ ( set_list_a2 @ Xs2 ) )
& ( Ys4 != Zs3 ) )
=> ( ( inf_inf_set_a @ ( set_a2 @ Ys4 ) @ ( set_a2 @ Zs3 ) )
= bot_bot_set_a ) ) ) ) ).
% distinct_concat_iff
thf(fact_1056_distinct__concat__iff,axiom,
! [Xs2: list_l1896549967257458927la_b_a] :
( ( distinct_formula_b_a @ ( concat_formula_b_a @ Xs2 ) )
= ( ( distin8544179423821613478la_b_a @ ( remove8334650765243156947la_b_a @ nil_formula_b_a @ Xs2 ) )
& ! [Ys4: list_formula_b_a] :
( ( member6997303959060339446la_b_a @ Ys4 @ ( set_list_formula_b_a2 @ Xs2 ) )
=> ( distinct_formula_b_a @ Ys4 ) )
& ! [Ys4: list_formula_b_a,Zs3: list_formula_b_a] :
( ( ( member6997303959060339446la_b_a @ Ys4 @ ( set_list_formula_b_a2 @ Xs2 ) )
& ( member6997303959060339446la_b_a @ Zs3 @ ( set_list_formula_b_a2 @ Xs2 ) )
& ( Ys4 != Zs3 ) )
=> ( ( inf_in5034913211621613591la_b_a @ ( set_formula_b_a2 @ Ys4 ) @ ( set_formula_b_a2 @ Zs3 ) )
= bot_bo7861856631361375769la_b_a ) ) ) ) ).
% distinct_concat_iff
thf(fact_1057_List_Oset__insert,axiom,
! [X: a,Xs2: list_a] :
( ( set_a2 @ ( insert_a @ X @ Xs2 ) )
= ( insert_a2 @ X @ ( set_a2 @ Xs2 ) ) ) ).
% List.set_insert
thf(fact_1058_in__set__insert,axiom,
! [X: formula_b_a,Xs2: list_formula_b_a] :
( ( member_formula_b_a2 @ X @ ( set_formula_b_a2 @ Xs2 ) )
=> ( ( insert_formula_b_a @ X @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_1059_in__set__insert,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( insert_nat @ X @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_1060_in__set__insert,axiom,
! [X: a,Xs2: list_a] :
( ( member_a2 @ X @ ( set_a2 @ Xs2 ) )
=> ( ( insert_a @ X @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_1061_not__in__set__insert,axiom,
! [X: formula_b_a,Xs2: list_formula_b_a] :
( ~ ( member_formula_b_a2 @ X @ ( set_formula_b_a2 @ Xs2 ) )
=> ( ( insert_formula_b_a @ X @ Xs2 )
= ( cons_formula_b_a @ X @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_1062_not__in__set__insert,axiom,
! [X: nat,Xs2: list_nat] :
( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( insert_nat @ X @ Xs2 )
= ( cons_nat @ X @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_1063_not__in__set__insert,axiom,
! [X: a,Xs2: list_a] :
( ~ ( member_a2 @ X @ ( set_a2 @ Xs2 ) )
=> ( ( insert_a @ X @ Xs2 )
= ( cons_a @ X @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_1064_List_Oinsert__def,axiom,
( insert_formula_b_a
= ( ^ [X4: formula_b_a,Xs3: list_formula_b_a] : ( if_list_formula_b_a @ ( member_formula_b_a2 @ X4 @ ( set_formula_b_a2 @ Xs3 ) ) @ Xs3 @ ( cons_formula_b_a @ X4 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_1065_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X4: nat,Xs3: list_nat] : ( if_list_nat @ ( member_nat2 @ X4 @ ( set_nat2 @ Xs3 ) ) @ Xs3 @ ( cons_nat @ X4 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_1066_List_Oinsert__def,axiom,
( insert_a
= ( ^ [X4: a,Xs3: list_a] : ( if_list_a @ ( member_a2 @ X4 @ ( set_a2 @ Xs3 ) ) @ Xs3 @ ( cons_a @ X4 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_1067_distinct__concat,axiom,
! [Xs2: list_list_a] :
( ( distinct_list_a @ Xs2 )
=> ( ! [Ys3: list_a] :
( ( member_list_a @ Ys3 @ ( set_list_a2 @ Xs2 ) )
=> ( distinct_a @ Ys3 ) )
=> ( ! [Ys3: list_a,Zs2: list_a] :
( ( member_list_a @ Ys3 @ ( set_list_a2 @ Xs2 ) )
=> ( ( member_list_a @ Zs2 @ ( set_list_a2 @ Xs2 ) )
=> ( ( Ys3 != Zs2 )
=> ( ( inf_inf_set_a @ ( set_a2 @ Ys3 ) @ ( set_a2 @ Zs2 ) )
= bot_bot_set_a ) ) ) )
=> ( distinct_a @ ( concat_a @ Xs2 ) ) ) ) ) ).
% distinct_concat
thf(fact_1068_distinct__concat,axiom,
! [Xs2: list_l1896549967257458927la_b_a] :
( ( distin8544179423821613478la_b_a @ Xs2 )
=> ( ! [Ys3: list_formula_b_a] :
( ( member6997303959060339446la_b_a @ Ys3 @ ( set_list_formula_b_a2 @ Xs2 ) )
=> ( distinct_formula_b_a @ Ys3 ) )
=> ( ! [Ys3: list_formula_b_a,Zs2: list_formula_b_a] :
( ( member6997303959060339446la_b_a @ Ys3 @ ( set_list_formula_b_a2 @ Xs2 ) )
=> ( ( member6997303959060339446la_b_a @ Zs2 @ ( set_list_formula_b_a2 @ Xs2 ) )
=> ( ( Ys3 != Zs2 )
=> ( ( inf_in5034913211621613591la_b_a @ ( set_formula_b_a2 @ Ys3 ) @ ( set_formula_b_a2 @ Zs2 ) )
= bot_bo7861856631361375769la_b_a ) ) ) )
=> ( distinct_formula_b_a @ ( concat_formula_b_a @ Xs2 ) ) ) ) ) ).
% distinct_concat
thf(fact_1069_finite__enum__subset,axiom,
! [X6: set_nat,Y6: set_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( finite_card_nat @ X6 ) )
=> ( ( infini8530281810654367211te_nat @ X6 @ I2 )
= ( infini8530281810654367211te_nat @ Y6 @ I2 ) ) )
=> ( ( finite_finite_nat @ X6 )
=> ( ( finite_finite_nat @ Y6 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ X6 ) @ ( finite_card_nat @ Y6 ) )
=> ( ord_less_eq_set_nat @ X6 @ Y6 ) ) ) ) ) ).
% finite_enum_subset
thf(fact_1070_card__Diff1__less,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a2 @ X @ A2 )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a2 @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1071_card__Diff1__less,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1072_card__Diff1__less,axiom,
! [A2: set_formula_b_a,X: formula_b_a] :
( ( finite4096952451150804198la_b_a @ A2 )
=> ( ( member_formula_b_a2 @ X @ A2 )
=> ( ord_less_nat @ ( finite7932102720334033959la_b_a @ ( minus_2577195155700852062la_b_a @ A2 @ ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a ) ) ) @ ( finite7932102720334033959la_b_a @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1073_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
! [A2: set_nat] :
( ( size_size_list_nat @ ( linord2614967742042102400et_nat @ A2 ) )
= ( finite_card_nat @ A2 ) ) ).
% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_1074_finite__enumerate__mono__iff,axiom,
! [S: set_nat,M: nat,N: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ M @ ( finite_card_nat @ S ) )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
=> ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ) ) ).
% finite_enumerate_mono_iff
thf(fact_1075_card__insert__le,axiom,
! [A2: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ ( insert_nat2 @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_1076_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A2: set_a,R: a > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A4: a] :
( ( member_a2 @ A4 @ A2 )
=> ? [B9: a] :
( ( member_a2 @ B9 @ B2 )
& ( R @ A4 @ B9 ) ) )
=> ( ! [A1: a,A22: a,B4: a] :
( ( member_a2 @ A1 @ A2 )
=> ( ( member_a2 @ A22 @ A2 )
=> ( ( member_a2 @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1077_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A2: set_formula_b_a,R: formula_b_a > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A4: formula_b_a] :
( ( member_formula_b_a2 @ A4 @ A2 )
=> ? [B9: a] :
( ( member_a2 @ B9 @ B2 )
& ( R @ A4 @ B9 ) ) )
=> ( ! [A1: formula_b_a,A22: formula_b_a,B4: a] :
( ( member_formula_b_a2 @ A1 @ A2 )
=> ( ( member_formula_b_a2 @ A22 @ A2 )
=> ( ( member_a2 @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite7932102720334033959la_b_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1078_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A2: set_nat,R: nat > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A4: nat] :
( ( member_nat2 @ A4 @ A2 )
=> ? [B9: a] :
( ( member_a2 @ B9 @ B2 )
& ( R @ A4 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B4: a] :
( ( member_nat2 @ A1 @ A2 )
=> ( ( member_nat2 @ A22 @ A2 )
=> ( ( member_a2 @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1079_card__le__if__inj__on__rel,axiom,
! [B2: set_formula_b_a,A2: set_a,R: a > formula_b_a > $o] :
( ( finite4096952451150804198la_b_a @ B2 )
=> ( ! [A4: a] :
( ( member_a2 @ A4 @ A2 )
=> ? [B9: formula_b_a] :
( ( member_formula_b_a2 @ B9 @ B2 )
& ( R @ A4 @ B9 ) ) )
=> ( ! [A1: a,A22: a,B4: formula_b_a] :
( ( member_a2 @ A1 @ A2 )
=> ( ( member_a2 @ A22 @ A2 )
=> ( ( member_formula_b_a2 @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite7932102720334033959la_b_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1080_card__le__if__inj__on__rel,axiom,
! [B2: set_formula_b_a,A2: set_formula_b_a,R: formula_b_a > formula_b_a > $o] :
( ( finite4096952451150804198la_b_a @ B2 )
=> ( ! [A4: formula_b_a] :
( ( member_formula_b_a2 @ A4 @ A2 )
=> ? [B9: formula_b_a] :
( ( member_formula_b_a2 @ B9 @ B2 )
& ( R @ A4 @ B9 ) ) )
=> ( ! [A1: formula_b_a,A22: formula_b_a,B4: formula_b_a] :
( ( member_formula_b_a2 @ A1 @ A2 )
=> ( ( member_formula_b_a2 @ A22 @ A2 )
=> ( ( member_formula_b_a2 @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite7932102720334033959la_b_a @ A2 ) @ ( finite7932102720334033959la_b_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1081_card__le__if__inj__on__rel,axiom,
! [B2: set_formula_b_a,A2: set_nat,R: nat > formula_b_a > $o] :
( ( finite4096952451150804198la_b_a @ B2 )
=> ( ! [A4: nat] :
( ( member_nat2 @ A4 @ A2 )
=> ? [B9: formula_b_a] :
( ( member_formula_b_a2 @ B9 @ B2 )
& ( R @ A4 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B4: formula_b_a] :
( ( member_nat2 @ A1 @ A2 )
=> ( ( member_nat2 @ A22 @ A2 )
=> ( ( member_formula_b_a2 @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite7932102720334033959la_b_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1082_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_a,R: a > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A4: a] :
( ( member_a2 @ A4 @ A2 )
=> ? [B9: nat] :
( ( member_nat2 @ B9 @ B2 )
& ( R @ A4 @ B9 ) ) )
=> ( ! [A1: a,A22: a,B4: nat] :
( ( member_a2 @ A1 @ A2 )
=> ( ( member_a2 @ A22 @ A2 )
=> ( ( member_nat2 @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1083_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_formula_b_a,R: formula_b_a > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A4: formula_b_a] :
( ( member_formula_b_a2 @ A4 @ A2 )
=> ? [B9: nat] :
( ( member_nat2 @ B9 @ B2 )
& ( R @ A4 @ B9 ) ) )
=> ( ! [A1: formula_b_a,A22: formula_b_a,B4: nat] :
( ( member_formula_b_a2 @ A1 @ A2 )
=> ( ( member_formula_b_a2 @ A22 @ A2 )
=> ( ( member_nat2 @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite7932102720334033959la_b_a @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1084_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A4: nat] :
( ( member_nat2 @ A4 @ A2 )
=> ? [B9: nat] :
( ( member_nat2 @ B9 @ B2 )
& ( R @ A4 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B4: nat] :
( ( member_nat2 @ A1 @ A2 )
=> ( ( member_nat2 @ A22 @ A2 )
=> ( ( member_nat2 @ B4 @ B2 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1085_infinite__arbitrarily__large,axiom,
! [A2: set_formula_b_a,N: nat] :
( ~ ( finite4096952451150804198la_b_a @ A2 )
=> ? [B8: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ B8 )
& ( ( finite7932102720334033959la_b_a @ B8 )
= N )
& ( ord_le5472159299058833381la_b_a @ B8 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_1086_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B8: set_nat] :
( ( finite_finite_nat @ B8 )
& ( ( finite_card_nat @ B8 )
= N )
& ( ord_less_eq_set_nat @ B8 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_1087_card__subset__eq,axiom,
! [B2: set_formula_b_a,A2: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ B2 )
=> ( ( ord_le5472159299058833381la_b_a @ A2 @ B2 )
=> ( ( ( finite7932102720334033959la_b_a @ A2 )
= ( finite7932102720334033959la_b_a @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_1088_card__subset__eq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_1089_card__mono,axiom,
! [B2: set_formula_b_a,A2: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ B2 )
=> ( ( ord_le5472159299058833381la_b_a @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite7932102720334033959la_b_a @ A2 ) @ ( finite7932102720334033959la_b_a @ B2 ) ) ) ) ).
% card_mono
thf(fact_1090_card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_1091_card__seteq,axiom,
! [B2: set_formula_b_a,A2: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ B2 )
=> ( ( ord_le5472159299058833381la_b_a @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite7932102720334033959la_b_a @ B2 ) @ ( finite7932102720334033959la_b_a @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_1092_card__seteq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_1093_exists__subset__between,axiom,
! [A2: set_formula_b_a,N: nat,C3: set_formula_b_a] :
( ( ord_less_eq_nat @ ( finite7932102720334033959la_b_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite7932102720334033959la_b_a @ C3 ) )
=> ( ( ord_le5472159299058833381la_b_a @ A2 @ C3 )
=> ( ( finite4096952451150804198la_b_a @ C3 )
=> ? [B8: set_formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ A2 @ B8 )
& ( ord_le5472159299058833381la_b_a @ B8 @ C3 )
& ( ( finite7932102720334033959la_b_a @ B8 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_1094_exists__subset__between,axiom,
! [A2: set_nat,N: nat,C3: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C3 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ( finite_finite_nat @ C3 )
=> ? [B8: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B8 )
& ( ord_less_eq_set_nat @ B8 @ C3 )
& ( ( finite_card_nat @ B8 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_1095_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_formula_b_a] :
( ( ord_less_eq_nat @ N @ ( finite7932102720334033959la_b_a @ S ) )
=> ~ ! [T5: set_formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ T5 @ S )
=> ( ( ( finite7932102720334033959la_b_a @ T5 )
= N )
=> ~ ( finite4096952451150804198la_b_a @ T5 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_1096_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T5: set_nat] :
( ( ord_less_eq_set_nat @ T5 @ S )
=> ( ( ( finite_card_nat @ T5 )
= N )
=> ~ ( finite_finite_nat @ T5 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_1097_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_formula_b_a,C3: nat] :
( ! [G2: set_formula_b_a] :
( ( ord_le5472159299058833381la_b_a @ G2 @ F2 )
=> ( ( finite4096952451150804198la_b_a @ G2 )
=> ( ord_less_eq_nat @ ( finite7932102720334033959la_b_a @ G2 ) @ C3 ) ) )
=> ( ( finite4096952451150804198la_b_a @ F2 )
& ( ord_less_eq_nat @ ( finite7932102720334033959la_b_a @ F2 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_1098_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C3: nat] :
( ! [G2: set_nat] :
( ( ord_less_eq_set_nat @ G2 @ F2 )
=> ( ( finite_finite_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C3 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_1099_card__less__sym__Diff,axiom,
! [A2: set_formula_b_a,B2: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ A2 )
=> ( ( finite4096952451150804198la_b_a @ B2 )
=> ( ( ord_less_nat @ ( finite7932102720334033959la_b_a @ A2 ) @ ( finite7932102720334033959la_b_a @ B2 ) )
=> ( ord_less_nat @ ( finite7932102720334033959la_b_a @ ( minus_2577195155700852062la_b_a @ A2 @ B2 ) ) @ ( finite7932102720334033959la_b_a @ ( minus_2577195155700852062la_b_a @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1100_card__less__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1101_card__le__sym__Diff,axiom,
! [A2: set_formula_b_a,B2: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ A2 )
=> ( ( finite4096952451150804198la_b_a @ B2 )
=> ( ( ord_less_eq_nat @ ( finite7932102720334033959la_b_a @ A2 ) @ ( finite7932102720334033959la_b_a @ B2 ) )
=> ( ord_less_eq_nat @ ( finite7932102720334033959la_b_a @ ( minus_2577195155700852062la_b_a @ A2 @ B2 ) ) @ ( finite7932102720334033959la_b_a @ ( minus_2577195155700852062la_b_a @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1102_card__le__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1103_card__length,axiom,
! [Xs2: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs2 ) ) @ ( size_size_list_nat @ Xs2 ) ) ).
% card_length
thf(fact_1104_card__length,axiom,
! [Xs2: list_a] : ( ord_less_eq_nat @ ( finite_card_a @ ( set_a2 @ Xs2 ) ) @ ( size_size_list_a @ Xs2 ) ) ).
% card_length
thf(fact_1105_psubset__card__mono,axiom,
! [B2: set_formula_b_a,A2: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ B2 )
=> ( ( ord_le976137276181116377la_b_a @ A2 @ B2 )
=> ( ord_less_nat @ ( finite7932102720334033959la_b_a @ A2 ) @ ( finite7932102720334033959la_b_a @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_1106_psubset__card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_1107_distinct__card,axiom,
! [Xs2: list_nat] :
( ( distinct_nat @ Xs2 )
=> ( ( finite_card_nat @ ( set_nat2 @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ) ).
% distinct_card
thf(fact_1108_distinct__card,axiom,
! [Xs2: list_a] :
( ( distinct_a @ Xs2 )
=> ( ( finite_card_a @ ( set_a2 @ Xs2 ) )
= ( size_size_list_a @ Xs2 ) ) ) ).
% distinct_card
thf(fact_1109_card__distinct,axiom,
! [Xs2: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) )
=> ( distinct_nat @ Xs2 ) ) ).
% card_distinct
thf(fact_1110_card__distinct,axiom,
! [Xs2: list_a] :
( ( ( finite_card_a @ ( set_a2 @ Xs2 ) )
= ( size_size_list_a @ Xs2 ) )
=> ( distinct_a @ Xs2 ) ) ).
% card_distinct
thf(fact_1111_finite__enum__ext,axiom,
! [X6: set_nat,Y6: set_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( finite_card_nat @ X6 ) )
=> ( ( infini8530281810654367211te_nat @ X6 @ I2 )
= ( infini8530281810654367211te_nat @ Y6 @ I2 ) ) )
=> ( ( finite_finite_nat @ X6 )
=> ( ( finite_finite_nat @ Y6 )
=> ( ( ( finite_card_nat @ X6 )
= ( finite_card_nat @ Y6 ) )
=> ( X6 = Y6 ) ) ) ) ) ).
% finite_enum_ext
thf(fact_1112_finite__enumerate__Ex,axiom,
! [S: set_nat,S2: nat] :
( ( finite_finite_nat @ S )
=> ( ( member_nat2 @ S2 @ S )
=> ? [N4: nat] :
( ( ord_less_nat @ N4 @ ( finite_card_nat @ S ) )
& ( ( infini8530281810654367211te_nat @ S @ N4 )
= S2 ) ) ) ) ).
% finite_enumerate_Ex
thf(fact_1113_finite__enumerate__in__set,axiom,
! [S: set_nat,N: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
=> ( member_nat2 @ ( infini8530281810654367211te_nat @ S @ N ) @ S ) ) ) ).
% finite_enumerate_in_set
thf(fact_1114_card__Diff1__le,axiom,
! [A2: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ).
% card_Diff1_le
thf(fact_1115_card__Diff1__le,axiom,
! [A2: set_formula_b_a,X: formula_b_a] : ( ord_less_eq_nat @ ( finite7932102720334033959la_b_a @ ( minus_2577195155700852062la_b_a @ A2 @ ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a ) ) ) @ ( finite7932102720334033959la_b_a @ A2 ) ) ).
% card_Diff1_le
thf(fact_1116_card__Diff__subset,axiom,
! [B2: set_formula_b_a,A2: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ B2 )
=> ( ( ord_le5472159299058833381la_b_a @ B2 @ A2 )
=> ( ( finite7932102720334033959la_b_a @ ( minus_2577195155700852062la_b_a @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite7932102720334033959la_b_a @ A2 ) @ ( finite7932102720334033959la_b_a @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1117_card__Diff__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1118_diff__card__le__card__Diff,axiom,
! [B2: set_formula_b_a,A2: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite7932102720334033959la_b_a @ A2 ) @ ( finite7932102720334033959la_b_a @ B2 ) ) @ ( finite7932102720334033959la_b_a @ ( minus_2577195155700852062la_b_a @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1119_diff__card__le__card__Diff,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1120_card__psubset,axiom,
! [B2: set_formula_b_a,A2: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ B2 )
=> ( ( ord_le5472159299058833381la_b_a @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite7932102720334033959la_b_a @ A2 ) @ ( finite7932102720334033959la_b_a @ B2 ) )
=> ( ord_le976137276181116377la_b_a @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_1121_card__psubset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_1122_card__Diff__subset__Int,axiom,
! [A2: set_formula_b_a,B2: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ ( inf_in5034913211621613591la_b_a @ A2 @ B2 ) )
=> ( ( finite7932102720334033959la_b_a @ ( minus_2577195155700852062la_b_a @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite7932102720334033959la_b_a @ A2 ) @ ( finite7932102720334033959la_b_a @ ( inf_in5034913211621613591la_b_a @ A2 @ B2 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_1123_card__Diff__subset__Int,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_1124_finite__enumerate__mono,axiom,
! [M: nat,N: nat,S: set_nat] :
( ( ord_less_nat @ M @ N )
=> ( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ) ).
% finite_enumerate_mono
thf(fact_1125_finite__le__enumerate,axiom,
! [S: set_nat,N: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ).
% finite_le_enumerate
thf(fact_1126_card__Diff1__less__iff,axiom,
! [A2: set_a,X: a] :
( ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a2 @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) )
= ( ( finite_finite_a @ A2 )
& ( member_a2 @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1127_card__Diff1__less__iff,axiom,
! [A2: set_nat,X: nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) )
= ( ( finite_finite_nat @ A2 )
& ( member_nat2 @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1128_card__Diff1__less__iff,axiom,
! [A2: set_formula_b_a,X: formula_b_a] :
( ( ord_less_nat @ ( finite7932102720334033959la_b_a @ ( minus_2577195155700852062la_b_a @ A2 @ ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a ) ) ) @ ( finite7932102720334033959la_b_a @ A2 ) )
= ( ( finite4096952451150804198la_b_a @ A2 )
& ( member_formula_b_a2 @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1129_card__Diff2__less,axiom,
! [A2: set_a,X: a,Y: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a2 @ X @ A2 )
=> ( ( member_a2 @ Y @ A2 )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a2 @ X @ bot_bot_set_a ) ) @ ( insert_a2 @ Y @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1130_card__Diff2__less,axiom,
! [A2: set_nat,X: nat,Y: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( ( member_nat2 @ Y @ A2 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ ( insert_nat2 @ Y @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1131_card__Diff2__less,axiom,
! [A2: set_formula_b_a,X: formula_b_a,Y: formula_b_a] :
( ( finite4096952451150804198la_b_a @ A2 )
=> ( ( member_formula_b_a2 @ X @ A2 )
=> ( ( member_formula_b_a2 @ Y @ A2 )
=> ( ord_less_nat @ ( finite7932102720334033959la_b_a @ ( minus_2577195155700852062la_b_a @ ( minus_2577195155700852062la_b_a @ A2 @ ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a ) ) @ ( insert_formula_b_a2 @ Y @ bot_bo7861856631361375769la_b_a ) ) ) @ ( finite7932102720334033959la_b_a @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1132_card__Diff__singleton__if,axiom,
! [X: a,A2: set_a] :
( ( ( member_a2 @ X @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a2 @ X @ bot_bot_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_a2 @ X @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a2 @ X @ bot_bot_set_a ) ) )
= ( finite_card_a @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1133_card__Diff__singleton__if,axiom,
! [X: nat,A2: set_nat] :
( ( ( member_nat2 @ X @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_nat2 @ X @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
= ( finite_card_nat @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1134_card__Diff__singleton__if,axiom,
! [X: formula_b_a,A2: set_formula_b_a] :
( ( ( member_formula_b_a2 @ X @ A2 )
=> ( ( finite7932102720334033959la_b_a @ ( minus_2577195155700852062la_b_a @ A2 @ ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a ) ) )
= ( minus_minus_nat @ ( finite7932102720334033959la_b_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_formula_b_a2 @ X @ A2 )
=> ( ( finite7932102720334033959la_b_a @ ( minus_2577195155700852062la_b_a @ A2 @ ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a ) ) )
= ( finite7932102720334033959la_b_a @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1135_card__Diff__singleton,axiom,
! [X: a,A2: set_a] :
( ( member_a2 @ X @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a2 @ X @ bot_bot_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1136_card__Diff__singleton,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat2 @ X @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1137_card__Diff__singleton,axiom,
! [X: formula_b_a,A2: set_formula_b_a] :
( ( member_formula_b_a2 @ X @ A2 )
=> ( ( finite7932102720334033959la_b_a @ ( minus_2577195155700852062la_b_a @ A2 @ ( insert_formula_b_a2 @ X @ bot_bo7861856631361375769la_b_a ) ) )
= ( minus_minus_nat @ ( finite7932102720334033959la_b_a @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1138_card__Diff__insert,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( member_a2 @ A @ A2 )
=> ( ~ ( member_a2 @ A @ B2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a2 @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1139_card__Diff__insert,axiom,
! [A: formula_b_a,A2: set_formula_b_a,B2: set_formula_b_a] :
( ( member_formula_b_a2 @ A @ A2 )
=> ( ~ ( member_formula_b_a2 @ A @ B2 )
=> ( ( finite7932102720334033959la_b_a @ ( minus_2577195155700852062la_b_a @ A2 @ ( insert_formula_b_a2 @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite7932102720334033959la_b_a @ ( minus_2577195155700852062la_b_a @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1140_card__Diff__insert,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( member_nat2 @ A @ A2 )
=> ( ~ ( member_nat2 @ A @ B2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1141_is__singleton__altdef,axiom,
( is_singleton_nat
= ( ^ [A5: set_nat] :
( ( finite_card_nat @ A5 )
= one_one_nat ) ) ) ).
% is_singleton_altdef
thf(fact_1142_card__1__singletonE,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= one_one_nat )
=> ~ ! [X2: nat] :
( A2
!= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ).
% card_1_singletonE
thf(fact_1143_card__1__singletonE,axiom,
! [A2: set_formula_b_a] :
( ( ( finite7932102720334033959la_b_a @ A2 )
= one_one_nat )
=> ~ ! [X2: formula_b_a] :
( A2
!= ( insert_formula_b_a2 @ X2 @ bot_bo7861856631361375769la_b_a ) ) ) ).
% card_1_singletonE
thf(fact_1144_length__remove1,axiom,
! [X: formula_b_a,Xs2: list_formula_b_a] :
( ( ( member_formula_b_a2 @ X @ ( set_formula_b_a2 @ Xs2 ) )
=> ( ( size_s6861460340215666547la_b_a @ ( remove1_formula_b_a @ X @ Xs2 ) )
= ( minus_minus_nat @ ( size_s6861460340215666547la_b_a @ Xs2 ) @ one_one_nat ) ) )
& ( ~ ( member_formula_b_a2 @ X @ ( set_formula_b_a2 @ Xs2 ) )
=> ( ( size_s6861460340215666547la_b_a @ ( remove1_formula_b_a @ X @ Xs2 ) )
= ( size_s6861460340215666547la_b_a @ Xs2 ) ) ) ) ).
% length_remove1
thf(fact_1145_length__remove1,axiom,
! [X: nat,Xs2: list_nat] :
( ( ( member_nat2 @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( size_size_list_nat @ ( remove1_nat @ X @ Xs2 ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) ) )
& ( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( size_size_list_nat @ ( remove1_nat @ X @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ) ) ).
% length_remove1
thf(fact_1146_length__remove1,axiom,
! [X: a,Xs2: list_a] :
( ( ( member_a2 @ X @ ( set_a2 @ Xs2 ) )
=> ( ( size_size_list_a @ ( remove1_a @ X @ Xs2 ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs2 ) @ one_one_nat ) ) )
& ( ~ ( member_a2 @ X @ ( set_a2 @ Xs2 ) )
=> ( ( size_size_list_a @ ( remove1_a @ X @ Xs2 ) )
= ( size_size_list_a @ Xs2 ) ) ) ) ).
% length_remove1
thf(fact_1147_card__insert__le__m1,axiom,
! [N: nat,Y: set_nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat2 @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_1148_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_1149_max__nat_Oeq__neutr__iff,axiom,
! [A: nat,B: nat] :
( ( ( ord_max_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_1150_max__nat_Oleft__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ zero_zero_nat @ A )
= A ) ).
% max_nat.left_neutral
thf(fact_1151_max__nat_Oneutr__eq__iff,axiom,
! [A: nat,B: nat] :
( ( zero_zero_nat
= ( ord_max_nat @ A @ B ) )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_1152_max__nat_Oright__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ zero_zero_nat )
= A ) ).
% max_nat.right_neutral
thf(fact_1153_max__0L,axiom,
! [N: nat] :
( ( ord_max_nat @ zero_zero_nat @ N )
= N ) ).
% max_0L
thf(fact_1154_max__0R,axiom,
! [N: nat] :
( ( ord_max_nat @ N @ zero_zero_nat )
= N ) ).
% max_0R
thf(fact_1155_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1156_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_1157_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1158_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1159_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1160_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1161_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1162_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1163_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1164_min__0R,axiom,
! [N: nat] :
( ( ord_min_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0R
thf(fact_1165_min__0L,axiom,
! [N: nat] :
( ( ord_min_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% min_0L
thf(fact_1166_min__0__1_I1_J,axiom,
( ( ord_min_nat @ zero_zero_nat @ one_one_nat )
= zero_zero_nat ) ).
% min_0_1(1)
thf(fact_1167_min__0__1_I2_J,axiom,
( ( ord_min_nat @ one_one_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0_1(2)
thf(fact_1168_max__0__1_I2_J,axiom,
( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
= one_one_nat ) ).
% max_0_1(2)
thf(fact_1169_max__0__1_I1_J,axiom,
( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
= one_one_nat ) ).
% max_0_1(1)
thf(fact_1170_length__0__conv,axiom,
! [Xs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_a ) ) ).
% length_0_conv
thf(fact_1171_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_1172_card_Oempty,axiom,
( ( finite7932102720334033959la_b_a @ bot_bo7861856631361375769la_b_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_1173_card_Oinfinite,axiom,
! [A2: set_formula_b_a] :
( ~ ( finite4096952451150804198la_b_a @ A2 )
=> ( ( finite7932102720334033959la_b_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_1174_card_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_1175_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1176_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1177_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1178_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1179_card__0__eq,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_1180_card__0__eq,axiom,
! [A2: set_formula_b_a] :
( ( finite4096952451150804198la_b_a @ A2 )
=> ( ( ( finite7932102720334033959la_b_a @ A2 )
= zero_zero_nat )
= ( A2 = bot_bo7861856631361375769la_b_a ) ) ) ).
% card_0_eq
thf(fact_1181_length__greater__0__conv,axiom,
! [Xs2: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs2 ) )
= ( Xs2 != nil_a ) ) ).
% length_greater_0_conv
thf(fact_1182_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_1183_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_1184_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_1185_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_1186_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_1187_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_1188_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_1189_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_1190_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1191_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1192_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1193_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1194_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1195_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1196_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1197_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1198_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1199_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1200_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1201_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1202_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1203_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1204_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1205_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1206_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1207_card__eq__0__iff,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_1208_card__eq__0__iff,axiom,
! [A2: set_formula_b_a] :
( ( ( finite7932102720334033959la_b_a @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bo7861856631361375769la_b_a )
| ~ ( finite4096952451150804198la_b_a @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_1209_card__ge__0__finite,axiom,
! [A2: set_formula_b_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite7932102720334033959la_b_a @ A2 ) )
=> ( finite4096952451150804198la_b_a @ A2 ) ) ).
% card_ge_0_finite
thf(fact_1210_card__ge__0__finite,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ( finite_finite_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_1211_length__pos__if__in__set,axiom,
! [X: formula_b_a,Xs2: list_formula_b_a] :
( ( member_formula_b_a2 @ X @ ( set_formula_b_a2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s6861460340215666547la_b_a @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_1212_length__pos__if__in__set,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_1213_length__pos__if__in__set,axiom,
! [X: a,Xs2: list_a] :
( ( member_a2 @ X @ ( set_a2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_1214_card__gt__0__iff,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
= ( ( A2 != bot_bot_set_nat )
& ( finite_finite_nat @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_1215_card__gt__0__iff,axiom,
! [A2: set_formula_b_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite7932102720334033959la_b_a @ A2 ) )
= ( ( A2 != bot_bo7861856631361375769la_b_a )
& ( finite4096952451150804198la_b_a @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_1216_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1217_timestamp__tfin__le__not__tfin,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( member_nat2 @ A @ tfin_tfin_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ~ ( member_nat2 @ B @ tfin_tfin_nat )
=> ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% timestamp_tfin_le_not_tfin
thf(fact_1218_sup__nat__def,axiom,
sup_sup_nat = ord_max_nat ).
% sup_nat_def
thf(fact_1219_timestamp__total,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
| ( ord_less_eq_nat @ B @ A ) ) ).
% timestamp_total
thf(fact_1220_length__n__lists__elem,axiom,
! [Ys: list_a,N: nat,Xs2: list_a] :
( ( member_list_a @ Ys @ ( set_list_a2 @ ( n_lists_a @ N @ Xs2 ) ) )
=> ( ( size_size_list_a @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_1221_zero__tfin,axiom,
member_nat2 @ zero_zero_nat @ tfin_tfin_nat ).
% zero_tfin
thf(fact_1222_zero__tfin,axiom,
member_a2 @ zero_zero_a @ tfin_tfin_a ).
% zero_tfin
thf(fact_1223_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1224_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1225_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1226_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1227_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_1228_nth__equalityI,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs2 ) )
=> ( ( nth_a @ Xs2 @ I2 )
= ( nth_a @ Ys @ I2 ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_1229_Skolem__list__nth,axiom,
! [K: nat,P: nat > a > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X7: a] : ( P @ I4 @ X7 ) ) )
= ( ? [Xs3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_a @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_1230_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_a,Z: list_a] : ( Y4 = Z ) )
= ( ^ [Xs3: list_a,Ys4: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs3 ) )
=> ( ( nth_a @ Xs3 @ I4 )
= ( nth_a @ Ys4 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_1231_nth__mem,axiom,
! [N: nat,Xs2: list_formula_b_a] :
( ( ord_less_nat @ N @ ( size_s6861460340215666547la_b_a @ Xs2 ) )
=> ( member_formula_b_a2 @ ( nth_formula_b_a @ Xs2 @ N ) @ ( set_formula_b_a2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_1232_nth__mem,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( member_nat2 @ ( nth_nat @ Xs2 @ N ) @ ( set_nat2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_1233_nth__mem,axiom,
! [N: nat,Xs2: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs2 ) )
=> ( member_a2 @ ( nth_a @ Xs2 @ N ) @ ( set_a2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_1234_list__ball__nth,axiom,
! [N: nat,Xs2: list_a,P: a > $o] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs2 ) )
=> ( ! [X2: a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs2 ) )
=> ( P @ X2 ) )
=> ( P @ ( nth_a @ Xs2 @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1235_in__set__conv__nth,axiom,
! [X: formula_b_a,Xs2: list_formula_b_a] :
( ( member_formula_b_a2 @ X @ ( set_formula_b_a2 @ Xs2 ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s6861460340215666547la_b_a @ Xs2 ) )
& ( ( nth_formula_b_a @ Xs2 @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1236_in__set__conv__nth,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs2 ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
& ( ( nth_nat @ Xs2 @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1237_in__set__conv__nth,axiom,
! [X: a,Xs2: list_a] :
( ( member_a2 @ X @ ( set_a2 @ Xs2 ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs2 ) )
& ( ( nth_a @ Xs2 @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1238_all__nth__imp__all__set,axiom,
! [Xs2: list_formula_b_a,P: formula_b_a > $o,X: formula_b_a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s6861460340215666547la_b_a @ Xs2 ) )
=> ( P @ ( nth_formula_b_a @ Xs2 @ I2 ) ) )
=> ( ( member_formula_b_a2 @ X @ ( set_formula_b_a2 @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1239_all__nth__imp__all__set,axiom,
! [Xs2: list_nat,P: nat > $o,X: nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ ( nth_nat @ Xs2 @ I2 ) ) )
=> ( ( member_nat2 @ X @ ( set_nat2 @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1240_all__nth__imp__all__set,axiom,
! [Xs2: list_a,P: a > $o,X: a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs2 ) )
=> ( P @ ( nth_a @ Xs2 @ I2 ) ) )
=> ( ( member_a2 @ X @ ( set_a2 @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1241_all__set__conv__all__nth,axiom,
! [Xs2: list_a,P: a > $o] :
( ( ! [X4: a] :
( ( member_a2 @ X4 @ ( set_a2 @ Xs2 ) )
=> ( P @ X4 ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs2 ) )
=> ( P @ ( nth_a @ Xs2 @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1242_nth__eq__iff__index__eq,axiom,
! [Xs2: list_a,I: nat,J: nat] :
( ( distinct_a @ Xs2 )
=> ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs2 ) )
=> ( ( ( nth_a @ Xs2 @ I )
= ( nth_a @ Xs2 @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_1243_distinct__conv__nth,axiom,
( distinct_a
= ( ^ [Xs3: list_a] :
! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs3 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_size_list_a @ Xs3 ) )
=> ( ( I4 != J3 )
=> ( ( nth_a @ Xs3 @ I4 )
!= ( nth_a @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_1244_distinct__Ex1,axiom,
! [Xs2: list_formula_b_a,X: formula_b_a] :
( ( distinct_formula_b_a @ Xs2 )
=> ( ( member_formula_b_a2 @ X @ ( set_formula_b_a2 @ Xs2 ) )
=> ? [X2: nat] :
( ( ord_less_nat @ X2 @ ( size_s6861460340215666547la_b_a @ Xs2 ) )
& ( ( nth_formula_b_a @ Xs2 @ X2 )
= X )
& ! [Y3: nat] :
( ( ( ord_less_nat @ Y3 @ ( size_s6861460340215666547la_b_a @ Xs2 ) )
& ( ( nth_formula_b_a @ Xs2 @ Y3 )
= X ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_1245_distinct__Ex1,axiom,
! [Xs2: list_nat,X: nat] :
( ( distinct_nat @ Xs2 )
=> ( ( member_nat2 @ X @ ( set_nat2 @ Xs2 ) )
=> ? [X2: nat] :
( ( ord_less_nat @ X2 @ ( size_size_list_nat @ Xs2 ) )
& ( ( nth_nat @ Xs2 @ X2 )
= X )
& ! [Y3: nat] :
( ( ( ord_less_nat @ Y3 @ ( size_size_list_nat @ Xs2 ) )
& ( ( nth_nat @ Xs2 @ Y3 )
= X ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_1246_distinct__Ex1,axiom,
! [Xs2: list_a,X: a] :
( ( distinct_a @ Xs2 )
=> ( ( member_a2 @ X @ ( set_a2 @ Xs2 ) )
=> ? [X2: nat] :
( ( ord_less_nat @ X2 @ ( size_size_list_a @ Xs2 ) )
& ( ( nth_a @ Xs2 @ X2 )
= X )
& ! [Y3: nat] :
( ( ( ord_less_nat @ Y3 @ ( size_size_list_a @ Xs2 ) )
& ( ( nth_a @ Xs2 @ Y3 )
= X ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_1247_nth__equal__first__eq,axiom,
! [X: formula_b_a,Xs2: list_formula_b_a,N: nat] :
( ~ ( member_formula_b_a2 @ X @ ( set_formula_b_a2 @ Xs2 ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s6861460340215666547la_b_a @ Xs2 ) )
=> ( ( ( nth_formula_b_a @ ( cons_formula_b_a @ X @ Xs2 ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1248_nth__equal__first__eq,axiom,
! [X: nat,Xs2: list_nat,N: nat] :
( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1249_nth__equal__first__eq,axiom,
! [X: a,Xs2: list_a,N: nat] :
( ~ ( member_a2 @ X @ ( set_a2 @ Xs2 ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_a @ Xs2 ) )
=> ( ( ( nth_a @ ( cons_a @ X @ Xs2 ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1250_set__update__distinct,axiom,
! [Xs2: list_a,N: nat,X: a] :
( ( distinct_a @ Xs2 )
=> ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs2 ) )
=> ( ( set_a2 @ ( list_update_a @ Xs2 @ N @ X ) )
= ( insert_a2 @ X @ ( minus_minus_set_a @ ( set_a2 @ Xs2 ) @ ( insert_a2 @ ( nth_a @ Xs2 @ N ) @ bot_bot_set_a ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_1251_set__update__distinct,axiom,
! [Xs2: list_formula_b_a,N: nat,X: formula_b_a] :
( ( distinct_formula_b_a @ Xs2 )
=> ( ( ord_less_nat @ N @ ( size_s6861460340215666547la_b_a @ Xs2 ) )
=> ( ( set_formula_b_a2 @ ( list_u3614645878711143009la_b_a @ Xs2 @ N @ X ) )
= ( insert_formula_b_a2 @ X @ ( minus_2577195155700852062la_b_a @ ( set_formula_b_a2 @ Xs2 ) @ ( insert_formula_b_a2 @ ( nth_formula_b_a @ Xs2 @ N ) @ bot_bo7861856631361375769la_b_a ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_1252_distinct__list__update,axiom,
! [Xs2: list_nat,A: nat,I: nat] :
( ( distinct_nat @ Xs2 )
=> ( ~ ( member_nat2 @ A @ ( minus_minus_set_nat @ ( set_nat2 @ Xs2 ) @ ( insert_nat2 @ ( nth_nat @ Xs2 @ I ) @ bot_bot_set_nat ) ) )
=> ( distinct_nat @ ( list_update_nat @ Xs2 @ I @ A ) ) ) ) ).
% distinct_list_update
thf(fact_1253_distinct__list__update,axiom,
! [Xs2: list_a,A: a,I: nat] :
( ( distinct_a @ Xs2 )
=> ( ~ ( member_a2 @ A @ ( minus_minus_set_a @ ( set_a2 @ Xs2 ) @ ( insert_a2 @ ( nth_a @ Xs2 @ I ) @ bot_bot_set_a ) ) )
=> ( distinct_a @ ( list_update_a @ Xs2 @ I @ A ) ) ) ) ).
% distinct_list_update
thf(fact_1254_distinct__list__update,axiom,
! [Xs2: list_formula_b_a,A: formula_b_a,I: nat] :
( ( distinct_formula_b_a @ Xs2 )
=> ( ~ ( member_formula_b_a2 @ A @ ( minus_2577195155700852062la_b_a @ ( set_formula_b_a2 @ Xs2 ) @ ( insert_formula_b_a2 @ ( nth_formula_b_a @ Xs2 @ I ) @ bot_bo7861856631361375769la_b_a ) ) )
=> ( distinct_formula_b_a @ ( list_u3614645878711143009la_b_a @ Xs2 @ I @ A ) ) ) ) ).
% distinct_list_update
thf(fact_1255_length__list__update,axiom,
! [Xs2: list_a,I: nat,X: a] :
( ( size_size_list_a @ ( list_update_a @ Xs2 @ I @ X ) )
= ( size_size_list_a @ Xs2 ) ) ).
% length_list_update
thf(fact_1256_list__update__beyond,axiom,
! [Xs2: list_a,I: nat,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ I )
=> ( ( list_update_a @ Xs2 @ I @ X )
= Xs2 ) ) ).
% list_update_beyond
thf(fact_1257_nth__list__update__eq,axiom,
! [I: nat,Xs2: list_a,X: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs2 ) )
=> ( ( nth_a @ ( list_update_a @ Xs2 @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_1258_set__swap,axiom,
! [I: nat,Xs2: list_a,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs2 ) )
=> ( ( set_a2 @ ( list_update_a @ ( list_update_a @ Xs2 @ I @ ( nth_a @ Xs2 @ J ) ) @ J @ ( nth_a @ Xs2 @ I ) ) )
= ( set_a2 @ Xs2 ) ) ) ) ).
% set_swap
thf(fact_1259_distinct__swap,axiom,
! [I: nat,Xs2: list_a,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs2 ) )
=> ( ( distinct_a @ ( list_update_a @ ( list_update_a @ Xs2 @ I @ ( nth_a @ Xs2 @ J ) ) @ J @ ( nth_a @ Xs2 @ I ) ) )
= ( distinct_a @ Xs2 ) ) ) ) ).
% distinct_swap
thf(fact_1260_list__update__same__conv,axiom,
! [I: nat,Xs2: list_a,X: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs2 ) )
=> ( ( ( list_update_a @ Xs2 @ I @ X )
= Xs2 )
= ( ( nth_a @ Xs2 @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_1261_nth__list__update,axiom,
! [I: nat,Xs2: list_a,J: nat,X: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs2 ) )
=> ( ( ( I = J )
=> ( ( nth_a @ ( list_update_a @ Xs2 @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth_a @ ( list_update_a @ Xs2 @ I @ X ) @ J )
= ( nth_a @ Xs2 @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_1262_set__update__subset__insert,axiom,
! [Xs2: list_a,I: nat,X: a] : ( ord_less_eq_set_a @ ( set_a2 @ ( list_update_a @ Xs2 @ I @ X ) ) @ ( insert_a2 @ X @ ( set_a2 @ Xs2 ) ) ) ).
% set_update_subset_insert
thf(fact_1263_set__update__memI,axiom,
! [N: nat,Xs2: list_formula_b_a,X: formula_b_a] :
( ( ord_less_nat @ N @ ( size_s6861460340215666547la_b_a @ Xs2 ) )
=> ( member_formula_b_a2 @ X @ ( set_formula_b_a2 @ ( list_u3614645878711143009la_b_a @ Xs2 @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_1264_set__update__memI,axiom,
! [N: nat,Xs2: list_nat,X: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( member_nat2 @ X @ ( set_nat2 @ ( list_update_nat @ Xs2 @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_1265_set__update__memI,axiom,
! [N: nat,Xs2: list_a,X: a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs2 ) )
=> ( member_a2 @ X @ ( set_a2 @ ( list_update_a @ Xs2 @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_1266_set__update__subsetI,axiom,
! [Xs2: list_formula_b_a,A2: set_formula_b_a,X: formula_b_a,I: nat] :
( ( ord_le5472159299058833381la_b_a @ ( set_formula_b_a2 @ Xs2 ) @ A2 )
=> ( ( member_formula_b_a2 @ X @ A2 )
=> ( ord_le5472159299058833381la_b_a @ ( set_formula_b_a2 @ ( list_u3614645878711143009la_b_a @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_1267_set__update__subsetI,axiom,
! [Xs2: list_nat,A2: set_nat,X: nat,I: nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
=> ( ( member_nat2 @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_1268_set__update__subsetI,axiom,
! [Xs2: list_a,A2: set_a,X: a,I: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ A2 )
=> ( ( member_a2 @ X @ A2 )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( list_update_a @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
% Helper facts (9)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__MDL__Oformula_Itf__b_Mtf__a_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__MDL__Oformula_Itf__b_Mtf__a_J_J_T,axiom,
! [X: list_formula_b_a,Y: list_formula_b_a] :
( ( if_list_formula_b_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__MDL__Oformula_Itf__b_Mtf__a_J_J_T,axiom,
! [X: list_formula_b_a,Y: list_formula_b_a] :
( ( if_list_formula_b_a @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_nat @ ( progress_b_a @ ( matchP_a_b @ i @ r ) @ tsa ) @ ( size_size_list_a @ tsa ) ).
%------------------------------------------------------------------------------