TPTP Problem File: SLH0384^1.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_01429_064152__5912936_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1473 ( 634 unt; 198 typ; 0 def)
% Number of atoms : 3532 (1229 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10990 ( 353 ~; 28 |; 167 &;9028 @)
% ( 0 <=>;1414 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 20 ( 19 usr)
% Number of type conns : 1187 (1187 >; 0 *; 0 +; 0 <<)
% Number of symbols : 182 ( 179 usr; 27 con; 0-4 aty)
% Number of variables : 3816 ( 414 ^;3331 !; 71 ?;3816 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:46:44.839
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_nat_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_nat_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
set_nat_nat_nat2: $tType ).
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_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
set_set_nat_nat2: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
set_nat_set_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_I_Eo_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
set_o_set_nat: $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_I_062_I_Eo_Mt__Nat__Onat_J_J,type,
set_o_nat: $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 (179)
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_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
disjoi831272138528337257at_nat: ( ( nat > nat ) > set_nat ) > set_nat_nat > $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_001t__Nat__Onat_001t__Nat__Onat,type,
disjoi6798895846410478970at_nat: ( nat > set_nat ) > 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_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
piE_nat_nat_nat: set_nat_nat > ( ( nat > nat ) > set_nat ) > set_nat_nat_nat2 ).
thf(sy_c_FuncSet_OPiE_001_Eo_001_Eo,type,
piE_o_o: set_o > ( $o > set_o ) > set_o_o ).
thf(sy_c_FuncSet_OPiE_001_Eo_001t__Nat__Onat,type,
piE_o_nat: set_o > ( $o > set_nat ) > set_o_nat ).
thf(sy_c_FuncSet_OPiE_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
piE_o_set_nat: set_o > ( $o > set_set_nat ) > set_o_set_nat ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
piE_nat_nat_nat2: set_nat > ( nat > set_nat_nat ) > set_nat_nat_nat ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001_Eo,type,
piE_nat_o: set_nat > ( nat > set_o ) > set_nat_o ).
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_OPiE_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
piE_nat_set_nat: set_nat > ( nat > set_set_nat ) > set_nat_set_nat ).
thf(sy_c_FuncSet_OPiE_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
piE_set_nat_o: set_set_nat > ( set_nat > set_o ) > set_set_nat_o ).
thf(sy_c_FuncSet_OPiE_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
piE_set_nat_nat: set_set_nat > ( set_nat > set_nat ) > set_set_nat_nat2 ).
thf(sy_c_FuncSet_OPiE_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
piE_set_nat_set_nat: set_set_nat > ( set_nat > set_set_nat ) > set_set_nat_set_nat ).
thf(sy_c_FuncSet_Orestrict_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
restrict_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat > ( nat > nat ) > nat ).
thf(sy_c_FuncSet_Orestrict_001_Eo_001t__Nat__Onat,type,
restrict_o_nat: ( $o > nat ) > set_o > $o > nat ).
thf(sy_c_FuncSet_Orestrict_001t__Nat__Onat_001t__Nat__Onat,type,
restrict_nat_nat: ( nat > nat ) > set_nat > 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_FuncSet_Orestrict_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
restrict_set_nat_nat: ( set_nat > nat ) > set_set_nat > set_nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
minus_167519014754328503_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > ( nat > nat ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_Eo_M_Eo_J,type,
minus_minus_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
minus_6910147592129066416_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_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_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
undefined_nat_nat: nat > nat ).
thf(sy_c_HOL_Oundefined_001_Eo,type,
undefined_o: $o ).
thf(sy_c_HOL_Oundefined_001t__Nat__Onat,type,
undefined_nat: nat ).
thf(sy_c_HOL_Oundefined_001t__Set__Oset_It__Nat__Onat_J,type,
undefined_set_nat: set_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_It__Nat__Onat_J_J,type,
inf_inf_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
sup_sup_nat_nat_o: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > $o ) > ( nat > nat ) > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_I_Eo_M_Eo_J,type,
sup_sup_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
sup_sup_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).
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_Nat_OSuc,type,
suc: nat > 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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
bot_bo945813143650711160at_nat: set_nat_nat_nat2 ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_Eo_Mt__Nat__Onat_J_J,type,
bot_bot_set_o_nat: set_o_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_062_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
bot_bo7208697003875722815at_nat: set_set_nat_nat2 ).
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_Odisjnt_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
disjnt_nat_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Set_Odisjnt_001_Eo,type,
disjnt_o: set_o > set_o > $o ).
thf(sy_c_Set_Odisjnt_001t__Nat__Onat,type,
disjnt_nat: set_nat > set_nat > $o ).
thf(sy_c_Set_Odisjnt_001t__Set__Oset_It__Nat__Onat_J,type,
disjnt_set_nat: set_set_nat > set_set_nat > $o ).
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_I_Eo_J,type,
image_nat_nat_set_o: ( ( nat > nat ) > set_o ) > set_nat_nat > set_set_o ).
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_M_Eo_J,type,
image_nat_nat_o2: ( nat > nat > $o ) > set_nat > set_nat_o ).
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_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_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__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_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
member_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat_nat2 > $o ).
thf(sy_c_member_001_062_I_Eo_M_Eo_J,type,
member_o_o: ( $o > $o ) > set_o_o > $o ).
thf(sy_c_member_001_062_I_Eo_Mt__Nat__Onat_J,type,
member_o_nat: ( $o > nat ) > set_o_nat > $o ).
thf(sy_c_member_001_062_I_Eo_Mt__Set__Oset_It__Nat__Onat_J_J,type,
member_o_set_nat: ( $o > set_nat ) > set_o_set_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member_nat_nat_nat2: ( nat > nat > nat ) > set_nat_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_Eo_J,type,
member_nat_o: ( nat > $o ) > set_nat_o > $o ).
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_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
member_nat_set_nat: ( nat > set_nat ) > set_nat_set_nat > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
member_set_nat_o: ( set_nat > $o ) > set_set_nat_o > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
member_set_nat_nat: ( set_nat > nat ) > set_set_nat_nat2 > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
member1686471427249568706et_nat: ( set_nat > set_nat ) > set_set_nat_set_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_nat2: 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_nat____,type,
nat2: nat ).
thf(sy_v_r,type,
r: nat ).
thf(sy_v_t,type,
t: nat ).
% Relevant facts (1269)
thf(fact_0_Suc,axiom,
( i
= ( suc @ nat2 ) ) ).
% Suc
thf(fact_1_Bvar__def,axiom,
( bvar
= ( ^ [I: nat] : ( if_set_nat @ ( I = zero_zero_nat ) @ ( bl @ zero_zero_nat ) @ ( hales_set_incr @ n2 @ ( bs @ ( minus_minus_nat @ I @ one_one_nat ) ) ) ) ) ) ).
% Bvar_def
thf(fact_2_n__def,axiom,
( n2
= ( plus_plus_nat @ n @ d ) ) ).
% n_def
thf(fact_3__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_4_set__incr__disjnt,axiom,
! [A: set_nat,B: set_nat,N: nat] :
( ( disjnt_nat @ A @ B )
=> ( disjnt_nat @ ( hales_set_incr @ N @ A ) @ ( hales_set_incr @ N @ B ) ) ) ).
% set_incr_disjnt
thf(fact_5_Bstat__def,axiom,
( bstat
= ( sup_sup_set_nat @ ( hales_set_incr @ n2 @ ( bs @ k ) ) @ ( bl @ one_one_nat ) ) ) ).
% Bstat_def
thf(fact_6_fact1,axiom,
( ( inf_inf_set_nat @ ( hales_set_incr @ n2 @ ( bs @ k ) ) @ ( bl @ one_one_nat ) )
= bot_bot_set_nat ) ).
% fact1
thf(fact_7_a,axiom,
member_nat @ i @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ k @ one_one_nat ) ) ).
% a
thf(fact_8_BfS__props_I1_J,axiom,
disjoi6798895846410478970at_nat @ bs @ ( set_ord_atMost_nat @ k ) ).
% BfS_props(1)
thf(fact_9_fT__def,axiom,
( fT
= ( ^ [X: nat] : ( if_nat @ ( member_nat @ X @ ( bl @ one_one_nat ) ) @ ( fL @ X ) @ ( if_nat @ ( member_nat @ X @ ( hales_set_incr @ n2 @ ( bs @ k ) ) ) @ ( fS @ ( minus_minus_nat @ X @ n2 ) ) @ undefined_nat ) ) ) ) ).
% fT_def
thf(fact_10__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_11_fact3,axiom,
! [X2: nat] :
( ( member_nat @ X2 @ ( set_ord_lessThan_nat @ k ) )
=> ( ( inf_inf_set_nat @ ( bl @ zero_zero_nat ) @ ( hales_set_incr @ n2 @ ( bs @ X2 ) ) )
= bot_bot_set_nat ) ) ).
% fact3
thf(fact_12_fact4,axiom,
! [X2: nat] :
( ( member_nat @ X2 @ ( 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 ) ) )
=> ( ( X2 != Xa )
=> ( ( inf_inf_set_nat @ ( hales_set_incr @ n2 @ ( bs @ X2 ) ) @ ( hales_set_incr @ n2 @ ( bs @ Xa ) ) )
= bot_bot_set_nat ) ) ) ) ).
% fact4
thf(fact_13_assms_I2_J,axiom,
ord_less_eq_nat @ one_one_nat @ k ).
% assms(2)
thf(fact_14_BfL__props_I1_J,axiom,
disjoi6798895846410478970at_nat @ bl @ ( set_ord_atMost_nat @ one_one_nat ) ).
% BfL_props(1)
thf(fact_15_M_H__prop,axiom,
ord_less_eq_nat @ ( plus_plus_nat @ n @ m2 ) @ m ).
% M'_prop
thf(fact_16_d__def,axiom,
( d
= ( minus_minus_nat @ m @ ( plus_plus_nat @ n @ m2 ) ) ) ).
% d_def
thf(fact_17_disjoint__family__onI,axiom,
! [S: set_set_nat,A: set_nat > set_nat] :
( ! [M: set_nat,N2: set_nat] :
( ( member_set_nat @ M @ S )
=> ( ( member_set_nat @ N2 @ S )
=> ( ( M != N2 )
=> ( ( inf_inf_set_nat @ ( A @ M ) @ ( A @ N2 ) )
= bot_bot_set_nat ) ) ) )
=> ( disjoi2115144663756723504at_nat @ A @ S ) ) ).
% disjoint_family_onI
thf(fact_18_disjoint__family__onI,axiom,
! [S: set_o,A: $o > set_nat] :
( ! [M: $o,N2: $o] :
( ( member_o @ M @ S )
=> ( ( member_o @ N2 @ S )
=> ( ( M != N2 )
=> ( ( inf_inf_set_nat @ ( A @ M ) @ ( A @ N2 ) )
= bot_bot_set_nat ) ) ) )
=> ( disjoi7928754725229124240_o_nat @ A @ S ) ) ).
% disjoint_family_onI
thf(fact_19_disjoint__family__onI,axiom,
! [S: set_nat_nat,A: ( nat > nat ) > set_nat] :
( ! [M: nat > nat,N2: nat > nat] :
( ( member_nat_nat @ M @ S )
=> ( ( member_nat_nat @ N2 @ S )
=> ( ( M != N2 )
=> ( ( inf_inf_set_nat @ ( A @ M ) @ ( A @ N2 ) )
= bot_bot_set_nat ) ) ) )
=> ( disjoi831272138528337257at_nat @ A @ S ) ) ).
% disjoint_family_onI
thf(fact_20_disjoint__family__onI,axiom,
! [S: set_nat,A: nat > set_nat] :
( ! [M: nat,N2: nat] :
( ( member_nat @ M @ S )
=> ( ( member_nat @ N2 @ S )
=> ( ( M != N2 )
=> ( ( inf_inf_set_nat @ ( A @ M ) @ ( A @ N2 ) )
= bot_bot_set_nat ) ) ) )
=> ( disjoi6798895846410478970at_nat @ A @ S ) ) ).
% disjoint_family_onI
thf(fact_21_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_22_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_23_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_24_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I2 )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_25_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_26_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_27_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_28_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_29_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_30_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_31_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_32_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_33_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_34_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_35_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_36_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_37_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_38_lessThan__eq__iff,axiom,
! [X3: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X3 )
= ( set_ord_lessThan_nat @ Y ) )
= ( X3 = Y ) ) ).
% lessThan_eq_iff
thf(fact_39_atMost__eq__iff,axiom,
! [X3: nat,Y: nat] :
( ( ( set_ord_atMost_nat @ X3 )
= ( set_ord_atMost_nat @ Y ) )
= ( X3 = Y ) ) ).
% atMost_eq_iff
thf(fact_40_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_41_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_42_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_43_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_44_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_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_46_mem__Collect__eq,axiom,
! [A2: $o,P: $o > $o] :
( ( member_o @ A2 @ ( collect_o @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_47_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_48_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X: set_nat] : ( member_set_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A: set_o] :
( ( collect_o
@ ^ [X: $o] : ( member_o @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
! [A: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_52_zero__eq__add__iff__both__eq__0,axiom,
! [X3: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X3 @ Y ) )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_53_add__eq__0__iff__both__eq__0,axiom,
! [X3: nat,Y: nat] :
( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_54_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_55_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_56_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_57_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_58_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_59_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_60_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_61_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_62_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_63_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_64_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_65_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_66_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_67_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_68_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_69_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_70_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_71_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_72_lessThan__subset__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X3 ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X3 @ Y ) ) ).
% lessThan_subset_iff
thf(fact_73_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_74_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_75_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_76_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_77_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_78_atMost__subset__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X3 ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X3 @ Y ) ) ).
% atMost_subset_iff
thf(fact_79_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_80_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_81_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_82_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_83_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_84_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_85_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_86_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_87_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_88_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_89_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_90_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_91_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_92_zero__reorient,axiom,
! [X3: nat] :
( ( zero_zero_nat = X3 )
= ( X3 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_93_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_94_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_95_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_96_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_97_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_98_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_99_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_100_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_101_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_102_one__reorient,axiom,
! [X3: nat] :
( ( one_one_nat = X3 )
= ( X3 = one_one_nat ) ) ).
% one_reorient
thf(fact_103_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_104_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_105_Suc__inject,axiom,
! [X3: nat,Y: nat] :
( ( ( suc @ X3 )
= ( suc @ Y ) )
=> ( X3 = Y ) ) ).
% Suc_inject
thf(fact_106_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M3: nat] :
( ( P @ X3 )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M3 ) )
=> ~ ! [M: nat] :
( ( P @ M )
=> ~ ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_107_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B2 ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_108_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_109_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_110_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_111_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_112_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_113_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_114_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_115_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_116_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_117_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_118_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C2: nat] :
( B3
= ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_119_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_120_less__eqE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ! [C3: nat] :
( B2
!= ( plus_plus_nat @ A2 @ C3 ) ) ) ).
% less_eqE
thf(fact_121_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_122_add__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_123_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_124_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_125_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_126_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_127_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_128_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_129_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_130_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_131_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_132_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M: nat] :
( N
= ( suc @ M ) ) ) ).
% not0_implies_Suc
thf(fact_133_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_134_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_135_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_136_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_137_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X4: nat,Y3: nat] :
( ( P @ X4 @ Y3 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_138_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_139_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_140_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_141_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_142_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_143_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_144_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_145_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_146_add__Suc__shift,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_147_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_148_nat__arith_Osuc1,axiom,
! [A: nat,K: nat,A2: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_149_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_150_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_151_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_152_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_153_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X4: nat] : ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y3: nat,Z: nat] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z )
=> ( R @ X4 @ Z ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_154_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P @ M2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_155_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_156_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_157_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_158_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_159_Suc__le__D,axiom,
! [N: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M5 )
=> ? [M: nat] :
( M5
= ( suc @ M ) ) ) ).
% Suc_le_D
thf(fact_160_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_161_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_162_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_163_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_164_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_165_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I2: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_166_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus_nat @ M6 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_167_trans__le__add2,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_168_trans__le__add1,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_169_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_170_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_171_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_172_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_173_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_174_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_175_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_176_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_177_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_178_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_179_diff__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_cancel2
thf(fact_180_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% Nat.diff_cancel
thf(fact_181_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_182_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_183_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_184_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_185_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_186_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_187_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_188_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_189_add__nonpos__eq__0__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_190_add__nonneg__eq__0__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_191_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_192_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_193_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_194_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_195_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_196_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_197_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_198_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_199_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_200_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_201_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_202_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_203_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_204_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_205_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_206_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_207_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_208_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_209_add__le__add__imp__diff__le,axiom,
! [I2: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_210_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_211_add__le__imp__le__diff,axiom,
! [I2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_212_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_213_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_214_one__is__add,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_215_add__is__1,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_216_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_217_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_218_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_219_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_220_Iio__eq__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = bot_bot_nat ) ) ).
% Iio_eq_empty_iff
thf(fact_221_diff__add__0,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_222_Suc__diff__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_223_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_224_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_225_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_226_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_227_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_228_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_229_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( set_ord_atMost_nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_230_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_231_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M6: nat,N3: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_232_image__add__0,axiom,
! [S: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S )
= S ) ).
% image_add_0
thf(fact_233_disjnt__Un2,axiom,
! [C4: set_nat,A: set_nat,B: set_nat] :
( ( disjnt_nat @ C4 @ ( sup_sup_set_nat @ A @ B ) )
= ( ( disjnt_nat @ C4 @ A )
& ( disjnt_nat @ C4 @ B ) ) ) ).
% disjnt_Un2
thf(fact_234_disjnt__Un1,axiom,
! [A: set_nat,B: set_nat,C4: set_nat] :
( ( disjnt_nat @ ( sup_sup_set_nat @ A @ B ) @ C4 )
= ( ( disjnt_nat @ A @ C4 )
& ( disjnt_nat @ B @ C4 ) ) ) ).
% disjnt_Un1
thf(fact_235_disjnt__self__iff__empty,axiom,
! [S: set_nat] :
( ( disjnt_nat @ S @ S )
= ( S = bot_bot_set_nat ) ) ).
% disjnt_self_iff_empty
thf(fact_236_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_237_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_238_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_239_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_240_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_241_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_242_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_243_image__eqI,axiom,
! [B2: nat,F: nat > nat,X3: nat,A: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_244_image__eqI,axiom,
! [B2: $o,F: nat > $o,X3: nat,A: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member_o @ B2 @ ( image_nat_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_245_image__eqI,axiom,
! [B2: nat,F: $o > nat,X3: $o,A: set_o] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_o @ X3 @ A )
=> ( member_nat @ B2 @ ( image_o_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_246_image__eqI,axiom,
! [B2: $o,F: $o > $o,X3: $o,A: set_o] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_o @ X3 @ A )
=> ( member_o @ B2 @ ( image_o_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_247_image__eqI,axiom,
! [B2: set_nat,F: nat > set_nat,X3: nat,A: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member_set_nat @ B2 @ ( image_nat_set_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_248_image__eqI,axiom,
! [B2: nat,F: set_nat > nat,X3: set_nat,A: set_set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_set_nat @ X3 @ A )
=> ( member_nat @ B2 @ ( image_set_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_249_image__eqI,axiom,
! [B2: $o,F: set_nat > $o,X3: set_nat,A: set_set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_set_nat @ X3 @ A )
=> ( member_o @ B2 @ ( image_set_nat_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_250_image__eqI,axiom,
! [B2: set_nat,F: $o > set_nat,X3: $o,A: set_o] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_o @ X3 @ A )
=> ( member_set_nat @ B2 @ ( image_o_set_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_251_image__eqI,axiom,
! [B2: nat > nat,F: nat > nat > nat,X3: nat,A: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat_nat @ B2 @ ( image_nat_nat_nat2 @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_252_image__eqI,axiom,
! [B2: set_nat,F: set_nat > set_nat,X3: set_nat,A: set_set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_set_nat @ X3 @ A )
=> ( member_set_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_253_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_254_empty__iff,axiom,
! [C: $o] :
~ ( member_o @ C @ bot_bot_set_o ) ).
% empty_iff
thf(fact_255_empty__iff,axiom,
! [C: nat > nat] :
~ ( member_nat_nat @ C @ bot_bot_set_nat_nat ) ).
% empty_iff
thf(fact_256_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_257_all__not__in__conv,axiom,
! [A: set_set_nat] :
( ( ! [X: set_nat] :
~ ( member_set_nat @ X @ A ) )
= ( A = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_258_all__not__in__conv,axiom,
! [A: set_o] :
( ( ! [X: $o] :
~ ( member_o @ X @ A ) )
= ( A = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_259_all__not__in__conv,axiom,
! [A: set_nat_nat] :
( ( ! [X: nat > nat] :
~ ( member_nat_nat @ X @ A ) )
= ( A = bot_bot_set_nat_nat ) ) ).
% all_not_in_conv
thf(fact_260_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X: nat] :
~ ( member_nat @ X @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_261_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_262_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_263_Diff__empty,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Diff_empty
thf(fact_264_empty__Diff,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_265_Diff__cancel,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ A )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_266_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ X4 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_267_subsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( member_set_nat @ X4 @ B ) )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% subsetI
thf(fact_268_subsetI,axiom,
! [A: set_o,B: set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( member_o @ X4 @ B ) )
=> ( ord_less_eq_set_o @ A @ B ) ) ).
% subsetI
thf(fact_269_subsetI,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
=> ( member_nat_nat @ X4 @ B ) )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% subsetI
thf(fact_270_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_271_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_272_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_273_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_274_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_275_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_276_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_277_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_278_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_279_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_280_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_281_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_282_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_283_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_284_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_285_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_286_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_287_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_288_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_289_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_290_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_291_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_292_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_293_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_294_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_295_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_296_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_297_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_298_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_299_Int__subset__iff,axiom,
! [C4: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ ( inf_inf_set_nat @ A @ B ) )
= ( ( ord_less_eq_set_nat @ C4 @ A )
& ( ord_less_eq_set_nat @ C4 @ B ) ) ) ).
% Int_subset_iff
thf(fact_300_Un__subset__iff,axiom,
! [A: set_nat,B: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C4 )
= ( ( ord_less_eq_set_nat @ A @ C4 )
& ( ord_less_eq_set_nat @ B @ C4 ) ) ) ).
% Un_subset_iff
thf(fact_301_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_302_Sup__atMost,axiom,
! [Y: set_nat] :
( ( comple7399068483239264473et_nat @ ( set_or4236626031148496127et_nat @ Y ) )
= Y ) ).
% Sup_atMost
thf(fact_303_Sup__atMost,axiom,
! [Y: $o] :
( ( complete_Sup_Sup_o @ ( set_ord_atMost_o @ Y ) )
= Y ) ).
% Sup_atMost
thf(fact_304_in__mono,axiom,
! [A: set_nat,B: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_305_in__mono,axiom,
! [A: set_set_nat,B: set_set_nat,X3: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ X3 @ A )
=> ( member_set_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_306_in__mono,axiom,
! [A: set_o,B: set_o,X3: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ X3 @ A )
=> ( member_o @ X3 @ B ) ) ) ).
% in_mono
thf(fact_307_in__mono,axiom,
! [A: set_nat_nat,B: set_nat_nat,X3: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( member_nat_nat @ X3 @ A )
=> ( member_nat_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_308_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_309_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_310_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_311_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_312_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( member_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_313_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
! [X: set_nat] :
( ( member_set_nat @ X @ A4 )
=> ( member_set_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_314_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A4: set_o,B4: set_o] :
! [X: $o] :
( ( member_o @ X @ A4 )
=> ( member_o @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_315_subset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
! [X: nat > nat] :
( ( member_nat_nat @ X @ A4 )
=> ( member_nat_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_316_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_317_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_318_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_319_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_320_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_321_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_322_zero__notin__Suc__image,axiom,
! [A: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A ) ) ).
% zero_notin_Suc_image
thf(fact_323_set__incr__altdef,axiom,
( hales_set_incr
= ( ^ [N3: nat] : ( image_nat_nat @ ( plus_plus_nat @ N3 ) ) ) ) ).
% set_incr_altdef
thf(fact_324_imageI,axiom,
! [X3: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_325_imageI,axiom,
! [X3: nat,A: set_nat,F: nat > $o] :
( ( member_nat @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ ( image_nat_o @ F @ A ) ) ) ).
% imageI
thf(fact_326_imageI,axiom,
! [X3: $o,A: set_o,F: $o > nat] :
( ( member_o @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ ( image_o_nat @ F @ A ) ) ) ).
% imageI
thf(fact_327_imageI,axiom,
! [X3: $o,A: set_o,F: $o > $o] :
( ( member_o @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ ( image_o_o @ F @ A ) ) ) ).
% imageI
thf(fact_328_imageI,axiom,
! [X3: nat,A: set_nat,F: nat > set_nat] :
( ( member_nat @ X3 @ A )
=> ( member_set_nat @ ( F @ X3 ) @ ( image_nat_set_nat @ F @ A ) ) ) ).
% imageI
thf(fact_329_imageI,axiom,
! [X3: set_nat,A: set_set_nat,F: set_nat > nat] :
( ( member_set_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ ( image_set_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_330_imageI,axiom,
! [X3: set_nat,A: set_set_nat,F: set_nat > $o] :
( ( member_set_nat @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ ( image_set_nat_o @ F @ A ) ) ) ).
% imageI
thf(fact_331_imageI,axiom,
! [X3: $o,A: set_o,F: $o > set_nat] :
( ( member_o @ X3 @ A )
=> ( member_set_nat @ ( F @ X3 ) @ ( image_o_set_nat @ F @ A ) ) ) ).
% imageI
thf(fact_332_imageI,axiom,
! [X3: nat,A: set_nat,F: nat > nat > nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat_nat @ ( F @ X3 ) @ ( image_nat_nat_nat2 @ F @ A ) ) ) ).
% imageI
thf(fact_333_imageI,axiom,
! [X3: set_nat,A: set_set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( member_set_nat @ ( F @ X3 ) @ ( image_7916887816326733075et_nat @ F @ A ) ) ) ).
% imageI
thf(fact_334_image__iff,axiom,
! [Z2: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Z2 @ ( image_nat_nat @ F @ A ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_335_image__iff,axiom,
! [Z2: set_nat,F: nat > set_nat,A: set_nat] :
( ( member_set_nat @ Z2 @ ( image_nat_set_nat @ F @ A ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_336_bex__imageD,axiom,
! [F: nat > set_nat,A: set_nat,P: set_nat > $o] :
( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ ( image_nat_set_nat @ F @ A ) )
& ( P @ X2 ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_337_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A ) )
& ( P @ X2 ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_338_image__cong,axiom,
! [M3: set_nat,N5: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( M3 = N5 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ N5 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_nat_set_nat @ F @ M3 )
= ( image_nat_set_nat @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_339_image__cong,axiom,
! [M3: set_nat,N5: set_nat,F: nat > nat,G: nat > nat] :
( ( M3 = N5 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ N5 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_nat_nat @ F @ M3 )
= ( image_nat_nat @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_340_ball__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 ) )
=> ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( P @ ( F @ X2 ) ) ) ) ).
% ball_imageD
thf(fact_341_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A ) )
=> ( P @ X4 ) )
=> ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( P @ ( F @ X2 ) ) ) ) ).
% ball_imageD
thf(fact_342_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B2: nat,F: nat > nat] :
( ( member_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_343_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B2: $o,F: nat > $o] :
( ( member_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_o @ B2 @ ( image_nat_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_344_rev__image__eqI,axiom,
! [X3: $o,A: set_o,B2: nat,F: $o > nat] :
( ( member_o @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat @ B2 @ ( image_o_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_345_rev__image__eqI,axiom,
! [X3: $o,A: set_o,B2: $o,F: $o > $o] :
( ( member_o @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_o @ B2 @ ( image_o_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_346_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B2: set_nat,F: nat > set_nat] :
( ( member_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_set_nat @ B2 @ ( image_nat_set_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_347_rev__image__eqI,axiom,
! [X3: set_nat,A: set_set_nat,B2: nat,F: set_nat > nat] :
( ( member_set_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat @ B2 @ ( image_set_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_348_rev__image__eqI,axiom,
! [X3: set_nat,A: set_set_nat,B2: $o,F: set_nat > $o] :
( ( member_set_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_o @ B2 @ ( image_set_nat_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_349_rev__image__eqI,axiom,
! [X3: $o,A: set_o,B2: set_nat,F: $o > set_nat] :
( ( member_o @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_set_nat @ B2 @ ( image_o_set_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_350_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B2: nat > nat,F: nat > nat > nat] :
( ( member_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat_nat @ B2 @ ( image_nat_nat_nat2 @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_351_rev__image__eqI,axiom,
! [X3: set_nat,A: set_set_nat,B2: set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_set_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_352_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_353_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_354_subset__image__iff,axiom,
! [B: set_set_nat,F: nat > set_nat,A: set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ ( image_nat_set_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_set_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_355_subset__image__iff,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_356_image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_357_image__subset__iff,axiom,
! [F: nat > set_nat,A: set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ B )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_set_nat @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_358_subset__imageE,axiom,
! [B: set_set_nat,F: nat > set_nat,A: set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ ( image_nat_set_nat @ F @ A ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A )
=> ( B
!= ( image_nat_set_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_359_subset__imageE,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A )
=> ( B
!= ( image_nat_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_360_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_361_image__subsetI,axiom,
! [A: set_nat,F: nat > $o,B: set_o] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_o @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_362_image__subsetI,axiom,
! [A: set_o,F: $o > nat,B: set_nat] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_o_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_363_image__subsetI,axiom,
! [A: set_o,F: $o > $o,B: set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( member_o @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_364_image__subsetI,axiom,
! [A: set_nat,F: nat > set_nat,B: set_set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_set_nat @ ( F @ X4 ) @ B ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_365_image__subsetI,axiom,
! [A: set_set_nat,F: set_nat > nat,B: set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_366_image__subsetI,axiom,
! [A: set_set_nat,F: set_nat > $o,B: set_o] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( member_o @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_set_nat_o @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_367_image__subsetI,axiom,
! [A: set_o,F: $o > set_nat,B: set_set_nat] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( member_set_nat @ ( F @ X4 ) @ B ) )
=> ( ord_le6893508408891458716et_nat @ ( image_o_set_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_368_image__subsetI,axiom,
! [A: set_nat,F: nat > nat > nat,B: set_nat_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat_nat @ ( F @ X4 ) @ B ) )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_369_image__subsetI,axiom,
! [A: set_set_nat,F: set_nat > set_nat,B: set_set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( member_set_nat @ ( F @ X4 ) @ B ) )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_370_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_371_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_372_emptyE,axiom,
! [A2: set_nat] :
~ ( member_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_373_emptyE,axiom,
! [A2: $o] :
~ ( member_o @ A2 @ bot_bot_set_o ) ).
% emptyE
thf(fact_374_emptyE,axiom,
! [A2: nat > nat] :
~ ( member_nat_nat @ A2 @ bot_bot_set_nat_nat ) ).
% emptyE
thf(fact_375_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_376_equals0D,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( A = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_377_equals0D,axiom,
! [A: set_o,A2: $o] :
( ( A = bot_bot_set_o )
=> ~ ( member_o @ A2 @ A ) ) ).
% equals0D
thf(fact_378_equals0D,axiom,
! [A: set_nat_nat,A2: nat > nat] :
( ( A = bot_bot_set_nat_nat )
=> ~ ( member_nat_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_379_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_380_equals0I,axiom,
! [A: set_set_nat] :
( ! [Y3: set_nat] :
~ ( member_set_nat @ Y3 @ A )
=> ( A = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_381_equals0I,axiom,
! [A: set_o] :
( ! [Y3: $o] :
~ ( member_o @ Y3 @ A )
=> ( A = bot_bot_set_o ) ) ).
% equals0I
thf(fact_382_equals0I,axiom,
! [A: set_nat_nat] :
( ! [Y3: nat > nat] :
~ ( member_nat_nat @ Y3 @ A )
=> ( A = bot_bot_set_nat_nat ) ) ).
% equals0I
thf(fact_383_equals0I,axiom,
! [A: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_384_ex__in__conv,axiom,
! [A: set_set_nat] :
( ( ? [X: set_nat] : ( member_set_nat @ X @ A ) )
= ( A != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_385_ex__in__conv,axiom,
! [A: set_o] :
( ( ? [X: $o] : ( member_o @ X @ A ) )
= ( A != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_386_ex__in__conv,axiom,
! [A: set_nat_nat] :
( ( ? [X: nat > nat] : ( member_nat_nat @ X @ A ) )
= ( A != bot_bot_set_nat_nat ) ) ).
% ex_in_conv
thf(fact_387_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_388_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_389_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_390_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_391_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_392_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_393_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_394_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_395_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_396_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_397_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_398_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_399_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_400_Int__assoc,axiom,
! [A: set_nat,B: set_nat,C4: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C4 )
= ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C4 ) ) ) ).
% Int_assoc
thf(fact_401_Int__absorb,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_402_Int__commute,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( inf_inf_set_nat @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_403_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_404_Int__left__commute,axiom,
! [A: set_nat,B: set_nat,C4: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C4 ) )
= ( inf_inf_set_nat @ B @ ( inf_inf_set_nat @ A @ C4 ) ) ) ).
% Int_left_commute
thf(fact_405_Int__mono,axiom,
! [A: set_nat,C4: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C4 )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C4 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_406_Int__lower1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_407_Int__lower2,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_408_Int__absorb1,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( inf_inf_set_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_409_Int__absorb2,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( inf_inf_set_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_410_Int__greatest,axiom,
! [C4: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
=> ( ( ord_less_eq_set_nat @ C4 @ B )
=> ( ord_less_eq_set_nat @ C4 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_411_Int__Collect__mono,axiom,
! [A: set_set_nat,B: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ ( collect_set_nat @ P ) ) @ ( inf_inf_set_set_nat @ B @ ( collect_set_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_412_Int__Collect__mono,axiom,
! [A: set_o,B: set_o,P: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A @ ( collect_o @ P ) ) @ ( inf_inf_set_o @ B @ ( collect_o @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_413_Int__Collect__mono,axiom,
! [A: set_nat_nat,B: set_nat_nat,P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A @ ( collect_nat_nat @ P ) ) @ ( inf_inf_set_nat_nat @ B @ ( collect_nat_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_414_Int__Collect__mono,axiom,
! [A: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_415_Int__Diff,axiom,
! [A: set_nat,B: set_nat,C4: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C4 )
= ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B @ C4 ) ) ) ).
% Int_Diff
thf(fact_416_Diff__Int2,axiom,
! [A: set_nat,C4: set_nat,B: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C4 ) @ ( inf_inf_set_nat @ B @ C4 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C4 ) @ B ) ) ).
% Diff_Int2
thf(fact_417_Diff__Diff__Int,axiom,
! [A: set_nat,B: set_nat] :
( ( minus_minus_set_nat @ A @ ( minus_minus_set_nat @ A @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_418_Diff__Int__distrib,axiom,
! [C4: set_nat,A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ C4 @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ C4 @ A ) @ ( inf_inf_set_nat @ C4 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_419_Diff__Int__distrib2,axiom,
! [A: set_nat,B: set_nat,C4: set_nat] :
( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A @ B ) @ C4 )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C4 ) @ ( inf_inf_set_nat @ B @ C4 ) ) ) ).
% Diff_Int_distrib2
thf(fact_420_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_421_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_422_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_423_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_424_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_425_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_426_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_427_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_428_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_429_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_430_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_431_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_432_bex__Un,axiom,
! [A: set_nat,B: set_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( sup_sup_set_nat @ A @ B ) )
& ( P @ X ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) )
| ? [X: nat] :
( ( member_nat @ X @ B )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_433_ball__Un,axiom,
! [A: set_nat,B: set_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( sup_sup_set_nat @ A @ B ) )
=> ( P @ X ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( P @ X ) )
& ! [X: nat] :
( ( member_nat @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_434_Un__assoc,axiom,
! [A: set_nat,B: set_nat,C4: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C4 )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C4 ) ) ) ).
% Un_assoc
thf(fact_435_Un__absorb,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_436_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_437_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_438_Un__left__commute,axiom,
! [A: set_nat,B: set_nat,C4: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C4 ) )
= ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C4 ) ) ) ).
% Un_left_commute
thf(fact_439_Un__mono,axiom,
! [A: set_nat,C4: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C4 )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C4 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_440_Un__least,axiom,
! [A: set_nat,C4: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C4 )
=> ( ( ord_less_eq_set_nat @ B @ C4 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C4 ) ) ) ).
% Un_least
thf(fact_441_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_442_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_443_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_444_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_445_subset__UnE,axiom,
! [C4: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ ( sup_sup_set_nat @ A @ B ) )
=> ~ ! [A5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ A )
=> ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ B )
=> ( C4
!= ( sup_sup_set_nat @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_446_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_447_Un__Diff,axiom,
! [A: set_nat,B: set_nat,C4: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C4 )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ C4 ) @ ( minus_minus_set_nat @ B @ C4 ) ) ) ).
% Un_Diff
thf(fact_448_Diff__subset__conv,axiom,
! [A: set_nat,B: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ C4 )
= ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ B @ C4 ) ) ) ).
% Diff_subset_conv
thf(fact_449_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_450_disjnt__iff,axiom,
( disjnt_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [X: nat] :
~ ( ( member_nat @ X @ A4 )
& ( member_nat @ X @ B4 ) ) ) ) ).
% disjnt_iff
thf(fact_451_disjnt__iff,axiom,
( disjnt_set_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
! [X: set_nat] :
~ ( ( member_set_nat @ X @ A4 )
& ( member_set_nat @ X @ B4 ) ) ) ) ).
% disjnt_iff
thf(fact_452_disjnt__iff,axiom,
( disjnt_o
= ( ^ [A4: set_o,B4: set_o] :
! [X: $o] :
~ ( ( member_o @ X @ A4 )
& ( member_o @ X @ B4 ) ) ) ) ).
% disjnt_iff
thf(fact_453_disjnt__iff,axiom,
( disjnt_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
! [X: nat > nat] :
~ ( ( member_nat_nat @ X @ A4 )
& ( member_nat_nat @ X @ B4 ) ) ) ) ).
% disjnt_iff
thf(fact_454_disjnt__sym,axiom,
! [A: set_nat,B: set_nat] :
( ( disjnt_nat @ A @ B )
=> ( disjnt_nat @ B @ A ) ) ).
% disjnt_sym
thf(fact_455_disjnt__subset1,axiom,
! [X5: set_nat,Y5: set_nat,Z3: set_nat] :
( ( disjnt_nat @ X5 @ Y5 )
=> ( ( ord_less_eq_set_nat @ Z3 @ X5 )
=> ( disjnt_nat @ Z3 @ Y5 ) ) ) ).
% disjnt_subset1
thf(fact_456_disjnt__subset2,axiom,
! [X5: set_nat,Y5: set_nat,Z3: set_nat] :
( ( disjnt_nat @ X5 @ Y5 )
=> ( ( ord_less_eq_set_nat @ Z3 @ Y5 )
=> ( disjnt_nat @ X5 @ Z3 ) ) ) ).
% disjnt_subset2
thf(fact_457_image__Int__subset,axiom,
! [F: nat > set_nat,A: set_nat,B: set_nat] : ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ ( inf_inf_set_nat @ A @ B ) ) @ ( inf_inf_set_set_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_458_image__Int__subset,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( inf_inf_set_nat @ A @ B ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_459_Int__emptyI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ~ ( member_set_nat @ X4 @ B ) )
=> ( ( inf_inf_set_set_nat @ A @ B )
= bot_bot_set_set_nat ) ) ).
% Int_emptyI
thf(fact_460_Int__emptyI,axiom,
! [A: set_o,B: set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ~ ( member_o @ X4 @ B ) )
=> ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o ) ) ).
% Int_emptyI
thf(fact_461_Int__emptyI,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
=> ~ ( member_nat_nat @ X4 @ B ) )
=> ( ( inf_inf_set_nat_nat @ A @ B )
= bot_bot_set_nat_nat ) ) ).
% Int_emptyI
thf(fact_462_Int__emptyI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ~ ( member_nat @ X4 @ B ) )
=> ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_463_disjoint__iff,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ( inf_inf_set_set_nat @ A @ B )
= bot_bot_set_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ~ ( member_set_nat @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_464_disjoint__iff,axiom,
! [A: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o )
= ( ! [X: $o] :
( ( member_o @ X @ A )
=> ~ ( member_o @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_465_disjoint__iff,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ( inf_inf_set_nat_nat @ A @ B )
= bot_bot_set_nat_nat )
= ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A )
=> ~ ( member_nat_nat @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_466_disjoint__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ~ ( member_nat @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_467_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_468_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_469_disjoint__iff__not__equal,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ! [Y6: nat] :
( ( member_nat @ Y6 @ B )
=> ( X != Y6 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_470_Diff__triv,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_471_Int__Diff__disjoint,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ B ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_472_image__Un,axiom,
! [F: nat > set_nat,A: set_nat,B: set_nat] :
( ( image_nat_set_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_set_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_473_image__Un,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_474_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_475_Un__empty__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Un_empty_right
thf(fact_476_Un__Int__crazy,axiom,
! [A: set_nat,B: set_nat,C4: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ B @ C4 ) ) @ ( inf_inf_set_nat @ C4 @ A ) )
= ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ B @ C4 ) ) @ ( sup_sup_set_nat @ C4 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_477_Int__Un__distrib,axiom,
! [A: set_nat,B: set_nat,C4: set_nat] :
( ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B @ C4 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ A @ C4 ) ) ) ).
% Int_Un_distrib
thf(fact_478_Un__Int__distrib,axiom,
! [A: set_nat,B: set_nat,C4: set_nat] :
( ( sup_sup_set_nat @ A @ ( inf_inf_set_nat @ B @ C4 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ A @ C4 ) ) ) ).
% Un_Int_distrib
thf(fact_479_Int__Un__distrib2,axiom,
! [B: set_nat,C4: set_nat,A: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B @ C4 ) @ A )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ B @ A ) @ ( inf_inf_set_nat @ C4 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_480_Un__Int__distrib2,axiom,
! [B: set_nat,C4: set_nat,A: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B @ C4 ) @ A )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ B @ A ) @ ( sup_sup_set_nat @ C4 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_481_Un__Int__assoc__eq,axiom,
! [A: set_nat,B: set_nat,C4: set_nat] :
( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C4 )
= ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B @ C4 ) ) )
= ( ord_less_eq_set_nat @ C4 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_482_Diff__Un,axiom,
! [A: set_nat,B: set_nat,C4: set_nat] :
( ( minus_minus_set_nat @ A @ ( sup_sup_set_nat @ B @ C4 ) )
= ( inf_inf_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ C4 ) ) ) ).
% Diff_Un
thf(fact_483_Diff__Int,axiom,
! [A: set_nat,B: set_nat,C4: set_nat] :
( ( minus_minus_set_nat @ A @ ( inf_inf_set_nat @ B @ C4 ) )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ C4 ) ) ) ).
% Diff_Int
thf(fact_484_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_485_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_486_disjnt__empty1,axiom,
! [A: set_nat] : ( disjnt_nat @ bot_bot_set_nat @ A ) ).
% disjnt_empty1
thf(fact_487_disjnt__empty2,axiom,
! [A: set_nat] : ( disjnt_nat @ A @ bot_bot_set_nat ) ).
% disjnt_empty2
thf(fact_488_disjnt__def,axiom,
( disjnt_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( inf_inf_set_nat @ A4 @ B4 )
= bot_bot_set_nat ) ) ) ).
% disjnt_def
thf(fact_489_Sup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Sup_empty
thf(fact_490_Sup__empty,axiom,
( ( complete_Sup_Sup_o @ bot_bot_set_o )
= bot_bot_o ) ).
% Sup_empty
thf(fact_491_disjnt__Union1,axiom,
! [A6: set_set_nat,B: set_nat] :
( ( disjnt_nat @ ( comple7399068483239264473et_nat @ A6 ) @ B )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A6 )
=> ( disjnt_nat @ X @ B ) ) ) ) ).
% disjnt_Union1
thf(fact_492_disjnt__Union2,axiom,
! [B: set_nat,A6: set_set_nat] :
( ( disjnt_nat @ B @ ( comple7399068483239264473et_nat @ A6 ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A6 )
=> ( disjnt_nat @ B @ X ) ) ) ) ).
% disjnt_Union2
thf(fact_493_cSup__atMost,axiom,
! [X3: set_nat] :
( ( comple7399068483239264473et_nat @ ( set_or4236626031148496127et_nat @ X3 ) )
= X3 ) ).
% cSup_atMost
thf(fact_494_cSup__atMost,axiom,
! [X3: nat] :
( ( complete_Sup_Sup_nat @ ( set_ord_atMost_nat @ X3 ) )
= X3 ) ).
% cSup_atMost
thf(fact_495_cSup__atMost,axiom,
! [X3: $o] :
( ( complete_Sup_Sup_o @ ( set_ord_atMost_o @ X3 ) )
= X3 ) ).
% cSup_atMost
thf(fact_496_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_497_UN__ball__bex__simps_I4_J,axiom,
! [B: nat > set_nat,A: set_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
& ( P @ X ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A )
& ? [Y6: nat] :
( ( member_nat @ Y6 @ ( B @ X ) )
& ( P @ Y6 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_498_Union__iff,axiom,
! [A: set_nat,C4: set_set_set_nat] :
( ( member_set_nat @ A @ ( comple548664676211718543et_nat @ C4 ) )
= ( ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ C4 )
& ( member_set_nat @ A @ X ) ) ) ) ).
% Union_iff
thf(fact_499_Union__iff,axiom,
! [A: $o,C4: set_set_o] :
( ( member_o @ A @ ( comple90263536869209701_set_o @ C4 ) )
= ( ? [X: set_o] :
( ( member_set_o @ X @ C4 )
& ( member_o @ A @ X ) ) ) ) ).
% Union_iff
thf(fact_500_Union__iff,axiom,
! [A: nat > nat,C4: set_set_nat_nat] :
( ( member_nat_nat @ A @ ( comple5448282615319421384at_nat @ C4 ) )
= ( ? [X: set_nat_nat] :
( ( member_set_nat_nat2 @ X @ C4 )
& ( member_nat_nat @ A @ X ) ) ) ) ).
% Union_iff
thf(fact_501_Union__iff,axiom,
! [A: nat,C4: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C4 ) )
= ( ? [X: set_nat] :
( ( member_set_nat @ X @ C4 )
& ( member_nat @ A @ X ) ) ) ) ).
% Union_iff
thf(fact_502_UnionI,axiom,
! [X5: set_set_nat,C4: set_set_set_nat,A: set_nat] :
( ( member_set_set_nat @ X5 @ C4 )
=> ( ( member_set_nat @ A @ X5 )
=> ( member_set_nat @ A @ ( comple548664676211718543et_nat @ C4 ) ) ) ) ).
% UnionI
thf(fact_503_UnionI,axiom,
! [X5: set_o,C4: set_set_o,A: $o] :
( ( member_set_o @ X5 @ C4 )
=> ( ( member_o @ A @ X5 )
=> ( member_o @ A @ ( comple90263536869209701_set_o @ C4 ) ) ) ) ).
% UnionI
thf(fact_504_UnionI,axiom,
! [X5: set_nat_nat,C4: set_set_nat_nat,A: nat > nat] :
( ( member_set_nat_nat2 @ X5 @ C4 )
=> ( ( member_nat_nat @ A @ X5 )
=> ( member_nat_nat @ A @ ( comple5448282615319421384at_nat @ C4 ) ) ) ) ).
% UnionI
thf(fact_505_UnionI,axiom,
! [X5: set_nat,C4: set_set_nat,A: nat] :
( ( member_set_nat @ X5 @ C4 )
=> ( ( member_nat @ A @ X5 )
=> ( member_nat @ A @ ( comple7399068483239264473et_nat @ C4 ) ) ) ) ).
% UnionI
thf(fact_506_UN__ball__bex__simps_I1_J,axiom,
! [A: set_set_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ A ) )
=> ( P @ X ) ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ! [Y6: nat] :
( ( member_nat @ Y6 @ X )
=> ( P @ Y6 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_507_UN__ball__bex__simps_I3_J,axiom,
! [A: set_set_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ A ) )
& ( P @ X ) ) )
= ( ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ? [Y6: nat] :
( ( member_nat @ Y6 @ X )
& ( P @ Y6 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_508_DiffI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_509_DiffI,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 @ ( minus_2163939370556025621et_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_510_DiffI,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ A )
=> ( ~ ( member_o @ C @ B )
=> ( member_o @ C @ ( minus_minus_set_o @ A @ B ) ) ) ) ).
% DiffI
thf(fact_511_DiffI,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 @ ( minus_8121590178497047118at_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_512_Diff__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ~ ( member_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_513_Diff__iff,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
= ( ( member_set_nat @ C @ A )
& ~ ( member_set_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_514_Diff__iff,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A @ B ) )
= ( ( member_o @ C @ A )
& ~ ( member_o @ C @ B ) ) ) ).
% Diff_iff
thf(fact_515_Diff__iff,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A @ B ) )
= ( ( member_nat_nat @ C @ A )
& ~ ( member_nat_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_516_Sup__bot__conv_I1_J,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( X = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_517_Sup__bot__conv_I1_J,axiom,
! [A: set_o] :
( ( ( complete_Sup_Sup_o @ A )
= bot_bot_o )
= ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( X = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_518_Sup__bot__conv_I2_J,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( X = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_519_Sup__bot__conv_I2_J,axiom,
! [A: set_o] :
( ( bot_bot_o
= ( complete_Sup_Sup_o @ A ) )
= ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( X = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_520_ball__UN,axiom,
! [B: nat > set_nat,A: set_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
=> ( P @ X ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ! [Y6: nat] :
( ( member_nat @ Y6 @ ( B @ X ) )
=> ( P @ Y6 ) ) ) ) ) ).
% ball_UN
thf(fact_521_bex__UN,axiom,
! [B: nat > set_nat,A: set_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
& ( P @ X ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A )
& ? [Y6: nat] :
( ( member_nat @ Y6 @ ( B @ X ) )
& ( P @ Y6 ) ) ) ) ) ).
% bex_UN
thf(fact_522_UN__ball__bex__simps_I2_J,axiom,
! [B: nat > set_nat,A: set_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
=> ( P @ X ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ! [Y6: nat] :
( ( member_nat @ Y6 @ ( B @ X ) )
=> ( P @ Y6 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_523_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_524_DiffE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_525_DiffE,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
=> ~ ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_526_DiffE,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A @ B ) )
=> ~ ( ( member_o @ C @ A )
=> ( member_o @ C @ B ) ) ) ).
% DiffE
thf(fact_527_DiffE,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A @ B ) )
=> ~ ( ( member_nat_nat @ C @ A )
=> ( member_nat_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_528_DiffD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% DiffD1
thf(fact_529_DiffD1,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
=> ( member_set_nat @ C @ A ) ) ).
% DiffD1
thf(fact_530_DiffD1,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A @ B ) )
=> ( member_o @ C @ A ) ) ).
% DiffD1
thf(fact_531_DiffD1,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A @ B ) )
=> ( member_nat_nat @ C @ A ) ) ).
% DiffD1
thf(fact_532_DiffD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( member_nat @ C @ B ) ) ).
% DiffD2
thf(fact_533_DiffD2,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
=> ~ ( member_set_nat @ C @ B ) ) ).
% DiffD2
thf(fact_534_DiffD2,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A @ B ) )
=> ~ ( member_o @ C @ B ) ) ).
% DiffD2
thf(fact_535_DiffD2,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A @ B ) )
=> ~ ( member_nat_nat @ C @ B ) ) ).
% DiffD2
thf(fact_536_Sup_OSUP__cong,axiom,
! [A: set_nat,B: set_nat,C4: nat > set_nat,D2: nat > set_nat,Sup: set_set_nat > set_nat] :
( ( A = B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Sup @ ( image_nat_set_nat @ C4 @ A ) )
= ( Sup @ ( image_nat_set_nat @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_537_Sup_OSUP__cong,axiom,
! [A: set_nat,B: set_nat,C4: nat > nat,D2: nat > nat,Sup: set_nat > nat] :
( ( A = B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C4 @ A ) )
= ( Sup @ ( image_nat_nat @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_538_Inf_OINF__cong,axiom,
! [A: set_nat,B: set_nat,C4: nat > set_nat,D2: nat > set_nat,Inf: set_set_nat > set_nat] :
( ( A = B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Inf @ ( image_nat_set_nat @ C4 @ A ) )
= ( Inf @ ( image_nat_set_nat @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_539_Inf_OINF__cong,axiom,
! [A: set_nat,B: set_nat,C4: nat > nat,D2: nat > nat,Inf: set_nat > nat] :
( ( A = B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C4 @ A ) )
= ( Inf @ ( image_nat_nat @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_540_UnionE,axiom,
! [A: set_nat,C4: set_set_set_nat] :
( ( member_set_nat @ A @ ( comple548664676211718543et_nat @ C4 ) )
=> ~ ! [X6: set_set_nat] :
( ( member_set_nat @ A @ X6 )
=> ~ ( member_set_set_nat @ X6 @ C4 ) ) ) ).
% UnionE
thf(fact_541_UnionE,axiom,
! [A: $o,C4: set_set_o] :
( ( member_o @ A @ ( comple90263536869209701_set_o @ C4 ) )
=> ~ ! [X6: set_o] :
( ( member_o @ A @ X6 )
=> ~ ( member_set_o @ X6 @ C4 ) ) ) ).
% UnionE
thf(fact_542_UnionE,axiom,
! [A: nat > nat,C4: set_set_nat_nat] :
( ( member_nat_nat @ A @ ( comple5448282615319421384at_nat @ C4 ) )
=> ~ ! [X6: set_nat_nat] :
( ( member_nat_nat @ A @ X6 )
=> ~ ( member_set_nat_nat2 @ X6 @ C4 ) ) ) ).
% UnionE
thf(fact_543_UnionE,axiom,
! [A: nat,C4: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C4 ) )
=> ~ ! [X6: set_nat] :
( ( member_nat @ A @ X6 )
=> ~ ( member_set_nat @ X6 @ C4 ) ) ) ).
% UnionE
thf(fact_544_cSup__eq__maximum,axiom,
! [Z2: set_nat,X5: set_set_nat] :
( ( member_set_nat @ Z2 @ X5 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ X5 )
=> ( ord_less_eq_set_nat @ X4 @ Z2 ) )
=> ( ( comple7399068483239264473et_nat @ X5 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_545_cSup__eq__maximum,axiom,
! [Z2: nat,X5: set_nat] :
( ( member_nat @ Z2 @ X5 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ X5 )
=> ( ord_less_eq_nat @ X4 @ Z2 ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_546_cSup__eq__maximum,axiom,
! [Z2: $o,X5: set_o] :
( ( member_o @ Z2 @ X5 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ X5 )
=> ( ord_less_eq_o @ X4 @ Z2 ) )
=> ( ( complete_Sup_Sup_o @ X5 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_547_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_548_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_549_Sup__le__iff,axiom,
! [A: set_set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ B2 )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( ord_less_eq_set_nat @ X @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_550_Sup__le__iff,axiom,
! [A: set_o,B2: $o] :
( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ B2 )
= ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_o @ X @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_551_Sup__upper,axiom,
! [X3: set_nat,A: set_set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ X3 @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% Sup_upper
thf(fact_552_Sup__upper,axiom,
! [X3: $o,A: set_o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_o @ X3 @ ( complete_Sup_Sup_o @ A ) ) ) ).
% Sup_upper
thf(fact_553_Sup__least,axiom,
! [A: set_set_nat,Z2: set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ X4 @ Z2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_554_Sup__least,axiom,
! [A: set_o,Z2: $o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_o @ X4 @ Z2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_555_Sup__mono,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [A7: set_nat] :
( ( member_set_nat @ A7 @ A )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ B )
& ( ord_less_eq_set_nat @ A7 @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_mono
thf(fact_556_Sup__mono,axiom,
! [A: set_o,B: set_o] :
( ! [A7: $o] :
( ( member_o @ A7 @ A )
=> ? [X2: $o] :
( ( member_o @ X2 @ B )
& ( ord_less_eq_o @ A7 @ X2 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_mono
thf(fact_557_Sup__eqI,axiom,
! [A: set_set_nat,X3: set_nat] :
( ! [Y3: set_nat] :
( ( member_set_nat @ Y3 @ A )
=> ( ord_less_eq_set_nat @ Y3 @ X3 ) )
=> ( ! [Y3: set_nat] :
( ! [Z4: set_nat] :
( ( member_set_nat @ Z4 @ A )
=> ( ord_less_eq_set_nat @ Z4 @ Y3 ) )
=> ( ord_less_eq_set_nat @ X3 @ Y3 ) )
=> ( ( comple7399068483239264473et_nat @ A )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_558_Sup__eqI,axiom,
! [A: set_o,X3: $o] :
( ! [Y3: $o] :
( ( member_o @ Y3 @ A )
=> ( ord_less_eq_o @ Y3 @ X3 ) )
=> ( ! [Y3: $o] :
( ! [Z4: $o] :
( ( member_o @ Z4 @ A )
=> ( ord_less_eq_o @ Z4 @ Y3 ) )
=> ( ord_less_eq_o @ X3 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ A )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_559_SUP__cong,axiom,
! [A: set_nat,B: set_nat,C4: nat > nat,D2: nat > nat] :
( ( A = B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C4 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_560_SUP__cong,axiom,
! [A: set_o,B: set_o,C4: $o > nat,D2: $o > nat] :
( ( A = B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_o_nat @ C4 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_o_nat @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_561_SUP__cong,axiom,
! [A: set_nat,B: set_nat,C4: nat > $o,D2: nat > $o] :
( ( A = B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ C4 @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_562_SUP__cong,axiom,
! [A: set_o,B: set_o,C4: $o > $o,D2: $o > $o] :
( ( A = B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ C4 @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_563_SUP__cong,axiom,
! [A: set_nat,B: set_nat,C4: nat > set_nat,D2: nat > set_nat] :
( ( A = B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C4 @ A ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_564_SUP__cong,axiom,
! [A: set_o,B: set_o,C4: $o > set_nat,D2: $o > set_nat] :
( ( A = B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ C4 @ A ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_565_SUP__cong,axiom,
! [A: set_set_nat,B: set_set_nat,C4: set_nat > nat,D2: set_nat > nat] :
( ( A = B )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ B )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_set_nat_nat @ C4 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_set_nat_nat @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_566_SUP__cong,axiom,
! [A: set_set_nat,B: set_set_nat,C4: set_nat > $o,D2: set_nat > $o] :
( ( A = B )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ B )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_nat_o @ C4 @ A ) )
= ( complete_Sup_Sup_o @ ( image_set_nat_o @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_567_SUP__cong,axiom,
! [A: set_set_nat,B: set_set_nat,C4: set_nat > set_nat,D2: set_nat > set_nat] :
( ( A = B )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ B )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ C4 @ A ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_568_SUP__cong,axiom,
! [A: set_nat_nat,B: set_nat_nat,C4: ( nat > nat ) > nat,D2: ( nat > nat ) > nat] :
( ( A = B )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ B )
=> ( ( C4 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat_nat @ C4 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat_nat @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_569_empty__Union__conv,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( X = bot_bot_set_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_570_Union__empty__conv,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( X = bot_bot_set_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_571_Union__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Union_empty
thf(fact_572_Union__subsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ? [Y4: set_nat] :
( ( member_set_nat @ Y4 @ B )
& ( ord_less_eq_set_nat @ X4 @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_subsetI
thf(fact_573_Union__upper,axiom,
! [B: set_nat,A: set_set_nat] :
( ( member_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ B @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% Union_upper
thf(fact_574_Union__least,axiom,
! [A: set_set_nat,C4: set_nat] :
( ! [X6: set_nat] :
( ( member_set_nat @ X6 @ A )
=> ( ord_less_eq_set_nat @ X6 @ C4 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ C4 ) ) ).
% Union_least
thf(fact_575_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_576_SUP__eq,axiom,
! [A: set_nat,B: set_nat,F: nat > $o,G: nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X2 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X2 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_577_SUP__eq,axiom,
! [A: set_nat,B: set_o,F: nat > $o,G: $o > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X2: $o] :
( ( member_o @ X2 @ B )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X2 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X2 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_578_SUP__eq,axiom,
! [A: set_o,B: set_nat,F: $o > $o,G: nat > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X2 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B )
=> ? [X2: $o] :
( ( member_o @ X2 @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X2 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_579_SUP__eq,axiom,
! [A: set_o,B: set_o,F: $o > $o,G: $o > $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ? [X2: $o] :
( ( member_o @ X2 @ B )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X2 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X2: $o] :
( ( member_o @ X2 @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X2 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_580_SUP__eq,axiom,
! [A: set_nat,B: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( G @ X2 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_set_nat @ ( G @ J2 ) @ ( F @ X2 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_581_SUP__eq,axiom,
! [A: set_nat,B: set_o,F: nat > set_nat,G: $o > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X2: $o] :
( ( member_o @ X2 @ B )
& ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( G @ X2 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_set_nat @ ( G @ J2 ) @ ( F @ X2 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_582_SUP__eq,axiom,
! [A: set_o,B: set_nat,F: $o > set_nat,G: nat > set_nat] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( G @ X2 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B )
=> ? [X2: $o] :
( ( member_o @ X2 @ A )
& ( ord_less_eq_set_nat @ ( G @ J2 ) @ ( F @ X2 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_583_SUP__eq,axiom,
! [A: set_o,B: set_o,F: $o > set_nat,G: $o > set_nat] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ? [X2: $o] :
( ( member_o @ X2 @ B )
& ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( G @ X2 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X2: $o] :
( ( member_o @ X2 @ A )
& ( ord_less_eq_set_nat @ ( G @ J2 ) @ ( F @ X2 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_584_SUP__eq,axiom,
! [A: set_nat,B: set_set_nat,F: nat > $o,G: set_nat > $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ B )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X2 ) ) ) )
=> ( ! [J2: set_nat] :
( ( member_set_nat @ J2 @ B )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X2 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_set_nat_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_585_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 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X2 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X2 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_586_cSup__eq__non__empty,axiom,
! [X5: set_set_nat,A2: set_nat] :
( ( X5 != bot_bot_set_set_nat )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ X5 )
=> ( ord_less_eq_set_nat @ X4 @ A2 ) )
=> ( ! [Y3: set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ X5 )
=> ( ord_less_eq_set_nat @ X2 @ Y3 ) )
=> ( ord_less_eq_set_nat @ A2 @ Y3 ) )
=> ( ( comple7399068483239264473et_nat @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_587_cSup__eq__non__empty,axiom,
! [X5: set_nat,A2: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ X5 )
=> ( ord_less_eq_nat @ X4 @ A2 ) )
=> ( ! [Y3: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ X5 )
=> ( ord_less_eq_nat @ X2 @ Y3 ) )
=> ( ord_less_eq_nat @ A2 @ Y3 ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_588_cSup__eq__non__empty,axiom,
! [X5: set_o,A2: $o] :
( ( X5 != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ X5 )
=> ( ord_less_eq_o @ X4 @ A2 ) )
=> ( ! [Y3: $o] :
( ! [X2: $o] :
( ( member_o @ X2 @ X5 )
=> ( ord_less_eq_o @ X2 @ Y3 ) )
=> ( ord_less_eq_o @ A2 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_589_cSup__least,axiom,
! [X5: set_set_nat,Z2: set_nat] :
( ( X5 != bot_bot_set_set_nat )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ X5 )
=> ( ord_less_eq_set_nat @ X4 @ Z2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ X5 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_590_cSup__least,axiom,
! [X5: set_nat,Z2: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ X5 )
=> ( ord_less_eq_nat @ X4 @ Z2 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X5 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_591_cSup__least,axiom,
! [X5: set_o,Z2: $o] :
( ( X5 != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ X5 )
=> ( ord_less_eq_o @ X4 @ Z2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ X5 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_592_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_593_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_594_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_595_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_596_SUP__eq__const,axiom,
! [I4: set_set_nat,F: set_nat > set_nat,X3: set_nat] :
( ( I4 != bot_bot_set_set_nat )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
=> ( ( F @ I3 )
= X3 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ I4 ) )
= X3 ) ) ) ).
% SUP_eq_const
thf(fact_597_SUP__eq__const,axiom,
! [I4: set_o,F: $o > set_nat,X3: set_nat] :
( ( I4 != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I4 )
=> ( ( F @ I3 )
= X3 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ I4 ) )
= X3 ) ) ) ).
% SUP_eq_const
thf(fact_598_SUP__eq__const,axiom,
! [I4: set_nat_nat,F: ( nat > nat ) > set_nat,X3: set_nat] :
( ( I4 != bot_bot_set_nat_nat )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I4 )
=> ( ( F @ I3 )
= X3 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ I4 ) )
= X3 ) ) ) ).
% SUP_eq_const
thf(fact_599_SUP__eq__const,axiom,
! [I4: set_nat,F: nat > set_nat,X3: set_nat] :
( ( I4 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ( F @ I3 )
= X3 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ I4 ) )
= X3 ) ) ) ).
% SUP_eq_const
thf(fact_600_SUP__eq__const,axiom,
! [I4: set_set_nat,F: set_nat > $o,X3: $o] :
( ( I4 != bot_bot_set_set_nat )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
=> ( ( F @ I3 )
= X3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ I4 ) )
= X3 ) ) ) ).
% SUP_eq_const
thf(fact_601_SUP__eq__const,axiom,
! [I4: set_o,F: $o > $o,X3: $o] :
( ( I4 != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I4 )
=> ( ( F @ I3 )
= X3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ I4 ) )
= X3 ) ) ) ).
% SUP_eq_const
thf(fact_602_SUP__eq__const,axiom,
! [I4: set_nat_nat,F: ( nat > nat ) > $o,X3: $o] :
( ( I4 != bot_bot_set_nat_nat )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I4 )
=> ( ( F @ I3 )
= X3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_nat_o @ F @ I4 ) )
= X3 ) ) ) ).
% SUP_eq_const
thf(fact_603_SUP__eq__const,axiom,
! [I4: set_nat,F: nat > $o,X3: $o] :
( ( I4 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ( F @ I3 )
= X3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ I4 ) )
= X3 ) ) ) ).
% SUP_eq_const
thf(fact_604_Union__disjoint,axiom,
! [C4: set_set_nat,A: set_nat] :
( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ C4 ) @ A )
= bot_bot_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ C4 )
=> ( ( inf_inf_set_nat @ X @ A )
= bot_bot_set_nat ) ) ) ) ).
% Union_disjoint
thf(fact_605_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_606_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_607_Union__Int__subset,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 ) ) ) ).
% Union_Int_subset
thf(fact_608_cSUP__least,axiom,
! [A: set_o,F: $o > nat,M3: nat] :
( ( A != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ M3 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ A ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_609_cSUP__least,axiom,
! [A: set_nat,F: nat > nat,M3: nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ M3 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_610_cSUP__least,axiom,
! [A: set_o,F: $o > $o,M3: $o] :
( ( A != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_o @ ( F @ X4 ) @ M3 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_611_cSUP__least,axiom,
! [A: set_nat,F: nat > $o,M3: $o] :
( ( A != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_less_eq_o @ ( F @ X4 ) @ M3 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_612_cSUP__least,axiom,
! [A: set_o,F: $o > set_nat,M3: set_nat] :
( ( A != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ M3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_613_cSUP__least,axiom,
! [A: set_nat,F: nat > set_nat,M3: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ M3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_614_cSUP__least,axiom,
! [A: set_set_nat,F: set_nat > nat,M3: nat] :
( ( A != bot_bot_set_set_nat )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ M3 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_set_nat_nat @ F @ A ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_615_cSUP__least,axiom,
! [A: set_set_nat,F: set_nat > $o,M3: $o] :
( ( A != bot_bot_set_set_nat )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ord_less_eq_o @ ( F @ X4 ) @ M3 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ A ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_616_cSUP__least,axiom,
! [A: set_set_nat,F: set_nat > set_nat,M3: set_nat] :
( ( A != bot_bot_set_set_nat )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ M3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_617_cSUP__least,axiom,
! [A: set_nat_nat,F: ( nat > nat ) > nat,M3: nat] :
( ( A != bot_bot_set_nat_nat )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ M3 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat_nat @ F @ A ) ) @ M3 ) ) ) ).
% cSUP_least
thf(fact_618_SUP__eq__iff,axiom,
! [I4: set_set_nat,C: set_nat,F: set_nat > set_nat] :
( ( I4 != bot_bot_set_set_nat )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
=> ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ I4 ) )
= C )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ I4 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_619_SUP__eq__iff,axiom,
! [I4: set_o,C: set_nat,F: $o > set_nat] :
( ( I4 != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I4 )
=> ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ I4 ) )
= C )
= ( ! [X: $o] :
( ( member_o @ X @ I4 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_620_SUP__eq__iff,axiom,
! [I4: set_nat_nat,C: set_nat,F: ( nat > nat ) > set_nat] :
( ( I4 != bot_bot_set_nat_nat )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I4 )
=> ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ I4 ) )
= C )
= ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ I4 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_621_SUP__eq__iff,axiom,
! [I4: set_nat,C: set_nat,F: nat > set_nat] :
( ( I4 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ I4 ) )
= C )
= ( ! [X: nat] :
( ( member_nat @ X @ I4 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_622_SUP__eq__iff,axiom,
! [I4: set_set_nat,C: $o,F: set_nat > $o] :
( ( I4 != bot_bot_set_set_nat )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
=> ( ord_less_eq_o @ C @ ( F @ I3 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ I4 ) )
= C )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ I4 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_623_SUP__eq__iff,axiom,
! [I4: set_o,C: $o,F: $o > $o] :
( ( I4 != bot_bot_set_o )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I4 )
=> ( ord_less_eq_o @ C @ ( F @ I3 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ I4 ) )
= C )
= ( ! [X: $o] :
( ( member_o @ X @ I4 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_624_SUP__eq__iff,axiom,
! [I4: set_nat_nat,C: $o,F: ( nat > nat ) > $o] :
( ( I4 != bot_bot_set_nat_nat )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I4 )
=> ( ord_less_eq_o @ C @ ( F @ I3 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_nat_nat_o @ F @ I4 ) )
= C )
= ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ I4 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_625_SUP__eq__iff,axiom,
! [I4: set_nat,C: $o,F: nat > $o] :
( ( I4 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ord_less_eq_o @ C @ ( F @ I3 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ I4 ) )
= C )
= ( ! [X: nat] :
( ( member_nat @ X @ I4 )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_626_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_627_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_628_sup__inf__absorb,axiom,
! [X3: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ X3 @ Y ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_629_inf__sup__absorb,axiom,
! [X3: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_630_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_631_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_632_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_633_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_634_inf_Oidem,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_635_inf__idem,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ X3 @ X3 )
= X3 ) ).
% inf_idem
thf(fact_636_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_637_inf__left__idem,axiom,
! [X3: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ X3 @ Y ) )
= ( inf_inf_set_nat @ X3 @ Y ) ) ).
% inf_left_idem
thf(fact_638_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_639_inf__right__idem,axiom,
! [X3: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ Y )
= ( inf_inf_set_nat @ X3 @ Y ) ) ).
% inf_right_idem
thf(fact_640_sup_Oidem,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_641_sup__idem,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_642_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_643_sup__left__idem,axiom,
! [X3: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y ) )
= ( sup_sup_set_nat @ X3 @ Y ) ) ).
% sup_left_idem
thf(fact_644_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_645_le__inf__iff,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z2 ) )
= ( ( ord_less_eq_set_nat @ X3 @ Y )
& ( ord_less_eq_set_nat @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_646_le__inf__iff,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y @ Z2 ) )
= ( ( ord_less_eq_nat @ X3 @ Y )
& ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_647_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_648_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_649_le__sup__iff,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X3 @ Y ) @ Z2 )
= ( ( ord_less_eq_set_nat @ X3 @ Z2 )
& ( ord_less_eq_set_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_650_le__sup__iff,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X3 @ Z2 )
& ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_651_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_652_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_653_inf__bot__left,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X3 )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_654_inf__bot__right,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ X3 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_655_sup__bot__left,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_656_sup__bot__right,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ bot_bot_set_nat )
= X3 ) ).
% sup_bot_right
thf(fact_657_bot__eq__sup__iff,axiom,
! [X3: set_nat,Y: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X3 @ Y ) )
= ( ( X3 = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_658_sup__eq__bot__iff,axiom,
! [X3: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X3 @ Y )
= bot_bot_set_nat )
= ( ( X3 = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_659_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_660_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_661_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_662_le__cases3,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_663_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y7: nat,Z5: nat] : ( Y7 = Z5 ) )
= ( ^ [X: nat,Y6: nat] :
( ( ord_less_eq_nat @ X @ Y6 )
& ( ord_less_eq_nat @ Y6 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_664_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_665_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_666_order__antisym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_667_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_668_order__trans,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_669_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A7: nat,B6: nat] :
( ( ord_less_eq_nat @ A7 @ B6 )
=> ( P @ A7 @ B6 ) )
=> ( ! [A7: nat,B6: nat] :
( ( P @ B6 @ A7 )
=> ( P @ A7 @ B6 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_670_dual__order_Oeq__iff,axiom,
( ( ^ [Y7: nat,Z5: nat] : ( Y7 = Z5 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_671_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_672_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_673_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_674_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y7: nat,Z5: nat] : ( Y7 = Z5 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_675_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 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_676_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 )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_677_order__eq__refl,axiom,
! [X3: nat,Y: nat] :
( ( X3 = Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_678_linorder__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_679_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_680_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_681_linorder__le__cases,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_682_order__antisym__conv,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_683_inf__sup__aci_I4_J,axiom,
! [X3: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ X3 @ Y ) )
= ( inf_inf_set_nat @ X3 @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_684_inf__sup__aci_I3_J,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z2 ) )
= ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_685_inf__sup__aci_I2_J,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ Z2 )
= ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_686_inf__sup__aci_I1_J,axiom,
( inf_inf_set_nat
= ( ^ [X: set_nat,Y6: set_nat] : ( inf_inf_set_nat @ Y6 @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_687_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_688_inf__assoc,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ Z2 )
= ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z2 ) ) ) ).
% inf_assoc
thf(fact_689_inf_Ocommute,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( inf_inf_set_nat @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_690_inf__commute,axiom,
( inf_inf_set_nat
= ( ^ [X: set_nat,Y6: set_nat] : ( inf_inf_set_nat @ Y6 @ X ) ) ) ).
% inf_commute
thf(fact_691_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_692_inf__left__commute,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z2 ) )
= ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_693_inf__sup__aci_I8_J,axiom,
! [X3: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y ) )
= ( sup_sup_set_nat @ X3 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_694_inf__sup__aci_I7_J,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y @ Z2 ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_695_inf__sup__aci_I6_J,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X3 @ Y ) @ Z2 )
= ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_696_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X: set_nat,Y6: set_nat] : ( sup_sup_set_nat @ Y6 @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_697_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_698_sup__assoc,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X3 @ Y ) @ Z2 )
= ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_699_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_700_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X: set_nat,Y6: set_nat] : ( sup_sup_set_nat @ Y6 @ X ) ) ) ).
% sup_commute
thf(fact_701_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_702_sup__left__commute,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y @ Z2 ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_703_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_704_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_705_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_706_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_707_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_708_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_709_inf__sup__ord_I2_J,axiom,
! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_710_inf__sup__ord_I2_J,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_711_inf__sup__ord_I1_J,axiom,
! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_712_inf__sup__ord_I1_J,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_713_inf__le1,axiom,
! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ X3 ) ).
% inf_le1
thf(fact_714_inf__le1,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ X3 ) ).
% inf_le1
thf(fact_715_inf__le2,axiom,
! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_716_inf__le2,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_717_le__infE,axiom,
! [X3: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_nat @ X3 @ A2 )
=> ~ ( ord_less_eq_set_nat @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_718_le__infE,axiom,
! [X3: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X3 @ A2 )
=> ~ ( ord_less_eq_nat @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_719_le__infI,axiom,
! [X3: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ A2 )
=> ( ( ord_less_eq_set_nat @ X3 @ B2 )
=> ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_720_le__infI,axiom,
! [X3: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X3 @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ B2 )
=> ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_721_inf__mono,axiom,
! [A2: set_nat,C: set_nat,B2: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ D )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_722_inf__mono,axiom,
! [A2: nat,C: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_723_le__infI1,axiom,
! [A2: set_nat,X3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X3 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_724_le__infI1,axiom,
! [A2: nat,X3: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_725_le__infI2,axiom,
! [B2: set_nat,X3: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ X3 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_726_le__infI2,axiom,
! [B2: nat,X3: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ X3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_727_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_728_inf_OorderE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_729_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_730_inf_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_731_inf__unique,axiom,
! [F: set_nat > set_nat > set_nat,X3: set_nat,Y: set_nat] :
( ! [X4: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( F @ X4 @ Y3 ) @ X4 )
=> ( ! [X4: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( F @ X4 @ Y3 ) @ Y3 )
=> ( ! [X4: set_nat,Y3: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_set_nat @ X4 @ Z )
=> ( ord_less_eq_set_nat @ X4 @ ( F @ Y3 @ Z ) ) ) )
=> ( ( inf_inf_set_nat @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_732_inf__unique,axiom,
! [F: nat > nat > nat,X3: nat,Y: nat] :
( ! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X4 @ Y3 ) @ X4 )
=> ( ! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X4 @ Y3 ) @ Y3 )
=> ( ! [X4: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_nat @ X4 @ Z )
=> ( ord_less_eq_nat @ X4 @ ( F @ Y3 @ Z ) ) ) )
=> ( ( inf_inf_nat @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_733_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X: set_nat,Y6: set_nat] :
( ( inf_inf_set_nat @ X @ Y6 )
= X ) ) ) ).
% le_iff_inf
thf(fact_734_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y6: nat] :
( ( inf_inf_nat @ X @ Y6 )
= X ) ) ) ).
% le_iff_inf
thf(fact_735_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_736_inf_Oabsorb1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_737_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_738_inf_Oabsorb2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_739_inf__absorb1,axiom,
! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ( inf_inf_set_nat @ X3 @ Y )
= X3 ) ) ).
% inf_absorb1
thf(fact_740_inf__absorb1,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( inf_inf_nat @ X3 @ Y )
= X3 ) ) ).
% inf_absorb1
thf(fact_741_inf__absorb2,axiom,
! [Y: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X3 )
=> ( ( inf_inf_set_nat @ X3 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_742_inf__absorb2,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( inf_inf_nat @ X3 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_743_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_744_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_745_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_746_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_747_inf__greatest,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ( ord_less_eq_set_nat @ X3 @ Z2 )
=> ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_748_inf__greatest,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ X3 @ Z2 )
=> ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_749_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_750_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( A3
= ( inf_inf_nat @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_751_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_752_inf_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_753_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_754_inf_Ocobounded2,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_755_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_756_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( inf_inf_nat @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_757_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_758_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( inf_inf_nat @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_759_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_760_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_761_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_762_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_763_inf__sup__ord_I4_J,axiom,
! [Y: set_nat,X3: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_764_inf__sup__ord_I4_J,axiom,
! [Y: nat,X3: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_765_inf__sup__ord_I3_J,axiom,
! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_766_inf__sup__ord_I3_J,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_767_le__supE,axiom,
! [A2: set_nat,B2: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ X3 )
=> ~ ( ord_less_eq_set_nat @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_768_le__supE,axiom,
! [A2: nat,B2: nat,X3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_less_eq_nat @ A2 @ X3 )
=> ~ ( ord_less_eq_nat @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_769_le__supI,axiom,
! [A2: set_nat,X3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X3 )
=> ( ( ord_less_eq_set_nat @ B2 @ X3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_770_le__supI,axiom,
! [A2: nat,X3: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X3 )
=> ( ( ord_less_eq_nat @ B2 @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_771_sup__ge1,axiom,
! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_772_sup__ge1,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_773_sup__ge2,axiom,
! [Y: set_nat,X3: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_774_sup__ge2,axiom,
! [Y: nat,X3: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_775_le__supI1,axiom,
! [X3: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_776_le__supI1,axiom,
! [X3: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_777_le__supI2,axiom,
! [X3: set_nat,B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ B2 )
=> ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_778_le__supI2,axiom,
! [X3: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ X3 @ B2 )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_779_sup_Omono,axiom,
! [C: set_nat,A2: set_nat,D: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ( ord_less_eq_set_nat @ D @ B2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D ) @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_780_sup_Omono,axiom,
! [C: nat,A2: nat,D: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_781_sup__mono,axiom,
! [A2: set_nat,C: set_nat,B2: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_782_sup__mono,axiom,
! [A2: nat,C: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_783_sup__least,axiom,
! [Y: set_nat,X3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X3 )
=> ( ( ord_less_eq_set_nat @ Z2 @ X3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_784_sup__least,axiom,
! [Y: nat,X3: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ Z2 @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_785_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X: set_nat,Y6: set_nat] :
( ( sup_sup_set_nat @ X @ Y6 )
= Y6 ) ) ) ).
% le_iff_sup
thf(fact_786_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y6: nat] :
( ( sup_sup_nat @ X @ Y6 )
= Y6 ) ) ) ).
% le_iff_sup
thf(fact_787_sup_OorderE,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_788_sup_OorderE,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_789_sup_OorderI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2
= ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_790_sup_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_791_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X3: set_nat,Y: set_nat] :
( ! [X4: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X4 @ ( F @ X4 @ Y3 ) )
=> ( ! [X4: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( F @ X4 @ Y3 ) )
=> ( ! [X4: set_nat,Y3: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X4 )
=> ( ( ord_less_eq_set_nat @ Z @ X4 )
=> ( ord_less_eq_set_nat @ ( F @ Y3 @ Z ) @ X4 ) ) )
=> ( ( sup_sup_set_nat @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_792_sup__unique,axiom,
! [F: nat > nat > nat,X3: nat,Y: nat] :
( ! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ X4 @ ( F @ X4 @ Y3 ) )
=> ( ! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F @ X4 @ Y3 ) )
=> ( ! [X4: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ Y3 @ X4 )
=> ( ( ord_less_eq_nat @ Z @ X4 )
=> ( ord_less_eq_nat @ ( F @ Y3 @ Z ) @ X4 ) ) )
=> ( ( sup_sup_nat @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_793_sup_Oabsorb1,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_794_sup_Oabsorb1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_795_sup_Oabsorb2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_796_sup_Oabsorb2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_797_sup__absorb1,axiom,
! [Y: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X3 )
=> ( ( sup_sup_set_nat @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_798_sup__absorb1,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( sup_sup_nat @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_799_sup__absorb2,axiom,
! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ( sup_sup_set_nat @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_800_sup__absorb2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( sup_sup_nat @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_801_sup_OboundedE,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.boundedE
thf(fact_802_sup_OboundedE,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.boundedE
thf(fact_803_sup_OboundedI,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_804_sup_OboundedI,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_805_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( A3
= ( sup_sup_set_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_806_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( A3
= ( sup_sup_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_807_sup_Ocobounded1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_808_sup_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_809_sup_Ocobounded2,axiom,
! [B2: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_810_sup_Ocobounded2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_811_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_812_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( sup_sup_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_813_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_814_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( sup_sup_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_815_sup_OcoboundedI1,axiom,
! [C: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_816_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_817_sup_OcoboundedI2,axiom,
! [C: set_nat,B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_818_sup_OcoboundedI2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_819_distrib__imp1,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ! [X4: set_nat,Y3: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X4 @ ( sup_sup_set_nat @ Y3 @ Z ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X4 @ Y3 ) @ ( inf_inf_set_nat @ X4 @ Z ) ) )
=> ( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_820_distrib__imp2,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ! [X4: set_nat,Y3: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X4 @ ( inf_inf_set_nat @ Y3 @ Z ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X4 @ Y3 ) @ ( sup_sup_set_nat @ X4 @ Z ) ) )
=> ( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y @ Z2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_821_inf__sup__distrib1,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y @ Z2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_822_inf__sup__distrib2,axiom,
! [Y: set_nat,Z2: set_nat,X3: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ Z2 ) @ X3 )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ X3 ) @ ( inf_inf_set_nat @ Z2 @ X3 ) ) ) ).
% inf_sup_distrib2
thf(fact_823_sup__inf__distrib1,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_824_sup__inf__distrib2,axiom,
! [Y: set_nat,Z2: set_nat,X3: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ Z2 ) @ X3 )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ X3 ) @ ( sup_sup_set_nat @ Z2 @ X3 ) ) ) ).
% sup_inf_distrib2
thf(fact_825_distrib__inf__le,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) @ ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_826_distrib__inf__le,axiom,
! [X3: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X3 @ Y ) @ ( inf_inf_nat @ X3 @ Z2 ) ) @ ( inf_inf_nat @ X3 @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_827_distrib__sup__le,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z2 ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_828_distrib__sup__le,axiom,
! [X3: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ ( inf_inf_nat @ Y @ Z2 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X3 @ Y ) @ ( sup_sup_nat @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_829_boolean__algebra_Oconj__zero__left,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X3 )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_830_boolean__algebra_Oconj__zero__right,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ X3 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_831_Union__image__empty,axiom,
! [A: set_nat,F: nat > set_nat] :
( ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ bot_bot_set_nat ) ) )
= A ) ).
% Union_image_empty
thf(fact_832_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_833_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_834_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_835_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_836_boolean__algebra_Odisj__zero__right,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ bot_bot_set_nat )
= X3 ) ).
% boolean_algebra.disj_zero_right
thf(fact_837_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_nat,Z2: set_nat,X3: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ Z2 ) @ X3 )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ X3 ) @ ( sup_sup_set_nat @ Z2 @ X3 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_838_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_nat,Z2: set_nat,X3: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ Z2 ) @ X3 )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ X3 ) @ ( inf_inf_set_nat @ Z2 @ X3 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_839_boolean__algebra_Odisj__conj__distrib,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_840_boolean__algebra_Oconj__disj__distrib,axiom,
! [X3: set_nat,Y: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y @ Z2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_841_diff__shunt__var,axiom,
! [X3: set_nat,Y: set_nat] :
( ( ( minus_minus_set_nat @ X3 @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X3 @ Y ) ) ).
% diff_shunt_var
thf(fact_842_disjoint__family__on__bisimulation,axiom,
! [F: set_nat > set_nat,S: set_set_nat,G: set_nat > set_nat] :
( ( disjoi2115144663756723504at_nat @ F @ S )
=> ( ! [N2: set_nat,M: set_nat] :
( ( member_set_nat @ N2 @ S )
=> ( ( member_set_nat @ M @ S )
=> ( ( N2 != M )
=> ( ( ( inf_inf_set_nat @ ( F @ N2 ) @ ( F @ M ) )
= bot_bot_set_nat )
=> ( ( inf_inf_set_nat @ ( G @ N2 ) @ ( G @ M ) )
= bot_bot_set_nat ) ) ) ) )
=> ( disjoi2115144663756723504at_nat @ G @ S ) ) ) ).
% disjoint_family_on_bisimulation
thf(fact_843_disjoint__family__on__bisimulation,axiom,
! [F: $o > set_nat,S: set_o,G: $o > set_nat] :
( ( disjoi7928754725229124240_o_nat @ F @ S )
=> ( ! [N2: $o,M: $o] :
( ( member_o @ N2 @ S )
=> ( ( member_o @ M @ S )
=> ( ( N2 != M )
=> ( ( ( inf_inf_set_nat @ ( F @ N2 ) @ ( F @ M ) )
= bot_bot_set_nat )
=> ( ( inf_inf_set_nat @ ( G @ N2 ) @ ( G @ M ) )
= bot_bot_set_nat ) ) ) ) )
=> ( disjoi7928754725229124240_o_nat @ G @ S ) ) ) ).
% disjoint_family_on_bisimulation
thf(fact_844_disjoint__family__on__bisimulation,axiom,
! [F: ( nat > nat ) > set_nat,S: set_nat_nat,G: ( nat > nat ) > set_nat] :
( ( disjoi831272138528337257at_nat @ F @ S )
=> ( ! [N2: nat > nat,M: nat > nat] :
( ( member_nat_nat @ N2 @ S )
=> ( ( member_nat_nat @ M @ S )
=> ( ( N2 != M )
=> ( ( ( inf_inf_set_nat @ ( F @ N2 ) @ ( F @ M ) )
= bot_bot_set_nat )
=> ( ( inf_inf_set_nat @ ( G @ N2 ) @ ( G @ M ) )
= bot_bot_set_nat ) ) ) ) )
=> ( disjoi831272138528337257at_nat @ G @ S ) ) ) ).
% disjoint_family_on_bisimulation
thf(fact_845_disjoint__family__on__bisimulation,axiom,
! [F: nat > set_nat,S: set_nat,G: nat > set_nat] :
( ( disjoi6798895846410478970at_nat @ F @ S )
=> ( ! [N2: nat,M: nat] :
( ( member_nat @ N2 @ S )
=> ( ( member_nat @ M @ S )
=> ( ( N2 != M )
=> ( ( ( inf_inf_set_nat @ ( F @ N2 ) @ ( F @ M ) )
= bot_bot_set_nat )
=> ( ( inf_inf_set_nat @ ( G @ N2 ) @ ( G @ M ) )
= bot_bot_set_nat ) ) ) ) )
=> ( disjoi6798895846410478970at_nat @ G @ S ) ) ) ).
% disjoint_family_on_bisimulation
thf(fact_846_disjoint__family__on__def,axiom,
( disjoi6798895846410478970at_nat
= ( ^ [A4: nat > set_nat,S2: set_nat] :
! [X: nat] :
( ( member_nat @ X @ S2 )
=> ! [Y6: nat] :
( ( member_nat @ Y6 @ S2 )
=> ( ( X != Y6 )
=> ( ( inf_inf_set_nat @ ( A4 @ X ) @ ( A4 @ Y6 ) )
= bot_bot_set_nat ) ) ) ) ) ) ).
% disjoint_family_on_def
thf(fact_847_disjoint__family__onD,axiom,
! [A: set_nat > set_nat,I4: set_set_nat,I2: set_nat,J: set_nat] :
( ( disjoi2115144663756723504at_nat @ A @ I4 )
=> ( ( member_set_nat @ I2 @ I4 )
=> ( ( member_set_nat @ J @ I4 )
=> ( ( I2 != J )
=> ( ( inf_inf_set_nat @ ( A @ I2 ) @ ( A @ J ) )
= bot_bot_set_nat ) ) ) ) ) ).
% disjoint_family_onD
thf(fact_848_disjoint__family__onD,axiom,
! [A: $o > set_nat,I4: set_o,I2: $o,J: $o] :
( ( disjoi7928754725229124240_o_nat @ A @ I4 )
=> ( ( member_o @ I2 @ I4 )
=> ( ( member_o @ J @ I4 )
=> ( ( I2 != J )
=> ( ( inf_inf_set_nat @ ( A @ I2 ) @ ( A @ J ) )
= bot_bot_set_nat ) ) ) ) ) ).
% disjoint_family_onD
thf(fact_849_disjoint__family__onD,axiom,
! [A: ( nat > nat ) > set_nat,I4: set_nat_nat,I2: nat > nat,J: nat > nat] :
( ( disjoi831272138528337257at_nat @ A @ I4 )
=> ( ( member_nat_nat @ I2 @ I4 )
=> ( ( member_nat_nat @ J @ I4 )
=> ( ( I2 != J )
=> ( ( inf_inf_set_nat @ ( A @ I2 ) @ ( A @ J ) )
= bot_bot_set_nat ) ) ) ) ) ).
% disjoint_family_onD
thf(fact_850_disjoint__family__onD,axiom,
! [A: nat > set_nat,I4: set_nat,I2: nat,J: nat] :
( ( disjoi6798895846410478970at_nat @ A @ I4 )
=> ( ( member_nat @ I2 @ I4 )
=> ( ( member_nat @ J @ I4 )
=> ( ( I2 != J )
=> ( ( inf_inf_set_nat @ ( A @ I2 ) @ ( A @ J ) )
= bot_bot_set_nat ) ) ) ) ) ).
% disjoint_family_onD
thf(fact_851_disjoint__family__on__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( disjoi6798895846410478970at_nat @ F @ B )
=> ( disjoi6798895846410478970at_nat @ F @ A ) ) ) ).
% disjoint_family_on_mono
thf(fact_852_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_853_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_854_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 )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ B )
=> ( ( inf_inf_set_nat @ X @ A2 )
= bot_bot_set_nat ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_855_Sup__inf__eq__bot__iff,axiom,
! [B: set_o,A2: $o] :
( ( ( inf_inf_o @ ( complete_Sup_Sup_o @ B ) @ A2 )
= bot_bot_o )
= ( ! [X: $o] :
( ( member_o @ X @ B )
=> ( ( inf_inf_o @ X @ A2 )
= bot_bot_o ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_856_fact2,axiom,
( ( inf_inf_set_nat @ ( bl @ zero_zero_nat )
@ ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( hales_set_incr @ n2 @ ( bs @ I ) )
@ ( set_ord_lessThan_nat @ k ) ) ) )
= bot_bot_set_nat ) ).
% fact2
thf(fact_857_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_858_bot__empty__eq,axiom,
( bot_bot_set_nat_o
= ( ^ [X: set_nat] : ( member_set_nat @ X @ bot_bot_set_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_859_bot__empty__eq,axiom,
( bot_bot_o_o
= ( ^ [X: $o] : ( member_o @ X @ bot_bot_set_o ) ) ) ).
% bot_empty_eq
thf(fact_860_bot__empty__eq,axiom,
( bot_bot_nat_nat_o
= ( ^ [X: nat > nat] : ( member_nat_nat @ X @ bot_bot_set_nat_nat ) ) ) ).
% bot_empty_eq
thf(fact_861_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_862_image__ident,axiom,
! [Y5: set_nat] :
( ( image_nat_nat
@ ^ [X: nat] : X
@ Y5 )
= Y5 ) ).
% image_ident
thf(fact_863_SUP__identity__eq,axiom,
! [A: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [X: set_nat] : X
@ A ) )
= ( comple7399068483239264473et_nat @ A ) ) ).
% SUP_identity_eq
thf(fact_864_SUP__identity__eq,axiom,
! [A: set_nat] :
( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X: nat] : X
@ A ) )
= ( complete_Sup_Sup_nat @ A ) ) ).
% SUP_identity_eq
thf(fact_865_SUP__identity__eq,axiom,
! [A: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X: $o] : X
@ A ) )
= ( complete_Sup_Sup_o @ A ) ) ).
% SUP_identity_eq
thf(fact_866_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_867_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_868_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_869_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_870_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_871_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_872_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_873_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_874_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_875_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_876_UN__iff,axiom,
! [B2: nat,B: nat > set_nat,A: set_nat] :
( ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A )
& ( member_nat @ B2 @ ( B @ X ) ) ) ) ) ).
% UN_iff
thf(fact_877_SUP__bot,axiom,
! [A: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : bot_bot_set_nat
@ A ) )
= bot_bot_set_nat ) ).
% SUP_bot
thf(fact_878_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 )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( B @ X )
= bot_bot_set_nat ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_879_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 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( B @ X )
= bot_bot_set_nat ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_880_cSUP__const,axiom,
! [A: set_nat,C: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : C
@ A ) )
= C ) ) ).
% cSUP_const
thf(fact_881_cSUP__const,axiom,
! [A: set_nat,C: nat] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X: nat] : C
@ A ) )
= C ) ) ).
% cSUP_const
thf(fact_882_cSUP__const,axiom,
! [A: set_nat,C: $o] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [X: nat] : C
@ A ) )
= C ) ) ).
% cSUP_const
thf(fact_883_SUP__const,axiom,
! [A: set_nat,F: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_884_SUP__const,axiom,
! [A: set_nat,F: $o] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [I: nat] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_885_if__image__distrib,axiom,
! [P: nat > $o,F: nat > set_nat,G: nat > set_nat,S: set_nat] :
( ( image_nat_set_nat
@ ^ [X: nat] : ( if_set_nat @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ 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
@ ^ [X: nat] :
~ ( P @ X ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_886_if__image__distrib,axiom,
! [P: nat > $o,F: nat > nat,G: nat > nat,S: set_nat] :
( ( image_nat_nat
@ ^ [X: nat] : ( if_nat @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ 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
@ ^ [X: nat] :
~ ( P @ X ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_887_UN__constant,axiom,
! [A: set_nat,C: set_nat] :
( ( ( A = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y6: nat] : C
@ A ) )
= bot_bot_set_nat ) )
& ( ( A != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y6: nat] : C
@ A ) )
= C ) ) ) ).
% UN_constant
thf(fact_888_UN__Un,axiom,
! [M3: nat > set_nat,A: set_nat,B: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M3 @ ( sup_sup_set_nat @ A @ B ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M3 @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M3 @ B ) ) ) ) ).
% UN_Un
thf(fact_889_UN__simps_I2_J,axiom,
! [C4: set_nat,A: nat > set_nat,B: set_nat] :
( ( ( C4 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( sup_sup_set_nat @ ( A @ X ) @ B )
@ C4 ) )
= bot_bot_set_nat ) )
& ( ( C4 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( sup_sup_set_nat @ ( A @ X ) @ B )
@ C4 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ C4 ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_890_UN__simps_I3_J,axiom,
! [C4: set_nat,A: set_nat,B: nat > set_nat] :
( ( ( C4 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( sup_sup_set_nat @ A @ ( B @ X ) )
@ C4 ) )
= bot_bot_set_nat ) )
& ( ( C4 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( sup_sup_set_nat @ A @ ( B @ X ) )
@ C4 ) )
= ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ C4 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_891_SUP__inf,axiom,
! [F: nat > set_nat,B: set_nat,A2: set_nat] :
( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ B ) ) @ A2 )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [B3: nat] : ( inf_inf_set_nat @ ( F @ B3 ) @ A2 )
@ B ) ) ) ).
% SUP_inf
thf(fact_892_Sup__inf,axiom,
! [B: set_set_nat,A2: set_nat] :
( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ B ) @ A2 )
= ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [B3: set_nat] : ( inf_inf_set_nat @ B3 @ A2 )
@ B ) ) ) ).
% Sup_inf
thf(fact_893_Sup__inf,axiom,
! [B: set_o,A2: $o] :
( ( inf_inf_o @ ( complete_Sup_Sup_o @ B ) @ A2 )
= ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [B3: $o] : ( inf_inf_o @ B3 @ A2 )
@ B ) ) ) ).
% Sup_inf
thf(fact_894_inf__SUP,axiom,
! [A2: set_nat,F: nat > set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ B ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [B3: nat] : ( inf_inf_set_nat @ A2 @ ( F @ B3 ) )
@ B ) ) ) ).
% inf_SUP
thf(fact_895_SUP__inf__distrib2,axiom,
! [F: nat > set_nat,A: set_nat,G: nat > set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A3: nat] :
( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [B3: nat] : ( inf_inf_set_nat @ ( F @ A3 ) @ ( G @ B3 ) )
@ B ) )
@ A ) ) ) ).
% SUP_inf_distrib2
thf(fact_896_Int__Union,axiom,
! [A: set_nat,B: set_set_nat] :
( ( inf_inf_set_nat @ A @ ( comple7399068483239264473et_nat @ B ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( inf_inf_set_nat @ A ) @ B ) ) ) ).
% Int_Union
thf(fact_897_Int__Union2,axiom,
! [B: set_set_nat,A: set_nat] :
( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ B ) @ A )
= ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [C6: set_nat] : ( inf_inf_set_nat @ C6 @ A )
@ B ) ) ) ).
% Int_Union2
thf(fact_898_Collect__disj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_899_sup__Un__eq,axiom,
! [R: set_set_nat,S: set_set_nat] :
( ( sup_sup_set_nat_o
@ ^ [X: set_nat] : ( member_set_nat @ X @ R )
@ ^ [X: set_nat] : ( member_set_nat @ X @ S ) )
= ( ^ [X: set_nat] : ( member_set_nat @ X @ ( sup_sup_set_set_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_900_sup__Un__eq,axiom,
! [R: set_o,S: set_o] :
( ( sup_sup_o_o
@ ^ [X: $o] : ( member_o @ X @ R )
@ ^ [X: $o] : ( member_o @ X @ S ) )
= ( ^ [X: $o] : ( member_o @ X @ ( sup_sup_set_o @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_901_sup__Un__eq,axiom,
! [R: set_nat_nat,S: set_nat_nat] :
( ( sup_sup_nat_nat_o
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ R )
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ S ) )
= ( ^ [X: nat > nat] : ( member_nat_nat @ X @ ( sup_sup_set_nat_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_902_sup__Un__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( sup_sup_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R )
@ ^ [X: nat] : ( member_nat @ X @ S ) )
= ( ^ [X: nat] : ( member_nat @ X @ ( sup_sup_set_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_903_sup__set__def,axiom,
( sup_sup_set_set_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( collect_set_nat
@ ( sup_sup_set_nat_o
@ ^ [X: set_nat] : ( member_set_nat @ X @ A4 )
@ ^ [X: set_nat] : ( member_set_nat @ X @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_904_sup__set__def,axiom,
( sup_sup_set_o
= ( ^ [A4: set_o,B4: set_o] :
( collect_o
@ ( sup_sup_o_o
@ ^ [X: $o] : ( member_o @ X @ A4 )
@ ^ [X: $o] : ( member_o @ X @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_905_sup__set__def,axiom,
( sup_sup_set_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( collect_nat_nat
@ ( sup_sup_nat_nat_o
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ A4 )
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_906_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A4 )
@ ^ [X: nat] : ( member_nat @ X @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_907_Un__def,axiom,
( sup_sup_set_set_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ A4 )
| ( member_set_nat @ X @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_908_Un__def,axiom,
( sup_sup_set_o
= ( ^ [A4: set_o,B4: set_o] :
( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A4 )
| ( member_o @ X @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_909_Un__def,axiom,
( sup_sup_set_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ A4 )
| ( member_nat_nat @ X @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_910_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A4 )
| ( member_nat @ X @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_911_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X: nat] : ( ord_less_eq_nat @ X @ U2 ) ) ) ) ).
% atMost_def
thf(fact_912_inf__Int__eq,axiom,
! [R: set_set_nat,S: set_set_nat] :
( ( inf_inf_set_nat_o
@ ^ [X: set_nat] : ( member_set_nat @ X @ R )
@ ^ [X: set_nat] : ( member_set_nat @ X @ S ) )
= ( ^ [X: set_nat] : ( member_set_nat @ X @ ( inf_inf_set_set_nat @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_913_inf__Int__eq,axiom,
! [R: set_o,S: set_o] :
( ( inf_inf_o_o
@ ^ [X: $o] : ( member_o @ X @ R )
@ ^ [X: $o] : ( member_o @ X @ S ) )
= ( ^ [X: $o] : ( member_o @ X @ ( inf_inf_set_o @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_914_inf__Int__eq,axiom,
! [R: set_nat_nat,S: set_nat_nat] :
( ( inf_inf_nat_nat_o
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ R )
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ S ) )
= ( ^ [X: nat > nat] : ( member_nat_nat @ X @ ( inf_inf_set_nat_nat @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_915_inf__Int__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( inf_inf_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R )
@ ^ [X: nat] : ( member_nat @ X @ S ) )
= ( ^ [X: nat] : ( member_nat @ X @ ( inf_inf_set_nat @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_916_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R )
@ ^ [X: nat] : ( member_nat @ X @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_917_pred__subset__eq,axiom,
! [R: set_set_nat,S: set_set_nat] :
( ( ord_le3964352015994296041_nat_o
@ ^ [X: set_nat] : ( member_set_nat @ X @ R )
@ ^ [X: set_nat] : ( member_set_nat @ X @ S ) )
= ( ord_le6893508408891458716et_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_918_pred__subset__eq,axiom,
! [R: set_o,S: set_o] :
( ( ord_less_eq_o_o
@ ^ [X: $o] : ( member_o @ X @ R )
@ ^ [X: $o] : ( member_o @ X @ S ) )
= ( ord_less_eq_set_o @ R @ S ) ) ).
% pred_subset_eq
thf(fact_919_pred__subset__eq,axiom,
! [R: set_nat_nat,S: set_nat_nat] :
( ( ord_le7366121074344172400_nat_o
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ R )
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ S ) )
= ( ord_le9059583361652607317at_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_920_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A4 )
@ ^ [X: nat] : ( member_nat @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_921_less__eq__set__def,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ord_le3964352015994296041_nat_o
@ ^ [X: set_nat] : ( member_set_nat @ X @ A4 )
@ ^ [X: set_nat] : ( member_set_nat @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_922_less__eq__set__def,axiom,
( ord_less_eq_set_o
= ( ^ [A4: set_o,B4: set_o] :
( ord_less_eq_o_o
@ ^ [X: $o] : ( member_o @ X @ A4 )
@ ^ [X: $o] : ( member_o @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_923_less__eq__set__def,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ord_le7366121074344172400_nat_o
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ A4 )
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_924_Collect__conj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_925_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_926_Collect__subset,axiom,
! [A: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_927_Collect__subset,axiom,
! [A: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_928_Collect__subset,axiom,
! [A: set_nat_nat,P: ( nat > nat ) > $o] :
( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_929_inf__set__def,axiom,
( inf_inf_set_set_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( collect_set_nat
@ ( inf_inf_set_nat_o
@ ^ [X: set_nat] : ( member_set_nat @ X @ A4 )
@ ^ [X: set_nat] : ( member_set_nat @ X @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_930_inf__set__def,axiom,
( inf_inf_set_o
= ( ^ [A4: set_o,B4: set_o] :
( collect_o
@ ( inf_inf_o_o
@ ^ [X: $o] : ( member_o @ X @ A4 )
@ ^ [X: $o] : ( member_o @ X @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_931_inf__set__def,axiom,
( inf_inf_set_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( collect_nat_nat
@ ( inf_inf_nat_nat_o
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ A4 )
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_932_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A4 )
@ ^ [X: nat] : ( member_nat @ X @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_933_Int__Collect,axiom,
! [X3: set_nat,A: set_set_nat,P: set_nat > $o] :
( ( member_set_nat @ X3 @ ( inf_inf_set_set_nat @ A @ ( collect_set_nat @ P ) ) )
= ( ( member_set_nat @ X3 @ A )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_934_Int__Collect,axiom,
! [X3: $o,A: set_o,P: $o > $o] :
( ( member_o @ X3 @ ( inf_inf_set_o @ A @ ( collect_o @ P ) ) )
= ( ( member_o @ X3 @ A )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_935_Int__Collect,axiom,
! [X3: nat > nat,A: set_nat_nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ X3 @ ( inf_inf_set_nat_nat @ A @ ( collect_nat_nat @ P ) ) )
= ( ( member_nat_nat @ X3 @ A )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_936_Int__Collect,axiom,
! [X3: nat,A: set_nat,P: nat > $o] :
( ( member_nat @ X3 @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X3 @ A )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_937_Int__def,axiom,
( inf_inf_set_set_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ A4 )
& ( member_set_nat @ X @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_938_Int__def,axiom,
( inf_inf_set_o
= ( ^ [A4: set_o,B4: set_o] :
( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A4 )
& ( member_o @ X @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_939_Int__def,axiom,
( inf_inf_set_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ A4 )
& ( member_nat_nat @ X @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_940_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A4 )
& ( member_nat @ X @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_941_SUP__eqI,axiom,
! [A: set_nat,F: nat > set_nat,X3: set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y3: set_nat] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I5 ) @ Y3 ) )
=> ( ord_less_eq_set_nat @ X3 @ Y3 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_942_SUP__eqI,axiom,
! [A: set_set_nat,F: set_nat > set_nat,X3: set_nat] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y3: set_nat] :
( ! [I5: set_nat] :
( ( member_set_nat @ I5 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I5 ) @ Y3 ) )
=> ( ord_less_eq_set_nat @ X3 @ Y3 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_943_SUP__eqI,axiom,
! [A: set_o,F: $o > set_nat,X3: set_nat] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y3: set_nat] :
( ! [I5: $o] :
( ( member_o @ I5 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I5 ) @ Y3 ) )
=> ( ord_less_eq_set_nat @ X3 @ Y3 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_944_SUP__eqI,axiom,
! [A: set_nat_nat,F: ( nat > nat ) > set_nat,X3: set_nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y3: set_nat] :
( ! [I5: nat > nat] :
( ( member_nat_nat @ I5 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I5 ) @ Y3 ) )
=> ( ord_less_eq_set_nat @ X3 @ Y3 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_945_SUP__eqI,axiom,
! [A: set_nat,F: nat > $o,X3: $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y3: $o] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A )
=> ( ord_less_eq_o @ ( F @ I5 ) @ Y3 ) )
=> ( ord_less_eq_o @ X3 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_946_SUP__eqI,axiom,
! [A: set_set_nat,F: set_nat > $o,X3: $o] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y3: $o] :
( ! [I5: set_nat] :
( ( member_set_nat @ I5 @ A )
=> ( ord_less_eq_o @ ( F @ I5 ) @ Y3 ) )
=> ( ord_less_eq_o @ X3 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_947_SUP__eqI,axiom,
! [A: set_o,F: $o > $o,X3: $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y3: $o] :
( ! [I5: $o] :
( ( member_o @ I5 @ A )
=> ( ord_less_eq_o @ ( F @ I5 ) @ Y3 ) )
=> ( ord_less_eq_o @ X3 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_948_SUP__eqI,axiom,
! [A: set_nat_nat,F: ( nat > nat ) > $o,X3: $o] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ X3 ) )
=> ( ! [Y3: $o] :
( ! [I5: nat > nat] :
( ( member_nat_nat @ I5 @ A )
=> ( ord_less_eq_o @ ( F @ I5 ) @ Y3 ) )
=> ( ord_less_eq_o @ X3 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_nat_o @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_949_SUP__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ! [N2: nat] :
( ( member_nat @ N2 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( G @ X2 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_950_SUP__mono,axiom,
! [A: set_set_nat,B: set_nat,F: set_nat > set_nat,G: nat > set_nat] :
( ! [N2: set_nat] :
( ( member_set_nat @ N2 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( G @ X2 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_951_SUP__mono,axiom,
! [A: set_o,B: set_nat,F: $o > set_nat,G: nat > set_nat] :
( ! [N2: $o] :
( ( member_o @ N2 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( G @ X2 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_952_SUP__mono,axiom,
! [A: set_nat_nat,B: set_nat,F: ( nat > nat ) > set_nat,G: nat > set_nat] :
( ! [N2: nat > nat] :
( ( member_nat_nat @ N2 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( G @ X2 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_953_SUP__least,axiom,
! [A: set_nat,F: nat > set_nat,U: set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ U ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_954_SUP__least,axiom,
! [A: set_set_nat,F: set_nat > set_nat,U: set_nat] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ U ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_955_SUP__least,axiom,
! [A: set_o,F: $o > set_nat,U: set_nat] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ U ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_956_SUP__least,axiom,
! [A: set_nat_nat,F: ( nat > nat ) > set_nat,U: set_nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ U ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_957_SUP__least,axiom,
! [A: set_nat,F: nat > $o,U: $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_958_SUP__least,axiom,
! [A: set_set_nat,F: set_nat > $o,U: $o] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_959_SUP__least,axiom,
! [A: set_o,F: $o > $o,U: $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_960_SUP__least,axiom,
! [A: set_nat_nat,F: ( nat > nat ) > $o,U: $o] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_nat_o @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_961_SUP__mono_H,axiom,
! [F: nat > set_nat,G: nat > set_nat,A: set_nat] :
( ! [X4: nat] : ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A ) ) ) ) ).
% SUP_mono'
thf(fact_962_SUP__upper,axiom,
! [I2: nat,A: set_nat,F: nat > set_nat] :
( ( member_nat @ I2 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I2 ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_963_SUP__upper,axiom,
! [I2: set_nat,A: set_set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ I2 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I2 ) @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_964_SUP__upper,axiom,
! [I2: $o,A: set_o,F: $o > set_nat] :
( ( member_o @ I2 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I2 ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_965_SUP__upper,axiom,
! [I2: nat > nat,A: set_nat_nat,F: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ I2 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I2 ) @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_966_SUP__upper,axiom,
! [I2: nat,A: set_nat,F: nat > $o] :
( ( member_nat @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_967_SUP__upper,axiom,
! [I2: set_nat,A: set_set_nat,F: set_nat > $o] :
( ( member_set_nat @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_968_SUP__upper,axiom,
! [I2: $o,A: set_o,F: $o > $o] :
( ( member_o @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_969_SUP__upper,axiom,
! [I2: nat > nat,A: set_nat_nat,F: ( nat > nat ) > $o] :
( ( member_nat_nat @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ ( complete_Sup_Sup_o @ ( image_nat_nat_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_970_SUP__le__iff,axiom,
! [F: nat > set_nat,A: set_nat,U: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ U )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_971_SUP__upper2,axiom,
! [I2: nat,A: set_nat,U: set_nat,F: nat > set_nat] :
( ( member_nat @ I2 @ A )
=> ( ( ord_less_eq_set_nat @ U @ ( F @ I2 ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_972_SUP__upper2,axiom,
! [I2: set_nat,A: set_set_nat,U: set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ I2 @ A )
=> ( ( ord_less_eq_set_nat @ U @ ( F @ I2 ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_973_SUP__upper2,axiom,
! [I2: $o,A: set_o,U: set_nat,F: $o > set_nat] :
( ( member_o @ I2 @ A )
=> ( ( ord_less_eq_set_nat @ U @ ( F @ I2 ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_974_SUP__upper2,axiom,
! [I2: nat > nat,A: set_nat_nat,U: set_nat,F: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ I2 @ A )
=> ( ( ord_less_eq_set_nat @ U @ ( F @ I2 ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_975_SUP__upper2,axiom,
! [I2: nat,A: set_nat,U: $o,F: nat > $o] :
( ( member_nat @ I2 @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ I2 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_976_SUP__upper2,axiom,
! [I2: set_nat,A: set_set_nat,U: $o,F: set_nat > $o] :
( ( member_set_nat @ I2 @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ I2 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_977_SUP__upper2,axiom,
! [I2: $o,A: set_o,U: $o,F: $o > $o] :
( ( member_o @ I2 @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ I2 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_978_SUP__upper2,axiom,
! [I2: nat > nat,A: set_nat_nat,U: $o,F: ( nat > nat ) > $o] :
( ( member_nat_nat @ I2 @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ I2 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_nat_nat_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_979_SUP__absorb,axiom,
! [K: nat,I4: set_nat,A: nat > set_nat] :
( ( member_nat @ K @ I4 )
=> ( ( sup_sup_set_nat @ ( A @ K ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ I4 ) ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ I4 ) ) ) ) ).
% SUP_absorb
thf(fact_980_SUP__absorb,axiom,
! [K: set_nat,I4: set_set_nat,A: set_nat > set_nat] :
( ( member_set_nat @ K @ I4 )
=> ( ( sup_sup_set_nat @ ( A @ K ) @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ A @ I4 ) ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ A @ I4 ) ) ) ) ).
% SUP_absorb
thf(fact_981_SUP__absorb,axiom,
! [K: $o,I4: set_o,A: $o > set_nat] :
( ( member_o @ K @ I4 )
=> ( ( sup_sup_set_nat @ ( A @ K ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A @ I4 ) ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A @ I4 ) ) ) ) ).
% SUP_absorb
thf(fact_982_SUP__absorb,axiom,
! [K: nat > nat,I4: set_nat_nat,A: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ K @ I4 )
=> ( ( sup_sup_set_nat @ ( A @ K ) @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ A @ I4 ) ) )
= ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ A @ I4 ) ) ) ) ).
% SUP_absorb
thf(fact_983_SUP__absorb,axiom,
! [K: nat,I4: set_nat,A: nat > $o] :
( ( member_nat @ K @ I4 )
=> ( ( sup_sup_o @ ( A @ K ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ A @ I4 ) ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ A @ I4 ) ) ) ) ).
% SUP_absorb
thf(fact_984_SUP__absorb,axiom,
! [K: set_nat,I4: set_set_nat,A: set_nat > $o] :
( ( member_set_nat @ K @ I4 )
=> ( ( sup_sup_o @ ( A @ K ) @ ( complete_Sup_Sup_o @ ( image_set_nat_o @ A @ I4 ) ) )
= ( complete_Sup_Sup_o @ ( image_set_nat_o @ A @ I4 ) ) ) ) ).
% SUP_absorb
thf(fact_985_SUP__absorb,axiom,
! [K: $o,I4: set_o,A: $o > $o] :
( ( member_o @ K @ I4 )
=> ( ( sup_sup_o @ ( A @ K ) @ ( complete_Sup_Sup_o @ ( image_o_o @ A @ I4 ) ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ A @ I4 ) ) ) ) ).
% SUP_absorb
thf(fact_986_SUP__absorb,axiom,
! [K: nat > nat,I4: set_nat_nat,A: ( nat > nat ) > $o] :
( ( member_nat_nat @ K @ I4 )
=> ( ( sup_sup_o @ ( A @ K ) @ ( complete_Sup_Sup_o @ ( image_nat_nat_o @ A @ I4 ) ) )
= ( complete_Sup_Sup_o @ ( image_nat_nat_o @ A @ I4 ) ) ) ) ).
% SUP_absorb
thf(fact_987_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_988_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A4 )
@ ^ [X: nat] : ( member_nat @ X @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_989_minus__set__def,axiom,
( minus_2163939370556025621et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( collect_set_nat
@ ( minus_6910147592129066416_nat_o
@ ^ [X: set_nat] : ( member_set_nat @ X @ A4 )
@ ^ [X: set_nat] : ( member_set_nat @ X @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_990_minus__set__def,axiom,
( minus_minus_set_o
= ( ^ [A4: set_o,B4: set_o] :
( collect_o
@ ( minus_minus_o_o
@ ^ [X: $o] : ( member_o @ X @ A4 )
@ ^ [X: $o] : ( member_o @ X @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_991_minus__set__def,axiom,
( minus_8121590178497047118at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( collect_nat_nat
@ ( minus_167519014754328503_nat_o
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ A4 )
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_992_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A4 )
& ~ ( member_nat @ X @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_993_set__diff__eq,axiom,
( minus_2163939370556025621et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ A4 )
& ~ ( member_set_nat @ X @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_994_set__diff__eq,axiom,
( minus_minus_set_o
= ( ^ [A4: set_o,B4: set_o] :
( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A4 )
& ~ ( member_o @ X @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_995_set__diff__eq,axiom,
( minus_8121590178497047118at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ A4 )
& ~ ( member_nat_nat @ X @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_996_UN__extend__simps_I10_J,axiom,
! [B: set_nat > set_nat,F: nat > set_nat,A: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A3: nat] : ( B @ ( F @ A3 ) )
@ A ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B @ ( image_nat_set_nat @ F @ A ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_997_UN__extend__simps_I10_J,axiom,
! [B: nat > set_nat,F: nat > nat,A: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A3: nat] : ( B @ ( F @ A3 ) )
@ A ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_998_image__UN,axiom,
! [F: nat > set_nat,B: nat > set_nat,A: set_nat] :
( ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
= ( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [X: nat] : ( image_nat_set_nat @ F @ ( B @ X ) )
@ A ) ) ) ).
% image_UN
thf(fact_999_image__UN,axiom,
! [F: nat > nat,B: nat > set_nat,A: set_nat] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( image_nat_nat @ F @ ( B @ X ) )
@ A ) ) ) ).
% image_UN
thf(fact_1000_UN__empty2,axiom,
! [A: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : bot_bot_set_nat
@ A ) )
= bot_bot_set_nat ) ).
% UN_empty2
thf(fact_1001_UN__empty,axiom,
! [B: nat > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% UN_empty
thf(fact_1002_UNION__empty__conv_I1_J,axiom,
! [B: nat > set_nat,A: set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( B @ X )
= bot_bot_set_nat ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_1003_UNION__empty__conv_I2_J,axiom,
! [B: nat > set_nat,A: set_nat] :
( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( B @ X )
= bot_bot_set_nat ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_1004_UN__subset__iff,axiom,
! [A: nat > set_nat,I4: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ I4 ) ) @ B )
= ( ! [X: nat] :
( ( member_nat @ X @ I4 )
=> ( ord_less_eq_set_nat @ ( A @ X ) @ B ) ) ) ) ).
% UN_subset_iff
thf(fact_1005_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_1006_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_1007_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_1008_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_1009_UN__least,axiom,
! [A: set_nat,B: nat > set_nat,C4: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( B @ X4 ) @ C4 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) @ C4 ) ) ).
% UN_least
thf(fact_1010_UN__least,axiom,
! [A: set_set_nat,B: set_nat > set_nat,C4: set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( B @ X4 ) @ C4 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B @ A ) ) @ C4 ) ) ).
% UN_least
thf(fact_1011_UN__least,axiom,
! [A: set_o,B: $o > set_nat,C4: set_nat] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( B @ X4 ) @ C4 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ A ) ) @ C4 ) ) ).
% UN_least
thf(fact_1012_UN__least,axiom,
! [A: set_nat_nat,B: ( nat > nat ) > set_nat,C4: set_nat] :
( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( B @ X4 ) @ C4 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ B @ A ) ) @ C4 ) ) ).
% UN_least
thf(fact_1013_UN__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( 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_1014_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 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_1015_UN__mono,axiom,
! [A: set_o,B: set_o,F: $o > set_nat,G: $o > set_nat] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( 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_1016_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 )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_1017_Int__UN__distrib2,axiom,
! [A: nat > set_nat,I4: set_nat,B: nat > set_nat,J3: set_nat] :
( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ I4 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ J3 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] :
( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [J4: nat] : ( inf_inf_set_nat @ ( A @ I ) @ ( B @ J4 ) )
@ J3 ) )
@ I4 ) ) ) ).
% Int_UN_distrib2
thf(fact_1018_Int__UN__distrib,axiom,
! [B: set_nat,A: nat > set_nat,I4: set_nat] :
( ( inf_inf_set_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ I4 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( inf_inf_set_nat @ B @ ( A @ I ) )
@ I4 ) ) ) ).
% Int_UN_distrib
thf(fact_1019_UN__extend__simps_I4_J,axiom,
! [A: nat > set_nat,C4: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ C4 ) ) @ B )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( inf_inf_set_nat @ ( A @ X ) @ B )
@ C4 ) ) ) ).
% UN_extend_simps(4)
thf(fact_1020_UN__extend__simps_I5_J,axiom,
! [A: set_nat,B: nat > set_nat,C4: set_nat] :
( ( inf_inf_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ C4 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( inf_inf_set_nat @ A @ ( B @ X ) )
@ C4 ) ) ) ).
% UN_extend_simps(5)
thf(fact_1021_UN__extend__simps_I6_J,axiom,
! [A: nat > set_nat,C4: set_nat,B: set_nat] :
( ( minus_minus_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ C4 ) ) @ B )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( minus_minus_set_nat @ ( A @ X ) @ B )
@ C4 ) ) ) ).
% UN_extend_simps(6)
thf(fact_1022_Un__Union__image,axiom,
! [A: nat > set_nat,B: nat > set_nat,C4: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( sup_sup_set_nat @ ( A @ X ) @ ( B @ X ) )
@ C4 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ C4 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ C4 ) ) ) ) ).
% Un_Union_image
thf(fact_1023_UN__Un__distrib,axiom,
! [A: nat > set_nat,B: nat > set_nat,I4: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( sup_sup_set_nat @ ( A @ I ) @ ( B @ I ) )
@ I4 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ I4 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ I4 ) ) ) ) ).
% UN_Un_distrib
thf(fact_1024_UN__absorb,axiom,
! [K: nat,I4: set_nat,A: nat > set_nat] :
( ( member_nat @ K @ I4 )
=> ( ( sup_sup_set_nat @ ( A @ K ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ I4 ) ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ I4 ) ) ) ) ).
% UN_absorb
thf(fact_1025_UN__absorb,axiom,
! [K: set_nat,I4: set_set_nat,A: set_nat > set_nat] :
( ( member_set_nat @ K @ I4 )
=> ( ( sup_sup_set_nat @ ( A @ K ) @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ A @ I4 ) ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ A @ I4 ) ) ) ) ).
% UN_absorb
thf(fact_1026_UN__absorb,axiom,
! [K: $o,I4: set_o,A: $o > set_nat] :
( ( member_o @ K @ I4 )
=> ( ( sup_sup_set_nat @ ( A @ K ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A @ I4 ) ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A @ I4 ) ) ) ) ).
% UN_absorb
thf(fact_1027_UN__absorb,axiom,
! [K: nat > nat,I4: set_nat_nat,A: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ K @ I4 )
=> ( ( sup_sup_set_nat @ ( A @ K ) @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ A @ I4 ) ) )
= ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ A @ I4 ) ) ) ) ).
% UN_absorb
thf(fact_1028_disjoint__UN__iff,axiom,
! [A: set_nat,B: nat > set_nat,I4: set_nat] :
( ( disjnt_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ I4 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ I4 )
=> ( disjnt_nat @ A @ ( B @ X ) ) ) ) ) ).
% disjoint_UN_iff
thf(fact_1029_set__incr__def,axiom,
( hales_set_incr
= ( ^ [N3: nat] :
( image_nat_nat
@ ^ [A3: nat] : ( plus_plus_nat @ A3 @ N3 ) ) ) ) ).
% set_incr_def
thf(fact_1030_set__incr__disjoint__family,axiom,
! [B: nat > set_nat,K: nat,N: nat] :
( ( disjoi6798895846410478970at_nat @ B @ ( set_ord_atMost_nat @ K ) )
=> ( disjoi6798895846410478970at_nat
@ ^ [I: nat] : ( hales_set_incr @ N @ ( B @ I ) )
@ ( set_ord_atMost_nat @ K ) ) ) ).
% set_incr_disjoint_family
thf(fact_1031_SUP__subset__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1032_SUP__subset__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 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1033_SUP__subset__mono,axiom,
! [A: set_o,B: set_o,F: $o > set_nat,G: $o > set_nat] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1034_SUP__subset__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 )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1035_SUP__subset__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > $o,G: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1036_SUP__subset__mono,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_nat > $o,G: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_set_nat_o @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1037_SUP__subset__mono,axiom,
! [A: set_o,B: set_o,F: $o > $o,G: $o > $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_o_o @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1038_SUP__subset__mono,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: ( nat > nat ) > $o,G: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_nat_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_nat_nat_o @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1039_SUP__empty,axiom,
! [F: nat > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% SUP_empty
thf(fact_1040_SUP__empty,axiom,
! [F: nat > $o] :
( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ bot_bot_set_nat ) )
= bot_bot_o ) ).
% SUP_empty
thf(fact_1041_SUP__constant,axiom,
! [A: set_nat,C: set_nat] :
( ( ( A = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y6: nat] : C
@ A ) )
= bot_bot_set_nat ) )
& ( ( A != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y6: nat] : C
@ A ) )
= C ) ) ) ).
% SUP_constant
thf(fact_1042_SUP__constant,axiom,
! [C: $o,A: set_nat] :
( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y6: nat] : C
@ A ) )
= ( ( ( A = bot_bot_set_nat )
=> bot_bot_o )
& ( ( A != bot_bot_set_nat )
=> C ) ) ) ).
% SUP_constant
thf(fact_1043_Compr__image__eq,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1044_Compr__image__eq,axiom,
! [F: $o > nat,A: set_o,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_o_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_o_nat @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1045_Compr__image__eq,axiom,
! [F: nat > $o,A: set_nat,P: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_nat_o @ F @ A ) )
& ( P @ X ) ) )
= ( image_nat_o @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1046_Compr__image__eq,axiom,
! [F: $o > $o,A: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_o_o @ F @ A ) )
& ( P @ X ) ) )
= ( image_o_o @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1047_Compr__image__eq,axiom,
! [F: set_nat > nat,A: set_set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_set_nat_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_set_nat_nat @ F
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1048_Compr__image__eq,axiom,
! [F: nat > set_nat,A: set_nat,P: set_nat > $o] :
( ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ ( image_nat_set_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_nat_set_nat @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1049_Compr__image__eq,axiom,
! [F: $o > set_nat,A: set_o,P: set_nat > $o] :
( ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ ( image_o_set_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_o_set_nat @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1050_Compr__image__eq,axiom,
! [F: set_nat > $o,A: set_set_nat,P: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_set_nat_o @ F @ A ) )
& ( P @ X ) ) )
= ( image_set_nat_o @ F
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1051_Compr__image__eq,axiom,
! [F: ( nat > nat ) > nat,A: set_nat_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_nat_nat_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_nat_nat_nat @ F
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1052_Compr__image__eq,axiom,
! [F: set_nat > set_nat,A: set_set_nat,P: set_nat > $o] :
( ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ ( image_7916887816326733075et_nat @ F @ A ) )
& ( P @ X ) ) )
= ( image_7916887816326733075et_nat @ F
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1053_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
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_1054_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
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_1055_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
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_1056_image__image,axiom,
! [F: nat > nat,G: nat > nat,A: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A ) )
= ( image_nat_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A ) ) ).
% image_image
thf(fact_1057_imageE,axiom,
! [B2: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ~ ! [X4: nat] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member_nat @ X4 @ A ) ) ) ).
% imageE
thf(fact_1058_imageE,axiom,
! [B2: nat,F: $o > nat,A: set_o] :
( ( member_nat @ B2 @ ( image_o_nat @ F @ A ) )
=> ~ ! [X4: $o] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member_o @ X4 @ A ) ) ) ).
% imageE
thf(fact_1059_imageE,axiom,
! [B2: $o,F: nat > $o,A: set_nat] :
( ( member_o @ B2 @ ( image_nat_o @ F @ A ) )
=> ~ ! [X4: nat] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member_nat @ X4 @ A ) ) ) ).
% imageE
thf(fact_1060_imageE,axiom,
! [B2: $o,F: $o > $o,A: set_o] :
( ( member_o @ B2 @ ( image_o_o @ F @ A ) )
=> ~ ! [X4: $o] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member_o @ X4 @ A ) ) ) ).
% imageE
thf(fact_1061_imageE,axiom,
! [B2: nat,F: set_nat > nat,A: set_set_nat] :
( ( member_nat @ B2 @ ( image_set_nat_nat @ F @ A ) )
=> ~ ! [X4: set_nat] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member_set_nat @ X4 @ A ) ) ) ).
% imageE
thf(fact_1062_imageE,axiom,
! [B2: set_nat,F: nat > set_nat,A: set_nat] :
( ( member_set_nat @ B2 @ ( image_nat_set_nat @ F @ A ) )
=> ~ ! [X4: nat] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member_nat @ X4 @ A ) ) ) ).
% imageE
thf(fact_1063_imageE,axiom,
! [B2: set_nat,F: $o > set_nat,A: set_o] :
( ( member_set_nat @ B2 @ ( image_o_set_nat @ F @ A ) )
=> ~ ! [X4: $o] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member_o @ X4 @ A ) ) ) ).
% imageE
thf(fact_1064_imageE,axiom,
! [B2: $o,F: set_nat > $o,A: set_set_nat] :
( ( member_o @ B2 @ ( image_set_nat_o @ F @ A ) )
=> ~ ! [X4: set_nat] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member_set_nat @ X4 @ A ) ) ) ).
% imageE
thf(fact_1065_imageE,axiom,
! [B2: nat,F: ( nat > nat ) > nat,A: set_nat_nat] :
( ( member_nat @ B2 @ ( image_nat_nat_nat @ F @ A ) )
=> ~ ! [X4: nat > nat] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member_nat_nat @ X4 @ A ) ) ) ).
% imageE
thf(fact_1066_imageE,axiom,
! [B2: set_nat,F: set_nat > set_nat,A: set_set_nat] :
( ( member_set_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A ) )
=> ~ ! [X4: set_nat] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member_set_nat @ X4 @ A ) ) ) ).
% imageE
thf(fact_1067_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat > nat,A: set_nat] :
( ( Sup
@ ( image_nat_nat
@ ^ [X: nat] : X
@ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_identity_eq
thf(fact_1068_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat > nat,A: set_nat] :
( ( Inf
@ ( image_nat_nat
@ ^ [X: nat] : X
@ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_identity_eq
thf(fact_1069_UN__extend__simps_I9_J,axiom,
! [C4: nat > set_nat,B: nat > set_nat,A: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C4 @ ( B @ X ) ) )
@ A ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C4 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_1070_UN__extend__simps_I8_J,axiom,
! [B: nat > set_nat,A: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [Y6: set_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ Y6 ) )
@ A ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_1071_SUP__commute,axiom,
! [F: nat > nat > set_nat,B: set_nat,A: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F @ I ) @ B ) )
@ A ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [J4: nat] :
( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( F @ I @ J4 )
@ A ) )
@ B ) ) ) ).
% SUP_commute
thf(fact_1072_SUP__UN__eq,axiom,
! [R2: nat > set_nat,S: set_nat] :
( ( comple8317665133742190828_nat_o
@ ( image_nat_nat_o2
@ ^ [I: nat,X: nat] : ( member_nat @ X @ ( R2 @ I ) )
@ S ) )
= ( ^ [X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ R2 @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_1073_SUP__Sup__eq,axiom,
! [S: set_set_set_nat] :
( ( comple3806919086088850358_nat_o
@ ( image_4331731847045299910_nat_o
@ ^ [I: set_set_nat,X: set_nat] : ( member_set_nat @ X @ I )
@ S ) )
= ( ^ [X: set_nat] : ( member_set_nat @ X @ ( comple548664676211718543et_nat @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1074_SUP__Sup__eq,axiom,
! [S: set_set_o] :
( ( complete_Sup_Sup_o_o
@ ( image_set_o_o_o
@ ^ [I: set_o,X: $o] : ( member_o @ X @ I )
@ S ) )
= ( ^ [X: $o] : ( member_o @ X @ ( comple90263536869209701_set_o @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1075_SUP__Sup__eq,axiom,
! [S: set_set_nat_nat] :
( ( comple8312177224774716605_nat_o
@ ( image_1242417779249009364_nat_o
@ ^ [I: set_nat_nat,X: nat > nat] : ( member_nat_nat @ X @ I )
@ S ) )
= ( ^ [X: nat > nat] : ( member_nat_nat @ X @ ( comple5448282615319421384at_nat @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1076_SUP__Sup__eq,axiom,
! [S: set_set_nat] :
( ( comple8317665133742190828_nat_o
@ ( image_set_nat_nat_o2
@ ^ [I: set_nat,X: nat] : ( member_nat @ X @ I )
@ S ) )
= ( ^ [X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_1077_UN__E,axiom,
! [B2: $o,B: nat > set_o,A: set_nat] :
( ( member_o @ B2 @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B @ A ) ) )
=> ~ ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ~ ( member_o @ B2 @ ( B @ X4 ) ) ) ) ).
% UN_E
thf(fact_1078_UN__E,axiom,
! [B2: $o,B: $o > set_o,A: set_o] :
( ( member_o @ B2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A ) ) )
=> ~ ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ~ ( member_o @ B2 @ ( B @ X4 ) ) ) ) ).
% UN_E
thf(fact_1079_UN__E,axiom,
! [B2: nat,B: nat > set_nat,A: set_nat] :
( ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
=> ~ ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ~ ( member_nat @ B2 @ ( B @ X4 ) ) ) ) ).
% UN_E
thf(fact_1080_UN__E,axiom,
! [B2: nat,B: $o > set_nat,A: set_o] :
( ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ A ) ) )
=> ~ ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ~ ( member_nat @ B2 @ ( B @ X4 ) ) ) ) ).
% UN_E
thf(fact_1081_UN__E,axiom,
! [B2: set_nat,B: nat > set_set_nat,A: set_nat] :
( ( member_set_nat @ B2 @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ B @ A ) ) )
=> ~ ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ~ ( member_set_nat @ B2 @ ( B @ X4 ) ) ) ) ).
% UN_E
thf(fact_1082_UN__E,axiom,
! [B2: set_nat,B: $o > set_set_nat,A: set_o] :
( ( member_set_nat @ B2 @ ( comple548664676211718543et_nat @ ( image_o_set_set_nat @ B @ A ) ) )
=> ~ ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ~ ( member_set_nat @ B2 @ ( B @ X4 ) ) ) ) ).
% UN_E
thf(fact_1083_UN__E,axiom,
! [B2: $o,B: set_nat > set_o,A: set_set_nat] :
( ( member_o @ B2 @ ( comple90263536869209701_set_o @ ( image_set_nat_set_o @ B @ A ) ) )
=> ~ ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ~ ( member_o @ B2 @ ( B @ X4 ) ) ) ) ).
% UN_E
thf(fact_1084_UN__E,axiom,
! [B2: nat,B: set_nat > set_nat,A: set_set_nat] :
( ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B @ A ) ) )
=> ~ ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ~ ( member_nat @ B2 @ ( B @ X4 ) ) ) ) ).
% UN_E
thf(fact_1085_UN__E,axiom,
! [B2: set_nat,B: set_nat > set_set_nat,A: set_set_nat] :
( ( member_set_nat @ B2 @ ( comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ B @ A ) ) )
=> ~ ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ~ ( member_set_nat @ B2 @ ( B @ X4 ) ) ) ) ).
% UN_E
thf(fact_1086_UN__E,axiom,
! [B2: $o,B: ( nat > nat ) > set_o,A: set_nat_nat] :
( ( member_o @ B2 @ ( comple90263536869209701_set_o @ ( image_nat_nat_set_o @ B @ A ) ) )
=> ~ ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
=> ~ ( member_o @ B2 @ ( B @ X4 ) ) ) ) ).
% UN_E
thf(fact_1087_Sup__set__def,axiom,
( comple548664676211718543et_nat
= ( ^ [A4: set_set_set_nat] :
( collect_set_nat
@ ^ [X: set_nat] : ( complete_Sup_Sup_o @ ( image_set_set_nat_o @ ( member_set_nat @ X ) @ A4 ) ) ) ) ) ).
% Sup_set_def
thf(fact_1088_Sup__set__def,axiom,
( comple90263536869209701_set_o
= ( ^ [A4: set_set_o] :
( collect_o
@ ^ [X: $o] : ( complete_Sup_Sup_o @ ( image_set_o_o @ ( member_o @ X ) @ A4 ) ) ) ) ) ).
% Sup_set_def
thf(fact_1089_Sup__set__def,axiom,
( comple5448282615319421384at_nat
= ( ^ [A4: set_set_nat_nat] :
( collect_nat_nat
@ ^ [X: nat > nat] : ( complete_Sup_Sup_o @ ( image_set_nat_nat_o @ ( member_nat_nat @ X ) @ A4 ) ) ) ) ) ).
% Sup_set_def
thf(fact_1090_Sup__set__def,axiom,
( comple7399068483239264473et_nat
= ( ^ [A4: set_set_nat] :
( collect_nat
@ ^ [X: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X ) @ A4 ) ) ) ) ) ).
% Sup_set_def
thf(fact_1091_image__Union,axiom,
! [F: nat > set_nat,S: set_set_nat] :
( ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_1092_image__Union,axiom,
! [F: nat > nat,S: set_set_nat] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_1093_UN__UN__flatten,axiom,
! [C4: nat > set_nat,B: nat > set_nat,A: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C4 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y6: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C4 @ ( B @ Y6 ) ) )
@ A ) ) ) ).
% UN_UN_flatten
thf(fact_1094_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X: nat] : $false ) ) ).
% empty_def
thf(fact_1095_UN__extend__simps_I3_J,axiom,
! [C4: set_nat,A: set_nat,B: nat > set_nat] :
( ( ( C4 = bot_bot_set_nat )
=> ( ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ C4 ) ) )
= A ) )
& ( ( C4 != bot_bot_set_nat )
=> ( ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ C4 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( sup_sup_set_nat @ A @ ( B @ X ) )
@ C4 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_1096_UN__extend__simps_I2_J,axiom,
! [C4: set_nat,A: nat > set_nat,B: set_nat] :
( ( ( C4 = bot_bot_set_nat )
=> ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ C4 ) ) @ B )
= B ) )
& ( ( C4 != bot_bot_set_nat )
=> ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ C4 ) ) @ B )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( sup_sup_set_nat @ ( A @ X ) @ B )
@ C4 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_1097_SUP__UNION,axiom,
! [F: nat > set_nat,G: nat > set_nat,A: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y6: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( G @ Y6 ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_1098_SUP__UNION,axiom,
! [F: nat > $o,G: nat > set_nat,A: set_nat] :
( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y6: nat] : ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( G @ Y6 ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_1099_SUP__union,axiom,
! [M3: nat > set_nat,A: set_nat,B: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M3 @ ( sup_sup_set_nat @ A @ B ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M3 @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M3 @ B ) ) ) ) ).
% SUP_union
thf(fact_1100_SUP__union,axiom,
! [M3: nat > $o,A: set_nat,B: set_nat] :
( ( complete_Sup_Sup_o @ ( image_nat_o @ M3 @ ( sup_sup_set_nat @ A @ B ) ) )
= ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ M3 @ A ) ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ M3 @ B ) ) ) ) ).
% SUP_union
thf(fact_1101_Sup__SUP__eq,axiom,
( comple3806919086088850358_nat_o
= ( ^ [S2: set_set_nat_o,X: set_nat] : ( member_set_nat @ X @ ( comple548664676211718543et_nat @ ( image_4687162037615663680et_nat @ collect_set_nat @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1102_Sup__SUP__eq,axiom,
( complete_Sup_Sup_o_o
= ( ^ [S2: set_o_o,X: $o] : ( member_o @ X @ ( comple90263536869209701_set_o @ ( image_o_o_set_o @ collect_o @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1103_Sup__SUP__eq,axiom,
( comple8312177224774716605_nat_o
= ( ^ [S2: set_nat_nat_o,X: nat > nat] : ( member_nat_nat @ X @ ( comple5448282615319421384at_nat @ ( image_7977807581451749376at_nat @ collect_nat_nat @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1104_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S2: set_nat_o,X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_1105_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_1106_UN__constant__eq,axiom,
! [A2: nat,A: set_nat,F: nat > set_nat,C: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ( F @ X4 )
= C ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) )
= C ) ) ) ).
% UN_constant_eq
thf(fact_1107_UN__constant__eq,axiom,
! [A2: set_nat,A: set_set_nat,F: set_nat > set_nat,C: set_nat] :
( ( member_set_nat @ A2 @ A )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ( F @ X4 )
= C ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A ) )
= C ) ) ) ).
% UN_constant_eq
thf(fact_1108_UN__constant__eq,axiom,
! [A2: $o,A: set_o,F: $o > set_nat,C: set_nat] :
( ( member_o @ A2 @ A )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ( F @ X4 )
= C ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) )
= C ) ) ) ).
% UN_constant_eq
thf(fact_1109_UN__constant__eq,axiom,
! [A2: nat > nat,A: set_nat_nat,F: ( nat > nat ) > set_nat,C: set_nat] :
( ( member_nat_nat @ A2 @ A )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
=> ( ( F @ X4 )
= C ) )
=> ( ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ A ) )
= C ) ) ) ).
% UN_constant_eq
thf(fact_1110_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat,B: set_nat] :
( ! [X4: nat] :
( ( P @ X4 )
=> ( member_nat @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1111_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > set_nat,B: set_set_nat] :
( ! [X4: nat] :
( ( P @ X4 )
=> ( member_set_nat @ ( F @ X4 ) @ B ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1112_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_1113_prop__restrict,axiom,
! [X3: nat,Z3: set_nat,X5: set_nat,P: nat > $o] :
( ( member_nat @ X3 @ Z3 )
=> ( ( ord_less_eq_set_nat @ Z3
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X5 )
& ( P @ X ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_1114_prop__restrict,axiom,
! [X3: set_nat,Z3: set_set_nat,X5: set_set_nat,P: set_nat > $o] :
( ( member_set_nat @ X3 @ Z3 )
=> ( ( ord_le6893508408891458716et_nat @ Z3
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ X5 )
& ( P @ X ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_1115_prop__restrict,axiom,
! [X3: $o,Z3: set_o,X5: set_o,P: $o > $o] :
( ( member_o @ X3 @ Z3 )
=> ( ( ord_less_eq_set_o @ Z3
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ X5 )
& ( P @ X ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_1116_prop__restrict,axiom,
! [X3: nat > nat,Z3: set_nat_nat,X5: set_nat_nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ X3 @ Z3 )
=> ( ( ord_le9059583361652607317at_nat @ Z3
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ X5 )
& ( P @ X ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_1117_Collect__restrict,axiom,
! [X5: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X5 )
& ( P @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_1118_Collect__restrict,axiom,
! [X5: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ X5 )
& ( P @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_1119_Collect__restrict,axiom,
! [X5: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ X5 )
& ( P @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_1120_Collect__restrict,axiom,
! [X5: set_nat_nat,P: ( nat > nat ) > $o] :
( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ X5 )
& ( P @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_1121_subset__emptyI,axiom,
! [A: set_set_nat] :
( ! [X4: set_nat] :
~ ( member_set_nat @ X4 @ A )
=> ( ord_le6893508408891458716et_nat @ A @ bot_bot_set_set_nat ) ) ).
% subset_emptyI
thf(fact_1122_subset__emptyI,axiom,
! [A: set_o] :
( ! [X4: $o] :
~ ( member_o @ X4 @ A )
=> ( ord_less_eq_set_o @ A @ bot_bot_set_o ) ) ).
% subset_emptyI
thf(fact_1123_subset__emptyI,axiom,
! [A: set_nat_nat] :
( ! [X4: nat > nat] :
~ ( member_nat_nat @ X4 @ A )
=> ( ord_le9059583361652607317at_nat @ A @ bot_bot_set_nat_nat ) ) ).
% subset_emptyI
thf(fact_1124_subset__emptyI,axiom,
! [A: set_nat] :
( ! [X4: nat] :
~ ( member_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_1125_restrict__apply,axiom,
( restrict_nat_nat
= ( ^ [F2: nat > nat,A4: set_nat,X: nat] : ( if_nat @ ( member_nat @ X @ A4 ) @ ( F2 @ X ) @ undefined_nat ) ) ) ).
% restrict_apply
thf(fact_1126_restrict__apply,axiom,
( restrict_set_nat_nat
= ( ^ [F2: set_nat > nat,A4: set_set_nat,X: set_nat] : ( if_nat @ ( member_set_nat @ X @ A4 ) @ ( F2 @ X ) @ undefined_nat ) ) ) ).
% restrict_apply
thf(fact_1127_restrict__apply,axiom,
( restrict_o_nat
= ( ^ [F2: $o > nat,A4: set_o,X: $o] : ( if_nat @ ( member_o @ X @ A4 ) @ ( F2 @ X ) @ undefined_nat ) ) ) ).
% restrict_apply
thf(fact_1128_restrict__apply,axiom,
( restrict_nat_nat_nat
= ( ^ [F2: ( nat > nat ) > nat,A4: set_nat_nat,X: nat > nat] : ( if_nat @ ( member_nat_nat @ X @ A4 ) @ ( F2 @ X ) @ undefined_nat ) ) ) ).
% restrict_apply
thf(fact_1129_restrict__apply,axiom,
( restrict_nat_set_nat
= ( ^ [F2: nat > set_nat,A4: set_nat,X: nat] : ( if_set_nat @ ( member_nat @ X @ A4 ) @ ( F2 @ X ) @ undefined_set_nat ) ) ) ).
% restrict_apply
thf(fact_1130_FuncSet_Orestrict__restrict,axiom,
! [F: nat > set_nat,A: set_nat,B: set_nat] :
( ( restrict_nat_set_nat @ ( restrict_nat_set_nat @ F @ A ) @ B )
= ( restrict_nat_set_nat @ F @ ( inf_inf_set_nat @ A @ B ) ) ) ).
% FuncSet.restrict_restrict
thf(fact_1131_image__restrict__eq,axiom,
! [F: nat > nat,A: set_nat] :
( ( image_nat_nat @ ( restrict_nat_nat @ F @ A ) @ A )
= ( image_nat_nat @ F @ A ) ) ).
% image_restrict_eq
thf(fact_1132_image__restrict__eq,axiom,
! [F: nat > set_nat,A: set_nat] :
( ( image_nat_set_nat @ ( restrict_nat_set_nat @ F @ A ) @ A )
= ( image_nat_set_nat @ F @ A ) ) ).
% image_restrict_eq
thf(fact_1133_restrict__def,axiom,
( restrict_nat_nat
= ( ^ [F2: nat > nat,A4: set_nat,X: nat] : ( if_nat @ ( member_nat @ X @ A4 ) @ ( F2 @ X ) @ undefined_nat ) ) ) ).
% restrict_def
thf(fact_1134_restrict__def,axiom,
( restrict_set_nat_nat
= ( ^ [F2: set_nat > nat,A4: set_set_nat,X: set_nat] : ( if_nat @ ( member_set_nat @ X @ A4 ) @ ( F2 @ X ) @ undefined_nat ) ) ) ).
% restrict_def
thf(fact_1135_restrict__def,axiom,
( restrict_o_nat
= ( ^ [F2: $o > nat,A4: set_o,X: $o] : ( if_nat @ ( member_o @ X @ A4 ) @ ( F2 @ X ) @ undefined_nat ) ) ) ).
% restrict_def
thf(fact_1136_restrict__def,axiom,
( restrict_nat_nat_nat
= ( ^ [F2: ( nat > nat ) > nat,A4: set_nat_nat,X: nat > nat] : ( if_nat @ ( member_nat_nat @ X @ A4 ) @ ( F2 @ X ) @ undefined_nat ) ) ) ).
% restrict_def
thf(fact_1137_restrict__def,axiom,
( restrict_nat_set_nat
= ( ^ [F2: nat > set_nat,A4: set_nat,X: nat] : ( if_set_nat @ ( member_nat @ X @ A4 ) @ ( F2 @ X ) @ undefined_set_nat ) ) ) ).
% restrict_def
thf(fact_1138_set__incr__image,axiom,
! [B: nat > set_nat,K: nat,N: nat,M2: nat] :
( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ ( set_ord_atMost_nat @ K ) ) )
= ( set_ord_lessThan_nat @ N ) )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I: nat] : ( hales_set_incr @ M2 @ ( B @ I ) )
@ ( set_ord_atMost_nat @ K ) ) )
= ( set_or4665077453230672383an_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) ) ) ).
% set_incr_image
thf(fact_1139_BfS__props_I4_J,axiom,
( member_nat_nat @ fS
@ ( piE_nat_nat @ ( bs @ k )
@ ^ [I: nat] : ( set_ord_lessThan_nat @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) ).
% BfS_props(4)
thf(fact_1140_BfL__props_I4_J,axiom,
( member_nat_nat @ fL
@ ( piE_nat_nat @ ( bl @ one_one_nat )
@ ^ [I: nat] : ( set_ord_lessThan_nat @ ( plus_plus_nat @ t @ one_one_nat ) ) ) ) ).
% BfL_props(4)
thf(fact_1141_PiE__I,axiom,
! [A: set_nat,F: nat > $o,B: nat > set_o] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_o @ ( F @ X4 ) @ ( B @ X4 ) ) )
=> ( ! [X4: nat] :
( ~ ( member_nat @ X4 @ A )
=> ( ( F @ X4 )
= undefined_o ) )
=> ( member_nat_o @ F @ ( piE_nat_o @ A @ B ) ) ) ) ).
% PiE_I
thf(fact_1142_PiE__I,axiom,
! [A: set_o,F: $o > $o,B: $o > set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( member_o @ ( F @ X4 ) @ ( B @ X4 ) ) )
=> ( ! [X4: $o] :
( ~ ( member_o @ X4 @ A )
=> ( ( F @ X4 )
= undefined_o ) )
=> ( member_o_o @ F @ ( piE_o_o @ A @ B ) ) ) ) ).
% PiE_I
thf(fact_1143_PiE__I,axiom,
! [A: set_o,F: $o > nat,B: $o > set_nat] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ ( B @ X4 ) ) )
=> ( ! [X4: $o] :
( ~ ( member_o @ X4 @ A )
=> ( ( F @ X4 )
= undefined_nat ) )
=> ( member_o_nat @ F @ ( piE_o_nat @ A @ B ) ) ) ) ).
% PiE_I
thf(fact_1144_PiE__I,axiom,
! [A: set_nat,F: nat > nat,B: nat > set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ ( B @ X4 ) ) )
=> ( ! [X4: nat] :
( ~ ( member_nat @ X4 @ A )
=> ( ( F @ X4 )
= undefined_nat ) )
=> ( member_nat_nat @ F @ ( piE_nat_nat @ A @ B ) ) ) ) ).
% PiE_I
thf(fact_1145_PiE__I,axiom,
! [A: set_nat,F: nat > set_nat,B: nat > set_set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_set_nat @ ( F @ X4 ) @ ( B @ X4 ) ) )
=> ( ! [X4: nat] :
( ~ ( member_nat @ X4 @ A )
=> ( ( F @ X4 )
= undefined_set_nat ) )
=> ( member_nat_set_nat @ F @ ( piE_nat_set_nat @ A @ B ) ) ) ) ).
% PiE_I
thf(fact_1146_PiE__I,axiom,
! [A: set_set_nat,F: set_nat > $o,B: set_nat > set_o] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( member_o @ ( F @ X4 ) @ ( B @ X4 ) ) )
=> ( ! [X4: set_nat] :
( ~ ( member_set_nat @ X4 @ A )
=> ( ( F @ X4 )
= undefined_o ) )
=> ( member_set_nat_o @ F @ ( piE_set_nat_o @ A @ B ) ) ) ) ).
% PiE_I
thf(fact_1147_PiE__I,axiom,
! [A: set_o,F: $o > set_nat,B: $o > set_set_nat] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( member_set_nat @ ( F @ X4 ) @ ( B @ X4 ) ) )
=> ( ! [X4: $o] :
( ~ ( member_o @ X4 @ A )
=> ( ( F @ X4 )
= undefined_set_nat ) )
=> ( member_o_set_nat @ F @ ( piE_o_set_nat @ A @ B ) ) ) ) ).
% PiE_I
thf(fact_1148_PiE__I,axiom,
! [A: set_set_nat,F: set_nat > nat,B: set_nat > set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ ( B @ X4 ) ) )
=> ( ! [X4: set_nat] :
( ~ ( member_set_nat @ X4 @ A )
=> ( ( F @ X4 )
= undefined_nat ) )
=> ( member_set_nat_nat @ F @ ( piE_set_nat_nat @ A @ B ) ) ) ) ).
% PiE_I
thf(fact_1149_PiE__I,axiom,
! [A: set_nat,F: nat > nat > nat,B: nat > set_nat_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat_nat @ ( F @ X4 ) @ ( B @ X4 ) ) )
=> ( ! [X4: nat] :
( ~ ( member_nat @ X4 @ A )
=> ( ( F @ X4 )
= undefined_nat_nat ) )
=> ( member_nat_nat_nat2 @ F @ ( piE_nat_nat_nat2 @ A @ B ) ) ) ) ).
% PiE_I
thf(fact_1150_PiE__I,axiom,
! [A: set_set_nat,F: set_nat > set_nat,B: set_nat > set_set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( member_set_nat @ ( F @ X4 ) @ ( B @ X4 ) ) )
=> ( ! [X4: set_nat] :
( ~ ( member_set_nat @ X4 @ A )
=> ( ( F @ X4 )
= undefined_set_nat ) )
=> ( member1686471427249568706et_nat @ F @ ( piE_set_nat_set_nat @ A @ B ) ) ) ) ).
% PiE_I
thf(fact_1151_assms_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ t ).
% assms(1)
thf(fact_1152_PiE__empty__range,axiom,
! [I2: set_nat,I4: set_set_nat,F3: set_nat > set_nat] :
( ( member_set_nat @ I2 @ I4 )
=> ( ( ( F3 @ I2 )
= bot_bot_set_nat )
=> ( ( piE_set_nat_nat @ I4 @ F3 )
= bot_bo7208697003875722815at_nat ) ) ) ).
% PiE_empty_range
thf(fact_1153_PiE__empty__range,axiom,
! [I2: $o,I4: set_o,F3: $o > set_nat] :
( ( member_o @ I2 @ I4 )
=> ( ( ( F3 @ I2 )
= bot_bot_set_nat )
=> ( ( piE_o_nat @ I4 @ F3 )
= bot_bot_set_o_nat ) ) ) ).
% PiE_empty_range
thf(fact_1154_PiE__empty__range,axiom,
! [I2: nat > nat,I4: set_nat_nat,F3: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ I2 @ I4 )
=> ( ( ( F3 @ I2 )
= bot_bot_set_nat )
=> ( ( piE_nat_nat_nat @ I4 @ F3 )
= bot_bo945813143650711160at_nat ) ) ) ).
% PiE_empty_range
thf(fact_1155_PiE__empty__range,axiom,
! [I2: nat,I4: set_nat,F3: nat > set_nat] :
( ( member_nat @ I2 @ I4 )
=> ( ( ( F3 @ I2 )
= bot_bot_set_nat )
=> ( ( piE_nat_nat @ I4 @ F3 )
= bot_bot_set_nat_nat ) ) ) ).
% PiE_empty_range
thf(fact_1156_assms_I4_J,axiom,
! [K3: nat,R2: nat] :
( ( ord_less_eq_nat @ K3 @ k )
=> ( hales_lhj @ R2 @ t @ K3 ) ) ).
% assms(4)
thf(fact_1157_atLeastLessThan__empty,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( set_or4665077453230672383an_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ).
% atLeastLessThan_empty
thf(fact_1158_ivl__subset,axiom,
! [I2: nat,J: nat,M2: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ J ) @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ J @ I2 )
| ( ( ord_less_eq_nat @ M2 @ I2 )
& ( ord_less_eq_nat @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_1159_image__add__atLeastLessThan,axiom,
! [K: nat,I2: nat,J: nat] :
( ( image_nat_nat @ ( plus_plus_nat @ K ) @ ( set_or4665077453230672383an_nat @ I2 @ J ) )
= ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% image_add_atLeastLessThan
thf(fact_1160_ivl__diff,axiom,
! [I2: nat,N: nat,M2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ M2 ) @ ( set_or4665077453230672383an_nat @ I2 @ N ) )
= ( set_or4665077453230672383an_nat @ N @ M2 ) ) ) ).
% ivl_diff
thf(fact_1161_lessThan__minus__lessThan,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( set_ord_lessThan_nat @ M2 ) )
= ( set_or4665077453230672383an_nat @ M2 @ N ) ) ).
% lessThan_minus_lessThan
thf(fact_1162_image__Suc__atLeastLessThan,axiom,
! [I2: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I2 @ J ) )
= ( set_or4665077453230672383an_nat @ ( suc @ I2 ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_1163_image__add__atLeastLessThan_H,axiom,
! [K: nat,I2: nat,J: nat] :
( ( image_nat_nat
@ ^ [N3: nat] : ( plus_plus_nat @ N3 @ K )
@ ( set_or4665077453230672383an_nat @ I2 @ J ) )
= ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% image_add_atLeastLessThan'
thf(fact_1164_PiE__Int,axiom,
! [I4: set_nat,A: nat > set_nat,B: nat > set_nat] :
( ( inf_inf_set_nat_nat @ ( piE_nat_nat @ I4 @ A ) @ ( piE_nat_nat @ I4 @ B ) )
= ( piE_nat_nat @ I4
@ ^ [X: nat] : ( inf_inf_set_nat @ ( A @ X ) @ ( B @ X ) ) ) ) ).
% PiE_Int
thf(fact_1165_PiE__eq__subset,axiom,
! [I4: set_set_nat,F3: set_nat > set_nat,F4: set_nat > set_nat,I2: set_nat] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_set_nat_nat @ I4 @ F3 )
= ( piE_set_nat_nat @ I4 @ F4 ) )
=> ( ( member_set_nat @ I2 @ I4 )
=> ( ord_less_eq_set_nat @ ( F3 @ I2 ) @ ( F4 @ I2 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_1166_PiE__eq__subset,axiom,
! [I4: set_o,F3: $o > set_nat,F4: $o > set_nat,I2: $o] :
( ! [I3: $o] :
( ( member_o @ I3 @ I4 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I4 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_o_nat @ I4 @ F3 )
= ( piE_o_nat @ I4 @ F4 ) )
=> ( ( member_o @ I2 @ I4 )
=> ( ord_less_eq_set_nat @ ( F3 @ I2 ) @ ( F4 @ I2 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_1167_PiE__eq__subset,axiom,
! [I4: set_nat_nat,F3: ( nat > nat ) > set_nat,F4: ( nat > nat ) > set_nat,I2: nat > nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I4 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I4 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat_nat @ I4 @ F3 )
= ( piE_nat_nat_nat @ I4 @ F4 ) )
=> ( ( member_nat_nat @ I2 @ I4 )
=> ( ord_less_eq_set_nat @ ( F3 @ I2 ) @ ( F4 @ I2 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_1168_PiE__eq__subset,axiom,
! [I4: set_nat,F3: nat > set_nat,F4: nat > set_nat,I2: nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat @ I4 @ F3 )
= ( piE_nat_nat @ I4 @ F4 ) )
=> ( ( member_nat @ I2 @ I4 )
=> ( ord_less_eq_set_nat @ ( F3 @ I2 ) @ ( F4 @ I2 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_1169_PiE__mono,axiom,
! [A: set_nat,B: nat > set_nat,C4: nat > set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( B @ X4 ) @ ( C4 @ X4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( piE_nat_nat @ A @ B ) @ ( piE_nat_nat @ A @ C4 ) ) ) ).
% PiE_mono
thf(fact_1170_subset__PiE,axiom,
! [I4: set_nat,S: nat > set_nat,T: nat > set_nat] :
( ( ord_le9059583361652607317at_nat @ ( piE_nat_nat @ I4 @ S ) @ ( piE_nat_nat @ I4 @ T ) )
= ( ( ( piE_nat_nat @ I4 @ S )
= bot_bot_set_nat_nat )
| ! [X: nat] :
( ( member_nat @ X @ I4 )
=> ( ord_less_eq_set_nat @ ( S @ X ) @ ( T @ X ) ) ) ) ) ).
% subset_PiE
thf(fact_1171_PiE__arb,axiom,
! [F: set_nat > nat,S: set_set_nat,T: set_nat > set_nat,X3: set_nat] :
( ( member_set_nat_nat @ F @ ( piE_set_nat_nat @ S @ T ) )
=> ( ~ ( member_set_nat @ X3 @ S )
=> ( ( F @ X3 )
= undefined_nat ) ) ) ).
% PiE_arb
thf(fact_1172_PiE__arb,axiom,
! [F: $o > nat,S: set_o,T: $o > set_nat,X3: $o] :
( ( member_o_nat @ F @ ( piE_o_nat @ S @ T ) )
=> ( ~ ( member_o @ X3 @ S )
=> ( ( F @ X3 )
= undefined_nat ) ) ) ).
% PiE_arb
thf(fact_1173_PiE__arb,axiom,
! [F: ( nat > nat ) > nat,S: set_nat_nat,T: ( nat > nat ) > set_nat,X3: nat > nat] :
( ( member_nat_nat_nat @ F @ ( piE_nat_nat_nat @ S @ T ) )
=> ( ~ ( member_nat_nat @ X3 @ S )
=> ( ( F @ X3 )
= undefined_nat ) ) ) ).
% PiE_arb
thf(fact_1174_PiE__arb,axiom,
! [F: nat > nat,S: set_nat,T: nat > set_nat,X3: nat] :
( ( member_nat_nat @ F @ ( piE_nat_nat @ S @ T ) )
=> ( ~ ( member_nat @ X3 @ S )
=> ( ( F @ X3 )
= undefined_nat ) ) ) ).
% PiE_arb
thf(fact_1175_PiE__E,axiom,
! [F: nat > $o,A: set_nat,B: nat > set_o,X3: nat] :
( ( member_nat_o @ F @ ( piE_nat_o @ A @ B ) )
=> ( ( ( member_nat @ X3 @ A )
=> ~ ( member_o @ ( F @ X3 ) @ ( B @ X3 ) ) )
=> ~ ( ~ ( member_nat @ X3 @ A )
=> ( ( F @ X3 )
= ~ undefined_o ) ) ) ) ).
% PiE_E
thf(fact_1176_PiE__E,axiom,
! [F: $o > $o,A: set_o,B: $o > set_o,X3: $o] :
( ( member_o_o @ F @ ( piE_o_o @ A @ B ) )
=> ( ( ( member_o @ X3 @ A )
=> ~ ( member_o @ ( F @ X3 ) @ ( B @ X3 ) ) )
=> ~ ( ~ ( member_o @ X3 @ A )
=> ( ( F @ X3 )
= ~ undefined_o ) ) ) ) ).
% PiE_E
thf(fact_1177_PiE__E,axiom,
! [F: $o > nat,A: set_o,B: $o > set_nat,X3: $o] :
( ( member_o_nat @ F @ ( piE_o_nat @ A @ B ) )
=> ( ( ( member_o @ X3 @ A )
=> ~ ( member_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
=> ~ ( ~ ( member_o @ X3 @ A )
=> ( ( F @ X3 )
!= undefined_nat ) ) ) ) ).
% PiE_E
thf(fact_1178_PiE__E,axiom,
! [F: nat > nat,A: set_nat,B: nat > set_nat,X3: nat] :
( ( member_nat_nat @ F @ ( piE_nat_nat @ A @ B ) )
=> ( ( ( member_nat @ X3 @ A )
=> ~ ( member_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
=> ~ ( ~ ( member_nat @ X3 @ A )
=> ( ( F @ X3 )
!= undefined_nat ) ) ) ) ).
% PiE_E
thf(fact_1179_PiE__E,axiom,
! [F: nat > set_nat,A: set_nat,B: nat > set_set_nat,X3: nat] :
( ( member_nat_set_nat @ F @ ( piE_nat_set_nat @ A @ B ) )
=> ( ( ( member_nat @ X3 @ A )
=> ~ ( member_set_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
=> ~ ( ~ ( member_nat @ X3 @ A )
=> ( ( F @ X3 )
!= undefined_set_nat ) ) ) ) ).
% PiE_E
thf(fact_1180_PiE__E,axiom,
! [F: set_nat > $o,A: set_set_nat,B: set_nat > set_o,X3: set_nat] :
( ( member_set_nat_o @ F @ ( piE_set_nat_o @ A @ B ) )
=> ( ( ( member_set_nat @ X3 @ A )
=> ~ ( member_o @ ( F @ X3 ) @ ( B @ X3 ) ) )
=> ~ ( ~ ( member_set_nat @ X3 @ A )
=> ( ( F @ X3 )
= ~ undefined_o ) ) ) ) ).
% PiE_E
thf(fact_1181_PiE__E,axiom,
! [F: $o > set_nat,A: set_o,B: $o > set_set_nat,X3: $o] :
( ( member_o_set_nat @ F @ ( piE_o_set_nat @ A @ B ) )
=> ( ( ( member_o @ X3 @ A )
=> ~ ( member_set_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
=> ~ ( ~ ( member_o @ X3 @ A )
=> ( ( F @ X3 )
!= undefined_set_nat ) ) ) ) ).
% PiE_E
thf(fact_1182_PiE__E,axiom,
! [F: set_nat > nat,A: set_set_nat,B: set_nat > set_nat,X3: set_nat] :
( ( member_set_nat_nat @ F @ ( piE_set_nat_nat @ A @ B ) )
=> ( ( ( member_set_nat @ X3 @ A )
=> ~ ( member_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
=> ~ ( ~ ( member_set_nat @ X3 @ A )
=> ( ( F @ X3 )
!= undefined_nat ) ) ) ) ).
% PiE_E
thf(fact_1183_PiE__E,axiom,
! [F: nat > nat > nat,A: set_nat,B: nat > set_nat_nat,X3: nat] :
( ( member_nat_nat_nat2 @ F @ ( piE_nat_nat_nat2 @ A @ B ) )
=> ( ( ( member_nat @ X3 @ A )
=> ~ ( member_nat_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
=> ~ ( ~ ( member_nat @ X3 @ A )
=> ( ( F @ X3 )
!= undefined_nat_nat ) ) ) ) ).
% PiE_E
thf(fact_1184_PiE__E,axiom,
! [F: set_nat > set_nat,A: set_set_nat,B: set_nat > set_set_nat,X3: set_nat] :
( ( member1686471427249568706et_nat @ F @ ( piE_set_nat_set_nat @ A @ B ) )
=> ( ( ( member_set_nat @ X3 @ A )
=> ~ ( member_set_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
=> ~ ( ~ ( member_set_nat @ X3 @ A )
=> ( ( F @ X3 )
!= undefined_set_nat ) ) ) ) ).
% PiE_E
thf(fact_1185_PiE__eq__iff__not__empty,axiom,
! [I4: set_set_nat,F3: set_nat > set_nat,F4: set_nat > set_nat] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ I4 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_set_nat_nat @ I4 @ F3 )
= ( piE_set_nat_nat @ I4 @ F4 ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ I4 )
=> ( ( F3 @ X )
= ( F4 @ X ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_1186_PiE__eq__iff__not__empty,axiom,
! [I4: set_o,F3: $o > set_nat,F4: $o > set_nat] :
( ! [I3: $o] :
( ( member_o @ I3 @ I4 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I4 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_o_nat @ I4 @ F3 )
= ( piE_o_nat @ I4 @ F4 ) )
= ( ! [X: $o] :
( ( member_o @ X @ I4 )
=> ( ( F3 @ X )
= ( F4 @ X ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_1187_PiE__eq__iff__not__empty,axiom,
! [I4: set_nat_nat,F3: ( nat > nat ) > set_nat,F4: ( nat > nat ) > set_nat] :
( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I4 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: nat > nat] :
( ( member_nat_nat @ I3 @ I4 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat_nat @ I4 @ F3 )
= ( piE_nat_nat_nat @ I4 @ F4 ) )
= ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ I4 )
=> ( ( F3 @ X )
= ( F4 @ X ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_1188_PiE__eq__iff__not__empty,axiom,
! [I4: set_nat,F3: nat > set_nat,F4: nat > set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ( F3 @ I3 )
!= bot_bot_set_nat ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ( F4 @ I3 )
!= bot_bot_set_nat ) )
=> ( ( ( piE_nat_nat @ I4 @ F3 )
= ( piE_nat_nat @ I4 @ F4 ) )
= ( ! [X: nat] :
( ( member_nat @ X @ I4 )
=> ( ( F3 @ X )
= ( F4 @ X ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_1189_PiE__eq__empty__iff,axiom,
! [I4: set_nat,F3: nat > set_nat] :
( ( ( piE_nat_nat @ I4 @ F3 )
= bot_bot_set_nat_nat )
= ( ? [X: nat] :
( ( member_nat @ X @ I4 )
& ( ( F3 @ X )
= bot_bot_set_nat ) ) ) ) ).
% PiE_eq_empty_iff
thf(fact_1190_PiE__eq__iff,axiom,
! [I4: set_nat,F3: nat > set_nat,F4: nat > set_nat] :
( ( ( piE_nat_nat @ I4 @ F3 )
= ( piE_nat_nat @ I4 @ F4 ) )
= ( ! [X: nat] :
( ( member_nat @ X @ I4 )
=> ( ( F3 @ X )
= ( F4 @ X ) ) )
| ( ? [X: nat] :
( ( member_nat @ X @ I4 )
& ( ( F3 @ X )
= bot_bot_set_nat ) )
& ? [X: nat] :
( ( member_nat @ X @ I4 )
& ( ( F4 @ X )
= bot_bot_set_nat ) ) ) ) ) ).
% PiE_eq_iff
thf(fact_1191_all__PiE__elements,axiom,
! [I4: set_nat,S: nat > set_nat,P: nat > nat > $o] :
( ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ ( piE_nat_nat @ I4 @ S ) )
=> ! [Y6: nat] :
( ( member_nat @ Y6 @ I4 )
=> ( P @ Y6 @ ( X @ Y6 ) ) ) ) )
= ( ( ( piE_nat_nat @ I4 @ S )
= bot_bot_set_nat_nat )
| ! [X: nat] :
( ( member_nat @ X @ I4 )
=> ! [Y6: nat] :
( ( member_nat @ Y6 @ ( S @ X ) )
=> ( P @ X @ Y6 ) ) ) ) ) ).
% all_PiE_elements
thf(fact_1192_PiE__eq,axiom,
! [I4: set_nat,S: nat > set_nat,T: nat > set_nat] :
( ( ( piE_nat_nat @ I4 @ S )
= ( piE_nat_nat @ I4 @ T ) )
= ( ( ( ( piE_nat_nat @ I4 @ S )
= bot_bot_set_nat_nat )
& ( ( piE_nat_nat @ I4 @ T )
= bot_bot_set_nat_nat ) )
| ! [X: nat] :
( ( member_nat @ X @ I4 )
=> ( ( S @ X )
= ( T @ X ) ) ) ) ) ).
% PiE_eq
thf(fact_1193_atLeastLessThan0,axiom,
! [M2: nat] :
( ( set_or4665077453230672383an_nat @ M2 @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_1194_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_1195_ivl__disj__int__two_I3_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or4665077453230672383an_nat @ L @ M2 ) @ ( set_or4665077453230672383an_nat @ M2 @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(3)
thf(fact_1196_ivl__disj__un__two_I3_J,axiom,
! [L: nat,M2: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ U )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ M2 ) @ ( set_or4665077453230672383an_nat @ M2 @ U ) )
= ( set_or4665077453230672383an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_1197_atLeastLessThan__subset__iff,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A2 @ B2 ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
| ( ( ord_less_eq_nat @ C @ A2 )
& ( ord_less_eq_nat @ B2 @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_1198_fun__ex,axiom,
! [A2: nat,A: set_nat,B2: nat,B: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( member_nat @ B2 @ B )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4
@ ( piE_nat_nat @ A
@ ^ [I: nat] : B ) )
& ( ( X4 @ A2 )
= B2 ) ) ) ) ).
% fun_ex
thf(fact_1199_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_1200_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1201_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1202_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_1203_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_1204_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1205_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1206_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1207_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1208_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1209_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1210_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1211_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_1212_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1213_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1214_assms_I5_J,axiom,
ord_less_nat @ zero_zero_nat @ r ).
% assms(5)
thf(fact_1215_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1216_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_1217_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_1218_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1219_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1220_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N3: nat] :
( ( ord_less_nat @ M6 @ N3 )
| ( M6 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1221_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_1222_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M6: nat,N3: nat] :
( ( ord_less_eq_nat @ M6 @ N3 )
& ( M6 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_1223_linorder__neqE__nat,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_1224_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1225_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1226_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1227_less__not__refl3,axiom,
! [S3: nat,T3: nat] :
( ( ord_less_nat @ S3 @ T3 )
=> ( S3 != T3 ) ) ).
% less_not_refl3
thf(fact_1228_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_1229_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1230_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_1231_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_1232_trans__less__add2,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1233_trans__less__add1,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1234_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_1235_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_1236_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_1237_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_1238_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_1239_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1240_strict__inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_1241_less__Suc__induct,axiom,
! [I2: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K4: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K4 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K4 )
=> ( P @ I3 @ K4 ) ) ) ) )
=> ( P @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1242_less__trans__Suc,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1243_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_1244_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_1245_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M7: nat] :
( ( M2
= ( suc @ M7 ) )
& ( ord_less_nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1246_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P @ I ) ) )
= ( ( P @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P @ I ) ) ) ) ).
% All_less_Suc
thf(fact_1247_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_1248_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_1249_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P @ I ) ) )
= ( ( P @ N )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_1250_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1251_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_1252_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1253_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1254_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_1255_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1256_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1257_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1258_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1259_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1260_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1261_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1262_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1263_diff__less__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1264_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1265_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less_nat @ M2 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1266_less__diff__conv,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1267_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ ( F @ M ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1268_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $true @ X3 @ Y )
= X3 ) ).
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,
! [X3: set_nat,Y: set_nat] :
( ( if_set_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X3: set_nat,Y: set_nat] :
( ( if_set_nat @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( bvar @ i )
= ( hales_set_incr @ n2 @ ( bs @ nat2 ) ) ) ).
%------------------------------------------------------------------------------