TPTP Problem File: SLH0541^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Hales_Jewett/0002_Hales_Jewett/prob_01424_063840__5910838_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1452 ( 720 unt; 175 typ; 0 def)
% Number of atoms : 3327 (1427 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10114 ( 326 ~; 34 |; 186 &;8390 @)
% ( 0 <=>;1178 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 6 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 854 ( 854 >; 0 *; 0 +; 0 <<)
% Number of symbols : 166 ( 163 usr; 21 con; 0-4 aty)
% Number of variables : 3535 ( 310 ^;3160 !; 65 ?;3535 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:46:44.517
%------------------------------------------------------------------------------
% Could-be-implicit typings (12)
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
set_nat_nat_o: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J,type,
set_set_nat_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
set_o_o: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (163)
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
comple8312177224774716605_nat_o: set_nat_nat_o > ( nat > nat ) > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_I_Eo_M_Eo_J,type,
complete_Sup_Sup_o_o: set_o_o > $o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_Eo_J,type,
comple8317665133742190828_nat_o: set_nat_o > nat > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
comple3806919086088850358_nat_o: set_set_nat_o > set_nat > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
comple5448282615319421384at_nat: set_set_nat_nat > set_nat_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_Eo_J,type,
comple90263536869209701_set_o: set_set_o > set_o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
comple548664676211718543et_nat: set_set_set_nat > set_set_nat ).
thf(sy_c_Disjoint__Sets_Odisjoint__family__on_001_Eo_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
disjoi8955196976510109695at_nat: ( $o > set_nat_nat ) > set_o > $o ).
thf(sy_c_Disjoint__Sets_Odisjoint__family__on_001_Eo_001_Eo,type,
disjoi298098064294094040on_o_o: ( $o > set_o ) > set_o > $o ).
thf(sy_c_Disjoint__Sets_Odisjoint__family__on_001_Eo_001t__Nat__Onat,type,
disjoi7928754725229124240_o_nat: ( $o > set_nat ) > set_o > $o ).
thf(sy_c_Disjoint__Sets_Odisjoint__family__on_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
disjoi1536465572982621766et_nat: ( $o > set_set_nat ) > set_o > $o ).
thf(sy_c_Disjoint__Sets_Odisjoint__family__on_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
disjoi8598568060105092073at_nat: ( nat > set_nat_nat ) > set_nat > $o ).
thf(sy_c_Disjoint__Sets_Odisjoint__family__on_001t__Nat__Onat_001_Eo,type,
disjoi1808054049482533742_nat_o: ( nat > set_o ) > set_nat > $o ).
thf(sy_c_Disjoint__Sets_Odisjoint__family__on_001t__Nat__Onat_001t__Nat__Onat,type,
disjoi6798895846410478970at_nat: ( nat > set_nat ) > set_nat > $o ).
thf(sy_c_Disjoint__Sets_Odisjoint__family__on_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
disjoi5680779568412092720et_nat: ( nat > set_set_nat ) > set_nat > $o ).
thf(sy_c_Disjoint__Sets_Odisjoint__family__on_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
disjoi7862385731094200888_nat_o: ( set_nat > set_o ) > set_set_nat > $o ).
thf(sy_c_Disjoint__Sets_Odisjoint__family__on_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
disjoi2115144663756723504at_nat: ( set_nat > set_nat ) > set_set_nat > $o ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
fun_upd_nat_set_nat: ( nat > set_nat ) > nat > set_nat > nat > set_nat ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001t__Nat__Onat,type,
piE_nat_nat: set_nat > ( nat > set_nat ) > set_nat_nat ).
thf(sy_c_FuncSet_Orestrict_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
restrict_nat_set_nat: ( nat > set_nat ) > set_nat > nat > set_nat ).
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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
minus_8121590178497047118at_nat: set_nat_nat > set_nat_nat > set_nat_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_Eo_J,type,
minus_minus_set_o: set_o > set_o > set_o ).
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_It__Set__Oset_It__Nat__Onat_J_J,type,
minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_HOL_Oundefined_001t__Nat__Onat,type,
undefined_nat: nat ).
thf(sy_c_Hales__Jewett_Olhj,type,
hales_lhj: nat > nat > nat > $o ).
thf(sy_c_Hales__Jewett_Oset__incr,type,
hales_set_incr: nat > set_nat > set_nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
if_set_nat: $o > set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
inf_inf_nat_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > ( nat > nat ) > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_Eo_M_Eo_J,type,
inf_inf_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_Eo_J,type,
inf_inf_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
inf_inf_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_Eo,type,
inf_inf_o: $o > $o > $o ).
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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
inf_inf_set_nat_nat: set_nat_nat > set_nat_nat > set_nat_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_Eo_J,type,
inf_inf_set_o: set_o > set_o > set_o ).
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_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
inf_in710756014367367485at_nat: set_set_nat_nat > set_set_nat_nat > set_set_nat_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
inf_inf_set_set_o: set_set_o > set_set_o > set_set_o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
inf_inf_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
inf_in5711780100303410308et_nat: set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001_Eo,type,
sup_sup_o: $o > $o > $o ).
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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
sup_sup_set_nat_nat: set_nat_nat > set_nat_nat > set_nat_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_Eo_J,type,
sup_sup_set_o: set_o > set_o > set_o ).
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_It__Set__Oset_It__Nat__Onat_J_J,type,
sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
bot_bot_nat_nat_o: ( nat > nat ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_Eo_M_Eo_J,type,
bot_bot_o_o: $o > $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_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
bot_bot_set_nat_o: set_nat > $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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bot_bot_set_nat_nat: set_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
bot_bot_set_o: set_o ).
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_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le7366121074344172400_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_Eo_J,type,
ord_less_eq_o_o: ( $o > $o ) > ( $o > $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_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_less_eq_nat_nat: ( nat > nat ) > ( nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
ord_le3964352015994296041_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
ord_less_eq_o: $o > $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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le9059583361652607317at_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $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_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
collect_nat_nat: ( ( nat > nat ) > $o ) > set_nat_nat ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_7977807581451749376at_nat: ( ( ( nat > nat ) > $o ) > set_nat_nat ) > set_nat_nat_o > set_set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_I_Eo_M_Eo_J_001t__Set__Oset_I_Eo_J,type,
image_o_o_set_o: ( ( $o > $o ) > set_o ) > set_o_o > set_set_o ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_Eo_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_o_set_nat: ( ( nat > $o ) > set_nat ) > set_nat_o > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo,type,
image_nat_nat_o: ( ( nat > nat ) > $o ) > set_nat_nat > set_o ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat > set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7432509271690132940et_nat: ( ( nat > nat ) > set_nat ) > set_nat_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_4687162037615663680et_nat: ( ( set_nat > $o ) > set_set_nat ) > set_set_nat_o > set_set_set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Nat__Onat,type,
image_o_nat: ( $o > nat ) > set_o > set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_I_Eo_J,type,
image_o_set_o: ( $o > set_o ) > set_o > set_set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
image_o_set_nat: ( $o > set_nat ) > set_o > set_set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_o_set_set_nat: ( $o > set_set_nat ) > set_o > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_nat_nat_nat2: ( nat > nat > nat ) > set_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
image_nat_o: ( nat > $o ) > set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_7301343469591561292at_nat: ( nat > set_nat_nat ) > set_nat > set_set_nat_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_I_Eo_J,type,
image_nat_set_o: ( nat > set_o ) > set_nat > set_set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_2194112158459175443et_nat: ( nat > set_set_nat ) > set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
image_1242417779249009364_nat_o: ( set_nat_nat > ( nat > nat ) > $o ) > set_set_nat_nat > set_nat_nat_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_Eo,type,
image_set_nat_nat_o: ( set_nat_nat > $o ) > set_set_nat_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_3832368097948589297at_nat: ( set_nat_nat > set_nat_nat ) > set_set_nat_nat > set_set_nat_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001_062_I_Eo_M_Eo_J,type,
image_set_o_o_o: ( set_o > $o > $o ) > set_set_o > set_o_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001_Eo,type,
image_set_o_o: ( set_o > $o ) > set_set_o > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
image_set_o_set_o: ( set_o > set_o ) > set_set_o > set_set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Nat__Onat_M_Eo_J,type,
image_set_nat_nat_o2: ( set_nat > nat > $o ) > set_set_nat > set_nat_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
image_set_nat_o: ( set_nat > $o ) > set_set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
image_set_nat_nat: ( set_nat > nat ) > set_set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_I_Eo_J,type,
image_set_nat_set_o: ( set_nat > set_o ) > set_set_nat > set_set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_6725021117256019401et_nat: ( set_nat > set_set_nat ) > set_set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
image_4331731847045299910_nat_o: ( set_set_nat > set_nat > $o ) > set_set_set_nat > set_set_nat_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001_Eo,type,
image_set_set_nat_o: ( set_set_nat > $o ) > set_set_set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_7884819252390400639et_nat: ( set_set_nat > set_set_nat ) > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
insert_nat_nat: ( nat > nat ) > set_nat_nat > set_nat_nat ).
thf(sy_c_Set_Oinsert_001_Eo,type,
insert_o: $o > set_o > set_o ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
insert_set_nat_nat: set_nat_nat > set_set_nat_nat > set_set_nat_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_I_Eo_J,type,
insert_set_o: set_o > set_set_o > set_set_o ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
insert_set_set_nat: set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: set_nat > $o ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_It__Nat__Onat_J,type,
the_elem_set_nat: set_set_nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
set_or9140604705432621368at_nat: ( nat > nat ) > set_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_Eo,type,
set_ord_atMost_o: $o > set_o ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
set_or3470937268066497519at_nat: ( nat > nat ) > ( nat > nat ) > set_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001_Eo,type,
set_or8254209520273421544Most_o: $o > $o > set_o ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or7074010630789208630et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
set_or1731194346131681939at_nat: ( nat > nat ) > ( nat > nat ) > set_nat_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001_Eo,type,
set_or1716231572884733764Than_o: $o > $o > set_o ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or8625682525731655386et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member_set_nat_nat: set_nat_nat > set_set_nat_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_v_BL____,type,
bl: nat > set_nat ).
thf(sy_v_BS____,type,
bs: nat > set_nat ).
thf(sy_v_BT____,type,
bt: nat > set_nat ).
thf(sy_v_Bstat____,type,
bstat: set_nat ).
thf(sy_v_Bvar____,type,
bvar: nat > set_nat ).
thf(sy_v_M_H____,type,
m: nat ).
thf(sy_v_d____,type,
d: nat ).
thf(sy_v_fL____,type,
fL: nat > nat ).
thf(sy_v_fS____,type,
fS: nat > nat ).
thf(sy_v_fT____,type,
fT: nat > nat ).
thf(sy_v_i____,type,
i: nat ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_m____,type,
m2: nat ).
thf(sy_v_n_H____,type,
n: nat ).
thf(sy_v_n____,type,
n2: nat ).
thf(sy_v_r,type,
r: nat ).
thf(sy_v_t,type,
t: nat ).
% Relevant facts (1271)
thf(fact_0__C0_C,axiom,
i = zero_zero_nat ).
% "0"
thf(fact_1_inf__bot__left,axiom,
! [X: set_nat_nat] :
( ( inf_inf_set_nat_nat @ bot_bot_set_nat_nat @ X )
= bot_bot_set_nat_nat ) ).
% inf_bot_left
thf(fact_2_inf__bot__left,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ X )
= bot_bot_set_o ) ).
% inf_bot_left
thf(fact_3_inf__bot__left,axiom,
! [X: set_set_nat] :
( ( inf_inf_set_set_nat @ bot_bot_set_set_nat @ X )
= bot_bot_set_set_nat ) ).
% inf_bot_left
thf(fact_4_inf__bot__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_5_inf__bot__right,axiom,
! [X: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ bot_bot_set_nat_nat )
= bot_bot_set_nat_nat ) ).
% inf_bot_right
thf(fact_6_inf__bot__right,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ X @ bot_bot_set_o )
= bot_bot_set_o ) ).
% inf_bot_right
thf(fact_7_inf__bot__right,axiom,
! [X: set_set_nat] :
( ( inf_inf_set_set_nat @ X @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% inf_bot_right
thf(fact_8_inf__bot__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_9_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_nat_nat] :
( ( inf_inf_set_nat_nat @ bot_bot_set_nat_nat @ X )
= bot_bot_set_nat_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_10_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ X )
= bot_bot_set_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_11_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_set_nat] :
( ( inf_inf_set_set_nat @ bot_bot_set_set_nat @ X )
= bot_bot_set_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_12_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_13_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ bot_bot_set_nat_nat )
= bot_bot_set_nat_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_14_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ X @ bot_bot_set_o )
= bot_bot_set_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_15_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_set_nat] :
( ( inf_inf_set_set_nat @ X @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_16_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_17_calculation,axiom,
( ( bvar @ i )
= ( bl @ zero_zero_nat ) ) ).
% calculation
thf(fact_18_IntI,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ A )
=> ( ( member_set_nat @ C @ B )
=> ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_19_IntI,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ A )
=> ( ( member_o @ C @ B )
=> ( member_o @ C @ ( inf_inf_set_o @ A @ B ) ) ) ) ).
% IntI
thf(fact_20_IntI,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ A )
=> ( ( member_nat_nat @ C @ B )
=> ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_21_IntI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_22_Int__iff,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B ) )
= ( ( member_set_nat @ C @ A )
& ( member_set_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_23_Int__iff,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A @ B ) )
= ( ( member_o @ C @ A )
& ( member_o @ C @ B ) ) ) ).
% Int_iff
thf(fact_24_Int__iff,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A @ B ) )
= ( ( member_nat_nat @ C @ A )
& ( member_nat_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_25_Int__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ( member_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_26_inf_Oidem,axiom,
! [A2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_27_inf_Oidem,axiom,
! [A2: set_o] :
( ( inf_inf_set_o @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_28_inf_Oidem,axiom,
! [A2: set_set_nat] :
( ( inf_inf_set_set_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_29_inf_Oidem,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_30_inf__idem,axiom,
! [X: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_31_inf__idem,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ X @ X )
= X ) ).
% inf_idem
thf(fact_32_inf__idem,axiom,
! [X: set_set_nat] :
( ( inf_inf_set_set_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_33_inf__idem,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_34_inf_Oleft__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_35_inf_Oleft__idem,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_36_inf_Oleft__idem,axiom,
! [A2: set_o,B2: set_o] :
( ( inf_inf_set_o @ A2 @ ( inf_inf_set_o @ A2 @ B2 ) )
= ( inf_inf_set_o @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_37_inf_Oleft__idem,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( inf_inf_set_set_nat @ A2 @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_set_nat @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_38_inf__left__idem,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
= ( inf_inf_set_nat @ X @ Y ) ) ).
% inf_left_idem
thf(fact_39_inf__left__idem,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ X @ Y ) )
= ( inf_inf_set_nat_nat @ X @ Y ) ) ).
% inf_left_idem
thf(fact_40_inf__left__idem,axiom,
! [X: set_o,Y: set_o] :
( ( inf_inf_set_o @ X @ ( inf_inf_set_o @ X @ Y ) )
= ( inf_inf_set_o @ X @ Y ) ) ).
% inf_left_idem
thf(fact_41_inf__left__idem,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( inf_inf_set_set_nat @ X @ ( inf_inf_set_set_nat @ X @ Y ) )
= ( inf_inf_set_set_nat @ X @ Y ) ) ).
% inf_left_idem
thf(fact_42_inf_Oright__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ B2 )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_43_inf_Oright__idem,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ B2 )
= ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_44_inf_Oright__idem,axiom,
! [A2: set_o,B2: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ A2 @ B2 ) @ B2 )
= ( inf_inf_set_o @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_45_inf_Oright__idem,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ B2 )
= ( inf_inf_set_set_nat @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_46_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_47_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_48_all__not__in__conv,axiom,
! [A: set_set_nat] :
( ( ! [X2: set_nat] :
~ ( member_set_nat @ X2 @ A ) )
= ( A = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_49_all__not__in__conv,axiom,
! [A: set_o] :
( ( ! [X2: $o] :
~ ( member_o @ X2 @ A ) )
= ( A = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_50_all__not__in__conv,axiom,
! [A: set_nat_nat] :
( ( ! [X2: nat > nat] :
~ ( member_nat_nat @ X2 @ A ) )
= ( A = bot_bot_set_nat_nat ) ) ).
% all_not_in_conv
thf(fact_51_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_52_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_53_empty__iff,axiom,
! [C: $o] :
~ ( member_o @ C @ bot_bot_set_o ) ).
% empty_iff
thf(fact_54_empty__iff,axiom,
! [C: nat > nat] :
~ ( member_nat_nat @ C @ bot_bot_set_nat_nat ) ).
% empty_iff
thf(fact_55_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_56_inf__right__idem,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y )
= ( inf_inf_set_nat @ X @ Y ) ) ).
% inf_right_idem
thf(fact_57_inf__right__idem,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ X @ Y ) @ Y )
= ( inf_inf_set_nat_nat @ X @ Y ) ) ).
% inf_right_idem
thf(fact_58_inf__right__idem,axiom,
! [X: set_o,Y: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ X @ Y ) @ Y )
= ( inf_inf_set_o @ X @ Y ) ) ).
% inf_right_idem
thf(fact_59_inf__right__idem,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ X @ Y ) @ Y )
= ( inf_inf_set_set_nat @ X @ Y ) ) ).
% inf_right_idem
thf(fact_60_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_61_ex__in__conv,axiom,
! [A: set_set_nat] :
( ( ? [X2: set_nat] : ( member_set_nat @ X2 @ A ) )
= ( A != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_62_ex__in__conv,axiom,
! [A: set_o] :
( ( ? [X2: $o] : ( member_o @ X2 @ A ) )
= ( A != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_63_ex__in__conv,axiom,
! [A: set_nat_nat] :
( ( ? [X2: nat > nat] : ( member_nat_nat @ X2 @ A ) )
= ( A != bot_bot_set_nat_nat ) ) ).
% ex_in_conv
thf(fact_64_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_65_equals0I,axiom,
! [A: set_set_nat] :
( ! [Y2: set_nat] :
~ ( member_set_nat @ Y2 @ A )
=> ( A = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_66_equals0I,axiom,
! [A: set_o] :
( ! [Y2: $o] :
~ ( member_o @ Y2 @ A )
=> ( A = bot_bot_set_o ) ) ).
% equals0I
thf(fact_67_equals0I,axiom,
! [A: set_nat_nat] :
( ! [Y2: nat > nat] :
~ ( member_nat_nat @ Y2 @ A )
=> ( A = bot_bot_set_nat_nat ) ) ).
% equals0I
thf(fact_68_equals0I,axiom,
! [A: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_69_equals0D,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( A = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_70_equals0D,axiom,
! [A: set_o,A2: $o] :
( ( A = bot_bot_set_o )
=> ~ ( member_o @ A2 @ A ) ) ).
% equals0D
thf(fact_71_equals0D,axiom,
! [A: set_nat_nat,A2: nat > nat] :
( ( A = bot_bot_set_nat_nat )
=> ~ ( member_nat_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_72_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_73_emptyE,axiom,
! [A2: set_nat] :
~ ( member_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_74_emptyE,axiom,
! [A2: $o] :
~ ( member_o @ A2 @ bot_bot_set_o ) ).
% emptyE
thf(fact_75_emptyE,axiom,
! [A2: nat > nat] :
~ ( member_nat_nat @ A2 @ bot_bot_set_nat_nat ) ).
% emptyE
thf(fact_76_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_77_inf__left__commute,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_78_inf__left__commute,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ Y @ Z ) )
= ( inf_inf_set_nat_nat @ Y @ ( inf_inf_set_nat_nat @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_79_inf__left__commute,axiom,
! [X: set_o,Y: set_o,Z: set_o] :
( ( inf_inf_set_o @ X @ ( inf_inf_set_o @ Y @ Z ) )
= ( inf_inf_set_o @ Y @ ( inf_inf_set_o @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_80_inf__left__commute,axiom,
! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( inf_inf_set_set_nat @ X @ ( inf_inf_set_set_nat @ Y @ Z ) )
= ( inf_inf_set_set_nat @ Y @ ( inf_inf_set_set_nat @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_81_inf_Oleft__commute,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ B2 @ ( inf_inf_set_nat @ A2 @ C ) )
= ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_82_inf_Oleft__commute,axiom,
! [B2: set_nat_nat,A2: set_nat_nat,C: set_nat_nat] :
( ( inf_inf_set_nat_nat @ B2 @ ( inf_inf_set_nat_nat @ A2 @ C ) )
= ( inf_inf_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_83_inf_Oleft__commute,axiom,
! [B2: set_o,A2: set_o,C: set_o] :
( ( inf_inf_set_o @ B2 @ ( inf_inf_set_o @ A2 @ C ) )
= ( inf_inf_set_o @ A2 @ ( inf_inf_set_o @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_84_inf_Oleft__commute,axiom,
! [B2: set_set_nat,A2: set_set_nat,C: set_set_nat] :
( ( inf_inf_set_set_nat @ B2 @ ( inf_inf_set_set_nat @ A2 @ C ) )
= ( inf_inf_set_set_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_85_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_nat,K: set_nat,B2: set_nat,A2: set_nat] :
( ( B
= ( inf_inf_set_nat @ K @ B2 ) )
=> ( ( inf_inf_set_nat @ A2 @ B )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_86_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_nat_nat,K: set_nat_nat,B2: set_nat_nat,A2: set_nat_nat] :
( ( B
= ( inf_inf_set_nat_nat @ K @ B2 ) )
=> ( ( inf_inf_set_nat_nat @ A2 @ B )
= ( inf_inf_set_nat_nat @ K @ ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_87_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_o,K: set_o,B2: set_o,A2: set_o] :
( ( B
= ( inf_inf_set_o @ K @ B2 ) )
=> ( ( inf_inf_set_o @ A2 @ B )
= ( inf_inf_set_o @ K @ ( inf_inf_set_o @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_88_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_set_nat,K: set_set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( B
= ( inf_inf_set_set_nat @ K @ B2 ) )
=> ( ( inf_inf_set_set_nat @ A2 @ B )
= ( inf_inf_set_set_nat @ K @ ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_89_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_nat,K: set_nat,A2: set_nat,B2: set_nat] :
( ( A
= ( inf_inf_set_nat @ K @ A2 ) )
=> ( ( inf_inf_set_nat @ A @ B2 )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_90_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_nat_nat,K: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( A
= ( inf_inf_set_nat_nat @ K @ A2 ) )
=> ( ( inf_inf_set_nat_nat @ A @ B2 )
= ( inf_inf_set_nat_nat @ K @ ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_91_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_o,K: set_o,A2: set_o,B2: set_o] :
( ( A
= ( inf_inf_set_o @ K @ A2 ) )
=> ( ( inf_inf_set_o @ A @ B2 )
= ( inf_inf_set_o @ K @ ( inf_inf_set_o @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_92_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_set_nat,K: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( A
= ( inf_inf_set_set_nat @ K @ A2 ) )
=> ( ( inf_inf_set_set_nat @ A @ B2 )
= ( inf_inf_set_set_nat @ K @ ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_93_inf__commute,axiom,
( inf_inf_set_nat
= ( ^ [X2: set_nat,Y3: set_nat] : ( inf_inf_set_nat @ Y3 @ X2 ) ) ) ).
% inf_commute
thf(fact_94_inf__commute,axiom,
( inf_inf_set_nat_nat
= ( ^ [X2: set_nat_nat,Y3: set_nat_nat] : ( inf_inf_set_nat_nat @ Y3 @ X2 ) ) ) ).
% inf_commute
thf(fact_95_inf__commute,axiom,
( inf_inf_set_o
= ( ^ [X2: set_o,Y3: set_o] : ( inf_inf_set_o @ Y3 @ X2 ) ) ) ).
% inf_commute
thf(fact_96_inf__commute,axiom,
( inf_inf_set_set_nat
= ( ^ [X2: set_set_nat,Y3: set_set_nat] : ( inf_inf_set_set_nat @ Y3 @ X2 ) ) ) ).
% inf_commute
thf(fact_97_inf_Ocommute,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( inf_inf_set_nat @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_98_inf_Ocommute,axiom,
( inf_inf_set_nat_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] : ( inf_inf_set_nat_nat @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_99_inf_Ocommute,axiom,
( inf_inf_set_o
= ( ^ [A3: set_o,B3: set_o] : ( inf_inf_set_o @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_100_inf_Ocommute,axiom,
( inf_inf_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] : ( inf_inf_set_set_nat @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_101_inf__assoc,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z )
= ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_102_inf__assoc,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ X @ Y ) @ Z )
= ( inf_inf_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_103_inf__assoc,axiom,
! [X: set_o,Y: set_o,Z: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ X @ Y ) @ Z )
= ( inf_inf_set_o @ X @ ( inf_inf_set_o @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_104_inf__assoc,axiom,
! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ X @ Y ) @ Z )
= ( inf_inf_set_set_nat @ X @ ( inf_inf_set_set_nat @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_105_inf_Oassoc,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C )
= ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_106_inf_Oassoc,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ C )
= ( inf_inf_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_107_inf_Oassoc,axiom,
! [A2: set_o,B2: set_o,C: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ A2 @ B2 ) @ C )
= ( inf_inf_set_o @ A2 @ ( inf_inf_set_o @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_108_inf_Oassoc,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ C )
= ( inf_inf_set_set_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_109_inf__sup__aci_I1_J,axiom,
( inf_inf_set_nat
= ( ^ [X2: set_nat,Y3: set_nat] : ( inf_inf_set_nat @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_110_inf__sup__aci_I1_J,axiom,
( inf_inf_set_nat_nat
= ( ^ [X2: set_nat_nat,Y3: set_nat_nat] : ( inf_inf_set_nat_nat @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_111_inf__sup__aci_I1_J,axiom,
( inf_inf_set_o
= ( ^ [X2: set_o,Y3: set_o] : ( inf_inf_set_o @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_112_inf__sup__aci_I1_J,axiom,
( inf_inf_set_set_nat
= ( ^ [X2: set_set_nat,Y3: set_set_nat] : ( inf_inf_set_set_nat @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_113_inf__sup__aci_I2_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z )
= ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_114_inf__sup__aci_I2_J,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ X @ Y ) @ Z )
= ( inf_inf_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_115_inf__sup__aci_I2_J,axiom,
! [X: set_o,Y: set_o,Z: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ X @ Y ) @ Z )
= ( inf_inf_set_o @ X @ ( inf_inf_set_o @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_116_inf__sup__aci_I2_J,axiom,
! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ X @ Y ) @ Z )
= ( inf_inf_set_set_nat @ X @ ( inf_inf_set_set_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_117_inf__sup__aci_I3_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_118_inf__sup__aci_I3_J,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ Y @ Z ) )
= ( inf_inf_set_nat_nat @ Y @ ( inf_inf_set_nat_nat @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_119_inf__sup__aci_I3_J,axiom,
! [X: set_o,Y: set_o,Z: set_o] :
( ( inf_inf_set_o @ X @ ( inf_inf_set_o @ Y @ Z ) )
= ( inf_inf_set_o @ Y @ ( inf_inf_set_o @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_120_inf__sup__aci_I3_J,axiom,
! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( inf_inf_set_set_nat @ X @ ( inf_inf_set_set_nat @ Y @ Z ) )
= ( inf_inf_set_set_nat @ Y @ ( inf_inf_set_set_nat @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_121_inf__sup__aci_I4_J,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
= ( inf_inf_set_nat @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_122_inf__sup__aci_I4_J,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ X @ Y ) )
= ( inf_inf_set_nat_nat @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_123_inf__sup__aci_I4_J,axiom,
! [X: set_o,Y: set_o] :
( ( inf_inf_set_o @ X @ ( inf_inf_set_o @ X @ Y ) )
= ( inf_inf_set_o @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_124_inf__sup__aci_I4_J,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( inf_inf_set_set_nat @ X @ ( inf_inf_set_set_nat @ X @ Y ) )
= ( inf_inf_set_set_nat @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_125_Int__left__commute,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) )
= ( inf_inf_set_nat @ B @ ( inf_inf_set_nat @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_126_Int__left__commute,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A @ ( inf_inf_set_nat_nat @ B @ C2 ) )
= ( inf_inf_set_nat_nat @ B @ ( inf_inf_set_nat_nat @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_127_Int__left__commute,axiom,
! [A: set_o,B: set_o,C2: set_o] :
( ( inf_inf_set_o @ A @ ( inf_inf_set_o @ B @ C2 ) )
= ( inf_inf_set_o @ B @ ( inf_inf_set_o @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_128_Int__left__commute,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( inf_inf_set_set_nat @ A @ ( inf_inf_set_set_nat @ B @ C2 ) )
= ( inf_inf_set_set_nat @ B @ ( inf_inf_set_set_nat @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_129_Int__left__absorb,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ A @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ).
% Int_left_absorb
thf(fact_130_Int__left__absorb,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A @ ( inf_inf_set_nat_nat @ A @ B ) )
= ( inf_inf_set_nat_nat @ A @ B ) ) ).
% Int_left_absorb
thf(fact_131_Int__left__absorb,axiom,
! [A: set_o,B: set_o] :
( ( inf_inf_set_o @ A @ ( inf_inf_set_o @ A @ B ) )
= ( inf_inf_set_o @ A @ B ) ) ).
% Int_left_absorb
thf(fact_132_Int__left__absorb,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( inf_inf_set_set_nat @ A @ ( inf_inf_set_set_nat @ A @ B ) )
= ( inf_inf_set_set_nat @ A @ B ) ) ).
% Int_left_absorb
thf(fact_133_Int__commute,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( inf_inf_set_nat @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_134_Int__commute,axiom,
( inf_inf_set_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] : ( inf_inf_set_nat_nat @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_135_Int__commute,axiom,
( inf_inf_set_o
= ( ^ [A4: set_o,B4: set_o] : ( inf_inf_set_o @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_136_Int__commute,axiom,
( inf_inf_set_set_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] : ( inf_inf_set_set_nat @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_137_Int__absorb,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_138_Int__absorb,axiom,
! [A: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_139_Int__absorb,axiom,
! [A: set_o] :
( ( inf_inf_set_o @ A @ A )
= A ) ).
% Int_absorb
thf(fact_140_Int__absorb,axiom,
! [A: set_set_nat] :
( ( inf_inf_set_set_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_141_Int__assoc,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_142_Int__assoc,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat_nat @ A @ ( inf_inf_set_nat_nat @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_143_Int__assoc,axiom,
! [A: set_o,B: set_o,C2: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ A @ B ) @ C2 )
= ( inf_inf_set_o @ A @ ( inf_inf_set_o @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_144_Int__assoc,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ A @ B ) @ C2 )
= ( inf_inf_set_set_nat @ A @ ( inf_inf_set_set_nat @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_145_IntD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ B ) ) ).
% IntD2
thf(fact_146_IntD2,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A @ B ) )
=> ( member_nat_nat @ C @ B ) ) ).
% IntD2
thf(fact_147_IntD2,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A @ B ) )
=> ( member_o @ C @ B ) ) ).
% IntD2
thf(fact_148_IntD2,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B ) )
=> ( member_set_nat @ C @ B ) ) ).
% IntD2
thf(fact_149_IntD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% IntD1
thf(fact_150_IntD1,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A @ B ) )
=> ( member_nat_nat @ C @ A ) ) ).
% IntD1
thf(fact_151_IntD1,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A @ B ) )
=> ( member_o @ C @ A ) ) ).
% IntD1
thf(fact_152_IntD1,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B ) )
=> ( member_set_nat @ C @ A ) ) ).
% IntD1
thf(fact_153_IntE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ~ ( member_nat @ C @ B ) ) ) ).
% IntE
thf(fact_154_IntE,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A @ B ) )
=> ~ ( ( member_nat_nat @ C @ A )
=> ~ ( member_nat_nat @ C @ B ) ) ) ).
% IntE
thf(fact_155_IntE,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A @ B ) )
=> ~ ( ( member_o @ C @ A )
=> ~ ( member_o @ C @ B ) ) ) ).
% IntE
thf(fact_156_IntE,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B ) )
=> ~ ( ( member_set_nat @ C @ A )
=> ~ ( member_set_nat @ C @ B ) ) ) ).
% IntE
thf(fact_157_mem__Collect__eq,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A2 @ ( collect_set_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_158_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_159_mem__Collect__eq,axiom,
! [A2: $o,P: $o > $o] :
( ( member_o @ A2 @ ( collect_o @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_160_mem__Collect__eq,axiom,
! [A2: nat > nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ A2 @ ( collect_nat_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_161_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_162_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_163_Collect__mem__eq,axiom,
! [A: set_o] :
( ( collect_o
@ ^ [X2: $o] : ( member_o @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_164_Collect__mem__eq,axiom,
! [A: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_165_disjoint__iff__not__equal,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ( inf_inf_set_nat_nat @ A @ B )
= bot_bot_set_nat_nat )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A )
=> ! [Y3: nat > nat] :
( ( member_nat_nat @ Y3 @ B )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_166_disjoint__iff__not__equal,axiom,
! [A: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ! [Y3: $o] :
( ( member_o @ Y3 @ B )
=> ( X2 = (~ Y3) ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_167_disjoint__iff__not__equal,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ( inf_inf_set_set_nat @ A @ B )
= bot_bot_set_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ! [Y3: set_nat] :
( ( member_set_nat @ Y3 @ B )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_168_disjoint__iff__not__equal,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ B )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_169_Int__empty__right,axiom,
! [A: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A @ bot_bot_set_nat_nat )
= bot_bot_set_nat_nat ) ).
% Int_empty_right
thf(fact_170_Int__empty__right,axiom,
! [A: set_o] :
( ( inf_inf_set_o @ A @ bot_bot_set_o )
= bot_bot_set_o ) ).
% Int_empty_right
thf(fact_171_Int__empty__right,axiom,
! [A: set_set_nat] :
( ( inf_inf_set_set_nat @ A @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% Int_empty_right
thf(fact_172_Int__empty__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_173_Int__empty__left,axiom,
! [B: set_nat_nat] :
( ( inf_inf_set_nat_nat @ bot_bot_set_nat_nat @ B )
= bot_bot_set_nat_nat ) ).
% Int_empty_left
thf(fact_174_Int__empty__left,axiom,
! [B: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ B )
= bot_bot_set_o ) ).
% Int_empty_left
thf(fact_175_Int__empty__left,axiom,
! [B: set_set_nat] :
( ( inf_inf_set_set_nat @ bot_bot_set_set_nat @ B )
= bot_bot_set_set_nat ) ).
% Int_empty_left
thf(fact_176_Int__empty__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_177_disjoint__iff,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ( inf_inf_set_nat_nat @ A @ B )
= bot_bot_set_nat_nat )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A )
=> ~ ( member_nat_nat @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_178_disjoint__iff,axiom,
! [A: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ~ ( member_o @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_179_disjoint__iff,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ( inf_inf_set_set_nat @ A @ B )
= bot_bot_set_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ~ ( member_set_nat @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_180_disjoint__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ~ ( member_nat @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_181_Int__emptyI,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ~ ( member_nat_nat @ X3 @ B ) )
=> ( ( inf_inf_set_nat_nat @ A @ B )
= bot_bot_set_nat_nat ) ) ).
% Int_emptyI
thf(fact_182_Int__emptyI,axiom,
! [A: set_o,B: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ~ ( member_o @ X3 @ B ) )
=> ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o ) ) ).
% Int_emptyI
thf(fact_183_Int__emptyI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ~ ( member_set_nat @ X3 @ B ) )
=> ( ( inf_inf_set_set_nat @ A @ B )
= bot_bot_set_set_nat ) ) ).
% Int_emptyI
thf(fact_184_Int__emptyI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ~ ( member_nat @ X3 @ B ) )
=> ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_185_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_186_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A4: set_nat] : ( A4 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_187_BfL__props_I3_J,axiom,
~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ bl @ ( set_ord_lessThan_nat @ one_one_nat ) ) ) ).
% BfL_props(3)
thf(fact_188_BfL__props_I1_J,axiom,
disjoi6798895846410478970at_nat @ bl @ ( set_ord_atMost_nat @ one_one_nat ) ).
% BfL_props(1)
thf(fact_189_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_190_disjoint__family__onI,axiom,
! [S: set_nat,A: nat > set_o] :
( ! [M: nat,N: nat] :
( ( member_nat @ M @ S )
=> ( ( member_nat @ N @ S )
=> ( ( M != N )
=> ( ( inf_inf_set_o @ ( A @ M ) @ ( A @ N ) )
= bot_bot_set_o ) ) ) )
=> ( disjoi1808054049482533742_nat_o @ A @ S ) ) ).
% disjoint_family_onI
thf(fact_191_disjoint__family__onI,axiom,
! [S: set_o,A: $o > set_o] :
( ! [M: $o,N: $o] :
( ( member_o @ M @ S )
=> ( ( member_o @ N @ S )
=> ( ( M != N )
=> ( ( inf_inf_set_o @ ( A @ M ) @ ( A @ N ) )
= bot_bot_set_o ) ) ) )
=> ( disjoi298098064294094040on_o_o @ A @ S ) ) ).
% disjoint_family_onI
thf(fact_192_disjoint__family__onI,axiom,
! [S: set_o,A: $o > set_nat] :
( ! [M: $o,N: $o] :
( ( member_o @ M @ S )
=> ( ( member_o @ N @ S )
=> ( ( M != N )
=> ( ( inf_inf_set_nat @ ( A @ M ) @ ( A @ N ) )
= bot_bot_set_nat ) ) ) )
=> ( disjoi7928754725229124240_o_nat @ A @ S ) ) ).
% disjoint_family_onI
thf(fact_193_disjoint__family__onI,axiom,
! [S: set_nat,A: nat > set_nat] :
( ! [M: nat,N: nat] :
( ( member_nat @ M @ S )
=> ( ( member_nat @ N @ S )
=> ( ( M != N )
=> ( ( inf_inf_set_nat @ ( A @ M ) @ ( A @ N ) )
= bot_bot_set_nat ) ) ) )
=> ( disjoi6798895846410478970at_nat @ A @ S ) ) ).
% disjoint_family_onI
thf(fact_194_disjoint__family__onI,axiom,
! [S: set_set_nat,A: set_nat > set_o] :
( ! [M: set_nat,N: set_nat] :
( ( member_set_nat @ M @ S )
=> ( ( member_set_nat @ N @ S )
=> ( ( M != N )
=> ( ( inf_inf_set_o @ ( A @ M ) @ ( A @ N ) )
= bot_bot_set_o ) ) ) )
=> ( disjoi7862385731094200888_nat_o @ A @ S ) ) ).
% disjoint_family_onI
thf(fact_195_disjoint__family__onI,axiom,
! [S: set_nat,A: nat > set_set_nat] :
( ! [M: nat,N: nat] :
( ( member_nat @ M @ S )
=> ( ( member_nat @ N @ S )
=> ( ( M != N )
=> ( ( inf_inf_set_set_nat @ ( A @ M ) @ ( A @ N ) )
= bot_bot_set_set_nat ) ) ) )
=> ( disjoi5680779568412092720et_nat @ A @ S ) ) ).
% disjoint_family_onI
thf(fact_196_disjoint__family__onI,axiom,
! [S: set_o,A: $o > set_set_nat] :
( ! [M: $o,N: $o] :
( ( member_o @ M @ S )
=> ( ( member_o @ N @ S )
=> ( ( M != N )
=> ( ( inf_inf_set_set_nat @ ( A @ M ) @ ( A @ N ) )
= bot_bot_set_set_nat ) ) ) )
=> ( disjoi1536465572982621766et_nat @ A @ S ) ) ).
% disjoint_family_onI
thf(fact_197_disjoint__family__onI,axiom,
! [S: set_set_nat,A: set_nat > set_nat] :
( ! [M: set_nat,N: set_nat] :
( ( member_set_nat @ M @ S )
=> ( ( member_set_nat @ N @ S )
=> ( ( M != N )
=> ( ( inf_inf_set_nat @ ( A @ M ) @ ( A @ N ) )
= bot_bot_set_nat ) ) ) )
=> ( disjoi2115144663756723504at_nat @ A @ S ) ) ).
% disjoint_family_onI
thf(fact_198_disjoint__family__onI,axiom,
! [S: set_nat,A: nat > set_nat_nat] :
( ! [M: nat,N: nat] :
( ( member_nat @ M @ S )
=> ( ( member_nat @ N @ S )
=> ( ( M != N )
=> ( ( inf_inf_set_nat_nat @ ( A @ M ) @ ( A @ N ) )
= bot_bot_set_nat_nat ) ) ) )
=> ( disjoi8598568060105092073at_nat @ A @ S ) ) ).
% disjoint_family_onI
thf(fact_199_disjoint__family__onI,axiom,
! [S: set_o,A: $o > set_nat_nat] :
( ! [M: $o,N: $o] :
( ( member_o @ M @ S )
=> ( ( member_o @ N @ S )
=> ( ( M != N )
=> ( ( inf_inf_set_nat_nat @ ( A @ M ) @ ( A @ N ) )
= bot_bot_set_nat_nat ) ) ) )
=> ( disjoi8955196976510109695at_nat @ A @ S ) ) ).
% disjoint_family_onI
thf(fact_200_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_201_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_202_image__eqI,axiom,
! [B2: nat,F: nat > nat,X: nat,A: set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_203_image__eqI,axiom,
! [B2: $o,F: nat > $o,X: nat,A: set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_o @ B2 @ ( image_nat_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_204_image__eqI,axiom,
! [B2: nat,F: $o > nat,X: $o,A: set_o] :
( ( B2
= ( F @ X ) )
=> ( ( member_o @ X @ A )
=> ( member_nat @ B2 @ ( image_o_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_205_image__eqI,axiom,
! [B2: $o,F: $o > $o,X: $o,A: set_o] :
( ( B2
= ( F @ X ) )
=> ( ( member_o @ X @ A )
=> ( member_o @ B2 @ ( image_o_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_206_image__eqI,axiom,
! [B2: nat,F: set_nat > nat,X: set_nat,A: set_set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_set_nat @ X @ A )
=> ( member_nat @ B2 @ ( image_set_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_207_image__eqI,axiom,
! [B2: $o,F: set_nat > $o,X: set_nat,A: set_set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_set_nat @ X @ A )
=> ( member_o @ B2 @ ( image_set_nat_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_208_image__eqI,axiom,
! [B2: set_nat,F: nat > set_nat,X: nat,A: set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_set_nat @ B2 @ ( image_nat_set_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_209_image__eqI,axiom,
! [B2: set_nat,F: $o > set_nat,X: $o,A: set_o] :
( ( B2
= ( F @ X ) )
=> ( ( member_o @ X @ A )
=> ( member_set_nat @ B2 @ ( image_o_set_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_210_image__eqI,axiom,
! [B2: set_nat,F: set_nat > set_nat,X: set_nat,A: set_set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_set_nat @ X @ A )
=> ( member_set_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_211_image__eqI,axiom,
! [B2: nat > nat,F: nat > nat > nat,X: nat,A: set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_nat_nat @ B2 @ ( image_nat_nat_nat2 @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_212_image__empty,axiom,
! [F: nat > set_nat] :
( ( image_nat_set_nat @ F @ bot_bot_set_nat )
= bot_bot_set_set_nat ) ).
% image_empty
thf(fact_213_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_214_empty__is__image,axiom,
! [F: nat > set_nat,A: set_nat] :
( ( bot_bot_set_set_nat
= ( image_nat_set_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_215_empty__is__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_216_image__is__empty,axiom,
! [F: nat > set_nat,A: set_nat] :
( ( ( image_nat_set_nat @ F @ A )
= bot_bot_set_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_217_image__is__empty,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_218_imageI,axiom,
! [X: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_219_imageI,axiom,
! [X: nat,A: set_nat,F: nat > $o] :
( ( member_nat @ X @ A )
=> ( member_o @ ( F @ X ) @ ( image_nat_o @ F @ A ) ) ) ).
% imageI
thf(fact_220_imageI,axiom,
! [X: $o,A: set_o,F: $o > nat] :
( ( member_o @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_o_nat @ F @ A ) ) ) ).
% imageI
thf(fact_221_imageI,axiom,
! [X: $o,A: set_o,F: $o > $o] :
( ( member_o @ X @ A )
=> ( member_o @ ( F @ X ) @ ( image_o_o @ F @ A ) ) ) ).
% imageI
thf(fact_222_imageI,axiom,
! [X: set_nat,A: set_set_nat,F: set_nat > nat] :
( ( member_set_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_set_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_223_imageI,axiom,
! [X: set_nat,A: set_set_nat,F: set_nat > $o] :
( ( member_set_nat @ X @ A )
=> ( member_o @ ( F @ X ) @ ( image_set_nat_o @ F @ A ) ) ) ).
% imageI
thf(fact_224_imageI,axiom,
! [X: nat,A: set_nat,F: nat > set_nat] :
( ( member_nat @ X @ A )
=> ( member_set_nat @ ( F @ X ) @ ( image_nat_set_nat @ F @ A ) ) ) ).
% imageI
thf(fact_225_imageI,axiom,
! [X: $o,A: set_o,F: $o > set_nat] :
( ( member_o @ X @ A )
=> ( member_set_nat @ ( F @ X ) @ ( image_o_set_nat @ F @ A ) ) ) ).
% imageI
thf(fact_226_imageI,axiom,
! [X: set_nat,A: set_set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X @ A )
=> ( member_set_nat @ ( F @ X ) @ ( image_7916887816326733075et_nat @ F @ A ) ) ) ).
% imageI
thf(fact_227_imageI,axiom,
! [X: nat,A: set_nat,F: nat > nat > nat] :
( ( member_nat @ X @ A )
=> ( member_nat_nat @ ( F @ X ) @ ( image_nat_nat_nat2 @ F @ A ) ) ) ).
% imageI
thf(fact_228_image__iff,axiom,
! [Z: set_nat,F: nat > set_nat,A: set_nat] :
( ( member_set_nat @ Z @ ( image_nat_set_nat @ F @ A ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_229_image__iff,axiom,
! [Z: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F @ A ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_230_bex__imageD,axiom,
! [F: nat > set_nat,A: set_nat,P: set_nat > $o] :
( ? [X4: set_nat] :
( ( member_set_nat @ X4 @ ( image_nat_set_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_231_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_232_image__cong,axiom,
! [M2: set_nat,N2: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( M2 = N2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_set_nat @ F @ M2 )
= ( image_nat_set_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_233_image__cong,axiom,
! [M2: set_nat,N2: set_nat,F: nat > nat,G: nat > nat] :
( ( M2 = N2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_nat @ F @ M2 )
= ( image_nat_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_234_ball__imageD,axiom,
! [F: nat > set_nat,A: set_nat,P: set_nat > $o] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( image_nat_set_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_235_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_236_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B2: nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_237_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B2: $o,F: nat > $o] :
( ( member_nat @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_o @ B2 @ ( image_nat_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_238_rev__image__eqI,axiom,
! [X: $o,A: set_o,B2: nat,F: $o > nat] :
( ( member_o @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_nat @ B2 @ ( image_o_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_239_rev__image__eqI,axiom,
! [X: $o,A: set_o,B2: $o,F: $o > $o] :
( ( member_o @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_o @ B2 @ ( image_o_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_240_rev__image__eqI,axiom,
! [X: set_nat,A: set_set_nat,B2: nat,F: set_nat > nat] :
( ( member_set_nat @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_nat @ B2 @ ( image_set_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_241_rev__image__eqI,axiom,
! [X: set_nat,A: set_set_nat,B2: $o,F: set_nat > $o] :
( ( member_set_nat @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_o @ B2 @ ( image_set_nat_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_242_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B2: set_nat,F: nat > set_nat] :
( ( member_nat @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_set_nat @ B2 @ ( image_nat_set_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_243_rev__image__eqI,axiom,
! [X: $o,A: set_o,B2: set_nat,F: $o > set_nat] :
( ( member_o @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_set_nat @ B2 @ ( image_o_set_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_244_rev__image__eqI,axiom,
! [X: set_nat,A: set_set_nat,B2: set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_set_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_245_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B2: nat > nat,F: nat > nat > nat] :
( ( member_nat @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_nat_nat @ B2 @ ( image_nat_nat_nat2 @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_246_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_247_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_248_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_249_lessThan__empty__iff,axiom,
! [N3: nat] :
( ( ( set_ord_lessThan_nat @ N3 )
= bot_bot_set_nat )
= ( N3 = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_250_disjoint__family__onD,axiom,
! [A: nat > set_o,I: set_nat,I2: nat,J: nat] :
( ( disjoi1808054049482533742_nat_o @ A @ I )
=> ( ( member_nat @ I2 @ I )
=> ( ( member_nat @ J @ I )
=> ( ( I2 != J )
=> ( ( inf_inf_set_o @ ( A @ I2 ) @ ( A @ J ) )
= bot_bot_set_o ) ) ) ) ) ).
% disjoint_family_onD
thf(fact_251_disjoint__family__onD,axiom,
! [A: $o > set_o,I: set_o,I2: $o,J: $o] :
( ( disjoi298098064294094040on_o_o @ A @ I )
=> ( ( member_o @ I2 @ I )
=> ( ( member_o @ J @ I )
=> ( ( I2 != J )
=> ( ( inf_inf_set_o @ ( A @ I2 ) @ ( A @ J ) )
= bot_bot_set_o ) ) ) ) ) ).
% disjoint_family_onD
thf(fact_252_disjoint__family__onD,axiom,
! [A: $o > set_nat,I: set_o,I2: $o,J: $o] :
( ( disjoi7928754725229124240_o_nat @ A @ I )
=> ( ( member_o @ I2 @ I )
=> ( ( member_o @ J @ I )
=> ( ( I2 != J )
=> ( ( inf_inf_set_nat @ ( A @ I2 ) @ ( A @ J ) )
= bot_bot_set_nat ) ) ) ) ) ).
% disjoint_family_onD
thf(fact_253_disjoint__family__onD,axiom,
! [A: nat > set_nat,I: set_nat,I2: nat,J: nat] :
( ( disjoi6798895846410478970at_nat @ A @ I )
=> ( ( member_nat @ I2 @ I )
=> ( ( member_nat @ J @ I )
=> ( ( I2 != J )
=> ( ( inf_inf_set_nat @ ( A @ I2 ) @ ( A @ J ) )
= bot_bot_set_nat ) ) ) ) ) ).
% disjoint_family_onD
thf(fact_254_disjoint__family__onD,axiom,
! [A: set_nat > set_o,I: set_set_nat,I2: set_nat,J: set_nat] :
( ( disjoi7862385731094200888_nat_o @ A @ I )
=> ( ( member_set_nat @ I2 @ I )
=> ( ( member_set_nat @ J @ I )
=> ( ( I2 != J )
=> ( ( inf_inf_set_o @ ( A @ I2 ) @ ( A @ J ) )
= bot_bot_set_o ) ) ) ) ) ).
% disjoint_family_onD
thf(fact_255_disjoint__family__onD,axiom,
! [A: nat > set_set_nat,I: set_nat,I2: nat,J: nat] :
( ( disjoi5680779568412092720et_nat @ A @ I )
=> ( ( member_nat @ I2 @ I )
=> ( ( member_nat @ J @ I )
=> ( ( I2 != J )
=> ( ( inf_inf_set_set_nat @ ( A @ I2 ) @ ( A @ J ) )
= bot_bot_set_set_nat ) ) ) ) ) ).
% disjoint_family_onD
thf(fact_256_disjoint__family__onD,axiom,
! [A: $o > set_set_nat,I: set_o,I2: $o,J: $o] :
( ( disjoi1536465572982621766et_nat @ A @ I )
=> ( ( member_o @ I2 @ I )
=> ( ( member_o @ J @ I )
=> ( ( I2 != J )
=> ( ( inf_inf_set_set_nat @ ( A @ I2 ) @ ( A @ J ) )
= bot_bot_set_set_nat ) ) ) ) ) ).
% disjoint_family_onD
thf(fact_257_disjoint__family__onD,axiom,
! [A: set_nat > set_nat,I: set_set_nat,I2: set_nat,J: set_nat] :
( ( disjoi2115144663756723504at_nat @ A @ I )
=> ( ( member_set_nat @ I2 @ I )
=> ( ( member_set_nat @ J @ I )
=> ( ( I2 != J )
=> ( ( inf_inf_set_nat @ ( A @ I2 ) @ ( A @ J ) )
= bot_bot_set_nat ) ) ) ) ) ).
% disjoint_family_onD
thf(fact_258_disjoint__family__onD,axiom,
! [A: nat > set_nat_nat,I: set_nat,I2: nat,J: nat] :
( ( disjoi8598568060105092073at_nat @ A @ I )
=> ( ( member_nat @ I2 @ I )
=> ( ( member_nat @ J @ I )
=> ( ( I2 != J )
=> ( ( inf_inf_set_nat_nat @ ( A @ I2 ) @ ( A @ J ) )
= bot_bot_set_nat_nat ) ) ) ) ) ).
% disjoint_family_onD
thf(fact_259_disjoint__family__onD,axiom,
! [A: $o > set_nat_nat,I: set_o,I2: $o,J: $o] :
( ( disjoi8955196976510109695at_nat @ A @ I )
=> ( ( member_o @ I2 @ I )
=> ( ( member_o @ J @ I )
=> ( ( I2 != J )
=> ( ( inf_inf_set_nat_nat @ ( A @ I2 ) @ ( A @ J ) )
= bot_bot_set_nat_nat ) ) ) ) ) ).
% disjoint_family_onD
thf(fact_260_disjoint__family__on__def,axiom,
( disjoi6798895846410478970at_nat
= ( ^ [A4: nat > set_nat,S2: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ S2 )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ S2 )
=> ( ( X2 != Y3 )
=> ( ( inf_inf_set_nat @ ( A4 @ X2 ) @ ( A4 @ Y3 ) )
= bot_bot_set_nat ) ) ) ) ) ) ).
% disjoint_family_on_def
thf(fact_261_disjoint__family__on__bisimulation,axiom,
! [F: nat > set_o,S: set_nat,G: nat > set_o] :
( ( disjoi1808054049482533742_nat_o @ F @ S )
=> ( ! [N: nat,M: nat] :
( ( member_nat @ N @ S )
=> ( ( member_nat @ M @ S )
=> ( ( N != M )
=> ( ( ( inf_inf_set_o @ ( F @ N ) @ ( F @ M ) )
= bot_bot_set_o )
=> ( ( inf_inf_set_o @ ( G @ N ) @ ( G @ M ) )
= bot_bot_set_o ) ) ) ) )
=> ( disjoi1808054049482533742_nat_o @ G @ S ) ) ) ).
% disjoint_family_on_bisimulation
thf(fact_262_disjoint__family__on__bisimulation,axiom,
! [F: $o > set_o,S: set_o,G: $o > set_o] :
( ( disjoi298098064294094040on_o_o @ F @ S )
=> ( ! [N: $o,M: $o] :
( ( member_o @ N @ S )
=> ( ( member_o @ M @ S )
=> ( ( N != M )
=> ( ( ( inf_inf_set_o @ ( F @ N ) @ ( F @ M ) )
= bot_bot_set_o )
=> ( ( inf_inf_set_o @ ( G @ N ) @ ( G @ M ) )
= bot_bot_set_o ) ) ) ) )
=> ( disjoi298098064294094040on_o_o @ G @ S ) ) ) ).
% disjoint_family_on_bisimulation
thf(fact_263_disjoint__family__on__bisimulation,axiom,
! [F: $o > set_o,S: set_o,G: $o > set_nat] :
( ( disjoi298098064294094040on_o_o @ F @ S )
=> ( ! [N: $o,M: $o] :
( ( member_o @ N @ S )
=> ( ( member_o @ M @ S )
=> ( ( N != M )
=> ( ( ( inf_inf_set_o @ ( F @ N ) @ ( F @ M ) )
= bot_bot_set_o )
=> ( ( inf_inf_set_nat @ ( G @ N ) @ ( G @ M ) )
= bot_bot_set_nat ) ) ) ) )
=> ( disjoi7928754725229124240_o_nat @ G @ S ) ) ) ).
% disjoint_family_on_bisimulation
thf(fact_264_disjoint__family__on__bisimulation,axiom,
! [F: $o > set_nat,S: set_o,G: $o > set_o] :
( ( disjoi7928754725229124240_o_nat @ F @ S )
=> ( ! [N: $o,M: $o] :
( ( member_o @ N @ S )
=> ( ( member_o @ M @ S )
=> ( ( N != M )
=> ( ( ( inf_inf_set_nat @ ( F @ N ) @ ( F @ M ) )
= bot_bot_set_nat )
=> ( ( inf_inf_set_o @ ( G @ N ) @ ( G @ M ) )
= bot_bot_set_o ) ) ) ) )
=> ( disjoi298098064294094040on_o_o @ G @ S ) ) ) ).
% disjoint_family_on_bisimulation
thf(fact_265_disjoint__family__on__bisimulation,axiom,
! [F: $o > set_nat,S: set_o,G: $o > set_nat] :
( ( disjoi7928754725229124240_o_nat @ F @ S )
=> ( ! [N: $o,M: $o] :
( ( member_o @ N @ S )
=> ( ( member_o @ M @ S )
=> ( ( N != M )
=> ( ( ( inf_inf_set_nat @ ( F @ N ) @ ( F @ M ) )
= bot_bot_set_nat )
=> ( ( inf_inf_set_nat @ ( G @ N ) @ ( G @ M ) )
= bot_bot_set_nat ) ) ) ) )
=> ( disjoi7928754725229124240_o_nat @ G @ S ) ) ) ).
% disjoint_family_on_bisimulation
thf(fact_266_disjoint__family__on__bisimulation,axiom,
! [F: nat > set_o,S: set_nat,G: nat > set_nat] :
( ( disjoi1808054049482533742_nat_o @ F @ S )
=> ( ! [N: nat,M: nat] :
( ( member_nat @ N @ S )
=> ( ( member_nat @ M @ S )
=> ( ( N != M )
=> ( ( ( inf_inf_set_o @ ( F @ N ) @ ( F @ M ) )
= bot_bot_set_o )
=> ( ( inf_inf_set_nat @ ( G @ N ) @ ( G @ M ) )
= bot_bot_set_nat ) ) ) ) )
=> ( disjoi6798895846410478970at_nat @ G @ S ) ) ) ).
% disjoint_family_on_bisimulation
thf(fact_267_disjoint__family__on__bisimulation,axiom,
! [F: nat > set_nat,S: set_nat,G: nat > set_o] :
( ( disjoi6798895846410478970at_nat @ F @ S )
=> ( ! [N: nat,M: nat] :
( ( member_nat @ N @ S )
=> ( ( member_nat @ M @ S )
=> ( ( N != M )
=> ( ( ( inf_inf_set_nat @ ( F @ N ) @ ( F @ M ) )
= bot_bot_set_nat )
=> ( ( inf_inf_set_o @ ( G @ N ) @ ( G @ M ) )
= bot_bot_set_o ) ) ) ) )
=> ( disjoi1808054049482533742_nat_o @ G @ S ) ) ) ).
% disjoint_family_on_bisimulation
thf(fact_268_disjoint__family__on__bisimulation,axiom,
! [F: nat > set_nat,S: set_nat,G: nat > set_nat] :
( ( disjoi6798895846410478970at_nat @ F @ S )
=> ( ! [N: nat,M: nat] :
( ( member_nat @ N @ S )
=> ( ( member_nat @ M @ S )
=> ( ( N != M )
=> ( ( ( inf_inf_set_nat @ ( F @ N ) @ ( F @ M ) )
= bot_bot_set_nat )
=> ( ( inf_inf_set_nat @ ( G @ N ) @ ( G @ M ) )
= bot_bot_set_nat ) ) ) ) )
=> ( disjoi6798895846410478970at_nat @ G @ S ) ) ) ).
% disjoint_family_on_bisimulation
thf(fact_269_disjoint__family__on__bisimulation,axiom,
! [F: set_nat > set_o,S: set_set_nat,G: set_nat > set_o] :
( ( disjoi7862385731094200888_nat_o @ F @ S )
=> ( ! [N: set_nat,M: set_nat] :
( ( member_set_nat @ N @ S )
=> ( ( member_set_nat @ M @ S )
=> ( ( N != M )
=> ( ( ( inf_inf_set_o @ ( F @ N ) @ ( F @ M ) )
= bot_bot_set_o )
=> ( ( inf_inf_set_o @ ( G @ N ) @ ( G @ M ) )
= bot_bot_set_o ) ) ) ) )
=> ( disjoi7862385731094200888_nat_o @ G @ S ) ) ) ).
% disjoint_family_on_bisimulation
thf(fact_270_disjoint__family__on__bisimulation,axiom,
! [F: nat > set_o,S: set_nat,G: nat > set_set_nat] :
( ( disjoi1808054049482533742_nat_o @ F @ S )
=> ( ! [N: nat,M: nat] :
( ( member_nat @ N @ S )
=> ( ( member_nat @ M @ S )
=> ( ( N != M )
=> ( ( ( inf_inf_set_o @ ( F @ N ) @ ( F @ M ) )
= bot_bot_set_o )
=> ( ( inf_inf_set_set_nat @ ( G @ N ) @ ( G @ M ) )
= bot_bot_set_set_nat ) ) ) ) )
=> ( disjoi5680779568412092720et_nat @ G @ S ) ) ) ).
% disjoint_family_on_bisimulation
thf(fact_271_Iio__eq__empty__iff,axiom,
! [N3: nat] :
( ( ( set_ord_lessThan_nat @ N3 )
= bot_bot_set_nat )
= ( N3 = bot_bot_nat ) ) ).
% Iio_eq_empty_iff
thf(fact_272_atMost__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_atMost_nat @ X )
= ( set_ord_atMost_nat @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_273_lessThan__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_274_bot__empty__eq,axiom,
( bot_bot_set_nat_o
= ( ^ [X2: set_nat] : ( member_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_275_bot__empty__eq,axiom,
( bot_bot_o_o
= ( ^ [X2: $o] : ( member_o @ X2 @ bot_bot_set_o ) ) ) ).
% bot_empty_eq
thf(fact_276_bot__empty__eq,axiom,
( bot_bot_nat_nat_o
= ( ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ bot_bot_set_nat_nat ) ) ) ).
% bot_empty_eq
thf(fact_277_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X2: nat] : ( member_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_278_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_279_not__empty__eq__Iic__eq__empty,axiom,
! [H: nat] :
( bot_bot_set_nat
!= ( set_ord_atMost_nat @ H ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_280_BfL__props_I2_J,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ bl @ ( set_ord_atMost_nat @ one_one_nat ) ) )
= ( set_ord_lessThan_nat @ n2 ) ) ).
% BfL_props(2)
thf(fact_281_the__elem__image__unique,axiom,
! [A: set_nat,F: nat > set_nat,X: nat] :
( ( A != bot_bot_set_nat )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ A )
=> ( ( F @ Y2 )
= ( F @ X ) ) )
=> ( ( the_elem_set_nat @ ( image_nat_set_nat @ F @ A ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_282_the__elem__image__unique,axiom,
! [A: set_nat,F: nat > nat,X: nat] :
( ( A != bot_bot_set_nat )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ A )
=> ( ( F @ Y2 )
= ( F @ X ) ) )
=> ( ( the_elem_nat @ ( image_nat_nat @ F @ A ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_283_a,axiom,
member_nat @ i @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ k @ one_one_nat ) ) ).
% a
thf(fact_284_atMost__0,axiom,
( ( set_ord_atMost_nat @ zero_zero_nat )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% atMost_0
thf(fact_285_ivl__disj__int__one_I1_J,axiom,
! [L: nat > nat,U: nat > nat] :
( ( inf_inf_set_nat_nat @ ( set_or9140604705432621368at_nat @ L ) @ ( set_or1731194346131681939at_nat @ L @ U ) )
= bot_bot_set_nat_nat ) ).
% ivl_disj_int_one(1)
thf(fact_286_ivl__disj__int__one_I1_J,axiom,
! [L: $o,U: $o] :
( ( inf_inf_set_o @ ( set_ord_atMost_o @ L ) @ ( set_or1716231572884733764Than_o @ L @ U ) )
= bot_bot_set_o ) ).
% ivl_disj_int_one(1)
thf(fact_287_ivl__disj__int__one_I1_J,axiom,
! [L: set_nat,U: set_nat] :
( ( inf_inf_set_set_nat @ ( set_or4236626031148496127et_nat @ L ) @ ( set_or8625682525731655386et_nat @ L @ U ) )
= bot_bot_set_set_nat ) ).
% ivl_disj_int_one(1)
thf(fact_288_ivl__disj__int__one_I1_J,axiom,
! [L: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_ord_atMost_nat @ L ) @ ( set_or5834768355832116004an_nat @ L @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_one(1)
thf(fact_289_ivl__disj__int__one_I3_J,axiom,
! [L: nat > nat,U: nat > nat] :
( ( inf_inf_set_nat_nat @ ( set_or9140604705432621368at_nat @ L ) @ ( set_or3470937268066497519at_nat @ L @ U ) )
= bot_bot_set_nat_nat ) ).
% ivl_disj_int_one(3)
thf(fact_290_ivl__disj__int__one_I3_J,axiom,
! [L: $o,U: $o] :
( ( inf_inf_set_o @ ( set_ord_atMost_o @ L ) @ ( set_or8254209520273421544Most_o @ L @ U ) )
= bot_bot_set_o ) ).
% ivl_disj_int_one(3)
thf(fact_291_ivl__disj__int__one_I3_J,axiom,
! [L: set_nat,U: set_nat] :
( ( inf_inf_set_set_nat @ ( set_or4236626031148496127et_nat @ L ) @ ( set_or7074010630789208630et_nat @ L @ U ) )
= bot_bot_set_set_nat ) ).
% ivl_disj_int_one(3)
thf(fact_292_ivl__disj__int__one_I3_J,axiom,
! [L: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_ord_atMost_nat @ L ) @ ( set_or6659071591806873216st_nat @ L @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_one(3)
thf(fact_293_BfS__props_I3_J,axiom,
~ ( member_set_nat @ bot_bot_set_nat @ ( image_nat_set_nat @ bs @ ( set_ord_lessThan_nat @ k ) ) ) ).
% BfS_props(3)
thf(fact_294_Inf_OINF__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > set_nat,D: nat > set_nat,Inf: set_set_nat > set_nat] :
( ( A = B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_nat_set_nat @ C2 @ A ) )
= ( Inf @ ( image_nat_set_nat @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_295_Inf_OINF__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > nat,D: nat > nat,Inf: set_nat > nat] :
( ( A = B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C2 @ A ) )
= ( Inf @ ( image_nat_nat @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_296_Sup_OSUP__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > set_nat,D: nat > set_nat,Sup: set_set_nat > set_nat] :
( ( A = B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_nat_set_nat @ C2 @ A ) )
= ( Sup @ ( image_nat_set_nat @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_297_Sup_OSUP__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > nat,D: nat > nat,Sup: set_nat > nat] :
( ( A = B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C2 @ A ) )
= ( Sup @ ( image_nat_nat @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_298_BfS__props_I1_J,axiom,
disjoi6798895846410478970at_nat @ bs @ ( set_ord_atMost_nat @ k ) ).
% BfS_props(1)
thf(fact_299_add__left__cancel,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ A2 @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_300_add__right__cancel,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_301_insertCI,axiom,
! [A2: set_nat,B: set_set_nat,B2: set_nat] :
( ( ~ ( member_set_nat @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_set_nat @ A2 @ ( insert_set_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_302_insertCI,axiom,
! [A2: nat,B: set_nat,B2: nat] :
( ( ~ ( member_nat @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_nat @ A2 @ ( insert_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_303_insertCI,axiom,
! [A2: $o,B: set_o,B2: $o] :
( ( ~ ( member_o @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_o @ A2 @ ( insert_o @ B2 @ B ) ) ) ).
% insertCI
thf(fact_304_insertCI,axiom,
! [A2: nat > nat,B: set_nat_nat,B2: nat > nat] :
( ( ~ ( member_nat_nat @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_nat_nat @ A2 @ ( insert_nat_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_305_insert__iff,axiom,
! [A2: set_nat,B2: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ ( insert_set_nat @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_set_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_306_insert__iff,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_307_insert__iff,axiom,
! [A2: $o,B2: $o,A: set_o] :
( ( member_o @ A2 @ ( insert_o @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_o @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_308_insert__iff,axiom,
! [A2: nat > nat,B2: nat > nat,A: set_nat_nat] :
( ( member_nat_nat @ A2 @ ( insert_nat_nat @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_nat_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_309_insert__absorb2,axiom,
! [X: nat,A: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ X @ A ) )
= ( insert_nat @ X @ A ) ) ).
% insert_absorb2
thf(fact_310_Union__iff,axiom,
! [A: set_nat,C2: set_set_set_nat] :
( ( member_set_nat @ A @ ( comple548664676211718543et_nat @ C2 ) )
= ( ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ C2 )
& ( member_set_nat @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_311_Union__iff,axiom,
! [A: $o,C2: set_set_o] :
( ( member_o @ A @ ( comple90263536869209701_set_o @ C2 ) )
= ( ? [X2: set_o] :
( ( member_set_o @ X2 @ C2 )
& ( member_o @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_312_Union__iff,axiom,
! [A: nat > nat,C2: set_set_nat_nat] :
( ( member_nat_nat @ A @ ( comple5448282615319421384at_nat @ C2 ) )
= ( ? [X2: set_nat_nat] :
( ( member_set_nat_nat @ X2 @ C2 )
& ( member_nat_nat @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_313_Union__iff,axiom,
! [A: nat,C2: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) )
= ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ C2 )
& ( member_nat @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_314_UnionI,axiom,
! [X5: set_set_nat,C2: set_set_set_nat,A: set_nat] :
( ( member_set_set_nat @ X5 @ C2 )
=> ( ( member_set_nat @ A @ X5 )
=> ( member_set_nat @ A @ ( comple548664676211718543et_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_315_UnionI,axiom,
! [X5: set_o,C2: set_set_o,A: $o] :
( ( member_set_o @ X5 @ C2 )
=> ( ( member_o @ A @ X5 )
=> ( member_o @ A @ ( comple90263536869209701_set_o @ C2 ) ) ) ) ).
% UnionI
thf(fact_316_UnionI,axiom,
! [X5: set_nat_nat,C2: set_set_nat_nat,A: nat > nat] :
( ( member_set_nat_nat @ X5 @ C2 )
=> ( ( member_nat_nat @ A @ X5 )
=> ( member_nat_nat @ A @ ( comple5448282615319421384at_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_317_UnionI,axiom,
! [X5: set_nat,C2: set_set_nat,A: nat] :
( ( member_set_nat @ X5 @ C2 )
=> ( ( member_nat @ A @ X5 )
=> ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_318_UN__ball__bex__simps_I1_J,axiom,
! [A: set_set_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ A ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ X2 )
=> ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_319_UN__ball__bex__simps_I3_J,axiom,
! [A: set_set_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ A ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ? [Y3: nat] :
( ( member_nat @ Y3 @ X2 )
& ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_320_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_321_add__cancel__left__left,axiom,
! [B2: nat,A2: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= A2 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_322_add__cancel__left__right,axiom,
! [A2: nat,B2: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= A2 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_323_add__cancel__right__left,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ A2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_324_add__cancel__right__right,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_325_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_326_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_327_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_328_image__insert,axiom,
! [F: nat > set_nat,A2: nat,B: set_nat] :
( ( image_nat_set_nat @ F @ ( insert_nat @ A2 @ B ) )
= ( insert_set_nat @ ( F @ A2 ) @ ( image_nat_set_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_329_image__insert,axiom,
! [F: nat > nat,A2: nat,B: set_nat] :
( ( image_nat_nat @ F @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ ( F @ A2 ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_330_insert__image,axiom,
! [X: set_nat,A: set_set_nat,F: set_nat > nat] :
( ( member_set_nat @ X @ A )
=> ( ( insert_nat @ ( F @ X ) @ ( image_set_nat_nat @ F @ A ) )
= ( image_set_nat_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_331_insert__image,axiom,
! [X: nat,A: set_nat,F: nat > set_nat] :
( ( member_nat @ X @ A )
=> ( ( insert_set_nat @ ( F @ X ) @ ( image_nat_set_nat @ F @ A ) )
= ( image_nat_set_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_332_insert__image,axiom,
! [X: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( insert_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A ) )
= ( image_nat_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_333_insert__image,axiom,
! [X: $o,A: set_o,F: $o > nat] :
( ( member_o @ X @ A )
=> ( ( insert_nat @ ( F @ X ) @ ( image_o_nat @ F @ A ) )
= ( image_o_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_334_insert__image,axiom,
! [X: nat > nat,A: set_nat_nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X @ A )
=> ( ( insert_nat @ ( F @ X ) @ ( image_nat_nat_nat @ F @ A ) )
= ( image_nat_nat_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_335_Sup__bot__conv_I2_J,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_336_Sup__bot__conv_I2_J,axiom,
! [A: set_o] :
( ( bot_bot_o
= ( complete_Sup_Sup_o @ A ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( X2 = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_337_Sup__bot__conv_I1_J,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_338_Sup__bot__conv_I1_J,axiom,
! [A: set_o] :
( ( ( complete_Sup_Sup_o @ A )
= bot_bot_o )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( X2 = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_339_singletonI,axiom,
! [A2: set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% singletonI
thf(fact_340_singletonI,axiom,
! [A2: $o] : ( member_o @ A2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) ).
% singletonI
thf(fact_341_singletonI,axiom,
! [A2: nat > nat] : ( member_nat_nat @ A2 @ ( insert_nat_nat @ A2 @ bot_bot_set_nat_nat ) ) ).
% singletonI
thf(fact_342_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_343_add__is__0,axiom,
! [M3: nat,N3: nat] :
( ( ( plus_plus_nat @ M3 @ N3 )
= zero_zero_nat )
= ( ( M3 = zero_zero_nat )
& ( N3 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_344_Nat_Oadd__0__right,axiom,
! [M3: nat] :
( ( plus_plus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% Nat.add_0_right
thf(fact_345_Int__insert__left__if0,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_nat @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_346_Int__insert__left__if0,axiom,
! [A2: nat > nat,C2: set_nat_nat,B: set_nat_nat] :
( ~ ( member_nat_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat_nat @ ( insert_nat_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_nat_nat @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_347_Int__insert__left__if0,axiom,
! [A2: $o,C2: set_o,B: set_o] :
( ~ ( member_o @ A2 @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A2 @ B ) @ C2 )
= ( inf_inf_set_o @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_348_Int__insert__left__if0,axiom,
! [A2: set_nat,C2: set_set_nat,B: set_set_nat] :
( ~ ( member_set_nat @ A2 @ C2 )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_set_nat @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_349_Int__insert__left__if1,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_350_Int__insert__left__if1,axiom,
! [A2: nat > nat,C2: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat_nat @ ( insert_nat_nat @ A2 @ B ) @ C2 )
= ( insert_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_351_Int__insert__left__if1,axiom,
! [A2: $o,C2: set_o,B: set_o] :
( ( member_o @ A2 @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A2 @ B ) @ C2 )
= ( insert_o @ A2 @ ( inf_inf_set_o @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_352_Int__insert__left__if1,axiom,
! [A2: set_nat,C2: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ A2 @ C2 )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ B ) @ C2 )
= ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_353_insert__inter__insert,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_354_insert__inter__insert,axiom,
! [A2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( insert_nat_nat @ A2 @ A ) @ ( insert_nat_nat @ A2 @ B ) )
= ( insert_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_355_insert__inter__insert,axiom,
! [A2: $o,A: set_o,B: set_o] :
( ( inf_inf_set_o @ ( insert_o @ A2 @ A ) @ ( insert_o @ A2 @ B ) )
= ( insert_o @ A2 @ ( inf_inf_set_o @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_356_insert__inter__insert,axiom,
! [A2: set_nat,A: set_set_nat,B: set_set_nat] :
( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ A ) @ ( insert_set_nat @ A2 @ B ) )
= ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_357_Int__insert__right__if0,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_358_Int__insert__right__if0,axiom,
! [A2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ~ ( member_nat_nat @ A2 @ A )
=> ( ( inf_inf_set_nat_nat @ A @ ( insert_nat_nat @ A2 @ B ) )
= ( inf_inf_set_nat_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_359_Int__insert__right__if0,axiom,
! [A2: $o,A: set_o,B: set_o] :
( ~ ( member_o @ A2 @ A )
=> ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B ) )
= ( inf_inf_set_o @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_360_Int__insert__right__if0,axiom,
! [A2: set_nat,A: set_set_nat,B: set_set_nat] :
( ~ ( member_set_nat @ A2 @ A )
=> ( ( inf_inf_set_set_nat @ A @ ( insert_set_nat @ A2 @ B ) )
= ( inf_inf_set_set_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_361_Int__insert__right__if1,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_362_Int__insert__right__if1,axiom,
! [A2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ A2 @ A )
=> ( ( inf_inf_set_nat_nat @ A @ ( insert_nat_nat @ A2 @ B ) )
= ( insert_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_363_Int__insert__right__if1,axiom,
! [A2: $o,A: set_o,B: set_o] :
( ( member_o @ A2 @ A )
=> ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B ) )
= ( insert_o @ A2 @ ( inf_inf_set_o @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_364_Int__insert__right__if1,axiom,
! [A2: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ A2 @ A )
=> ( ( inf_inf_set_set_nat @ A @ ( insert_set_nat @ A2 @ B ) )
= ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_365_ball__UN,axiom,
! [B: nat > set_nat,A: set_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( B @ X2 ) )
=> ( P @ Y3 ) ) ) ) ) ).
% ball_UN
thf(fact_366_bex__UN,axiom,
! [B: nat > set_nat,A: set_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ? [Y3: nat] :
( ( member_nat @ Y3 @ ( B @ X2 ) )
& ( P @ Y3 ) ) ) ) ) ).
% bex_UN
thf(fact_367_UN__ball__bex__simps_I2_J,axiom,
! [B: nat > set_nat,A: set_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( B @ X2 ) )
=> ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_368_UN__ball__bex__simps_I4_J,axiom,
! [B: nat > set_nat,A: set_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ? [Y3: nat] :
( ( member_nat @ Y3 @ ( B @ X2 ) )
& ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_369_Sup__atMost,axiom,
! [Y: set_nat] :
( ( comple7399068483239264473et_nat @ ( set_or4236626031148496127et_nat @ Y ) )
= Y ) ).
% Sup_atMost
thf(fact_370_Sup__atMost,axiom,
! [Y: $o] :
( ( complete_Sup_Sup_o @ ( set_ord_atMost_o @ Y ) )
= Y ) ).
% Sup_atMost
thf(fact_371_assms_I2_J,axiom,
ord_less_eq_nat @ one_one_nat @ k ).
% assms(2)
thf(fact_372_image__add__0,axiom,
! [S: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S )
= S ) ).
% image_add_0
thf(fact_373_Sup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Sup_empty
thf(fact_374_Sup__empty,axiom,
( ( complete_Sup_Sup_o @ bot_bot_set_o )
= bot_bot_o ) ).
% Sup_empty
thf(fact_375_disjoint__insert_I2_J,axiom,
! [A: set_nat_nat,B2: nat > nat,B: set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( inf_inf_set_nat_nat @ A @ ( insert_nat_nat @ B2 @ B ) ) )
= ( ~ ( member_nat_nat @ B2 @ A )
& ( bot_bot_set_nat_nat
= ( inf_inf_set_nat_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_376_disjoint__insert_I2_J,axiom,
! [A: set_o,B2: $o,B: set_o] :
( ( bot_bot_set_o
= ( inf_inf_set_o @ A @ ( insert_o @ B2 @ B ) ) )
= ( ~ ( member_o @ B2 @ A )
& ( bot_bot_set_o
= ( inf_inf_set_o @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_377_disjoint__insert_I2_J,axiom,
! [A: set_set_nat,B2: set_nat,B: set_set_nat] :
( ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ A @ ( insert_set_nat @ B2 @ B ) ) )
= ( ~ ( member_set_nat @ B2 @ A )
& ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_378_disjoint__insert_I2_J,axiom,
! [A: set_nat,B2: nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ ( insert_nat @ B2 @ B ) ) )
= ( ~ ( member_nat @ B2 @ A )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_379_disjoint__insert_I1_J,axiom,
! [B: set_nat_nat,A2: nat > nat,A: set_nat_nat] :
( ( ( inf_inf_set_nat_nat @ B @ ( insert_nat_nat @ A2 @ A ) )
= bot_bot_set_nat_nat )
= ( ~ ( member_nat_nat @ A2 @ B )
& ( ( inf_inf_set_nat_nat @ B @ A )
= bot_bot_set_nat_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_380_disjoint__insert_I1_J,axiom,
! [B: set_o,A2: $o,A: set_o] :
( ( ( inf_inf_set_o @ B @ ( insert_o @ A2 @ A ) )
= bot_bot_set_o )
= ( ~ ( member_o @ A2 @ B )
& ( ( inf_inf_set_o @ B @ A )
= bot_bot_set_o ) ) ) ).
% disjoint_insert(1)
thf(fact_381_disjoint__insert_I1_J,axiom,
! [B: set_set_nat,A2: set_nat,A: set_set_nat] :
( ( ( inf_inf_set_set_nat @ B @ ( insert_set_nat @ A2 @ A ) )
= bot_bot_set_set_nat )
= ( ~ ( member_set_nat @ A2 @ B )
& ( ( inf_inf_set_set_nat @ B @ A )
= bot_bot_set_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_382_disjoint__insert_I1_J,axiom,
! [B: set_nat,A2: nat,A: set_nat] :
( ( ( inf_inf_set_nat @ B @ ( insert_nat @ A2 @ A ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B )
& ( ( inf_inf_set_nat @ B @ A )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_383_insert__disjoint_I2_J,axiom,
! [A2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( inf_inf_set_nat_nat @ ( insert_nat_nat @ A2 @ A ) @ B ) )
= ( ~ ( member_nat_nat @ A2 @ B )
& ( bot_bot_set_nat_nat
= ( inf_inf_set_nat_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_384_insert__disjoint_I2_J,axiom,
! [A2: $o,A: set_o,B: set_o] :
( ( bot_bot_set_o
= ( inf_inf_set_o @ ( insert_o @ A2 @ A ) @ B ) )
= ( ~ ( member_o @ A2 @ B )
& ( bot_bot_set_o
= ( inf_inf_set_o @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_385_insert__disjoint_I2_J,axiom,
! [A2: set_nat,A: set_set_nat,B: set_set_nat] :
( ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ A ) @ B ) )
= ( ~ ( member_set_nat @ A2 @ B )
& ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_386_insert__disjoint_I2_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B ) )
= ( ~ ( member_nat @ A2 @ B )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_387_insert__disjoint_I1_J,axiom,
! [A2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( ( inf_inf_set_nat_nat @ ( insert_nat_nat @ A2 @ A ) @ B )
= bot_bot_set_nat_nat )
= ( ~ ( member_nat_nat @ A2 @ B )
& ( ( inf_inf_set_nat_nat @ A @ B )
= bot_bot_set_nat_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_388_insert__disjoint_I1_J,axiom,
! [A2: $o,A: set_o,B: set_o] :
( ( ( inf_inf_set_o @ ( insert_o @ A2 @ A ) @ B )
= bot_bot_set_o )
= ( ~ ( member_o @ A2 @ B )
& ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o ) ) ) ).
% insert_disjoint(1)
thf(fact_389_insert__disjoint_I1_J,axiom,
! [A2: set_nat,A: set_set_nat,B: set_set_nat] :
( ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ A ) @ B )
= bot_bot_set_set_nat )
= ( ~ ( member_set_nat @ A2 @ B )
& ( ( inf_inf_set_set_nat @ A @ B )
= bot_bot_set_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_390_insert__disjoint_I1_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B )
& ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_391_image__add__greaterThanAtMost,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( image_nat_nat @ ( plus_plus_nat @ C ) @ ( set_or6659071591806873216st_nat @ A2 @ B2 ) )
= ( set_or6659071591806873216st_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% image_add_greaterThanAtMost
thf(fact_392_the__elem__eq,axiom,
! [X: nat] :
( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% the_elem_eq
thf(fact_393_n__def,axiom,
( n2
= ( plus_plus_nat @ n @ d ) ) ).
% n_def
thf(fact_394_insertE,axiom,
! [A2: set_nat,B2: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ ( insert_set_nat @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_set_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_395_insertE,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_396_insertE,axiom,
! [A2: $o,B2: $o,A: set_o] :
( ( member_o @ A2 @ ( insert_o @ B2 @ A ) )
=> ( ( A2 = (~ B2) )
=> ( member_o @ A2 @ A ) ) ) ).
% insertE
thf(fact_397_insertE,axiom,
! [A2: nat > nat,B2: nat > nat,A: set_nat_nat] :
( ( member_nat_nat @ A2 @ ( insert_nat_nat @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_nat_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_398_insertI1,axiom,
! [A2: set_nat,B: set_set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_399_insertI1,axiom,
! [A2: nat,B: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_400_insertI1,axiom,
! [A2: $o,B: set_o] : ( member_o @ A2 @ ( insert_o @ A2 @ B ) ) ).
% insertI1
thf(fact_401_insertI1,axiom,
! [A2: nat > nat,B: set_nat_nat] : ( member_nat_nat @ A2 @ ( insert_nat_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_402_insertI2,axiom,
! [A2: set_nat,B: set_set_nat,B2: set_nat] :
( ( member_set_nat @ A2 @ B )
=> ( member_set_nat @ A2 @ ( insert_set_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_403_insertI2,axiom,
! [A2: nat,B: set_nat,B2: nat] :
( ( member_nat @ A2 @ B )
=> ( member_nat @ A2 @ ( insert_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_404_insertI2,axiom,
! [A2: $o,B: set_o,B2: $o] :
( ( member_o @ A2 @ B )
=> ( member_o @ A2 @ ( insert_o @ B2 @ B ) ) ) ).
% insertI2
thf(fact_405_insertI2,axiom,
! [A2: nat > nat,B: set_nat_nat,B2: nat > nat] :
( ( member_nat_nat @ A2 @ B )
=> ( member_nat_nat @ A2 @ ( insert_nat_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_406_Set_Oset__insert,axiom,
! [X: set_nat,A: set_set_nat] :
( ( member_set_nat @ X @ A )
=> ~ ! [B5: set_set_nat] :
( ( A
= ( insert_set_nat @ X @ B5 ) )
=> ( member_set_nat @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_407_Set_Oset__insert,axiom,
! [X: nat,A: set_nat] :
( ( member_nat @ X @ A )
=> ~ ! [B5: set_nat] :
( ( A
= ( insert_nat @ X @ B5 ) )
=> ( member_nat @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_408_Set_Oset__insert,axiom,
! [X: $o,A: set_o] :
( ( member_o @ X @ A )
=> ~ ! [B5: set_o] :
( ( A
= ( insert_o @ X @ B5 ) )
=> ( member_o @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_409_Set_Oset__insert,axiom,
! [X: nat > nat,A: set_nat_nat] :
( ( member_nat_nat @ X @ A )
=> ~ ! [B5: set_nat_nat] :
( ( A
= ( insert_nat_nat @ X @ B5 ) )
=> ( member_nat_nat @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_410_insert__ident,axiom,
! [X: set_nat,A: set_set_nat,B: set_set_nat] :
( ~ ( member_set_nat @ X @ A )
=> ( ~ ( member_set_nat @ X @ B )
=> ( ( ( insert_set_nat @ X @ A )
= ( insert_set_nat @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_411_insert__ident,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ~ ( member_nat @ X @ B )
=> ( ( ( insert_nat @ X @ A )
= ( insert_nat @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_412_insert__ident,axiom,
! [X: $o,A: set_o,B: set_o] :
( ~ ( member_o @ X @ A )
=> ( ~ ( member_o @ X @ B )
=> ( ( ( insert_o @ X @ A )
= ( insert_o @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_413_insert__ident,axiom,
! [X: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ~ ( member_nat_nat @ X @ A )
=> ( ~ ( member_nat_nat @ X @ B )
=> ( ( ( insert_nat_nat @ X @ A )
= ( insert_nat_nat @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_414_insert__absorb,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ A )
=> ( ( insert_set_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_415_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_416_insert__absorb,axiom,
! [A2: $o,A: set_o] :
( ( member_o @ A2 @ A )
=> ( ( insert_o @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_417_insert__absorb,axiom,
! [A2: nat > nat,A: set_nat_nat] :
( ( member_nat_nat @ A2 @ A )
=> ( ( insert_nat_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_418_insert__eq__iff,axiom,
! [A2: set_nat,A: set_set_nat,B2: set_nat,B: set_set_nat] :
( ~ ( member_set_nat @ A2 @ A )
=> ( ~ ( member_set_nat @ B2 @ B )
=> ( ( ( insert_set_nat @ A2 @ A )
= ( insert_set_nat @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_set_nat] :
( ( A
= ( insert_set_nat @ B2 @ C3 ) )
& ~ ( member_set_nat @ B2 @ C3 )
& ( B
= ( insert_set_nat @ A2 @ C3 ) )
& ~ ( member_set_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_419_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B2: nat,B: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ B2 @ B )
=> ( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_nat] :
( ( A
= ( insert_nat @ B2 @ C3 ) )
& ~ ( member_nat @ B2 @ C3 )
& ( B
= ( insert_nat @ A2 @ C3 ) )
& ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_420_insert__eq__iff,axiom,
! [A2: $o,A: set_o,B2: $o,B: set_o] :
( ~ ( member_o @ A2 @ A )
=> ( ~ ( member_o @ B2 @ B )
=> ( ( ( insert_o @ A2 @ A )
= ( insert_o @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 = (~ B2) )
=> ? [C3: set_o] :
( ( A
= ( insert_o @ B2 @ C3 ) )
& ~ ( member_o @ B2 @ C3 )
& ( B
= ( insert_o @ A2 @ C3 ) )
& ~ ( member_o @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_421_insert__eq__iff,axiom,
! [A2: nat > nat,A: set_nat_nat,B2: nat > nat,B: set_nat_nat] :
( ~ ( member_nat_nat @ A2 @ A )
=> ( ~ ( member_nat_nat @ B2 @ B )
=> ( ( ( insert_nat_nat @ A2 @ A )
= ( insert_nat_nat @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_nat_nat] :
( ( A
= ( insert_nat_nat @ B2 @ C3 ) )
& ~ ( member_nat_nat @ B2 @ C3 )
& ( B
= ( insert_nat_nat @ A2 @ C3 ) )
& ~ ( member_nat_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_422_insert__commute,axiom,
! [X: nat,Y: nat,A: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ Y @ A ) )
= ( insert_nat @ Y @ ( insert_nat @ X @ A ) ) ) ).
% insert_commute
thf(fact_423_mk__disjoint__insert,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ A )
=> ? [B5: set_set_nat] :
( ( A
= ( insert_set_nat @ A2 @ B5 ) )
& ~ ( member_set_nat @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_424_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ? [B5: set_nat] :
( ( A
= ( insert_nat @ A2 @ B5 ) )
& ~ ( member_nat @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_425_mk__disjoint__insert,axiom,
! [A2: $o,A: set_o] :
( ( member_o @ A2 @ A )
=> ? [B5: set_o] :
( ( A
= ( insert_o @ A2 @ B5 ) )
& ~ ( member_o @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_426_mk__disjoint__insert,axiom,
! [A2: nat > nat,A: set_nat_nat] :
( ( member_nat_nat @ A2 @ A )
=> ? [B5: set_nat_nat] :
( ( A
= ( insert_nat_nat @ A2 @ B5 ) )
& ~ ( member_nat_nat @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_427_UnionE,axiom,
! [A: set_nat,C2: set_set_set_nat] :
( ( member_set_nat @ A @ ( comple548664676211718543et_nat @ C2 ) )
=> ~ ! [X6: set_set_nat] :
( ( member_set_nat @ A @ X6 )
=> ~ ( member_set_set_nat @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_428_UnionE,axiom,
! [A: $o,C2: set_set_o] :
( ( member_o @ A @ ( comple90263536869209701_set_o @ C2 ) )
=> ~ ! [X6: set_o] :
( ( member_o @ A @ X6 )
=> ~ ( member_set_o @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_429_UnionE,axiom,
! [A: nat > nat,C2: set_set_nat_nat] :
( ( member_nat_nat @ A @ ( comple5448282615319421384at_nat @ C2 ) )
=> ~ ! [X6: set_nat_nat] :
( ( member_nat_nat @ A @ X6 )
=> ~ ( member_set_nat_nat @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_430_UnionE,axiom,
! [A: nat,C2: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) )
=> ~ ! [X6: set_nat] :
( ( member_nat @ A @ X6 )
=> ~ ( member_set_nat @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_431_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_432_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I2 @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_433_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A2: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_434_group__cancel_Oadd2,axiom,
! [B: nat,K: nat,B2: nat,A2: nat] :
( ( B
= ( plus_plus_nat @ K @ B2 ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_435_add_Oassoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_436_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_437_add_Oleft__commute,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_438_add__left__imp__eq,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ A2 @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_439_add__right__imp__eq,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_440_SUP__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > nat,D: nat > nat] :
( ( A = B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_441_SUP__cong,axiom,
! [A: set_o,B: set_o,C2: $o > nat,D: $o > nat] :
( ( A = B )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_o_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_o_nat @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_442_SUP__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > $o,D: nat > $o] :
( ( A = B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_443_SUP__cong,axiom,
! [A: set_o,B: set_o,C2: $o > $o,D: $o > $o] :
( ( A = B )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_444_SUP__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > set_nat,D: nat > set_nat] :
( ( A = B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ A ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_445_SUP__cong,axiom,
! [A: set_o,B: set_o,C2: $o > set_nat,D: $o > set_nat] :
( ( A = B )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ B )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ C2 @ A ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_446_SUP__cong,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_nat > nat,D: set_nat > nat] :
( ( A = B )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ B )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_set_nat_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_set_nat_nat @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_447_SUP__cong,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_nat > $o,D: set_nat > $o] :
( ( A = B )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ B )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_nat_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_set_nat_o @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_448_SUP__cong,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_nat > set_nat,D: set_nat > set_nat] :
( ( A = B )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ B )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ C2 @ A ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_449_SUP__cong,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: ( nat > nat ) > nat,D: ( nat > nat ) > nat] :
( ( A = B )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B )
=> ( ( C2 @ X3 )
= ( D @ X3 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat_nat @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_450_Union__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Union_empty
thf(fact_451_Union__empty__conv,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_452_empty__Union__conv,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_453_ivl__disj__int__two_I2_J,axiom,
! [L: nat > nat,M3: nat > nat,U: nat > nat] :
( ( inf_inf_set_nat_nat @ ( set_or3470937268066497519at_nat @ L @ M3 ) @ ( set_or1731194346131681939at_nat @ M3 @ U ) )
= bot_bot_set_nat_nat ) ).
% ivl_disj_int_two(2)
thf(fact_454_ivl__disj__int__two_I2_J,axiom,
! [L: $o,M3: $o,U: $o] :
( ( inf_inf_set_o @ ( set_or8254209520273421544Most_o @ L @ M3 ) @ ( set_or1716231572884733764Than_o @ M3 @ U ) )
= bot_bot_set_o ) ).
% ivl_disj_int_two(2)
thf(fact_455_ivl__disj__int__two_I2_J,axiom,
! [L: set_nat,M3: set_nat,U: set_nat] :
( ( inf_inf_set_set_nat @ ( set_or7074010630789208630et_nat @ L @ M3 ) @ ( set_or8625682525731655386et_nat @ M3 @ U ) )
= bot_bot_set_set_nat ) ).
% ivl_disj_int_two(2)
thf(fact_456_ivl__disj__int__two_I2_J,axiom,
! [L: nat,M3: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or6659071591806873216st_nat @ L @ M3 ) @ ( set_or5834768355832116004an_nat @ M3 @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(2)
thf(fact_457_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_458_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_459_singletonD,axiom,
! [B2: set_nat,A2: set_nat] :
( ( member_set_nat @ B2 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_460_singletonD,axiom,
! [B2: $o,A2: $o] :
( ( member_o @ B2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_461_singletonD,axiom,
! [B2: nat > nat,A2: nat > nat] :
( ( member_nat_nat @ B2 @ ( insert_nat_nat @ A2 @ bot_bot_set_nat_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_462_singletonD,axiom,
! [B2: nat,A2: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_463_singleton__iff,axiom,
! [B2: set_nat,A2: set_nat] :
( ( member_set_nat @ B2 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_464_singleton__iff,axiom,
! [B2: $o,A2: $o] :
( ( member_o @ B2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_465_singleton__iff,axiom,
! [B2: nat > nat,A2: nat > nat] :
( ( member_nat_nat @ B2 @ ( insert_nat_nat @ A2 @ bot_bot_set_nat_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_466_singleton__iff,axiom,
! [B2: nat,A2: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_467_doubleton__eq__iff,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ( insert_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A2 = C )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_468_insert__not__empty,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ A )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_469_singleton__inject,axiom,
! [A2: nat,B2: nat] :
( ( ( insert_nat @ A2 @ bot_bot_set_nat )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_470_Int__insert__left,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B @ C2 ) ) ) )
& ( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_471_Int__insert__left,axiom,
! [A2: nat > nat,C2: set_nat_nat,B: set_nat_nat] :
( ( ( member_nat_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat_nat @ ( insert_nat_nat @ A2 @ B ) @ C2 )
= ( insert_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ B @ C2 ) ) ) )
& ( ~ ( member_nat_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat_nat @ ( insert_nat_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_nat_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_472_Int__insert__left,axiom,
! [A2: $o,C2: set_o,B: set_o] :
( ( ( member_o @ A2 @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A2 @ B ) @ C2 )
= ( insert_o @ A2 @ ( inf_inf_set_o @ B @ C2 ) ) ) )
& ( ~ ( member_o @ A2 @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A2 @ B ) @ C2 )
= ( inf_inf_set_o @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_473_Int__insert__left,axiom,
! [A2: set_nat,C2: set_set_nat,B: set_set_nat] :
( ( ( member_set_nat @ A2 @ C2 )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ B ) @ C2 )
= ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ B @ C2 ) ) ) )
& ( ~ ( member_set_nat @ A2 @ C2 )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_474_Int__insert__right,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) )
& ( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_475_Int__insert__right,axiom,
! [A2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( ( member_nat_nat @ A2 @ A )
=> ( ( inf_inf_set_nat_nat @ A @ ( insert_nat_nat @ A2 @ B ) )
= ( insert_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ A @ B ) ) ) )
& ( ~ ( member_nat_nat @ A2 @ A )
=> ( ( inf_inf_set_nat_nat @ A @ ( insert_nat_nat @ A2 @ B ) )
= ( inf_inf_set_nat_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_476_Int__insert__right,axiom,
! [A2: $o,A: set_o,B: set_o] :
( ( ( member_o @ A2 @ A )
=> ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B ) )
= ( insert_o @ A2 @ ( inf_inf_set_o @ A @ B ) ) ) )
& ( ~ ( member_o @ A2 @ A )
=> ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B ) )
= ( inf_inf_set_o @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_477_Int__insert__right,axiom,
! [A2: set_nat,A: set_set_nat,B: set_set_nat] :
( ( ( member_set_nat @ A2 @ A )
=> ( ( inf_inf_set_set_nat @ A @ ( insert_set_nat @ A2 @ B ) )
= ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ A @ B ) ) ) )
& ( ~ ( member_set_nat @ A2 @ A )
=> ( ( inf_inf_set_set_nat @ A @ ( insert_set_nat @ A2 @ B ) )
= ( inf_inf_set_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_478_plus__nat_Oadd__0,axiom,
! [N3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N3 )
= N3 ) ).
% plus_nat.add_0
thf(fact_479_add__eq__self__zero,axiom,
! [M3: nat,N3: nat] :
( ( ( plus_plus_nat @ M3 @ N3 )
= M3 )
=> ( N3 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_480_SUP__eq__const,axiom,
! [I: set_set_nat,F: set_nat > set_nat,X: set_nat] :
( ( I != bot_bot_set_set_nat )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ I ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_481_SUP__eq__const,axiom,
! [I: set_o,F: $o > set_nat,X: set_nat] :
( ( I != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ I ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_482_SUP__eq__const,axiom,
! [I: set_nat_nat,F: ( nat > nat ) > set_nat,X: set_nat] :
( ( I != bot_bot_set_nat_nat )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ I ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_483_SUP__eq__const,axiom,
! [I: set_nat,F: nat > set_nat,X: set_nat] :
( ( I != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ I ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_484_SUP__eq__const,axiom,
! [I: set_set_nat,F: set_nat > $o,X: $o] :
( ( I != bot_bot_set_set_nat )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ I ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_485_SUP__eq__const,axiom,
! [I: set_o,F: $o > $o,X: $o] :
( ( I != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ I ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_486_SUP__eq__const,axiom,
! [I: set_nat_nat,F: ( nat > nat ) > $o,X: $o] :
( ( I != bot_bot_set_nat_nat )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_nat_o @ F @ I ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_487_SUP__eq__const,axiom,
! [I: set_nat,F: nat > $o,X: $o] :
( ( I != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ I ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_488_Union__disjoint,axiom,
! [C2: set_set_nat_nat,A: set_nat_nat] :
( ( ( inf_inf_set_nat_nat @ ( comple5448282615319421384at_nat @ C2 ) @ A )
= bot_bot_set_nat_nat )
= ( ! [X2: set_nat_nat] :
( ( member_set_nat_nat @ X2 @ C2 )
=> ( ( inf_inf_set_nat_nat @ X2 @ A )
= bot_bot_set_nat_nat ) ) ) ) ).
% Union_disjoint
thf(fact_489_Union__disjoint,axiom,
! [C2: set_set_o,A: set_o] :
( ( ( inf_inf_set_o @ ( comple90263536869209701_set_o @ C2 ) @ A )
= bot_bot_set_o )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ C2 )
=> ( ( inf_inf_set_o @ X2 @ A )
= bot_bot_set_o ) ) ) ) ).
% Union_disjoint
thf(fact_490_Union__disjoint,axiom,
! [C2: set_set_set_nat,A: set_set_nat] :
( ( ( inf_inf_set_set_nat @ ( comple548664676211718543et_nat @ C2 ) @ A )
= bot_bot_set_set_nat )
= ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ C2 )
=> ( ( inf_inf_set_set_nat @ X2 @ A )
= bot_bot_set_set_nat ) ) ) ) ).
% Union_disjoint
thf(fact_491_Union__disjoint,axiom,
! [C2: set_set_nat,A: set_nat] :
( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ C2 ) @ A )
= bot_bot_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ C2 )
=> ( ( inf_inf_set_nat @ X2 @ A )
= bot_bot_set_nat ) ) ) ) ).
% Union_disjoint
thf(fact_492_ivl__disj__int__two_I6_J,axiom,
! [L: nat > nat,M3: nat > nat,U: nat > nat] :
( ( inf_inf_set_nat_nat @ ( set_or3470937268066497519at_nat @ L @ M3 ) @ ( set_or3470937268066497519at_nat @ M3 @ U ) )
= bot_bot_set_nat_nat ) ).
% ivl_disj_int_two(6)
thf(fact_493_ivl__disj__int__two_I6_J,axiom,
! [L: $o,M3: $o,U: $o] :
( ( inf_inf_set_o @ ( set_or8254209520273421544Most_o @ L @ M3 ) @ ( set_or8254209520273421544Most_o @ M3 @ U ) )
= bot_bot_set_o ) ).
% ivl_disj_int_two(6)
thf(fact_494_ivl__disj__int__two_I6_J,axiom,
! [L: set_nat,M3: set_nat,U: set_nat] :
( ( inf_inf_set_set_nat @ ( set_or7074010630789208630et_nat @ L @ M3 ) @ ( set_or7074010630789208630et_nat @ M3 @ U ) )
= bot_bot_set_set_nat ) ).
% ivl_disj_int_two(6)
thf(fact_495_ivl__disj__int__two_I6_J,axiom,
! [L: nat,M3: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or6659071591806873216st_nat @ L @ M3 ) @ ( set_or6659071591806873216st_nat @ M3 @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(6)
thf(fact_496_fact3,axiom,
! [X4: nat] :
( ( member_nat @ X4 @ ( set_ord_lessThan_nat @ k ) )
=> ( ( inf_inf_set_nat @ ( bl @ zero_zero_nat ) @ ( hales_set_incr @ n2 @ ( bs @ X4 ) ) )
= bot_bot_set_nat ) ) ).
% fact3
thf(fact_497_fact4,axiom,
! [X4: nat] :
( ( member_nat @ X4 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ k @ one_one_nat ) ) )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ k @ one_one_nat ) ) )
=> ( ( X4 != Xa )
=> ( ( inf_inf_set_nat @ ( hales_set_incr @ n2 @ ( bs @ X4 ) ) @ ( hales_set_incr @ n2 @ ( bs @ Xa ) ) )
= bot_bot_set_nat ) ) ) ) ).
% fact4
thf(fact_498_fact1,axiom,
( ( inf_inf_set_nat @ ( hales_set_incr @ n2 @ ( bs @ k ) ) @ ( bl @ one_one_nat ) )
= bot_bot_set_nat ) ).
% fact1
thf(fact_499_BfS__props_I2_J,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ bs @ ( set_ord_atMost_nat @ k ) ) )
= ( set_ord_lessThan_nat @ m2 ) ) ).
% BfS_props(2)
thf(fact_500_cSup__singleton,axiom,
! [X: set_nat] :
( ( comple7399068483239264473et_nat @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
= X ) ).
% cSup_singleton
thf(fact_501_cSup__singleton,axiom,
! [X: nat] :
( ( complete_Sup_Sup_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% cSup_singleton
thf(fact_502_cSup__singleton,axiom,
! [X: $o] :
( ( complete_Sup_Sup_o @ ( insert_o @ X @ bot_bot_set_o ) )
= X ) ).
% cSup_singleton
thf(fact_503_ccpo__Sup__singleton,axiom,
! [X: set_nat] :
( ( comple7399068483239264473et_nat @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
= X ) ).
% ccpo_Sup_singleton
thf(fact_504_ccpo__Sup__singleton,axiom,
! [X: $o] :
( ( complete_Sup_Sup_o @ ( insert_o @ X @ bot_bot_set_o ) )
= X ) ).
% ccpo_Sup_singleton
thf(fact_505_Bvar__def,axiom,
( bvar
= ( ^ [I4: nat] : ( if_set_nat @ ( I4 = zero_zero_nat ) @ ( bl @ zero_zero_nat ) @ ( hales_set_incr @ n2 @ ( bs @ ( minus_minus_nat @ I4 @ one_one_nat ) ) ) ) ) ) ).
% Bvar_def
thf(fact_506_cSup__atMost,axiom,
! [X: set_nat] :
( ( comple7399068483239264473et_nat @ ( set_or4236626031148496127et_nat @ X ) )
= X ) ).
% cSup_atMost
thf(fact_507_cSup__atMost,axiom,
! [X: nat] :
( ( complete_Sup_Sup_nat @ ( set_ord_atMost_nat @ X ) )
= X ) ).
% cSup_atMost
thf(fact_508_cSup__atMost,axiom,
! [X: $o] :
( ( complete_Sup_Sup_o @ ( set_ord_atMost_o @ X ) )
= X ) ).
% cSup_atMost
thf(fact_509_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_510_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_511__092_060open_062n_H_A_092_060le_062_An_092_060close_062,axiom,
ord_less_eq_nat @ n @ n2 ).
% \<open>n' \<le> n\<close>
thf(fact_512_M_H__prop,axiom,
ord_less_eq_nat @ ( plus_plus_nat @ n @ m2 ) @ m ).
% M'_prop
thf(fact_513_le__zero__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_514_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_515_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_516_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_517_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_518_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_519_inf_Obounded__iff,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) )
= ( ( ord_less_eq_set_nat @ A2 @ B2 )
& ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_520_inf_Obounded__iff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C ) )
= ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
& ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_521_inf_Obounded__iff,axiom,
! [A2: set_o,B2: set_o,C: set_o] :
( ( ord_less_eq_set_o @ A2 @ ( inf_inf_set_o @ B2 @ C ) )
= ( ( ord_less_eq_set_o @ A2 @ B2 )
& ( ord_less_eq_set_o @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_522_inf_Obounded__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C ) )
= ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
& ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_523_inf_Obounded__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_524_le__inf__iff,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( ( ord_less_eq_set_nat @ X @ Y )
& ( ord_less_eq_set_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_525_le__inf__iff,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ ( inf_inf_set_nat_nat @ Y @ Z ) )
= ( ( ord_le9059583361652607317at_nat @ X @ Y )
& ( ord_le9059583361652607317at_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_526_le__inf__iff,axiom,
! [X: set_o,Y: set_o,Z: set_o] :
( ( ord_less_eq_set_o @ X @ ( inf_inf_set_o @ Y @ Z ) )
= ( ( ord_less_eq_set_o @ X @ Y )
& ( ord_less_eq_set_o @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_527_le__inf__iff,axiom,
! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ ( inf_inf_set_set_nat @ Y @ Z ) )
= ( ( ord_le6893508408891458716et_nat @ X @ Y )
& ( ord_le6893508408891458716et_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_528_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_529_add__diff__cancel__right_H,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_530_add__diff__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( minus_minus_nat @ A2 @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_531_add__diff__cancel__left_H,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ A2 )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_532_add__diff__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( minus_minus_nat @ A2 @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_533_le0,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% le0
thf(fact_534_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_535_lessThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_536_diff__self__eq__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ M3 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_537_diff__0__eq__0,axiom,
! [N3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N3 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_538_nat__add__left__cancel__le,axiom,
! [K: nat,M3: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N3 ) )
= ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% nat_add_left_cancel_le
thf(fact_539_atMost__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_540_atMost__iff,axiom,
! [I2: set_nat,K: set_nat] :
( ( member_set_nat @ I2 @ ( set_or4236626031148496127et_nat @ K ) )
= ( ord_less_eq_set_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_541_atMost__iff,axiom,
! [I2: $o,K: $o] :
( ( member_o @ I2 @ ( set_ord_atMost_o @ K ) )
= ( ord_less_eq_o @ I2 @ K ) ) ).
% atMost_iff
thf(fact_542_atMost__iff,axiom,
! [I2: nat > nat,K: nat > nat] :
( ( member_nat_nat @ I2 @ ( set_or9140604705432621368at_nat @ K ) )
= ( ord_less_eq_nat_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_543_atMost__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat @ I2 @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_544_diff__diff__left,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_545_diff__diff__cancel,axiom,
! [I2: nat,N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( minus_minus_nat @ N3 @ ( minus_minus_nat @ N3 @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_546_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_547_d__def,axiom,
( d
= ( minus_minus_nat @ m @ ( plus_plus_nat @ n @ m2 ) ) ) ).
% d_def
thf(fact_548_add__le__same__cancel1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_549_add__le__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_550_le__add__same__cancel1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_551_le__add__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_552_le__add__diff__inverse2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_553_le__add__diff__inverse,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A2 @ B2 ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_554_diff__add__zero,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_555_diff__is__0__eq,axiom,
! [M3: nat,N3: nat] :
( ( ( minus_minus_nat @ M3 @ N3 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% diff_is_0_eq
thf(fact_556_diff__is__0__eq_H,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( minus_minus_nat @ M3 @ N3 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_557_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_558_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_559_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_560_greaterThanAtMost__empty,axiom,
! [L: nat,K: nat] :
( ( ord_less_eq_nat @ L @ K )
=> ( ( set_or6659071591806873216st_nat @ K @ L )
= bot_bot_set_nat ) ) ).
% greaterThanAtMost_empty
thf(fact_561_greaterThanLessThan__empty,axiom,
! [L: nat,K: nat] :
( ( ord_less_eq_nat @ L @ K )
=> ( ( set_or5834768355832116004an_nat @ K @ L )
= bot_bot_set_nat ) ) ).
% greaterThanLessThan_empty
thf(fact_562__092_060open_062n_A_L_Am_A_061_AM_H_092_060close_062,axiom,
( ( plus_plus_nat @ n2 @ m2 )
= m ) ).
% \<open>n + m = M'\<close>
thf(fact_563_Sup__subset__mono,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_subset_mono
thf(fact_564_Sup__subset__mono,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_subset_mono
thf(fact_565_add__le__imp__le__diff,axiom,
! [I2: nat,K: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N3 )
=> ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N3 @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_566_diff__add,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ A2 )
= B2 ) ) ).
% diff_add
thf(fact_567_add__le__add__imp__diff__le,axiom,
! [I2: nat,K: nat,N3: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N3 @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_568_le__add__diff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 ) ) ) ).
% le_add_diff
thf(fact_569_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_570_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_571_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A2 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_572_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_573_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 )
= ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_574_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_575_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B2 @ A2 ) )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_576_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( minus_minus_nat @ B2 @ A2 )
= C )
= ( B2
= ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_577_le__diff__conv,axiom,
! [J: nat,K: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_578_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_579_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_580_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_581_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( minus_minus_nat @ J @ I2 )
= K )
= ( J
= ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_582_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_583_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_584_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_585_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_586_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_587_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_588_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B2 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_589_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_590_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_591_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_592_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_593_nat__le__linear,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
| ( ord_less_eq_nat @ N3 @ M3 ) ) ).
% nat_le_linear
thf(fact_594_diff__le__mono2,axiom,
! [M3: nat,N3: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N3 ) @ ( minus_minus_nat @ L @ M3 ) ) ) ).
% diff_le_mono2
thf(fact_595_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_596_diff__le__self,axiom,
! [M3: nat,N3: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ N3 ) @ M3 ) ).
% diff_le_self
thf(fact_597_diff__le__mono,axiom,
! [M3: nat,N3: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ L ) @ ( minus_minus_nat @ N3 @ L ) ) ) ).
% diff_le_mono
thf(fact_598_Nat_Odiff__diff__eq,axiom,
! [K: nat,M3: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N3 @ K ) )
= ( minus_minus_nat @ M3 @ N3 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_599_diff__commute,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% diff_commute
thf(fact_600_le__diff__iff,axiom,
! [K: nat,M3: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N3 @ K ) )
= ( ord_less_eq_nat @ M3 @ N3 ) ) ) ) ).
% le_diff_iff
thf(fact_601_eq__diff__iff,axiom,
! [K: nat,M3: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( ( minus_minus_nat @ M3 @ K )
= ( minus_minus_nat @ N3 @ K ) )
= ( M3 = N3 ) ) ) ) ).
% eq_diff_iff
thf(fact_602_le__antisym,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ M3 )
=> ( M3 = N3 ) ) ) ).
% le_antisym
thf(fact_603_eq__imp__le,axiom,
! [M3: nat,N3: nat] :
( ( M3 = N3 )
=> ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% eq_imp_le
thf(fact_604_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_605_le__refl,axiom,
! [N3: nat] : ( ord_less_eq_nat @ N3 @ N3 ) ).
% le_refl
thf(fact_606_diff__right__commute,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C ) ) ).
% diff_right_commute
thf(fact_607_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_608_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_609_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_610_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: nat,B6: nat] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_611_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_612_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_613_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_614_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_615_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_616_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_617_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_618_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_619_diff__shunt__var,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( minus_minus_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_620_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M2: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M2 ) )
=> ~ ! [M: nat] :
( ( P @ M )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_621_Ioc__subset__iff,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_set_nat @ ( set_or6659071591806873216st_nat @ A2 @ B2 ) @ ( set_or6659071591806873216st_nat @ C @ D2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
| ( ( ord_less_eq_nat @ C @ A2 )
& ( ord_less_eq_nat @ B2 @ D2 ) ) ) ) ).
% Ioc_subset_iff
thf(fact_622_cSup__eq__maximum,axiom,
! [Z: set_nat,X5: set_set_nat] :
( ( member_set_nat @ Z @ X5 )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ X5 )
=> ( ord_less_eq_set_nat @ X3 @ Z ) )
=> ( ( comple7399068483239264473et_nat @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_623_cSup__eq__maximum,axiom,
! [Z: nat,X5: set_nat] :
( ( member_nat @ Z @ X5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ( ord_less_eq_nat @ X3 @ Z ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_624_cSup__eq__maximum,axiom,
! [Z: $o,X5: set_o] :
( ( member_o @ Z @ X5 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X5 )
=> ( ord_less_eq_o @ X3 @ Z ) )
=> ( ( complete_Sup_Sup_o @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_625_Sup__SUP__eq,axiom,
( comple3806919086088850358_nat_o
= ( ^ [S2: set_set_nat_o,X2: set_nat] : ( member_set_nat @ X2 @ ( comple548664676211718543et_nat @ ( image_4687162037615663680et_nat @ collect_set_nat @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_626_Sup__SUP__eq,axiom,
( complete_Sup_Sup_o_o
= ( ^ [S2: set_o_o,X2: $o] : ( member_o @ X2 @ ( comple90263536869209701_set_o @ ( image_o_o_set_o @ collect_o @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_627_Sup__SUP__eq,axiom,
( comple8312177224774716605_nat_o
= ( ^ [S2: set_nat_nat_o,X2: nat > nat] : ( member_nat_nat @ X2 @ ( comple5448282615319421384at_nat @ ( image_7977807581451749376at_nat @ collect_nat_nat @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_628_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S2: set_nat_o,X2: nat] : ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_629_diff__diff__eq,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C )
= ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_630_add__implies__diff,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ( plus_plus_nat @ C @ B2 )
= A2 )
=> ( C
= ( minus_minus_nat @ A2 @ B2 ) ) ) ).
% add_implies_diff
thf(fact_631_diffs0__imp__equal,axiom,
! [M3: nat,N3: nat] :
( ( ( minus_minus_nat @ M3 @ N3 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N3 @ M3 )
= zero_zero_nat )
=> ( M3 = N3 ) ) ) ).
% diffs0_imp_equal
thf(fact_632_minus__nat_Odiff__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% minus_nat.diff_0
thf(fact_633_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_634_diff__add__inverse2,axiom,
! [M3: nat,N3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ N3 ) @ N3 )
= M3 ) ).
% diff_add_inverse2
thf(fact_635_diff__add__inverse,axiom,
! [N3: nat,M3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N3 @ M3 ) @ N3 )
= M3 ) ).
% diff_add_inverse
thf(fact_636_diff__cancel2,axiom,
! [M3: nat,K: nat,N3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ K ) @ ( plus_plus_nat @ N3 @ K ) )
= ( minus_minus_nat @ M3 @ N3 ) ) ).
% diff_cancel2
thf(fact_637_Nat_Odiff__cancel,axiom,
! [K: nat,M3: nat,N3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N3 ) )
= ( minus_minus_nat @ M3 @ N3 ) ) ).
% Nat.diff_cancel
thf(fact_638_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_639_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_640_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C4: nat] :
( B3
= ( plus_plus_nat @ A3 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_641_add__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_642_less__eqE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ! [C5: nat] :
( B2
!= ( plus_plus_nat @ A2 @ C5 ) ) ) ).
% less_eqE
thf(fact_643_add__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_644_add__mono,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_mono
thf(fact_645_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_646_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_647_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_648_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_649_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_650_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_651_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_652_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_653_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_654_inf_OcoboundedI2,axiom,
! [B2: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_655_inf_OcoboundedI2,axiom,
! [B2: set_nat_nat,C: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ C )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_656_inf_OcoboundedI2,axiom,
! [B2: set_o,C: set_o,A2: set_o] :
( ( ord_less_eq_set_o @ B2 @ C )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_657_inf_OcoboundedI2,axiom,
! [B2: set_set_nat,C: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_658_inf_OcoboundedI2,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_659_inf_OcoboundedI1,axiom,
! [A2: set_nat,C: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_660_inf_OcoboundedI1,axiom,
! [A2: set_nat_nat,C: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_661_inf_OcoboundedI1,axiom,
! [A2: set_o,C: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A2 @ C )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_662_inf_OcoboundedI1,axiom,
! [A2: set_set_nat,C: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_663_inf_OcoboundedI1,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_664_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( inf_inf_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_665_inf_Oabsorb__iff2,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B3: set_nat_nat,A3: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_666_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_o
= ( ^ [B3: set_o,A3: set_o] :
( ( inf_inf_set_o @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_667_inf_Oabsorb__iff2,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B3: set_set_nat,A3: set_set_nat] :
( ( inf_inf_set_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_668_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( inf_inf_nat @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_669_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( inf_inf_set_nat @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_670_inf_Oabsorb__iff1,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_671_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
( ( inf_inf_set_o @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_672_inf_Oabsorb__iff1,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( inf_inf_set_set_nat @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_673_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( inf_inf_nat @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_674_inf_Ocobounded2,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_675_inf_Ocobounded2,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_676_inf_Ocobounded2,axiom,
! [A2: set_o,B2: set_o] : ( ord_less_eq_set_o @ ( inf_inf_set_o @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_677_inf_Ocobounded2,axiom,
! [A2: set_set_nat,B2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_678_inf_Ocobounded2,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_679_inf_Ocobounded1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_680_inf_Ocobounded1,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_681_inf_Ocobounded1,axiom,
! [A2: set_o,B2: set_o] : ( ord_less_eq_set_o @ ( inf_inf_set_o @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_682_inf_Ocobounded1,axiom,
! [A2: set_set_nat,B2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_683_inf_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_684_inf_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( A3
= ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_685_inf_Oorder__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( A3
= ( inf_inf_set_nat_nat @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_686_inf_Oorder__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
( A3
= ( inf_inf_set_o @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_687_inf_Oorder__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( A3
= ( inf_inf_set_set_nat @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_688_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( A3
= ( inf_inf_nat @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_689_inf__greatest,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Z )
=> ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_690_inf__greatest,axiom,
! [X: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y )
=> ( ( ord_le9059583361652607317at_nat @ X @ Z )
=> ( ord_le9059583361652607317at_nat @ X @ ( inf_inf_set_nat_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_691_inf__greatest,axiom,
! [X: set_o,Y: set_o,Z: set_o] :
( ( ord_less_eq_set_o @ X @ Y )
=> ( ( ord_less_eq_set_o @ X @ Z )
=> ( ord_less_eq_set_o @ X @ ( inf_inf_set_o @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_692_inf__greatest,axiom,
! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( ord_le6893508408891458716et_nat @ X @ Z )
=> ( ord_le6893508408891458716et_nat @ X @ ( inf_inf_set_set_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_693_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_694_inf_OboundedI,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_695_inf_OboundedI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ C )
=> ( ord_le9059583361652607317at_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_696_inf_OboundedI,axiom,
! [A2: set_o,B2: set_o,C: set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ( ord_less_eq_set_o @ A2 @ C )
=> ( ord_less_eq_set_o @ A2 @ ( inf_inf_set_o @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_697_inf_OboundedI,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ C )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_698_inf_OboundedI,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_699_inf_OboundedE,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_700_inf_OboundedE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C ) )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ~ ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_701_inf_OboundedE,axiom,
! [A2: set_o,B2: set_o,C: set_o] :
( ( ord_less_eq_set_o @ A2 @ ( inf_inf_set_o @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_o @ A2 @ B2 )
=> ~ ( ord_less_eq_set_o @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_702_inf_OboundedE,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C ) )
=> ~ ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ~ ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_703_inf_OboundedE,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_704_inf__absorb2,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( inf_inf_set_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_705_inf__absorb2,axiom,
! [Y: set_nat_nat,X: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y @ X )
=> ( ( inf_inf_set_nat_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_706_inf__absorb2,axiom,
! [Y: set_o,X: set_o] :
( ( ord_less_eq_set_o @ Y @ X )
=> ( ( inf_inf_set_o @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_707_inf__absorb2,axiom,
! [Y: set_set_nat,X: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X )
=> ( ( inf_inf_set_set_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_708_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_709_inf__absorb1,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( inf_inf_set_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_710_inf__absorb1,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y )
=> ( ( inf_inf_set_nat_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_711_inf__absorb1,axiom,
! [X: set_o,Y: set_o] :
( ( ord_less_eq_set_o @ X @ Y )
=> ( ( inf_inf_set_o @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_712_inf__absorb1,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( inf_inf_set_set_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_713_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_714_inf_Oabsorb2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_715_inf_Oabsorb2,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( ( inf_inf_set_nat_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_716_inf_Oabsorb2,axiom,
! [B2: set_o,A2: set_o] :
( ( ord_less_eq_set_o @ B2 @ A2 )
=> ( ( inf_inf_set_o @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_717_inf_Oabsorb2,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( inf_inf_set_set_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_718_inf_Oabsorb2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_719_inf_Oabsorb1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_720_inf_Oabsorb1,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( inf_inf_set_nat_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_721_inf_Oabsorb1,axiom,
! [A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ( inf_inf_set_o @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_722_inf_Oabsorb1,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( inf_inf_set_set_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_723_inf_Oabsorb1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_724_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X2: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_725_le__iff__inf,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [X2: set_nat_nat,Y3: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_726_le__iff__inf,axiom,
( ord_less_eq_set_o
= ( ^ [X2: set_o,Y3: set_o] :
( ( inf_inf_set_o @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_727_le__iff__inf,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [X2: set_set_nat,Y3: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_728_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( inf_inf_nat @ X2 @ Y3 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_729_inf__unique,axiom,
! [F: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
( ! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: set_nat,Y2: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_set_nat @ X3 @ Z3 )
=> ( ord_less_eq_set_nat @ X3 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_set_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_730_inf__unique,axiom,
! [F: set_nat_nat > set_nat_nat > set_nat_nat,X: set_nat_nat,Y: set_nat_nat] :
( ! [X3: set_nat_nat,Y2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: set_nat_nat,Y2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: set_nat_nat,Y2: set_nat_nat,Z3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y2 )
=> ( ( ord_le9059583361652607317at_nat @ X3 @ Z3 )
=> ( ord_le9059583361652607317at_nat @ X3 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_set_nat_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_731_inf__unique,axiom,
! [F: set_o > set_o > set_o,X: set_o,Y: set_o] :
( ! [X3: set_o,Y2: set_o] : ( ord_less_eq_set_o @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: set_o,Y2: set_o] : ( ord_less_eq_set_o @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: set_o,Y2: set_o,Z3: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y2 )
=> ( ( ord_less_eq_set_o @ X3 @ Z3 )
=> ( ord_less_eq_set_o @ X3 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_set_o @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_732_inf__unique,axiom,
! [F: set_set_nat > set_set_nat > set_set_nat,X: set_set_nat,Y: set_set_nat] :
( ! [X3: set_set_nat,Y2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: set_set_nat,Y2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: set_set_nat,Y2: set_set_nat,Z3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y2 )
=> ( ( ord_le6893508408891458716et_nat @ X3 @ Z3 )
=> ( ord_le6893508408891458716et_nat @ X3 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_set_set_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_733_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ X3 @ Z3 )
=> ( ord_less_eq_nat @ X3 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_734_inf_OorderI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2
= ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_735_inf_OorderI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2
= ( inf_inf_set_nat_nat @ A2 @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_736_inf_OorderI,axiom,
! [A2: set_o,B2: set_o] :
( ( A2
= ( inf_inf_set_o @ A2 @ B2 ) )
=> ( ord_less_eq_set_o @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_737_inf_OorderI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2
= ( inf_inf_set_set_nat @ A2 @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_738_inf_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_739_inf_OorderE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_740_inf_OorderE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_741_inf_OorderE,axiom,
! [A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_o @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_742_inf_OorderE,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_743_inf_OorderE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_744_le__infI2,axiom,
! [B2: set_nat,X: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ X )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_745_le__infI2,axiom,
! [B2: set_nat_nat,X: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ X )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_746_le__infI2,axiom,
! [B2: set_o,X: set_o,A2: set_o] :
( ( ord_less_eq_set_o @ B2 @ X )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_747_le__infI2,axiom,
! [B2: set_set_nat,X: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ X )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_748_le__infI2,axiom,
! [B2: nat,X: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_749_le__infI1,axiom,
! [A2: set_nat,X: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_750_le__infI1,axiom,
! [A2: set_nat_nat,X: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ X )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_751_le__infI1,axiom,
! [A2: set_o,X: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A2 @ X )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_752_le__infI1,axiom,
! [A2: set_set_nat,X: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ X )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_753_le__infI1,axiom,
! [A2: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_754_inf__mono,axiom,
! [A2: set_nat,C: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_755_inf__mono,axiom,
! [A2: set_nat_nat,C: set_nat_nat,B2: set_nat_nat,D2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ D2 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ ( inf_inf_set_nat_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_756_inf__mono,axiom,
! [A2: set_o,C: set_o,B2: set_o,D2: set_o] :
( ( ord_less_eq_set_o @ A2 @ C )
=> ( ( ord_less_eq_set_o @ B2 @ D2 )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A2 @ B2 ) @ ( inf_inf_set_o @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_757_inf__mono,axiom,
! [A2: set_set_nat,C: set_set_nat,B2: set_set_nat,D2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ D2 )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ ( inf_inf_set_set_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_758_inf__mono,axiom,
! [A2: nat,C: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_759_le__infI,axiom,
! [X: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ X @ B2 )
=> ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_760_le__infI,axiom,
! [X: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ X @ B2 )
=> ( ord_le9059583361652607317at_nat @ X @ ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_761_le__infI,axiom,
! [X: set_o,A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ X @ A2 )
=> ( ( ord_less_eq_set_o @ X @ B2 )
=> ( ord_less_eq_set_o @ X @ ( inf_inf_set_o @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_762_le__infI,axiom,
! [X: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X @ B2 )
=> ( ord_le6893508408891458716et_nat @ X @ ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_763_le__infI,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_764_le__infE,axiom,
! [X: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_nat @ X @ A2 )
=> ~ ( ord_less_eq_set_nat @ X @ B2 ) ) ) ).
% le_infE
thf(fact_765_le__infE,axiom,
! [X: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ ( inf_inf_set_nat_nat @ A2 @ B2 ) )
=> ~ ( ( ord_le9059583361652607317at_nat @ X @ A2 )
=> ~ ( ord_le9059583361652607317at_nat @ X @ B2 ) ) ) ).
% le_infE
thf(fact_766_le__infE,axiom,
! [X: set_o,A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ X @ ( inf_inf_set_o @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_o @ X @ A2 )
=> ~ ( ord_less_eq_set_o @ X @ B2 ) ) ) ).
% le_infE
thf(fact_767_le__infE,axiom,
! [X: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
=> ~ ( ( ord_le6893508408891458716et_nat @ X @ A2 )
=> ~ ( ord_le6893508408891458716et_nat @ X @ B2 ) ) ) ).
% le_infE
thf(fact_768_le__infE,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X @ A2 )
=> ~ ( ord_less_eq_nat @ X @ B2 ) ) ) ).
% le_infE
thf(fact_769_inf__le2,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_770_inf__le2,axiom,
! [X: set_nat_nat,Y: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_771_inf__le2,axiom,
! [X: set_o,Y: set_o] : ( ord_less_eq_set_o @ ( inf_inf_set_o @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_772_inf__le2,axiom,
! [X: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_773_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_774_inf__le1,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_775_inf__le1,axiom,
! [X: set_nat_nat,Y: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_776_inf__le1,axiom,
! [X: set_o,Y: set_o] : ( ord_less_eq_set_o @ ( inf_inf_set_o @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_777_inf__le1,axiom,
! [X: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_778_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_779_inf__sup__ord_I1_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_780_inf__sup__ord_I1_J,axiom,
! [X: set_nat_nat,Y: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_781_inf__sup__ord_I1_J,axiom,
! [X: set_o,Y: set_o] : ( ord_less_eq_set_o @ ( inf_inf_set_o @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_782_inf__sup__ord_I1_J,axiom,
! [X: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_783_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_784_inf__sup__ord_I2_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_785_inf__sup__ord_I2_J,axiom,
! [X: set_nat_nat,Y: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_786_inf__sup__ord_I2_J,axiom,
! [X: set_o,Y: set_o] : ( ord_less_eq_set_o @ ( inf_inf_set_o @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_787_inf__sup__ord_I2_J,axiom,
! [X: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_788_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_789_le__0__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_790_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_791_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_792_less__eq__nat_Osimps_I1_J,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% less_eq_nat.simps(1)
thf(fact_793_Sup__eqI,axiom,
! [A: set_set_nat,X: set_nat] :
( ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ A )
=> ( ord_less_eq_set_nat @ Y2 @ X ) )
=> ( ! [Y2: set_nat] :
( ! [Z4: set_nat] :
( ( member_set_nat @ Z4 @ A )
=> ( ord_less_eq_set_nat @ Z4 @ Y2 ) )
=> ( ord_less_eq_set_nat @ X @ Y2 ) )
=> ( ( comple7399068483239264473et_nat @ A )
= X ) ) ) ).
% Sup_eqI
thf(fact_794_Sup__eqI,axiom,
! [A: set_o,X: $o] :
( ! [Y2: $o] :
( ( member_o @ Y2 @ A )
=> ( ord_less_eq_o @ Y2 @ X ) )
=> ( ! [Y2: $o] :
( ! [Z4: $o] :
( ( member_o @ Z4 @ A )
=> ( ord_less_eq_o @ Z4 @ Y2 ) )
=> ( ord_less_eq_o @ X @ Y2 ) )
=> ( ( complete_Sup_Sup_o @ A )
= X ) ) ) ).
% Sup_eqI
thf(fact_795_Sup__mono,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [A5: set_nat] :
( ( member_set_nat @ A5 @ A )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ B )
& ( ord_less_eq_set_nat @ A5 @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_mono
thf(fact_796_Sup__mono,axiom,
! [A: set_o,B: set_o] :
( ! [A5: $o] :
( ( member_o @ A5 @ A )
=> ? [X4: $o] :
( ( member_o @ X4 @ B )
& ( ord_less_eq_o @ A5 @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_mono
thf(fact_797_Sup__least,axiom,
! [A: set_set_nat,Z: set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ X3 @ Z ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ Z ) ) ).
% Sup_least
thf(fact_798_Sup__least,axiom,
! [A: set_o,Z: $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_o @ X3 @ Z ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ Z ) ) ).
% Sup_least
thf(fact_799_Sup__upper,axiom,
! [X: set_nat,A: set_set_nat] :
( ( member_set_nat @ X @ A )
=> ( ord_less_eq_set_nat @ X @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% Sup_upper
thf(fact_800_Sup__upper,axiom,
! [X: $o,A: set_o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_o @ X @ ( complete_Sup_Sup_o @ A ) ) ) ).
% Sup_upper
thf(fact_801_Sup__le__iff,axiom,
! [A: set_set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ B2 )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ X2 @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_802_Sup__le__iff,axiom,
! [A: set_o,B2: $o] :
( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ B2 )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_o @ X2 @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_803_Sup__upper2,axiom,
! [U: set_nat,A: set_set_nat,V: set_nat] :
( ( member_set_nat @ U @ A )
=> ( ( ord_less_eq_set_nat @ V @ U )
=> ( ord_less_eq_set_nat @ V @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% Sup_upper2
thf(fact_804_Sup__upper2,axiom,
! [U: $o,A: set_o,V: $o] :
( ( member_o @ U @ A )
=> ( ( ord_less_eq_o @ V @ U )
=> ( ord_less_eq_o @ V @ ( complete_Sup_Sup_o @ A ) ) ) ) ).
% Sup_upper2
thf(fact_805_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
? [K2: nat] :
( N4
= ( plus_plus_nat @ M4 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_806_trans__le__add2,axiom,
! [I2: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_le_add2
thf(fact_807_trans__le__add1,axiom,
! [I2: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_le_add1
thf(fact_808_add__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_809_add__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_810_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N: nat] :
( L
= ( plus_plus_nat @ K @ N ) ) ) ).
% le_Suc_ex
thf(fact_811_add__leD2,axiom,
! [M3: nat,K: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N3 )
=> ( ord_less_eq_nat @ K @ N3 ) ) ).
% add_leD2
thf(fact_812_add__leD1,axiom,
! [M3: nat,K: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N3 )
=> ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% add_leD1
thf(fact_813_le__add2,axiom,
! [N3: nat,M3: nat] : ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ M3 @ N3 ) ) ).
% le_add2
thf(fact_814_le__add1,axiom,
! [N3: nat,M3: nat] : ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ N3 @ M3 ) ) ).
% le_add1
thf(fact_815_add__leE,axiom,
! [M3: nat,K: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N3 )
=> ~ ( ( ord_less_eq_nat @ M3 @ N3 )
=> ~ ( ord_less_eq_nat @ K @ N3 ) ) ) ).
% add_leE
thf(fact_816_Ioc__inj,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ( set_or6659071591806873216st_nat @ A2 @ B2 )
= ( set_or6659071591806873216st_nat @ C @ D2 ) )
= ( ( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ D2 @ C ) )
| ( ( A2 = C )
& ( B2 = D2 ) ) ) ) ).
% Ioc_inj
thf(fact_817_cSup__least,axiom,
! [X5: set_set_nat,Z: set_nat] :
( ( X5 != bot_bot_set_set_nat )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ X5 )
=> ( ord_less_eq_set_nat @ X3 @ Z ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_818_cSup__least,axiom,
! [X5: set_nat,Z: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ( ord_less_eq_nat @ X3 @ Z ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_819_cSup__least,axiom,
! [X5: set_o,Z: $o] :
( ( X5 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X5 )
=> ( ord_less_eq_o @ X3 @ Z ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ X5 ) @ Z ) ) ) ).
% cSup_least
thf(fact_820_cSup__eq__non__empty,axiom,
! [X5: set_set_nat,A2: set_nat] :
( ( X5 != bot_bot_set_set_nat )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ X5 )
=> ( ord_less_eq_set_nat @ X3 @ A2 ) )
=> ( ! [Y2: set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ X5 )
=> ( ord_less_eq_set_nat @ X4 @ Y2 ) )
=> ( ord_less_eq_set_nat @ A2 @ Y2 ) )
=> ( ( comple7399068483239264473et_nat @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_821_cSup__eq__non__empty,axiom,
! [X5: set_nat,A2: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ( ord_less_eq_nat @ X3 @ A2 ) )
=> ( ! [Y2: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ X5 )
=> ( ord_less_eq_nat @ X4 @ Y2 ) )
=> ( ord_less_eq_nat @ A2 @ Y2 ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_822_cSup__eq__non__empty,axiom,
! [X5: set_o,A2: $o] :
( ( X5 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X5 )
=> ( ord_less_eq_o @ X3 @ A2 ) )
=> ( ! [Y2: $o] :
( ! [X4: $o] :
( ( member_o @ X4 @ X5 )
=> ( ord_less_eq_o @ X4 @ Y2 ) )
=> ( ord_less_eq_o @ A2 @ Y2 ) )
=> ( ( complete_Sup_Sup_o @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_823_set__incr__altdef,axiom,
( hales_set_incr
= ( ^ [N4: nat] : ( image_nat_nat @ ( plus_plus_nat @ N4 ) ) ) ) ).
% set_incr_altdef
thf(fact_824_cSUP__least,axiom,
! [A: set_o,F: $o > nat,M2: nat] :
( ( A != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ M2 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ A ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_825_cSUP__least,axiom,
! [A: set_nat,F: nat > nat,M2: nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ M2 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_826_cSUP__least,axiom,
! [A: set_o,F: $o > $o,M2: $o] :
( ( A != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_o @ ( F @ X3 ) @ M2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_827_cSUP__least,axiom,
! [A: set_nat,F: nat > $o,M2: $o] :
( ( A != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_o @ ( F @ X3 ) @ M2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_828_cSUP__least,axiom,
! [A: set_o,F: $o > set_nat,M2: set_nat] :
( ( A != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ M2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_829_cSUP__least,axiom,
! [A: set_nat,F: nat > set_nat,M2: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ M2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_830_cSUP__least,axiom,
! [A: set_set_nat,F: set_nat > nat,M2: nat] :
( ( A != bot_bot_set_set_nat )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ M2 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_set_nat_nat @ F @ A ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_831_cSUP__least,axiom,
! [A: set_set_nat,F: set_nat > $o,M2: $o] :
( ( A != bot_bot_set_set_nat )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ord_less_eq_o @ ( F @ X3 ) @ M2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ A ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_832_cSUP__least,axiom,
! [A: set_set_nat,F: set_nat > set_nat,M2: set_nat] :
( ( A != bot_bot_set_set_nat )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ M2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_833_cSUP__least,axiom,
! [A: set_nat_nat,F: ( nat > nat ) > nat,M2: nat] :
( ( A != bot_bot_set_nat_nat )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ M2 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat_nat @ F @ A ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_834_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_835_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_836_add__nonpos__nonpos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_837_add__nonneg__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_838_add__increasing2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_839_add__decreasing2,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_840_add__increasing,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_841_add__decreasing,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_842_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_843_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_844_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_845_diff__add__0,axiom,
! [N3: nat,M3: nat] :
( ( minus_minus_nat @ N3 @ ( plus_plus_nat @ N3 @ M3 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_846_SUP__eq,axiom,
! [A: set_nat,B: set_nat,F: nat > $o,G: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_847_SUP__eq,axiom,
! [A: set_nat,B: set_o,F: nat > $o,G: $o > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X4: $o] :
( ( member_o @ X4 @ B )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_848_SUP__eq,axiom,
! [A: set_o,B: set_nat,F: $o > $o,G: nat > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_849_SUP__eq,axiom,
! [A: set_o,B: set_o,F: $o > $o,G: $o > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ? [X4: $o] :
( ( member_o @ X4 @ B )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_850_SUP__eq,axiom,
! [A: set_nat,B: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B )
& ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_set_nat @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_851_SUP__eq,axiom,
! [A: set_nat,B: set_o,F: nat > set_nat,G: $o > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X4: $o] :
( ( member_o @ X4 @ B )
& ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_set_nat @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_852_SUP__eq,axiom,
! [A: set_o,B: set_nat,F: $o > set_nat,G: nat > set_nat] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B )
& ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ( ord_less_eq_set_nat @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_853_SUP__eq,axiom,
! [A: set_o,B: set_o,F: $o > set_nat,G: $o > set_nat] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ? [X4: $o] :
( ( member_o @ X4 @ B )
& ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X4: $o] :
( ( member_o @ X4 @ A )
& ( ord_less_eq_set_nat @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_854_SUP__eq,axiom,
! [A: set_set_nat,B: set_nat,F: set_nat > $o,G: nat > $o] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_855_SUP__eq,axiom,
! [A: set_set_nat,B: set_o,F: set_nat > $o,G: $o > $o] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ A )
=> ? [X4: $o] :
( ( member_o @ X4 @ B )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_856_less__eq__Sup,axiom,
! [A: set_set_nat,U: set_nat] :
( ! [V2: set_nat] :
( ( member_set_nat @ V2 @ A )
=> ( ord_less_eq_set_nat @ U @ V2 ) )
=> ( ( A != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_857_less__eq__Sup,axiom,
! [A: set_o,U: $o] :
( ! [V2: $o] :
( ( member_o @ V2 @ A )
=> ( ord_less_eq_o @ U @ V2 ) )
=> ( ( A != bot_bot_set_o )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_858_SUP__eq__iff,axiom,
! [I: set_set_nat,C: set_nat,F: set_nat > set_nat] :
( ( I != bot_bot_set_set_nat )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I )
=> ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ I ) )
= C )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_859_SUP__eq__iff,axiom,
! [I: set_o,C: set_nat,F: $o > set_nat] :
( ( I != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I )
=> ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ I ) )
= C )
= ( ! [X2: $o] :
( ( member_o @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_860_SUP__eq__iff,axiom,
! [I: set_nat_nat,C: set_nat,F: ( nat > nat ) > set_nat] :
( ( I != bot_bot_set_nat_nat )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I )
=> ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ I ) )
= C )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_861_SUP__eq__iff,axiom,
! [I: set_nat,C: set_nat,F: nat > set_nat] :
( ( I != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I )
=> ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ I ) )
= C )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_862_SUP__eq__iff,axiom,
! [I: set_set_nat,C: $o,F: set_nat > $o] :
( ( I != bot_bot_set_set_nat )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I )
=> ( ord_less_eq_o @ C @ ( F @ I3 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ I ) )
= C )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_863_SUP__eq__iff,axiom,
! [I: set_o,C: $o,F: $o > $o] :
( ( I != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I )
=> ( ord_less_eq_o @ C @ ( F @ I3 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ I ) )
= C )
= ( ! [X2: $o] :
( ( member_o @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_864_SUP__eq__iff,axiom,
! [I: set_nat_nat,C: $o,F: ( nat > nat ) > $o] :
( ( I != bot_bot_set_nat_nat )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I )
=> ( ord_less_eq_o @ C @ ( F @ I3 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_nat_nat_o @ F @ I ) )
= C )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_865_SUP__eq__iff,axiom,
! [I: set_nat,C: $o,F: nat > $o] :
( ( I != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I )
=> ( ord_less_eq_o @ C @ ( F @ I3 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ I ) )
= C )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I )
=> ( ( F @ X2 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_866_Sup__inter__less__eq,axiom,
! [A: set_set_nat_nat,B: set_set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( inf_in710756014367367485at_nat @ A @ B ) ) @ ( inf_inf_set_nat_nat @ ( comple5448282615319421384at_nat @ A ) @ ( comple5448282615319421384at_nat @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_867_Sup__inter__less__eq,axiom,
! [A: set_set_o,B: set_set_o] : ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( inf_inf_set_set_o @ A @ B ) ) @ ( inf_inf_set_o @ ( comple90263536869209701_set_o @ A ) @ ( comple90263536869209701_set_o @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_868_Sup__inter__less__eq,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] : ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) ) @ ( inf_inf_set_set_nat @ ( comple548664676211718543et_nat @ A ) @ ( comple548664676211718543et_nat @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_869_Sup__inter__less__eq,axiom,
! [A: set_set_nat,B: set_set_nat] : ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( inf_inf_set_set_nat @ A @ B ) ) @ ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_870_Sup__inter__less__eq,axiom,
! [A: set_o,B: set_o] : ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( inf_inf_set_o @ A @ B ) ) @ ( inf_inf_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_871_Ioc__disjoint,axiom,
! [A2: $o,B2: $o,C: $o,D2: $o] :
( ( ( inf_inf_set_o @ ( set_or8254209520273421544Most_o @ A2 @ B2 ) @ ( set_or8254209520273421544Most_o @ C @ D2 ) )
= bot_bot_set_o )
= ( ( ord_less_eq_o @ B2 @ A2 )
| ( ord_less_eq_o @ D2 @ C )
| ( ord_less_eq_o @ B2 @ C )
| ( ord_less_eq_o @ D2 @ A2 ) ) ) ).
% Ioc_disjoint
thf(fact_872_Ioc__disjoint,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ( inf_inf_set_nat @ ( set_or6659071591806873216st_nat @ A2 @ B2 ) @ ( set_or6659071591806873216st_nat @ C @ D2 ) )
= bot_bot_set_nat )
= ( ( ord_less_eq_nat @ B2 @ A2 )
| ( ord_less_eq_nat @ D2 @ C )
| ( ord_less_eq_nat @ B2 @ C )
| ( ord_less_eq_nat @ D2 @ A2 ) ) ) ).
% Ioc_disjoint
thf(fact_873_Bstat__def,axiom,
( bstat
= ( sup_sup_set_nat @ ( hales_set_incr @ n2 @ ( bs @ k ) ) @ ( bl @ one_one_nat ) ) ) ).
% Bstat_def
thf(fact_874_fT__def,axiom,
( fT
= ( ^ [X2: nat] : ( if_nat @ ( member_nat @ X2 @ ( bl @ one_one_nat ) ) @ ( fL @ X2 ) @ ( if_nat @ ( member_nat @ X2 @ ( hales_set_incr @ n2 @ ( bs @ k ) ) ) @ ( fS @ ( minus_minus_nat @ X2 @ n2 ) ) @ undefined_nat ) ) ) ) ).
% fT_def
thf(fact_875_insert__partition,axiom,
! [X: set_nat_nat,F2: set_set_nat_nat] :
( ~ ( member_set_nat_nat @ X @ F2 )
=> ( ! [X3: set_nat_nat] :
( ( member_set_nat_nat @ X3 @ ( insert_set_nat_nat @ X @ F2 ) )
=> ! [Xa2: set_nat_nat] :
( ( member_set_nat_nat @ Xa2 @ ( insert_set_nat_nat @ X @ F2 ) )
=> ( ( X3 != Xa2 )
=> ( ( inf_inf_set_nat_nat @ X3 @ Xa2 )
= bot_bot_set_nat_nat ) ) ) )
=> ( ( inf_inf_set_nat_nat @ X @ ( comple5448282615319421384at_nat @ F2 ) )
= bot_bot_set_nat_nat ) ) ) ).
% insert_partition
thf(fact_876_insert__partition,axiom,
! [X: set_o,F2: set_set_o] :
( ~ ( member_set_o @ X @ F2 )
=> ( ! [X3: set_o] :
( ( member_set_o @ X3 @ ( insert_set_o @ X @ F2 ) )
=> ! [Xa2: set_o] :
( ( member_set_o @ Xa2 @ ( insert_set_o @ X @ F2 ) )
=> ( ( X3 != Xa2 )
=> ( ( inf_inf_set_o @ X3 @ Xa2 )
= bot_bot_set_o ) ) ) )
=> ( ( inf_inf_set_o @ X @ ( comple90263536869209701_set_o @ F2 ) )
= bot_bot_set_o ) ) ) ).
% insert_partition
thf(fact_877_insert__partition,axiom,
! [X: set_set_nat,F2: set_set_set_nat] :
( ~ ( member_set_set_nat @ X @ F2 )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ ( insert_set_set_nat @ X @ F2 ) )
=> ! [Xa2: set_set_nat] :
( ( member_set_set_nat @ Xa2 @ ( insert_set_set_nat @ X @ F2 ) )
=> ( ( X3 != Xa2 )
=> ( ( inf_inf_set_set_nat @ X3 @ Xa2 )
= bot_bot_set_set_nat ) ) ) )
=> ( ( inf_inf_set_set_nat @ X @ ( comple548664676211718543et_nat @ F2 ) )
= bot_bot_set_set_nat ) ) ) ).
% insert_partition
thf(fact_878_insert__partition,axiom,
! [X: set_nat,F2: set_set_nat] :
( ~ ( member_set_nat @ X @ F2 )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( insert_set_nat @ X @ F2 ) )
=> ! [Xa2: set_nat] :
( ( member_set_nat @ Xa2 @ ( insert_set_nat @ X @ F2 ) )
=> ( ( X3 != Xa2 )
=> ( ( inf_inf_set_nat @ X3 @ Xa2 )
= bot_bot_set_nat ) ) ) )
=> ( ( inf_inf_set_nat @ X @ ( comple7399068483239264473et_nat @ F2 ) )
= bot_bot_set_nat ) ) ) ).
% insert_partition
thf(fact_879_inf__Sup,axiom,
! [A2: set_nat_nat,B: set_set_nat_nat] :
( ( inf_inf_set_nat_nat @ A2 @ ( comple5448282615319421384at_nat @ B ) )
= ( comple5448282615319421384at_nat @ ( image_3832368097948589297at_nat @ ( inf_inf_set_nat_nat @ A2 ) @ B ) ) ) ).
% inf_Sup
thf(fact_880_inf__Sup,axiom,
! [A2: set_o,B: set_set_o] :
( ( inf_inf_set_o @ A2 @ ( comple90263536869209701_set_o @ B ) )
= ( comple90263536869209701_set_o @ ( image_set_o_set_o @ ( inf_inf_set_o @ A2 ) @ B ) ) ) ).
% inf_Sup
thf(fact_881_inf__Sup,axiom,
! [A2: set_set_nat,B: set_set_set_nat] :
( ( inf_inf_set_set_nat @ A2 @ ( comple548664676211718543et_nat @ B ) )
= ( comple548664676211718543et_nat @ ( image_7884819252390400639et_nat @ ( inf_inf_set_set_nat @ A2 ) @ B ) ) ) ).
% inf_Sup
thf(fact_882_inf__Sup,axiom,
! [A2: set_nat,B: set_set_nat] :
( ( inf_inf_set_nat @ A2 @ ( comple7399068483239264473et_nat @ B ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( inf_inf_set_nat @ A2 ) @ B ) ) ) ).
% inf_Sup
thf(fact_883_inf__Sup,axiom,
! [A2: $o,B: set_o] :
( ( inf_inf_o @ A2 @ ( complete_Sup_Sup_o @ B ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ ( inf_inf_o @ A2 ) @ B ) ) ) ).
% inf_Sup
thf(fact_884_Sup__inf__eq__bot__iff,axiom,
! [B: set_set_nat_nat,A2: set_nat_nat] :
( ( ( inf_inf_set_nat_nat @ ( comple5448282615319421384at_nat @ B ) @ A2 )
= bot_bot_set_nat_nat )
= ( ! [X2: set_nat_nat] :
( ( member_set_nat_nat @ X2 @ B )
=> ( ( inf_inf_set_nat_nat @ X2 @ A2 )
= bot_bot_set_nat_nat ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_885_Sup__inf__eq__bot__iff,axiom,
! [B: set_set_o,A2: set_o] :
( ( ( inf_inf_set_o @ ( comple90263536869209701_set_o @ B ) @ A2 )
= bot_bot_set_o )
= ( ! [X2: set_o] :
( ( member_set_o @ X2 @ B )
=> ( ( inf_inf_set_o @ X2 @ A2 )
= bot_bot_set_o ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_886_Sup__inf__eq__bot__iff,axiom,
! [B: set_set_set_nat,A2: set_set_nat] :
( ( ( inf_inf_set_set_nat @ ( comple548664676211718543et_nat @ B ) @ A2 )
= bot_bot_set_set_nat )
= ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ B )
=> ( ( inf_inf_set_set_nat @ X2 @ A2 )
= bot_bot_set_set_nat ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_887_Sup__inf__eq__bot__iff,axiom,
! [B: set_set_nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ B ) @ A2 )
= bot_bot_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ B )
=> ( ( inf_inf_set_nat @ X2 @ A2 )
= bot_bot_set_nat ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_888_Sup__inf__eq__bot__iff,axiom,
! [B: set_o,A2: $o] :
( ( ( inf_inf_o @ ( complete_Sup_Sup_o @ B ) @ A2 )
= bot_bot_o )
= ( ! [X2: $o] :
( ( member_o @ X2 @ B )
=> ( ( inf_inf_o @ X2 @ A2 )
= bot_bot_o ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_889_fact2,axiom,
( ( inf_inf_set_nat @ ( bl @ zero_zero_nat )
@ ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I4: nat] : ( hales_set_incr @ n2 @ ( bs @ I4 ) )
@ ( set_ord_lessThan_nat @ k ) ) ) )
= bot_bot_set_nat ) ).
% fact2
thf(fact_890_Diff__empty,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Diff_empty
thf(fact_891_empty__Diff,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_892_Diff__cancel,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ A )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_893_subsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( member_set_nat @ X3 @ B ) )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% subsetI
thf(fact_894_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_895_subsetI,axiom,
! [A: set_o,B: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_o @ X3 @ B ) )
=> ( ord_less_eq_set_o @ A @ B ) ) ).
% subsetI
thf(fact_896_subsetI,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ( member_nat_nat @ X3 @ B ) )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% subsetI
thf(fact_897_Diff__insert0,axiom,
! [X: set_nat,A: set_set_nat,B: set_set_nat] :
( ~ ( member_set_nat @ X @ A )
=> ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ B ) )
= ( minus_2163939370556025621et_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_898_Diff__insert0,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ B ) )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_899_Diff__insert0,axiom,
! [X: $o,A: set_o,B: set_o] :
( ~ ( member_o @ X @ A )
=> ( ( minus_minus_set_o @ A @ ( insert_o @ X @ B ) )
= ( minus_minus_set_o @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_900_Diff__insert0,axiom,
! [X: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ~ ( member_nat_nat @ X @ A )
=> ( ( minus_8121590178497047118at_nat @ A @ ( insert_nat_nat @ X @ B ) )
= ( minus_8121590178497047118at_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_901_insert__Diff1,axiom,
! [X: set_nat,B: set_set_nat,A: set_set_nat] :
( ( member_set_nat @ X @ B )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A ) @ B )
= ( minus_2163939370556025621et_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_902_insert__Diff1,axiom,
! [X: nat,B: set_nat,A: set_nat] :
( ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_903_insert__Diff1,axiom,
! [X: $o,B: set_o,A: set_o] :
( ( member_o @ X @ B )
=> ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ B )
= ( minus_minus_set_o @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_904_insert__Diff1,axiom,
! [X: nat > nat,B: set_nat_nat,A: set_nat_nat] :
( ( member_nat_nat @ X @ B )
=> ( ( minus_8121590178497047118at_nat @ ( insert_nat_nat @ X @ A ) @ B )
= ( minus_8121590178497047118at_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_905_sup_Oidem,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_906_sup__idem,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_907_sup_Oleft__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_908_sup__left__idem,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
= ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_909_sup_Oright__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_910_Un__Diff__cancel2,axiom,
! [B: set_nat,A: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B @ A ) @ A )
= ( sup_sup_set_nat @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_911_Un__Diff__cancel,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
= ( sup_sup_set_nat @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_912_UnCI,axiom,
! [C: set_nat,B: set_set_nat,A: set_set_nat] :
( ( ~ ( member_set_nat @ C @ B )
=> ( member_set_nat @ C @ A ) )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_913_UnCI,axiom,
! [C: $o,B: set_o,A: set_o] :
( ( ~ ( member_o @ C @ B )
=> ( member_o @ C @ A ) )
=> ( member_o @ C @ ( sup_sup_set_o @ A @ B ) ) ) ).
% UnCI
thf(fact_914_UnCI,axiom,
! [C: nat > nat,B: set_nat_nat,A: set_nat_nat] :
( ( ~ ( member_nat_nat @ C @ B )
=> ( member_nat_nat @ C @ A ) )
=> ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_915_UnCI,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ A ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_916_Un__iff,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) )
= ( ( member_set_nat @ C @ A )
| ( member_set_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_917_Un__iff,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( sup_sup_set_o @ A @ B ) )
= ( ( member_o @ C @ A )
| ( member_o @ C @ B ) ) ) ).
% Un_iff
thf(fact_918_Un__iff,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A @ B ) )
= ( ( member_nat_nat @ C @ A )
| ( member_nat_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_919_Un__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
| ( member_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_920_image__ident,axiom,
! [Y6: set_nat] :
( ( image_nat_nat
@ ^ [X2: nat] : X2
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_921_le__sup__iff,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_set_nat @ X @ Z )
& ( ord_less_eq_set_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_922_le__sup__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_923_sup_Obounded__iff,axiom,
! [B2: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_set_nat @ B2 @ A2 )
& ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_924_sup_Obounded__iff,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_925_Diff__eq__empty__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( minus_minus_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_926_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_927_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_928_sup__bot__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_929_sup__bot__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% sup_bot_right
thf(fact_930_bot__eq__sup__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X @ Y ) )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_931_sup__eq__bot__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_932_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_933_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_934_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_935_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_936_insert__Diff__single,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= ( insert_nat @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_937_insert__subset,axiom,
! [X: set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X @ A ) @ B )
= ( ( member_set_nat @ X @ B )
& ( ord_le6893508408891458716et_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_938_insert__subset,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A ) @ B )
= ( ( member_nat @ X @ B )
& ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_939_insert__subset,axiom,
! [X: $o,A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ ( insert_o @ X @ A ) @ B )
= ( ( member_o @ X @ B )
& ( ord_less_eq_set_o @ A @ B ) ) ) ).
% insert_subset
thf(fact_940_insert__subset,axiom,
! [X: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( insert_nat_nat @ X @ A ) @ B )
= ( ( member_nat_nat @ X @ B )
& ( ord_le9059583361652607317at_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_941_inf__sup__absorb,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_942_inf__sup__absorb,axiom,
! [X: set_o,Y: set_o] :
( ( inf_inf_set_o @ X @ ( sup_sup_set_o @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_943_inf__sup__absorb,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( inf_inf_set_set_nat @ X @ ( sup_sup_set_set_nat @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_944_inf__sup__absorb,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_945_sup__inf__absorb,axiom,
! [X: set_nat_nat,Y: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_946_sup__inf__absorb,axiom,
! [X: set_o,Y: set_o] :
( ( sup_sup_set_o @ X @ ( inf_inf_set_o @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_947_sup__inf__absorb,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( sup_sup_set_set_nat @ X @ ( inf_inf_set_set_nat @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_948_sup__inf__absorb,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_949_Diff__disjoint,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A @ ( minus_8121590178497047118at_nat @ B @ A ) )
= bot_bot_set_nat_nat ) ).
% Diff_disjoint
thf(fact_950_Diff__disjoint,axiom,
! [A: set_o,B: set_o] :
( ( inf_inf_set_o @ A @ ( minus_minus_set_o @ B @ A ) )
= bot_bot_set_o ) ).
% Diff_disjoint
thf(fact_951_Diff__disjoint,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( inf_inf_set_set_nat @ A @ ( minus_2163939370556025621et_nat @ B @ A ) )
= bot_bot_set_set_nat ) ).
% Diff_disjoint
thf(fact_952_Diff__disjoint,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_953_Un__empty,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_954_Int__subset__iff,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
= ( ( ord_less_eq_set_nat @ C2 @ A )
& ( ord_less_eq_set_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_955_Int__subset__iff,axiom,
! [C2: set_nat_nat,A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ ( inf_inf_set_nat_nat @ A @ B ) )
= ( ( ord_le9059583361652607317at_nat @ C2 @ A )
& ( ord_le9059583361652607317at_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_956_Int__subset__iff,axiom,
! [C2: set_o,A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ C2 @ ( inf_inf_set_o @ A @ B ) )
= ( ( ord_less_eq_set_o @ C2 @ A )
& ( ord_less_eq_set_o @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_957_Int__subset__iff,axiom,
! [C2: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ ( inf_inf_set_set_nat @ A @ B ) )
= ( ( ord_le6893508408891458716et_nat @ C2 @ A )
& ( ord_le6893508408891458716et_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_958_Un__subset__iff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_nat @ A @ C2 )
& ( ord_less_eq_set_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_959_Un__insert__left,axiom,
! [A2: nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( insert_nat @ A2 @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_960_Un__insert__right,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_961_Int__Un__eq_I4_J,axiom,
! [T: set_nat_nat,S: set_nat_nat] :
( ( sup_sup_set_nat_nat @ T @ ( inf_inf_set_nat_nat @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_962_Int__Un__eq_I4_J,axiom,
! [T: set_o,S: set_o] :
( ( sup_sup_set_o @ T @ ( inf_inf_set_o @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_963_Int__Un__eq_I4_J,axiom,
! [T: set_set_nat,S: set_set_nat] :
( ( sup_sup_set_set_nat @ T @ ( inf_inf_set_set_nat @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_964_Int__Un__eq_I4_J,axiom,
! [T: set_nat,S: set_nat] :
( ( sup_sup_set_nat @ T @ ( inf_inf_set_nat @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_965_Int__Un__eq_I3_J,axiom,
! [S: set_nat_nat,T: set_nat_nat] :
( ( sup_sup_set_nat_nat @ S @ ( inf_inf_set_nat_nat @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_966_Int__Un__eq_I3_J,axiom,
! [S: set_o,T: set_o] :
( ( sup_sup_set_o @ S @ ( inf_inf_set_o @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_967_Int__Un__eq_I3_J,axiom,
! [S: set_set_nat,T: set_set_nat] :
( ( sup_sup_set_set_nat @ S @ ( inf_inf_set_set_nat @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_968_Int__Un__eq_I3_J,axiom,
! [S: set_nat,T: set_nat] :
( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_969_Int__Un__eq_I2_J,axiom,
! [S: set_nat_nat,T: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_970_Int__Un__eq_I2_J,axiom,
! [S: set_o,T: set_o] :
( ( sup_sup_set_o @ ( inf_inf_set_o @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_971_Int__Un__eq_I2_J,axiom,
! [S: set_set_nat,T: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_972_Int__Un__eq_I2_J,axiom,
! [S: set_nat,T: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_973_Int__Un__eq_I1_J,axiom,
! [S: set_nat_nat,T: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_974_Int__Un__eq_I1_J,axiom,
! [S: set_o,T: set_o] :
( ( sup_sup_set_o @ ( inf_inf_set_o @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_975_Int__Un__eq_I1_J,axiom,
! [S: set_set_nat,T: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_976_Int__Un__eq_I1_J,axiom,
! [S: set_nat,T: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_977_Un__Int__eq_I4_J,axiom,
! [T: set_nat_nat,S: set_nat_nat] :
( ( inf_inf_set_nat_nat @ T @ ( sup_sup_set_nat_nat @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_978_Un__Int__eq_I4_J,axiom,
! [T: set_o,S: set_o] :
( ( inf_inf_set_o @ T @ ( sup_sup_set_o @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_979_Un__Int__eq_I4_J,axiom,
! [T: set_set_nat,S: set_set_nat] :
( ( inf_inf_set_set_nat @ T @ ( sup_sup_set_set_nat @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_980_Un__Int__eq_I4_J,axiom,
! [T: set_nat,S: set_nat] :
( ( inf_inf_set_nat @ T @ ( sup_sup_set_nat @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_981_Un__Int__eq_I3_J,axiom,
! [S: set_nat_nat,T: set_nat_nat] :
( ( inf_inf_set_nat_nat @ S @ ( sup_sup_set_nat_nat @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_982_Un__Int__eq_I3_J,axiom,
! [S: set_o,T: set_o] :
( ( inf_inf_set_o @ S @ ( sup_sup_set_o @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_983_Un__Int__eq_I3_J,axiom,
! [S: set_set_nat,T: set_set_nat] :
( ( inf_inf_set_set_nat @ S @ ( sup_sup_set_set_nat @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_984_Un__Int__eq_I3_J,axiom,
! [S: set_nat,T: set_nat] :
( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_985_Un__Int__eq_I2_J,axiom,
! [S: set_nat_nat,T: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_986_Un__Int__eq_I2_J,axiom,
! [S: set_o,T: set_o] :
( ( inf_inf_set_o @ ( sup_sup_set_o @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_987_Un__Int__eq_I2_J,axiom,
! [S: set_set_nat,T: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_988_Un__Int__eq_I2_J,axiom,
! [S: set_nat,T: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_989_Un__Int__eq_I1_J,axiom,
! [S: set_nat_nat,T: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_990_Un__Int__eq_I1_J,axiom,
! [S: set_o,T: set_o] :
( ( inf_inf_set_o @ ( sup_sup_set_o @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_991_Un__Int__eq_I1_J,axiom,
! [S: set_set_nat,T: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_992_Un__Int__eq_I1_J,axiom,
! [S: set_nat,T: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_993_Union__Un__distrib,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_Un_distrib
thf(fact_994_SUP__identity__eq,axiom,
! [A: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [X2: set_nat] : X2
@ A ) )
= ( comple7399068483239264473et_nat @ A ) ) ).
% SUP_identity_eq
thf(fact_995_SUP__identity__eq,axiom,
! [A: set_nat] :
( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A ) )
= ( complete_Sup_Sup_nat @ A ) ) ).
% SUP_identity_eq
thf(fact_996_SUP__identity__eq,axiom,
! [A: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X2: $o] : X2
@ A ) )
= ( complete_Sup_Sup_o @ A ) ) ).
% SUP_identity_eq
thf(fact_997_singleton__conv2,axiom,
! [A2: nat] :
( ( collect_nat
@ ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 )
@ A2 ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_998_singleton__conv,axiom,
! [A2: nat] :
( ( collect_nat
@ ^ [X2: nat] : ( X2 = A2 ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_999_UN__iff,axiom,
! [B2: nat,B: nat > set_nat,A: set_nat] :
( ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( member_nat @ B2 @ ( B @ X2 ) ) ) ) ) ).
% UN_iff
thf(fact_1000_UN__I,axiom,
! [A2: nat,A: set_nat,B2: $o,B: nat > set_o] :
( ( member_nat @ A2 @ A )
=> ( ( member_o @ B2 @ ( B @ A2 ) )
=> ( member_o @ B2 @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_1001_UN__I,axiom,
! [A2: $o,A: set_o,B2: $o,B: $o > set_o] :
( ( member_o @ A2 @ A )
=> ( ( member_o @ B2 @ ( B @ A2 ) )
=> ( member_o @ B2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_1002_UN__I,axiom,
! [A2: nat,A: set_nat,B2: nat,B: nat > set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( member_nat @ B2 @ ( B @ A2 ) )
=> ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_1003_UN__I,axiom,
! [A2: $o,A: set_o,B2: nat,B: $o > set_nat] :
( ( member_o @ A2 @ A )
=> ( ( member_nat @ B2 @ ( B @ A2 ) )
=> ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_1004_UN__I,axiom,
! [A2: set_nat,A: set_set_nat,B2: $o,B: set_nat > set_o] :
( ( member_set_nat @ A2 @ A )
=> ( ( member_o @ B2 @ ( B @ A2 ) )
=> ( member_o @ B2 @ ( comple90263536869209701_set_o @ ( image_set_nat_set_o @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_1005_UN__I,axiom,
! [A2: nat,A: set_nat,B2: set_nat,B: nat > set_set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( member_set_nat @ B2 @ ( B @ A2 ) )
=> ( member_set_nat @ B2 @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_1006_UN__I,axiom,
! [A2: $o,A: set_o,B2: set_nat,B: $o > set_set_nat] :
( ( member_o @ A2 @ A )
=> ( ( member_set_nat @ B2 @ ( B @ A2 ) )
=> ( member_set_nat @ B2 @ ( comple548664676211718543et_nat @ ( image_o_set_set_nat @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_1007_UN__I,axiom,
! [A2: set_nat,A: set_set_nat,B2: nat,B: set_nat > set_nat] :
( ( member_set_nat @ A2 @ A )
=> ( ( member_nat @ B2 @ ( B @ A2 ) )
=> ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_1008_UN__I,axiom,
! [A2: set_nat,A: set_set_nat,B2: set_nat,B: set_nat > set_set_nat] :
( ( member_set_nat @ A2 @ A )
=> ( ( member_set_nat @ B2 @ ( B @ A2 ) )
=> ( member_set_nat @ B2 @ ( comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_1009_UN__I,axiom,
! [A2: nat,A: set_nat,B2: nat > nat,B: nat > set_nat_nat] :
( ( member_nat @ A2 @ A )
=> ( ( member_nat_nat @ B2 @ ( B @ A2 ) )
=> ( member_nat_nat @ B2 @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_1010_singleton__insert__inj__eq_H,axiom,
! [A2: nat,A: set_nat,B2: nat] :
( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1011_singleton__insert__inj__eq,axiom,
! [B2: nat,A2: nat,A: set_nat] :
( ( ( insert_nat @ B2 @ bot_bot_set_nat )
= ( insert_nat @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1012_Sup__insert,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A2 @ A ) )
= ( sup_sup_set_nat @ A2 @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% Sup_insert
thf(fact_1013_Sup__insert,axiom,
! [A2: $o,A: set_o] :
( ( complete_Sup_Sup_o @ ( insert_o @ A2 @ A ) )
= ( sup_sup_o @ A2 @ ( complete_Sup_Sup_o @ A ) ) ) ).
% Sup_insert
thf(fact_1014_single__Diff__lessThan,axiom,
! [K: nat] :
( ( minus_minus_set_nat @ ( insert_nat @ K @ bot_bot_set_nat ) @ ( set_ord_lessThan_nat @ K ) )
= ( insert_nat @ K @ bot_bot_set_nat ) ) ).
% single_Diff_lessThan
thf(fact_1015_SUP__bot__conv_I2_J,axiom,
! [B: nat > set_nat,A: set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( B @ X2 )
= bot_bot_set_nat ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_1016_SUP__bot__conv_I1_J,axiom,
! [B: nat > set_nat,A: set_nat] :
( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( B @ X2 )
= bot_bot_set_nat ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_1017_SUP__bot,axiom,
! [A: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : bot_bot_set_nat
@ A ) )
= bot_bot_set_nat ) ).
% SUP_bot
thf(fact_1018_SUP__const,axiom,
! [A: set_nat,F: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I4: nat] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_1019_SUP__const,axiom,
! [A: set_nat,F: $o] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [I4: nat] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_1020_cSUP__const,axiom,
! [A: set_nat,C: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : C
@ A ) )
= C ) ) ).
% cSUP_const
thf(fact_1021_cSUP__const,axiom,
! [A: set_nat,C: nat] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X2: nat] : C
@ A ) )
= C ) ) ).
% cSUP_const
thf(fact_1022_cSUP__const,axiom,
! [A: set_nat,C: $o] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [X2: nat] : C
@ A ) )
= C ) ) ).
% cSUP_const
thf(fact_1023_if__image__distrib,axiom,
! [P: nat > $o,F: nat > set_nat,G: nat > set_nat,S: set_nat] :
( ( image_nat_set_nat
@ ^ [X2: nat] : ( if_set_nat @ ( P @ X2 ) @ ( F @ X2 ) @ ( G @ X2 ) )
@ S )
= ( sup_sup_set_set_nat @ ( image_nat_set_nat @ F @ ( inf_inf_set_nat @ S @ ( collect_nat @ P ) ) )
@ ( image_nat_set_nat @ G
@ ( inf_inf_set_nat @ S
@ ( collect_nat
@ ^ [X2: nat] :
~ ( P @ X2 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1024_if__image__distrib,axiom,
! [P: nat > $o,F: nat > nat,G: nat > nat,S: set_nat] :
( ( image_nat_nat
@ ^ [X2: nat] : ( if_nat @ ( P @ X2 ) @ ( F @ X2 ) @ ( G @ X2 ) )
@ S )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ ( inf_inf_set_nat @ S @ ( collect_nat @ P ) ) )
@ ( image_nat_nat @ G
@ ( inf_inf_set_nat @ S
@ ( collect_nat
@ ^ [X2: nat] :
~ ( P @ X2 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1025_if__image__distrib,axiom,
! [P: ( nat > nat ) > $o,F: ( nat > nat ) > nat,G: ( nat > nat ) > nat,S: set_nat_nat] :
( ( image_nat_nat_nat
@ ^ [X2: nat > nat] : ( if_nat @ ( P @ X2 ) @ ( F @ X2 ) @ ( G @ X2 ) )
@ S )
= ( sup_sup_set_nat @ ( image_nat_nat_nat @ F @ ( inf_inf_set_nat_nat @ S @ ( collect_nat_nat @ P ) ) )
@ ( image_nat_nat_nat @ G
@ ( inf_inf_set_nat_nat @ S
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
~ ( P @ X2 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1026_if__image__distrib,axiom,
! [P: $o > $o,F: $o > nat,G: $o > nat,S: set_o] :
( ( image_o_nat
@ ^ [X2: $o] : ( if_nat @ ( P @ X2 ) @ ( F @ X2 ) @ ( G @ X2 ) )
@ S )
= ( sup_sup_set_nat @ ( image_o_nat @ F @ ( inf_inf_set_o @ S @ ( collect_o @ P ) ) )
@ ( image_o_nat @ G
@ ( inf_inf_set_o @ S
@ ( collect_o
@ ^ [X2: $o] :
~ ( P @ X2 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1027_if__image__distrib,axiom,
! [P: set_nat > $o,F: set_nat > nat,G: set_nat > nat,S: set_set_nat] :
( ( image_set_nat_nat
@ ^ [X2: set_nat] : ( if_nat @ ( P @ X2 ) @ ( F @ X2 ) @ ( G @ X2 ) )
@ S )
= ( sup_sup_set_nat @ ( image_set_nat_nat @ F @ ( inf_inf_set_set_nat @ S @ ( collect_set_nat @ P ) ) )
@ ( image_set_nat_nat @ G
@ ( inf_inf_set_set_nat @ S
@ ( collect_set_nat
@ ^ [X2: set_nat] :
~ ( P @ X2 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1028_UN__constant,axiom,
! [A: set_nat,C: set_nat] :
( ( ( A = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y3: nat] : C
@ A ) )
= bot_bot_set_nat ) )
& ( ( A != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y3: nat] : C
@ A ) )
= C ) ) ) ).
% UN_constant
thf(fact_1029_UN__Un,axiom,
! [M2: nat > set_nat,A: set_nat,B: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M2 @ ( sup_sup_set_nat @ A @ B ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M2 @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M2 @ B ) ) ) ) ).
% UN_Un
thf(fact_1030_UN__simps_I1_J,axiom,
! [C2: set_nat,A2: nat,B: nat > set_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( insert_nat @ A2 @ ( B @ X2 ) )
@ C2 ) )
= bot_bot_set_nat ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( insert_nat @ A2 @ ( B @ X2 ) )
@ C2 ) )
= ( insert_nat @ A2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_1031_UN__singleton,axiom,
! [A: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( insert_nat @ X2 @ bot_bot_set_nat )
@ A ) )
= A ) ).
% UN_singleton
thf(fact_1032_UN__simps_I3_J,axiom,
! [C2: set_nat,A: set_nat,B: nat > set_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( sup_sup_set_nat @ A @ ( B @ X2 ) )
@ C2 ) )
= bot_bot_set_nat ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( sup_sup_set_nat @ A @ ( B @ X2 ) )
@ C2 ) )
= ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ C2 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1033_UN__simps_I2_J,axiom,
! [C2: set_nat,A: nat > set_nat,B: set_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( sup_sup_set_nat @ ( A @ X2 ) @ B )
@ C2 ) )
= bot_bot_set_nat ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( sup_sup_set_nat @ ( A @ X2 ) @ B )
@ C2 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ C2 ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_1034_UN__insert,axiom,
! [B: nat > set_nat,A2: nat,A: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ ( insert_nat @ A2 @ A ) ) )
= ( sup_sup_set_nat @ ( B @ A2 ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) ) ) ).
% UN_insert
thf(fact_1035_Sup__set__def,axiom,
( comple548664676211718543et_nat
= ( ^ [A4: set_set_set_nat] :
( collect_set_nat
@ ^ [X2: set_nat] : ( complete_Sup_Sup_o @ ( image_set_set_nat_o @ ( member_set_nat @ X2 ) @ A4 ) ) ) ) ) ).
% Sup_set_def
thf(fact_1036_Sup__set__def,axiom,
( comple90263536869209701_set_o
= ( ^ [A4: set_set_o] :
( collect_o
@ ^ [X2: $o] : ( complete_Sup_Sup_o @ ( image_set_o_o @ ( member_o @ X2 ) @ A4 ) ) ) ) ) ).
% Sup_set_def
thf(fact_1037_Sup__set__def,axiom,
( comple5448282615319421384at_nat
= ( ^ [A4: set_set_nat_nat] :
( collect_nat_nat
@ ^ [X2: nat > nat] : ( complete_Sup_Sup_o @ ( image_set_nat_nat_o @ ( member_nat_nat @ X2 ) @ A4 ) ) ) ) ) ).
% Sup_set_def
thf(fact_1038_Sup__set__def,axiom,
( comple7399068483239264473et_nat
= ( ^ [A4: set_set_nat] :
( collect_nat
@ ^ [X2: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X2 ) @ A4 ) ) ) ) ) ).
% Sup_set_def
thf(fact_1039_SUP__Sup__eq,axiom,
! [S: set_set_set_nat] :
( ( comple3806919086088850358_nat_o
@ ( image_4331731847045299910_nat_o
@ ^ [I4: set_set_nat,X2: set_nat] : ( member_set_nat @ X2 @ I4 )
@ S ) )
= ( ^ [X2: set_nat] : ( member_set_nat @ X2 @ ( comple548664676211718543et_nat @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1040_SUP__Sup__eq,axiom,
! [S: set_set_o] :
( ( complete_Sup_Sup_o_o
@ ( image_set_o_o_o
@ ^ [I4: set_o,X2: $o] : ( member_o @ X2 @ I4 )
@ S ) )
= ( ^ [X2: $o] : ( member_o @ X2 @ ( comple90263536869209701_set_o @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1041_SUP__Sup__eq,axiom,
! [S: set_set_nat_nat] :
( ( comple8312177224774716605_nat_o
@ ( image_1242417779249009364_nat_o
@ ^ [I4: set_nat_nat,X2: nat > nat] : ( member_nat_nat @ X2 @ I4 )
@ S ) )
= ( ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ ( comple5448282615319421384at_nat @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1042_SUP__Sup__eq,axiom,
! [S: set_set_nat] :
( ( comple8317665133742190828_nat_o
@ ( image_set_nat_nat_o2
@ ^ [I4: set_nat,X2: nat] : ( member_nat @ X2 @ I4 )
@ S ) )
= ( ^ [X2: nat] : ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1043_complete__lattice__class_OSUP__sup__distrib,axiom,
! [F: nat > set_nat,A: set_nat,G: nat > set_nat] :
( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A3: nat] : ( sup_sup_set_nat @ ( F @ A3 ) @ ( G @ A3 ) )
@ A ) ) ) ).
% complete_lattice_class.SUP_sup_distrib
thf(fact_1044_SUP__absorb,axiom,
! [K: set_nat,I: set_set_nat,A: set_nat > set_nat] :
( ( member_set_nat @ K @ I )
=> ( ( sup_sup_set_nat @ ( A @ K ) @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ A @ I ) ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ A @ I ) ) ) ) ).
% SUP_absorb
thf(fact_1045_SUP__absorb,axiom,
! [K: nat,I: set_nat,A: nat > set_nat] :
( ( member_nat @ K @ I )
=> ( ( sup_sup_set_nat @ ( A @ K ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ I ) ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ I ) ) ) ) ).
% SUP_absorb
thf(fact_1046_SUP__absorb,axiom,
! [K: $o,I: set_o,A: $o > set_nat] :
( ( member_o @ K @ I )
=> ( ( sup_sup_set_nat @ ( A @ K ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A @ I ) ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A @ I ) ) ) ) ).
% SUP_absorb
thf(fact_1047_SUP__absorb,axiom,
! [K: nat > nat,I: set_nat_nat,A: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ K @ I )
=> ( ( sup_sup_set_nat @ ( A @ K ) @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ A @ I ) ) )
= ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ A @ I ) ) ) ) ).
% SUP_absorb
thf(fact_1048_SUP__absorb,axiom,
! [K: set_nat,I: set_set_nat,A: set_nat > $o] :
( ( member_set_nat @ K @ I )
=> ( ( sup_sup_o @ ( A @ K ) @ ( complete_Sup_Sup_o @ ( image_set_nat_o @ A @ I ) ) )
= ( complete_Sup_Sup_o @ ( image_set_nat_o @ A @ I ) ) ) ) ).
% SUP_absorb
thf(fact_1049_SUP__absorb,axiom,
! [K: nat,I: set_nat,A: nat > $o] :
( ( member_nat @ K @ I )
=> ( ( sup_sup_o @ ( A @ K ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ A @ I ) ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ A @ I ) ) ) ) ).
% SUP_absorb
thf(fact_1050_SUP__absorb,axiom,
! [K: $o,I: set_o,A: $o > $o] :
( ( member_o @ K @ I )
=> ( ( sup_sup_o @ ( A @ K ) @ ( complete_Sup_Sup_o @ ( image_o_o @ A @ I ) ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ A @ I ) ) ) ) ).
% SUP_absorb
thf(fact_1051_SUP__absorb,axiom,
! [K: nat > nat,I: set_nat_nat,A: ( nat > nat ) > $o] :
( ( member_nat_nat @ K @ I )
=> ( ( sup_sup_o @ ( A @ K ) @ ( complete_Sup_Sup_o @ ( image_nat_nat_o @ A @ I ) ) )
= ( complete_Sup_Sup_o @ ( image_nat_nat_o @ A @ I ) ) ) ) ).
% SUP_absorb
thf(fact_1052_SUP__union,axiom,
! [M2: nat > set_nat,A: set_nat,B: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M2 @ ( sup_sup_set_nat @ A @ B ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M2 @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M2 @ B ) ) ) ) ).
% SUP_union
thf(fact_1053_SUP__union,axiom,
! [M2: nat > $o,A: set_nat,B: set_nat] :
( ( complete_Sup_Sup_o @ ( image_nat_o @ M2 @ ( sup_sup_set_nat @ A @ B ) ) )
= ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ M2 @ A ) ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ M2 @ B ) ) ) ) ).
% SUP_union
thf(fact_1054_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat > nat,A: set_nat] :
( ( Sup
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_identity_eq
thf(fact_1055_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat > nat,A: set_nat] :
( ( Inf
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_identity_eq
thf(fact_1056_insert__def,axiom,
( insert_nat
= ( ^ [A3: nat] :
( sup_sup_set_nat
@ ( collect_nat
@ ^ [X2: nat] : ( X2 = A3 ) ) ) ) ) ).
% insert_def
thf(fact_1057_insert__compr,axiom,
( insert_set_nat
= ( ^ [A3: set_nat,B4: set_set_nat] :
( collect_set_nat
@ ^ [X2: set_nat] :
( ( X2 = A3 )
| ( member_set_nat @ X2 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_1058_insert__compr,axiom,
( insert_nat
= ( ^ [A3: nat,B4: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( X2 = A3 )
| ( member_nat @ X2 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_1059_insert__compr,axiom,
( insert_o
= ( ^ [A3: $o,B4: set_o] :
( collect_o
@ ^ [X2: $o] :
( ( X2 = A3 )
| ( member_o @ X2 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_1060_insert__compr,axiom,
( insert_nat_nat
= ( ^ [A3: nat > nat,B4: set_nat_nat] :
( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( X2 = A3 )
| ( member_nat_nat @ X2 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_1061_insert__Collect,axiom,
! [A2: nat,P: nat > $o] :
( ( insert_nat @ A2 @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [U2: nat] :
( ( U2 != A2 )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_1062_subset__Diff__insert,axiom,
! [A: set_set_nat,B: set_set_nat,X: set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( minus_2163939370556025621et_nat @ B @ ( insert_set_nat @ X @ C2 ) ) )
= ( ( ord_le6893508408891458716et_nat @ A @ ( minus_2163939370556025621et_nat @ B @ C2 ) )
& ~ ( member_set_nat @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1063_subset__Diff__insert,axiom,
! [A: set_nat,B: set_nat,X: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ ( insert_nat @ X @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ C2 ) )
& ~ ( member_nat @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1064_subset__Diff__insert,axiom,
! [A: set_o,B: set_o,X: $o,C2: set_o] :
( ( ord_less_eq_set_o @ A @ ( minus_minus_set_o @ B @ ( insert_o @ X @ C2 ) ) )
= ( ( ord_less_eq_set_o @ A @ ( minus_minus_set_o @ B @ C2 ) )
& ~ ( member_o @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1065_subset__Diff__insert,axiom,
! [A: set_nat_nat,B: set_nat_nat,X: nat > nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ ( minus_8121590178497047118at_nat @ B @ ( insert_nat_nat @ X @ C2 ) ) )
= ( ( ord_le9059583361652607317at_nat @ A @ ( minus_8121590178497047118at_nat @ B @ C2 ) )
& ~ ( member_nat_nat @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1066_Union__mono,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_mono
thf(fact_1067_Sup__union__distrib,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_union_distrib
thf(fact_1068_Sup__union__distrib,axiom,
! [A: set_o,B: set_o] :
( ( complete_Sup_Sup_o @ ( sup_sup_set_o @ A @ B ) )
= ( sup_sup_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_union_distrib
thf(fact_1069_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X2: nat] : $false ) ) ).
% empty_def
thf(fact_1070_Diff__subset__conv,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ C2 )
= ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1071_less__eq__set__def,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ord_le3964352015994296041_nat_o
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A4 )
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_1072_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A4 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_1073_less__eq__set__def,axiom,
( ord_less_eq_set_o
= ( ^ [A4: set_o,B4: set_o] :
( ord_less_eq_o_o
@ ^ [X2: $o] : ( member_o @ X2 @ A4 )
@ ^ [X2: $o] : ( member_o @ X2 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_1074_less__eq__set__def,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ord_le7366121074344172400_nat_o
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A4 )
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_1075_Diff__partition,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_1076_Collect__subset,axiom,
! [A: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_1077_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_1078_Collect__subset,axiom,
! [A: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_1079_Collect__subset,axiom,
! [A: set_nat_nat,P: ( nat > nat ) > $o] :
( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_1080_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_1081_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
! [T2: set_nat] :
( ( member_set_nat @ T2 @ A4 )
=> ( member_set_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_1082_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A4 )
=> ( member_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_1083_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A4: set_o,B4: set_o] :
! [T2: $o] :
( ( member_o @ T2 @ A4 )
=> ( member_o @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_1084_subset__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
! [T2: nat > nat] :
( ( member_nat_nat @ T2 @ A4 )
=> ( member_nat_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_1085_subset__UnE,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) )
=> ~ ! [A6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ A )
=> ! [B7: set_nat] :
( ( ord_less_eq_set_nat @ B7 @ B )
=> ( C2
!= ( sup_sup_set_nat @ A6 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_1086_Un__absorb2,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_1087_Un__absorb1,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_1088_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
! [X2: set_nat] :
( ( member_set_nat @ X2 @ A4 )
=> ( member_set_nat @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_1089_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A4 )
=> ( member_nat @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_1090_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A4: set_o,B4: set_o] :
! [X2: $o] :
( ( member_o @ X2 @ A4 )
=> ( member_o @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_1091_subset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A4 )
=> ( member_nat_nat @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_1092_Un__upper2,axiom,
! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_1093_Un__upper1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_1094_Un__least,axiom,
! [A: set_nat,C2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_1095_subsetD,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_1096_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_1097_subsetD,axiom,
! [A: set_o,B: set_o,C: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ C @ A )
=> ( member_o @ C @ B ) ) ) ).
% subsetD
thf(fact_1098_subsetD,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( member_nat_nat @ C @ A )
=> ( member_nat_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_1099_in__mono,axiom,
! [A: set_set_nat,B: set_set_nat,X: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ X @ A )
=> ( member_set_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_1100_in__mono,axiom,
! [A: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_1101_in__mono,axiom,
! [A: set_o,B: set_o,X: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ X @ A )
=> ( member_o @ X @ B ) ) ) ).
% in_mono
thf(fact_1102_in__mono,axiom,
! [A: set_nat_nat,B: set_nat_nat,X: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( member_nat_nat @ X @ A )
=> ( member_nat_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_1103_Un__mono,axiom,
! [A: set_nat,C2: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_1104_Un__Diff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ C2 ) @ ( minus_minus_set_nat @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_1105_Un__Int__assoc__eq,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ B @ C2 ) ) )
= ( ord_le9059583361652607317at_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1106_Un__Int__assoc__eq,axiom,
! [A: set_o,B: set_o,C2: set_o] :
( ( ( sup_sup_set_o @ ( inf_inf_set_o @ A @ B ) @ C2 )
= ( inf_inf_set_o @ A @ ( sup_sup_set_o @ B @ C2 ) ) )
= ( ord_less_eq_set_o @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1107_Un__Int__assoc__eq,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A @ B ) @ C2 )
= ( inf_inf_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C2 ) ) )
= ( ord_le6893508408891458716et_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1108_Un__Int__assoc__eq,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) )
= ( ord_less_eq_set_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1109_Un__Diff__Int,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( minus_8121590178497047118at_nat @ A @ B ) @ ( inf_inf_set_nat_nat @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_1110_Un__Diff__Int,axiom,
! [A: set_o,B: set_o] :
( ( sup_sup_set_o @ ( minus_minus_set_o @ A @ B ) @ ( inf_inf_set_o @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_1111_Un__Diff__Int,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ A @ B ) @ ( inf_inf_set_set_nat @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_1112_Un__Diff__Int,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( inf_inf_set_nat @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_1113_Int__Diff__Un,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ ( minus_8121590178497047118at_nat @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_1114_Int__Diff__Un,axiom,
! [A: set_o,B: set_o] :
( ( sup_sup_set_o @ ( inf_inf_set_o @ A @ B ) @ ( minus_minus_set_o @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_1115_Int__Diff__Un,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A @ B ) @ ( minus_2163939370556025621et_nat @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_1116_Int__Diff__Un,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_1117_Diff__Int,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ A @ ( inf_inf_set_nat_nat @ B @ C2 ) )
= ( sup_sup_set_nat_nat @ ( minus_8121590178497047118at_nat @ A @ B ) @ ( minus_8121590178497047118at_nat @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1118_Diff__Int,axiom,
! [A: set_o,B: set_o,C2: set_o] :
( ( minus_minus_set_o @ A @ ( inf_inf_set_o @ B @ C2 ) )
= ( sup_sup_set_o @ ( minus_minus_set_o @ A @ B ) @ ( minus_minus_set_o @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1119_Diff__Int,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A @ ( inf_inf_set_set_nat @ B @ C2 ) )
= ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ A @ B ) @ ( minus_2163939370556025621et_nat @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1120_Diff__Int,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1121_Diff__Un,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ A @ ( sup_sup_set_nat_nat @ B @ C2 ) )
= ( inf_inf_set_nat_nat @ ( minus_8121590178497047118at_nat @ A @ B ) @ ( minus_8121590178497047118at_nat @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1122_Diff__Un,axiom,
! [A: set_o,B: set_o,C2: set_o] :
( ( minus_minus_set_o @ A @ ( sup_sup_set_o @ B @ C2 ) )
= ( inf_inf_set_o @ ( minus_minus_set_o @ A @ B ) @ ( minus_minus_set_o @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1123_Diff__Un,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A @ ( sup_sup_set_set_nat @ B @ C2 ) )
= ( inf_inf_set_set_nat @ ( minus_2163939370556025621et_nat @ A @ B ) @ ( minus_2163939370556025621et_nat @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1124_Diff__Un,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) )
= ( inf_inf_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1125_image__diff__subset,axiom,
! [F: nat > set_nat,A: set_nat,B: set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B ) ) @ ( image_nat_set_nat @ F @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_1126_image__diff__subset,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_1127_pred__subset__eq,axiom,
! [R: set_set_nat,S: set_set_nat] :
( ( ord_le3964352015994296041_nat_o
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ R )
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ S ) )
= ( ord_le6893508408891458716et_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_1128_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R )
@ ^ [X2: nat] : ( member_nat @ X2 @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_1129_pred__subset__eq,axiom,
! [R: set_o,S: set_o] :
( ( ord_less_eq_o_o
@ ^ [X2: $o] : ( member_o @ X2 @ R )
@ ^ [X2: $o] : ( member_o @ X2 @ S ) )
= ( ord_less_eq_set_o @ R @ S ) ) ).
% pred_subset_eq
thf(fact_1130_pred__subset__eq,axiom,
! [R: set_nat_nat,S: set_nat_nat] :
( ( ord_le7366121074344172400_nat_o
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ R )
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ S ) )
= ( ord_le9059583361652607317at_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_1131_Compr__image__eq,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1132_Compr__image__eq,axiom,
! [F: $o > nat,A: set_o,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_o_nat @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_o_nat @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1133_Compr__image__eq,axiom,
! [F: nat > $o,A: set_nat,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_nat_o @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_nat_o @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1134_Compr__image__eq,axiom,
! [F: $o > $o,A: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_o_o @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_o_o @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1135_Compr__image__eq,axiom,
! [F: nat > set_nat,A: set_nat,P: set_nat > $o] :
( ( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ ( image_nat_set_nat @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_nat_set_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1136_Compr__image__eq,axiom,
! [F: $o > set_nat,A: set_o,P: set_nat > $o] :
( ( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ ( image_o_set_nat @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_o_set_nat @ F
@ ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1137_Compr__image__eq,axiom,
! [F: set_nat > nat,A: set_set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_set_nat_nat @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_set_nat_nat @ F
@ ( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1138_Compr__image__eq,axiom,
! [F: set_nat > $o,A: set_set_nat,P: $o > $o] :
( ( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ ( image_set_nat_o @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_set_nat_o @ F
@ ( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1139_Compr__image__eq,axiom,
! [F: set_nat > set_nat,A: set_set_nat,P: set_nat > $o] :
( ( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ ( image_7916887816326733075et_nat @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_7916887816326733075et_nat @ F
@ ( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1140_Compr__image__eq,axiom,
! [F: ( nat > nat ) > nat,A: set_nat_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat_nat @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_nat_nat_nat @ F
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1141_image__image,axiom,
! [F: set_nat > set_nat,G: nat > set_nat,A: set_nat] :
( ( image_7916887816326733075et_nat @ F @ ( image_nat_set_nat @ G @ A ) )
= ( image_nat_set_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_1142_image__image,axiom,
! [F: set_nat > nat,G: nat > set_nat,A: set_nat] :
( ( image_set_nat_nat @ F @ ( image_nat_set_nat @ G @ A ) )
= ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_1143_image__image,axiom,
! [F: nat > set_nat,G: nat > nat,A: set_nat] :
( ( image_nat_set_nat @ F @ ( image_nat_nat @ G @ A ) )
= ( image_nat_set_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_1144_image__image,axiom,
! [F: nat > nat,G: nat > nat,A: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A ) )
= ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_1145_imageE,axiom,
! [B2: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_1146_imageE,axiom,
! [B2: nat,F: $o > nat,A: set_o] :
( ( member_nat @ B2 @ ( image_o_nat @ F @ A ) )
=> ~ ! [X3: $o] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% imageE
thf(fact_1147_imageE,axiom,
! [B2: $o,F: nat > $o,A: set_nat] :
( ( member_o @ B2 @ ( image_nat_o @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_1148_imageE,axiom,
! [B2: $o,F: $o > $o,A: set_o] :
( ( member_o @ B2 @ ( image_o_o @ F @ A ) )
=> ~ ! [X3: $o] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% imageE
thf(fact_1149_imageE,axiom,
! [B2: set_nat,F: nat > set_nat,A: set_nat] :
( ( member_set_nat @ B2 @ ( image_nat_set_nat @ F @ A ) )
=> ~ ! [X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_1150_imageE,axiom,
! [B2: set_nat,F: $o > set_nat,A: set_o] :
( ( member_set_nat @ B2 @ ( image_o_set_nat @ F @ A ) )
=> ~ ! [X3: $o] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_o @ X3 @ A ) ) ) ).
% imageE
thf(fact_1151_imageE,axiom,
! [B2: nat,F: set_nat > nat,A: set_set_nat] :
( ( member_nat @ B2 @ ( image_set_nat_nat @ F @ A ) )
=> ~ ! [X3: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_set_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_1152_imageE,axiom,
! [B2: $o,F: set_nat > $o,A: set_set_nat] :
( ( member_o @ B2 @ ( image_set_nat_o @ F @ A ) )
=> ~ ! [X3: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_set_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_1153_imageE,axiom,
! [B2: set_nat,F: set_nat > set_nat,A: set_set_nat] :
( ( member_set_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A ) )
=> ~ ! [X3: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_set_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_1154_imageE,axiom,
! [B2: nat,F: ( nat > nat ) > nat,A: set_nat_nat] :
( ( member_nat @ B2 @ ( image_nat_nat_nat @ F @ A ) )
=> ~ ! [X3: nat > nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_nat_nat @ X3 @ A ) ) ) ).
% imageE
thf(fact_1155_UN__absorb,axiom,
! [K: set_nat,I: set_set_nat,A: set_nat > set_nat] :
( ( member_set_nat @ K @ I )
=> ( ( sup_sup_set_nat @ ( A @ K ) @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ A @ I ) ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ A @ I ) ) ) ) ).
% UN_absorb
thf(fact_1156_UN__absorb,axiom,
! [K: nat,I: set_nat,A: nat > set_nat] :
( ( member_nat @ K @ I )
=> ( ( sup_sup_set_nat @ ( A @ K ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ I ) ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ I ) ) ) ) ).
% UN_absorb
thf(fact_1157_UN__absorb,axiom,
! [K: $o,I: set_o,A: $o > set_nat] :
( ( member_o @ K @ I )
=> ( ( sup_sup_set_nat @ ( A @ K ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A @ I ) ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A @ I ) ) ) ) ).
% UN_absorb
thf(fact_1158_UN__absorb,axiom,
! [K: nat > nat,I: set_nat_nat,A: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ K @ I )
=> ( ( sup_sup_set_nat @ ( A @ K ) @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ A @ I ) ) )
= ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ A @ I ) ) ) ) ).
% UN_absorb
thf(fact_1159_UN__Un__distrib,axiom,
! [A: nat > set_nat,B: nat > set_nat,I: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I4: nat] : ( sup_sup_set_nat @ ( A @ I4 ) @ ( B @ I4 ) )
@ I ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ I ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ I ) ) ) ) ).
% UN_Un_distrib
thf(fact_1160_Un__Union__image,axiom,
! [A: nat > set_nat,B: nat > set_nat,C2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( sup_sup_set_nat @ ( A @ X2 ) @ ( B @ X2 ) )
@ C2 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ C2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ C2 ) ) ) ) ).
% Un_Union_image
thf(fact_1161_UN__extend__simps_I6_J,axiom,
! [A: nat > set_nat,C2: set_nat,B: set_nat] :
( ( minus_minus_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ C2 ) ) @ B )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( minus_minus_set_nat @ ( A @ X2 ) @ B )
@ C2 ) ) ) ).
% UN_extend_simps(6)
thf(fact_1162_UN__mono,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_nat > set_nat,G: set_nat > set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_1163_UN__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_1164_UN__mono,axiom,
! [A: set_o,B: set_o,F: $o > set_nat,G: $o > set_nat] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_1165_UN__mono,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: ( nat > nat ) > set_nat,G: ( nat > nat ) > set_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_1166_UN__least,axiom,
! [A: set_set_nat,B: set_nat > set_nat,C2: set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( B @ X3 ) @ C2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_1167_UN__least,axiom,
! [A: set_nat,B: nat > set_nat,C2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( B @ X3 ) @ C2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_1168_UN__least,axiom,
! [A: set_o,B: $o > set_nat,C2: set_nat] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( B @ X3 ) @ C2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_1169_UN__least,axiom,
! [A: set_nat_nat,B: ( nat > nat ) > set_nat,C2: set_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( B @ X3 ) @ C2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_1170_UN__upper,axiom,
! [A2: set_nat,A: set_set_nat,B: set_nat > set_nat] :
( ( member_set_nat @ A2 @ A )
=> ( ord_less_eq_set_nat @ ( B @ A2 ) @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_1171_UN__upper,axiom,
! [A2: nat,A: set_nat,B: nat > set_nat] :
( ( member_nat @ A2 @ A )
=> ( ord_less_eq_set_nat @ ( B @ A2 ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_1172_UN__upper,axiom,
! [A2: $o,A: set_o,B: $o > set_nat] :
( ( member_o @ A2 @ A )
=> ( ord_less_eq_set_nat @ ( B @ A2 ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_1173_UN__upper,axiom,
! [A2: nat > nat,A: set_nat_nat,B: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ A2 @ A )
=> ( ord_less_eq_set_nat @ ( B @ A2 ) @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_1174_UN__subset__iff,axiom,
! [A: nat > set_nat,I: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ I ) ) @ B )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I )
=> ( ord_less_eq_set_nat @ ( A @ X2 ) @ B ) ) ) ) ).
% UN_subset_iff
thf(fact_1175_all__subset__image,axiom,
! [F: nat > set_nat,A: set_nat,P: set_set_nat > $o] :
( ( ! [B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ ( image_nat_set_nat @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( P @ ( image_nat_set_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1176_all__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( P @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1177_inf__Int__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( inf_inf_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R )
@ ^ [X2: nat] : ( member_nat @ X2 @ S ) )
= ( ^ [X2: nat] : ( member_nat @ X2 @ ( inf_inf_set_nat @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1178_inf__Int__eq,axiom,
! [R: set_nat_nat,S: set_nat_nat] :
( ( inf_inf_nat_nat_o
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ R )
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ S ) )
= ( ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1179_inf__Int__eq,axiom,
! [R: set_o,S: set_o] :
( ( inf_inf_o_o
@ ^ [X2: $o] : ( member_o @ X2 @ R )
@ ^ [X2: $o] : ( member_o @ X2 @ S ) )
= ( ^ [X2: $o] : ( member_o @ X2 @ ( inf_inf_set_o @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1180_inf__Int__eq,axiom,
! [R: set_set_nat,S: set_set_nat] :
( ( inf_inf_set_nat_o
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ R )
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ S ) )
= ( ^ [X2: set_nat] : ( member_set_nat @ X2 @ ( inf_inf_set_set_nat @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1181_inf__sup__aci_I8_J,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
= ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_1182_inf__sup__aci_I7_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_1183_inf__sup__aci_I6_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_1184_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y3: set_nat] : ( sup_sup_set_nat @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_1185_sup_Oassoc,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_1186_sup__assoc,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_1187_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_1188_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y3: set_nat] : ( sup_sup_set_nat @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_1189_boolean__algebra__cancel_Osup1,axiom,
! [A: set_nat,K: set_nat,A2: set_nat,B2: set_nat] :
( ( A
= ( sup_sup_set_nat @ K @ A2 ) )
=> ( ( sup_sup_set_nat @ A @ B2 )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_1190_boolean__algebra__cancel_Osup2,axiom,
! [B: set_nat,K: set_nat,B2: set_nat,A2: set_nat] :
( ( B
= ( sup_sup_set_nat @ K @ B2 ) )
=> ( ( sup_sup_set_nat @ A2 @ B )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_1191_sup_Oleft__commute,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ C ) )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_1192_sup__left__commute,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_1193_UnE,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) )
=> ( ~ ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B ) ) ) ).
% UnE
thf(fact_1194_UnE,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( sup_sup_set_o @ A @ B ) )
=> ( ~ ( member_o @ C @ A )
=> ( member_o @ C @ B ) ) ) ).
% UnE
thf(fact_1195_UnE,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A @ B ) )
=> ( ~ ( member_nat_nat @ C @ A )
=> ( member_nat_nat @ C @ B ) ) ) ).
% UnE
thf(fact_1196_UnE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
=> ( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% UnE
thf(fact_1197_UnI1,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_1198_UnI1,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ A )
=> ( member_o @ C @ ( sup_sup_set_o @ A @ B ) ) ) ).
% UnI1
thf(fact_1199_UnI1,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ A )
=> ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_1200_UnI1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_1201_UnI2,axiom,
! [C: set_nat,B: set_set_nat,A: set_set_nat] :
( ( member_set_nat @ C @ B )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_1202_UnI2,axiom,
! [C: $o,B: set_o,A: set_o] :
( ( member_o @ C @ B )
=> ( member_o @ C @ ( sup_sup_set_o @ A @ B ) ) ) ).
% UnI2
thf(fact_1203_UnI2,axiom,
! [C: nat > nat,B: set_nat_nat,A: set_nat_nat] :
( ( member_nat_nat @ C @ B )
=> ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_1204_UnI2,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_1205_Un__def,axiom,
( sup_sup_set_set_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ A4 )
| ( member_set_nat @ X2 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_1206_Un__def,axiom,
( sup_sup_set_o
= ( ^ [A4: set_o,B4: set_o] :
( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A4 )
| ( member_o @ X2 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_1207_Un__def,axiom,
( sup_sup_set_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A4 )
| ( member_nat_nat @ X2 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_1208_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A4 )
| ( member_nat @ X2 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_1209_bex__Un,axiom,
! [A: set_nat,B: set_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ X2 ) )
| ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_1210_ball__Un,axiom,
! [A: set_nat,B: set_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( P @ X2 ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_1211_Un__assoc,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_1212_Un__absorb,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_1213_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_1214_Un__left__absorb,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ A @ B ) ) ).
% Un_left_absorb
thf(fact_1215_Collect__disj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_1216_Un__left__commute,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) )
= ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_1217_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A4 )
& ( member_nat @ X2 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_1218_Int__def,axiom,
( inf_inf_set_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( collect_nat_nat
@ ^ [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A4 )
& ( member_nat_nat @ X2 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_1219_Int__def,axiom,
( inf_inf_set_o
= ( ^ [A4: set_o,B4: set_o] :
( collect_o
@ ^ [X2: $o] :
( ( member_o @ X2 @ A4 )
& ( member_o @ X2 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_1220_Int__def,axiom,
( inf_inf_set_set_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ A4 )
& ( member_set_nat @ X2 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_1221_Int__Collect,axiom,
! [X: set_nat,A: set_set_nat,P: set_nat > $o] :
( ( member_set_nat @ X @ ( inf_inf_set_set_nat @ A @ ( collect_set_nat @ P ) ) )
= ( ( member_set_nat @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_1222_set__incr__def,axiom,
( hales_set_incr
= ( ^ [N4: nat] :
( image_nat_nat
@ ^ [A3: nat] : ( plus_plus_nat @ A3 @ N4 ) ) ) ) ).
% set_incr_def
thf(fact_1223_BT__def,axiom,
( bt
= ( fun_upd_nat_set_nat @ ( restrict_nat_set_nat @ bvar @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ k @ one_one_nat ) ) ) @ ( plus_plus_nat @ k @ one_one_nat ) @ bstat ) ) ).
% BT_def
thf(fact_1224_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_1225_BfS__props_I4_J,axiom,
( member_nat_nat @ fS
@ ( piE_nat_nat @ ( bs @ k )
@ ^ [I4: nat] : ( set_ord_lessThan_nat @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) ).
% BfS_props(4)
thf(fact_1226_BfL__props_I4_J,axiom,
( member_nat_nat @ fL
@ ( piE_nat_nat @ ( bl @ one_one_nat )
@ ^ [I4: nat] : ( set_ord_lessThan_nat @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) ).
% BfL_props(4)
thf(fact_1227_assms_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ t ).
% assms(1)
thf(fact_1228_assms_I4_J,axiom,
! [K3: nat,R2: nat] :
( ( ord_less_eq_nat @ K3 @ k )
=> ( hales_lhj @ R2 @ t @ K3 ) ) ).
% assms(4)
thf(fact_1229_atLeastLessThan0,axiom,
! [M3: nat] :
( ( set_or4665077453230672383an_nat @ M3 @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_1230_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_1231_atLeastLessThan__add__Un,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( set_or4665077453230672383an_nat @ I2 @ ( plus_plus_nat @ J @ K ) )
= ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_1232_less__nat__zero__code,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1233_neq0__conv,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% neq0_conv
thf(fact_1234_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1235_nat__add__left__cancel__less,axiom,
! [K: nat,M3: nat,N3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N3 ) )
= ( ord_less_nat @ M3 @ N3 ) ) ).
% nat_add_left_cancel_less
thf(fact_1236_add__gr__0,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M3 @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M3 )
| ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% add_gr_0
thf(fact_1237_less__one,axiom,
! [N3: nat] :
( ( ord_less_nat @ N3 @ one_one_nat )
= ( N3 = zero_zero_nat ) ) ).
% less_one
thf(fact_1238_zero__less__diff,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N3 @ M3 ) )
= ( ord_less_nat @ M3 @ N3 ) ) ).
% zero_less_diff
thf(fact_1239_assms_I5_J,axiom,
ord_less_nat @ zero_zero_nat @ r ).
% assms(5)
thf(fact_1240_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
& ( M4 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_1241_less__imp__le__nat,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% less_imp_le_nat
thf(fact_1242_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
| ( M4 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1243_less__or__eq__imp__le,axiom,
! [M3: nat,N3: nat] :
( ( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) )
=> ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% less_or_eq_imp_le
thf(fact_1244_le__neq__implies__less,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( M3 != N3 )
=> ( ord_less_nat @ M3 @ N3 ) ) ) ).
% le_neq_implies_less
thf(fact_1245_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1246_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ? [K4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K4 )
& ( ( plus_plus_nat @ I2 @ K4 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1247_ex__least__nat__le,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ N3 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N3 )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K4 )
=> ~ ( P @ I5 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1248_mono__nat__linear__lb,axiom,
! [F: nat > nat,M3: nat,K: nat] :
( ! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( F @ M ) @ ( F @ N ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M3 ) @ K ) @ ( F @ ( plus_plus_nat @ M3 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1249_nat__neq__iff,axiom,
! [M3: nat,N3: nat] :
( ( M3 != N3 )
= ( ( ord_less_nat @ M3 @ N3 )
| ( ord_less_nat @ N3 @ M3 ) ) ) ).
% nat_neq_iff
thf(fact_1250_less__not__refl,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ N3 ) ).
% less_not_refl
thf(fact_1251_less__not__refl2,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_nat @ N3 @ M3 )
=> ( M3 != N3 ) ) ).
% less_not_refl2
thf(fact_1252_less__not__refl3,axiom,
! [S3: nat,T3: nat] :
( ( ord_less_nat @ S3 @ T3 )
=> ( S3 != T3 ) ) ).
% less_not_refl3
thf(fact_1253_less__irrefl__nat,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ N3 ) ).
% less_irrefl_nat
thf(fact_1254_nat__less__induct,axiom,
! [P: nat > $o,N3: nat] :
( ! [N: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N )
=> ( P @ M5 ) )
=> ( P @ N ) )
=> ( P @ N3 ) ) ).
% nat_less_induct
thf(fact_1255_infinite__descent,axiom,
! [P: nat > $o,N3: nat] :
( ! [N: nat] :
( ~ ( P @ N )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N )
& ~ ( P @ M5 ) ) )
=> ( P @ N3 ) ) ).
% infinite_descent
thf(fact_1256_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_1257_less__add__eq__less,axiom,
! [K: nat,L: nat,M3: nat,N3: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M3 @ L )
= ( plus_plus_nat @ K @ N3 ) )
=> ( ord_less_nat @ M3 @ N3 ) ) ) ).
% less_add_eq_less
thf(fact_1258_trans__less__add2,axiom,
! [I2: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_less_add2
thf(fact_1259_trans__less__add1,axiom,
! [I2: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_less_add1
thf(fact_1260_add__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1261_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_1262_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_1263_add__less__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1264_add__lessD1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_1265_infinite__descent0,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ~ ( P @ N )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N3 ) ) ) ).
% infinite_descent0
thf(fact_1266_gr__implies__not0,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( N3 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1267_less__zeroE,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1268_not__less0,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% not_less0
thf(fact_1269_not__gr0,axiom,
! [N3: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
= ( N3 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1270_gr0I,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% gr0I
% Helper facts (5)
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_3_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X: set_nat,Y: set_nat] :
( ( if_set_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X: set_nat,Y: set_nat] :
( ( if_set_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( inf_inf_set_nat @ ( bl @ zero_zero_nat ) @ ( bl @ one_one_nat ) )
= bot_bot_set_nat ) ).
%------------------------------------------------------------------------------