TPTP Problem File: SLH0739^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Eval_FO/0005_Ailamazyan/prob_00735_027033__15626144_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1542 ( 613 unt; 258 typ; 0 def)
% Number of atoms : 3586 (1196 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 11348 ( 243 ~; 9 |; 165 &;9386 @)
% ( 0 <=>;1545 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 36 ( 35 usr)
% Number of type conns : 1213 (1213 >; 0 *; 0 +; 0 <<)
% Number of symbols : 226 ( 223 usr; 30 con; 0-5 aty)
% Number of variables : 4000 ( 393 ^;3517 !; 90 ?;4000 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:06:05.470
%------------------------------------------------------------------------------
% Could-be-implicit typings (35)
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
set_Su4181735293145462043_a_nat: $tType ).
thf(ty_n_t__Sum____Type__Osum_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
sum_su4277537365653698277_a_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
set_Su4609323131145623387_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
set_Su2711871490478030048at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
set_Su1604132753588698630_a_nat: $tType ).
thf(ty_n_t__Sum____Type__Osum_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
sum_su4711128520861746048at_nat: $tType ).
thf(ty_n_t__Sum____Type__Osum_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
sum_su2907400405196879782_a_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
set_Su4807432515219581600at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
set_na3699693778330250182_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
set_se4904748513628223167_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Sum_sum_nat_nat: $tType ).
thf(ty_n_t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
list_Sum_sum_a_nat: $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_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
set_Sum_sum_a_nat: $tType ).
thf(ty_n_t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
sum_sum_nat_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__Sum____Type__Osum_I_Eo_M_Eo_J_J,type,
set_Sum_sum_o_o: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J,type,
product_prod_b_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__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
sum_sum_a_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__FO__Ofo____fmla_Itf__a_Mtf__b_J,type,
fo_fmla_a_b: $tType ).
thf(ty_n_t__Sum____Type__Osum_I_Eo_M_Eo_J,type,
sum_sum_o_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__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__List__Olist_I_Eo_J,type,
list_o: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (223)
thf(sy_c_Ailamazyan_OSP_001tf__a_001tf__b,type,
sP_a_b: fo_fmla_a_b > set_nat ).
thf(sy_c_Ailamazyan_OSP__list__rec_001tf__a_001tf__b,type,
sP_list_rec_a_b: fo_fmla_a_b > list_nat ).
thf(sy_c_Ailamazyan_Oact__edom_001tf__a_001tf__b,type,
act_edom_a_b: fo_fmla_a_b > ( product_prod_b_nat > set_list_a ) > set_a ).
thf(sy_c_Ailamazyan_Oad__agr_001tf__a_001tf__b_001t__Nat__Onat,type,
ad_agr_a_b_nat: fo_fmla_a_b > set_a > ( nat > sum_sum_a_nat ) > ( nat > sum_sum_a_nat ) > $o ).
thf(sy_c_Ailamazyan_Oad__agr__sets_001tf__a_001t__Nat__Onat,type,
ad_agr_sets_a_nat: set_nat > set_nat > set_a > ( nat > sum_sum_a_nat ) > ( nat > sum_sum_a_nat ) > $o ).
thf(sy_c_Ailamazyan_Od_001tf__a_001tf__b,type,
d_a_b: fo_fmla_a_b > nat ).
thf(sy_c_Ailamazyan_Oesat_001tf__a_001tf__b,type,
esat_a_b: fo_fmla_a_b > ( product_prod_b_nat > set_list_a ) > ( nat > sum_sum_a_nat ) > set_Sum_sum_a_nat > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_I_Eo_M_Eo_J,type,
complete_Inf_Inf_o_o: set_o_o > $o > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Nat__Onat_M_Eo_J,type,
comple6214475593288795910_nat_o: set_nat_o > nat > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001_Eo,type,
complete_Inf_Inf_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
complete_Inf_Inf_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_I_Eo_J,type,
comple3063163877087187839_set_o: set_set_o > set_o ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
comple7806235888213564991et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
comple1065008630642458357et_nat: set_set_set_nat > set_set_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
comple1528121977673479270_a_nat: set_se4904748513628223167_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_Itf__a_J,type,
comple6135023378680113637_set_a: set_set_a > set_a ).
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_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_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_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
comple1247738100258233164_a_nat: set_se4904748513628223167_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__a_J,type,
comple2307003609928055243_set_a: set_set_a > set_a ).
thf(sy_c_FO_Ofo__fmla_OExists_001tf__a_001tf__b,type,
fo_Exists_a_b: nat > fo_fmla_a_b > fo_fmla_a_b ).
thf(sy_c_FO_Ofo__fmla_OForall_001tf__a_001tf__b,type,
fo_Forall_a_b: nat > fo_fmla_a_b > fo_fmla_a_b ).
thf(sy_c_FO_Ofo__fmla_ONeg_001tf__a_001tf__b,type,
fo_Neg_a_b: fo_fmla_a_b > fo_fmla_a_b ).
thf(sy_c_FO_Ofo__fmla_Orel__fo__fmla_001tf__a_001tf__a_001tf__b_001tf__b,type,
fo_rel8103664758052585748_a_b_b: ( a > a > $o ) > ( b > b > $o ) > fo_fmla_a_b > fo_fmla_a_b > $o ).
thf(sy_c_FO_Ofv__fo__fmla_001tf__a_001tf__b,type,
fv_fo_fmla_a_b: fo_fmla_a_b > set_nat ).
thf(sy_c_FO_Ofv__fo__fmla__list__rec_001tf__a_001tf__b,type,
fv_fo_5581231672024362409ec_a_b: fo_fmla_a_b > list_nat ).
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001t__Set__Oset_It__Nat__Onat_J,type,
finite_Fpow_set_nat: set_set_nat > set_set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
finite171985660919647589_a_nat: set_Sum_sum_a_nat > set_se4904748513628223167_a_nat ).
thf(sy_c_Finite__Set_OFpow_001tf__a,type,
finite_Fpow_a: set_a > set_set_a ).
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__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
minus_1134630996077396038_a_nat: set_Sum_sum_a_nat > set_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
if_set_nat: $o > set_nat > set_nat > set_nat ).
thf(sy_c_If_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
if_set_Sum_sum_a_nat: $o > set_Sum_sum_a_nat > set_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_If_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
if_Sum_sum_a_nat: $o > sum_sum_a_nat > sum_sum_a_nat > sum_sum_a_nat ).
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_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_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
inf_in3446711685962385325_a_nat: set_se4904748513628223167_a_nat > set_se4904748513628223167_a_nat > set_se4904748513628223167_a_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
inf_inf_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
inf_in7084830621192376909_a_nat: set_Sum_sum_a_nat > set_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_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__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J,type,
sup_su491480579010597738_nat_o: ( sum_sum_a_nat > $o ) > ( sum_sum_a_nat > $o ) > sum_sum_a_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_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_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
sup_su2291686591051470483_a_nat: set_se4904748513628223167_a_nat > set_se4904748513628223167_a_nat > set_se4904748513628223167_a_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
sup_su3567568935942035937at_nat: set_Sum_sum_nat_nat > set_Sum_sum_nat_nat > set_Sum_sum_nat_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
sup_su2990397725382381530_a_nat: set_Su1604132753588698630_a_nat > set_Su1604132753588698630_a_nat > set_Su1604132753588698630_a_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Sum____Type__Osum_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
sup_su4098136462271712948at_nat: set_Su2711871490478030048at_nat > set_Su2711871490478030048at_nat > set_Su2711871490478030048at_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Sum____Type__Osum_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
sup_su5691376275764465223_a_nat: set_Su4181735293145462043_a_nat > set_Su4181735293145462043_a_nat > set_Su4181735293145462043_a_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
sup_su6804446743777130803_a_nat: set_Sum_sum_a_nat > set_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_List_Ocoset_001_Eo,type,
coset_o: list_o > set_o ).
thf(sy_c_List_Ocoset_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
coset_Sum_sum_a_nat: list_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_List_Ofilter_001_Eo,type,
filter_o: ( $o > $o ) > list_o > list_o ).
thf(sy_c_List_Ofilter_001t__Nat__Onat,type,
filter_nat: ( nat > $o ) > list_nat > list_nat ).
thf(sy_c_List_Ofilter_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
filter_Sum_sum_a_nat: ( sum_sum_a_nat > $o ) > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
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_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_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_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
bot_bo2635121477170169643_a_nat: set_se4904748513628223167_a_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
bot_bo3438331934148233675_a_nat: set_Sum_sum_a_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless__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__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le1477630214076318366_nat_o: ( sum_sum_a_nat > $o ) > ( sum_sum_a_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_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_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_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le9131159989063066194et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
ord_le7974500612278410847_a_nat: set_se4904748513628223167_a_nat > set_se4904748513628223167_a_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
ord_le1325389633284124927_a_nat: set_Sum_sum_a_nat > set_Sum_sum_a_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
order_2294890068632620472_a_nat: ( set_Sum_sum_a_nat > $o ) > set_Sum_sum_a_nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_Itf__a_J,type,
order_Greatest_set_a: ( set_a > $o ) > set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_Eo_M_Eo_J,type,
top_top_o_o: $o > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J,type,
top_to1565196397637005550_nat_o: sum_sum_a_nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_Eo,type,
top_top_o: $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_top_set_nat_nat: set_nat_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
top_to9106040778512017686_a_nat: set_na3699693778330250182_a_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
top_to990407478546573296at_nat: set_Su4807432515219581600at_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
top_to7766267419512669707_a_nat: set_Su4609323131145623387_a_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
top_top_set_o: set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
top_top_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
top_to9085961846241471503_a_nat: set_se4904748513628223167_a_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_to6661820994512907621at_nat: set_Sum_sum_nat_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
top_to7568329678976531286_a_nat: set_Su1604132753588698630_a_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
top_to8676068415865862704at_nat: set_Su2711871490478030048at_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
top_to599037537065133003_a_nat: set_Su4181735293145462043_a_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
top_to795618464972521135_a_nat: set_Sum_sum_a_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Relation_OPowp_001_Eo,type,
powp_o: ( $o > $o ) > set_o > $o ).
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_OCollect_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
collec4049389696321283146_a_nat: ( set_Sum_sum_a_nat > $o ) > set_se4904748513628223167_a_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
collec7073057861543223018_a_nat: ( sum_sum_a_nat > $o ) > set_Sum_sum_a_nat ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OPow_001_Eo,type,
pow_o: set_o > set_set_o ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_OPow_001t__Set__Oset_It__Nat__Onat_J,type,
pow_set_nat: set_set_nat > set_set_set_nat ).
thf(sy_c_Set_OPow_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
pow_Sum_sum_a_nat: set_Sum_sum_a_nat > set_se4904748513628223167_a_nat ).
thf(sy_c_Set_OPow_001tf__a,type,
pow_a: set_a > set_set_a ).
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__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__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001_Eo,type,
image_2376713081370839351_nat_o: ( ( nat > sum_sum_a_nat ) > $o ) > set_na3699693778330250182_a_nat > set_o ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_5790795286017029031et_nat: ( ( nat > sum_sum_a_nat ) > set_nat ) > set_na3699693778330250182_a_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001_Eo,type,
image_2859160068955298525_nat_o: ( ( sum_sum_a_nat > nat ) > $o ) > set_Su4807432515219581600at_nat > set_o ).
thf(sy_c_Set_Oimage_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7410052990245744513et_nat: ( ( sum_sum_a_nat > nat ) > set_nat ) > set_Su4807432515219581600at_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001_Eo,type,
image_1729793951705799532_nat_o: ( ( sum_sum_a_nat > sum_sum_a_nat ) > $o ) > set_Su4609323131145623387_a_nat > set_o ).
thf(sy_c_Set_Oimage_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_2259655797477813426et_nat: ( ( sum_sum_a_nat > sum_sum_a_nat ) > set_nat ) > set_Su4609323131145623387_a_nat > 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__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
image_3365592128754359116_a_nat: ( $o > set_Sum_sum_a_nat ) > set_o > set_se4904748513628223167_a_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_Itf__a_J,type,
image_o_set_a: ( $o > set_a ) > set_o > set_set_a ).
thf(sy_c_Set_Oimage_001_Eo_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
image_4139480514073730540_a_nat: ( $o > sum_sum_a_nat ) > set_o > set_Sum_sum_a_nat ).
thf(sy_c_Set_Oimage_001_Eo_001tf__a,type,
image_o_a: ( $o > a ) > set_o > set_a ).
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_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_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__Nat__Onat_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
image_4085361583586468296_a_nat: ( nat > set_Sum_sum_a_nat ) > set_nat > set_se4904748513628223167_a_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_678696785212003926at_nat: ( nat > sum_sum_nat_nat ) > set_nat > set_Sum_sum_nat_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
image_7108941391349317263_a_nat: ( nat > sum_su2907400405196879782_a_nat ) > set_nat > set_Su1604132753588698630_a_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Sum____Type__Osum_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
image_8912669507014183529at_nat: ( nat > sum_su4711128520861746048at_nat ) > set_nat > set_Su2711871490478030048at_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
image_7293268710728258664_a_nat: ( nat > sum_sum_a_nat ) > set_nat > set_Sum_sum_a_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
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_o: ( 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__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__Nat__Onat_J_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
image_3578472599065059474_a_nat: ( set_nat > set_Sum_sum_a_nat ) > set_set_nat > set_se4904748513628223167_a_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
image_6589623134797232946_a_nat: ( set_nat > sum_sum_a_nat ) > set_set_nat > set_Sum_sum_a_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
image_5599399343138760645_a_nat: ( set_Sum_sum_a_nat > set_Sum_sum_a_nat ) > set_se4904748513628223167_a_nat > set_se4904748513628223167_a_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
image_6715601112060939782_a_nat: ( set_a > set_Sum_sum_a_nat ) > set_set_a > set_se4904748513628223167_a_nat ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001_Eo,type,
image_6095136190293192542_nat_o: ( sum_sum_a_nat > $o ) > set_Sum_sum_a_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_2473878607534554506at_nat: ( sum_sum_a_nat > nat ) > set_Sum_sum_a_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_4589483402070311232et_nat: ( sum_sum_a_nat > set_nat ) > set_Sum_sum_a_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
image_7877458644602423589_a_nat: ( sum_sum_a_nat > set_Sum_sum_a_nat ) > set_Sum_sum_a_nat > set_se4904748513628223167_a_nat ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Set__Oset_Itf__a_J,type,
image_8809959208761443620_set_a: ( sum_sum_a_nat > set_a ) > set_Sum_sum_a_nat > set_set_a ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
image_2874716449350571628_a_nat: ( sum_sum_a_nat > sum_su2907400405196879782_a_nat ) > set_Sum_sum_a_nat > set_Su1604132753588698630_a_nat ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
image_4678444565015437894at_nat: ( sum_sum_a_nat > sum_su4711128520861746048at_nat ) > set_Sum_sum_a_nat > set_Su2711871490478030048at_nat ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
image_1339813895292368543_a_nat: ( sum_sum_a_nat > sum_su4277537365653698277_a_nat ) > set_Sum_sum_a_nat > set_Su4181735293145462043_a_nat ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
image_7142520692256960453_a_nat: ( sum_sum_a_nat > sum_sum_a_nat ) > set_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001tf__a,type,
image_6322530041254294468_nat_a: ( sum_sum_a_nat > a ) > set_Sum_sum_a_nat > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001_Eo,type,
image_a_o: ( a > $o ) > set_a > set_o ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Nat__Onat_J,type,
image_a_set_nat: ( a > set_nat ) > set_a > set_set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
image_1201892972954314598_a_nat: ( a > set_Sum_sum_a_nat ) > set_a > set_se4904748513628223167_a_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
image_a_set_a: ( a > set_a ) > set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
image_7873763678140191238_a_nat: ( a > sum_sum_a_nat ) > set_a > set_Sum_sum_a_nat ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001_Eo,type,
set_ord_atLeast_o: $o > set_o ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
set_ord_atLeast_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_It__Nat__Onat_J,type,
set_or1731685050470061051et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
set_or1144079512921665450_a_nat: set_Sum_sum_a_nat > set_se4904748513628223167_a_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_Itf__a_J,type,
set_or8362275514725411625_set_a: set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Sum__Type_OInl_001_Eo_001_Eo,type,
sum_Inl_o_o: $o > sum_sum_o_o ).
thf(sy_c_Sum__Type_OInl_001t__Nat__Onat_001t__Nat__Onat,type,
sum_Inl_nat_nat: nat > sum_sum_nat_nat ).
thf(sy_c_Sum__Type_OInl_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
sum_In6445040708324073535_a_nat: nat > sum_su2907400405196879782_a_nat ).
thf(sy_c_Sum__Type_OInl_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
sum_In1625650605130369377at_nat: sum_sum_a_nat > sum_su4711128520861746048at_nat ).
thf(sy_c_Sum__Type_OInl_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
sum_In4896129900671530542_a_nat: sum_sum_a_nat > sum_su4277537365653698277_a_nat ).
thf(sy_c_Sum__Type_OInl_001tf__a_001t__Nat__Onat,type,
sum_Inl_a_nat: a > sum_sum_a_nat ).
thf(sy_c_Sum__Type_OInr_001_Eo_001_Eo,type,
sum_Inr_o_o: $o > sum_sum_o_o ).
thf(sy_c_Sum__Type_OInr_001t__Nat__Onat_001t__Nat__Onat,type,
sum_Inr_nat_nat: nat > sum_sum_nat_nat ).
thf(sy_c_Sum__Type_OInr_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
sum_In1013769454558682809_a_nat: nat > sum_su4711128520861746048at_nat ).
thf(sy_c_Sum__Type_OInr_001t__Nat__Onat_001tf__a,type,
sum_Inr_nat_a: nat > sum_sum_a_nat ).
thf(sy_c_Sum__Type_OInr_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
sum_In5417751388219754459at_nat: sum_sum_a_nat > sum_su2907400405196879782_a_nat ).
thf(sy_c_Sum__Type_OInr_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
sum_In5020205417008236852_a_nat: sum_sum_a_nat > sum_su4277537365653698277_a_nat ).
thf(sy_c_Sum__Type_OPlus_001_Eo_001_Eo,type,
sum_Plus_o_o: set_o > set_o > set_Sum_sum_o_o ).
thf(sy_c_Sum__Type_OPlus_001t__Nat__Onat_001t__Nat__Onat,type,
sum_Plus_nat_nat: set_nat > set_nat > set_Sum_sum_nat_nat ).
thf(sy_c_Sum__Type_OPlus_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
sum_Pl4734085394118260730_a_nat: set_nat > set_Sum_sum_a_nat > set_Su1604132753588698630_a_nat ).
thf(sy_c_Sum__Type_OPlus_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
sum_Pl9138067327779332380at_nat: set_Sum_sum_a_nat > set_nat > set_Su2711871490478030048at_nat ).
thf(sy_c_Sum__Type_OPlus_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
sum_Pl1179408338895902643_a_nat: set_Sum_sum_a_nat > set_Sum_sum_a_nat > set_Su4181735293145462043_a_nat ).
thf(sy_c_Sum__Type_OPlus_001tf__a_001t__Nat__Onat,type,
sum_Plus_a_nat: set_a > set_nat > set_Sum_sum_a_nat ).
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_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__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
member8098812455498974984_a_nat: set_Sum_sum_a_nat > set_se4904748513628223167_a_nat > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001t__Sum____Type__Osum_I_Eo_M_Eo_J,type,
member_Sum_sum_o_o: sum_sum_o_o > set_Sum_sum_o_o > $o ).
thf(sy_c_member_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
member_Sum_sum_a_nat: sum_sum_a_nat > set_Sum_sum_a_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_AD,type,
ad: set_a ).
thf(sy_v_I,type,
i: product_prod_b_nat > set_list_a ).
thf(sy_v_X,type,
x: set_Sum_sum_a_nat ).
thf(sy_v__092_060phi_062_H____,type,
phi: fo_fmla_a_b ).
thf(sy_v__092_060sigma_062_H____,type,
sigma: nat > sum_sum_a_nat ).
thf(sy_v__092_060tau_062_H____,type,
tau: nat > sum_sum_a_nat ).
thf(sy_v_n____,type,
n: nat ).
% Relevant facts (1276)
thf(fact_0_fo__fmla_Oinject_I7_J,axiom,
! [X71: nat,X72: fo_fmla_a_b,Y71: nat,Y72: fo_fmla_a_b] :
( ( ( fo_Exists_a_b @ X71 @ X72 )
= ( fo_Exists_a_b @ Y71 @ Y72 ) )
= ( ( X71 = Y71 )
& ( X72 = Y72 ) ) ) ).
% fo_fmla.inject(7)
thf(fact_1_fo__fmla_Oinject_I4_J,axiom,
! [X4: fo_fmla_a_b,Y4: fo_fmla_a_b] :
( ( ( fo_Neg_a_b @ X4 )
= ( fo_Neg_a_b @ Y4 ) )
= ( X4 = Y4 ) ) ).
% fo_fmla.inject(4)
thf(fact_2_fo__fmla_Odistinct_I41_J,axiom,
! [X4: fo_fmla_a_b,X71: nat,X72: fo_fmla_a_b] :
( ( fo_Neg_a_b @ X4 )
!= ( fo_Exists_a_b @ X71 @ X72 ) ) ).
% fo_fmla.distinct(41)
thf(fact_3_Forall_Oprems_I1_J,axiom,
ad_agr_a_b_nat @ ( fo_Forall_a_b @ n @ phi ) @ ad @ sigma @ tau ).
% Forall.prems(1)
thf(fact_4_local_Ounfold_I3_J,axiom,
( ( fv_fo_fmla_a_b @ ( fo_Forall_a_b @ n @ phi ) )
= ( fv_fo_fmla_a_b @ ( fo_Exists_a_b @ n @ ( fo_Neg_a_b @ phi ) ) ) ) ).
% local.unfold(3)
thf(fact_5_local_Ounfold_I1_J,axiom,
( ( act_edom_a_b @ ( fo_Forall_a_b @ n @ phi ) @ i )
= ( act_edom_a_b @ ( fo_Exists_a_b @ n @ ( fo_Neg_a_b @ phi ) ) @ i ) ) ).
% local.unfold(1)
thf(fact_6_SP__list__rec_Osimps_I2_J,axiom,
! [Phi: fo_fmla_a_b] :
( ( sP_list_rec_a_b @ ( fo_Neg_a_b @ Phi ) )
= ( sP_list_rec_a_b @ Phi ) ) ).
% SP_list_rec.simps(2)
thf(fact_7_d_Osimps_I5_J,axiom,
! [N: nat,Phi: fo_fmla_a_b] :
( ( d_a_b @ ( fo_Exists_a_b @ N @ Phi ) )
= ( d_a_b @ Phi ) ) ).
% d.simps(5)
thf(fact_8_d_Osimps_I2_J,axiom,
! [Phi: fo_fmla_a_b] :
( ( d_a_b @ ( fo_Neg_a_b @ Phi ) )
= ( d_a_b @ Phi ) ) ).
% d.simps(2)
thf(fact_9_act__edom_Osimps_I7_J,axiom,
! [N: nat,Phi: fo_fmla_a_b,I: product_prod_b_nat > set_list_a] :
( ( act_edom_a_b @ ( fo_Exists_a_b @ N @ Phi ) @ I )
= ( act_edom_a_b @ Phi @ I ) ) ).
% act_edom.simps(7)
thf(fact_10_act__edom_Osimps_I4_J,axiom,
! [Phi: fo_fmla_a_b,I: product_prod_b_nat > set_list_a] :
( ( act_edom_a_b @ ( fo_Neg_a_b @ Phi ) @ I )
= ( act_edom_a_b @ Phi @ I ) ) ).
% act_edom.simps(4)
thf(fact_11_SP_Osimps_I2_J,axiom,
! [Phi: fo_fmla_a_b] :
( ( sP_a_b @ ( fo_Neg_a_b @ Phi ) )
= ( sP_a_b @ Phi ) ) ).
% SP.simps(2)
thf(fact_12_fo__fmla_Oinject_I8_J,axiom,
! [X81: nat,X82: fo_fmla_a_b,Y81: nat,Y82: fo_fmla_a_b] :
( ( ( fo_Forall_a_b @ X81 @ X82 )
= ( fo_Forall_a_b @ Y81 @ Y82 ) )
= ( ( X81 = Y81 )
& ( X82 = Y82 ) ) ) ).
% fo_fmla.inject(8)
thf(fact_13_Forall_Oprems_I2_J,axiom,
ord_less_eq_set_a @ ( act_edom_a_b @ ( fo_Forall_a_b @ n @ phi ) @ i ) @ ad ).
% Forall.prems(2)
thf(fact_14_d_Osimps_I6_J,axiom,
! [N: nat,Phi: fo_fmla_a_b] :
( ( d_a_b @ ( fo_Forall_a_b @ N @ Phi ) )
= ( d_a_b @ Phi ) ) ).
% d.simps(6)
thf(fact_15_act__edom_Osimps_I8_J,axiom,
! [N: nat,Phi: fo_fmla_a_b,I: product_prod_b_nat > set_list_a] :
( ( act_edom_a_b @ ( fo_Forall_a_b @ N @ Phi ) @ I )
= ( act_edom_a_b @ Phi @ I ) ) ).
% act_edom.simps(8)
thf(fact_16_fo__fmla_Odistinct_I43_J,axiom,
! [X4: fo_fmla_a_b,X81: nat,X82: fo_fmla_a_b] :
( ( fo_Neg_a_b @ X4 )
!= ( fo_Forall_a_b @ X81 @ X82 ) ) ).
% fo_fmla.distinct(43)
thf(fact_17_fo__fmla_Odistinct_I55_J,axiom,
! [X71: nat,X72: fo_fmla_a_b,X81: nat,X82: fo_fmla_a_b] :
( ( fo_Exists_a_b @ X71 @ X72 )
!= ( fo_Forall_a_b @ X81 @ X82 ) ) ).
% fo_fmla.distinct(55)
thf(fact_18_fv__fo__fmla_Osimps_I4_J,axiom,
! [Phi: fo_fmla_a_b] :
( ( fv_fo_fmla_a_b @ ( fo_Neg_a_b @ Phi ) )
= ( fv_fo_fmla_a_b @ Phi ) ) ).
% fv_fo_fmla.simps(4)
thf(fact_19_Forall_Oprems_I4_J,axiom,
ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ tau @ ( fv_fo_fmla_a_b @ ( fo_Forall_a_b @ n @ phi ) ) ) @ x ).
% Forall.prems(4)
thf(fact_20_ad__agr__def,axiom,
( ad_agr_a_b_nat
= ( ^ [Phi2: fo_fmla_a_b] : ( ad_agr_sets_a_nat @ ( fv_fo_fmla_a_b @ Phi2 ) @ ( sP_a_b @ Phi2 ) ) ) ) ).
% ad_agr_def
thf(fact_21_SP__fv,axiom,
! [Phi: fo_fmla_a_b] : ( ord_less_eq_set_nat @ ( sP_a_b @ Phi ) @ ( fv_fo_fmla_a_b @ Phi ) ) ).
% SP_fv
thf(fact_22_SP__list__rec_Osimps_I6_J,axiom,
! [N: nat,Phi: fo_fmla_a_b] :
( ( sP_list_rec_a_b @ ( fo_Forall_a_b @ N @ Phi ) )
= ( filter_nat
@ ^ [M: nat] : ( N != M )
@ ( sP_list_rec_a_b @ Phi ) ) ) ).
% SP_list_rec.simps(6)
thf(fact_23_SP__list__rec_Osimps_I5_J,axiom,
! [N: nat,Phi: fo_fmla_a_b] :
( ( sP_list_rec_a_b @ ( fo_Exists_a_b @ N @ Phi ) )
= ( filter_nat
@ ^ [M: nat] : ( N != M )
@ ( sP_list_rec_a_b @ Phi ) ) ) ).
% SP_list_rec.simps(5)
thf(fact_24_fo__fmla_Orel__distinct_I56_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y81: nat,Y82: fo_fmla_a_b,X71: nat,X72: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Forall_a_b @ Y81 @ Y82 ) @ ( fo_Exists_a_b @ X71 @ X72 ) ) ).
% fo_fmla.rel_distinct(56)
thf(fact_25_fo__fmla_Orel__distinct_I55_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X71: nat,X72: fo_fmla_a_b,Y81: nat,Y82: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Exists_a_b @ X71 @ X72 ) @ ( fo_Forall_a_b @ Y81 @ Y82 ) ) ).
% fo_fmla.rel_distinct(55)
thf(fact_26_fo__fmla_Orel__distinct_I44_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y81: nat,Y82: fo_fmla_a_b,X4: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Forall_a_b @ Y81 @ Y82 ) @ ( fo_Neg_a_b @ X4 ) ) ).
% fo_fmla.rel_distinct(44)
thf(fact_27_fo__fmla_Orel__distinct_I43_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X4: fo_fmla_a_b,Y81: nat,Y82: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Neg_a_b @ X4 ) @ ( fo_Forall_a_b @ Y81 @ Y82 ) ) ).
% fo_fmla.rel_distinct(43)
thf(fact_28_fo__fmla_Orel__distinct_I41_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X4: fo_fmla_a_b,Y71: nat,Y72: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Neg_a_b @ X4 ) @ ( fo_Exists_a_b @ Y71 @ Y72 ) ) ).
% fo_fmla.rel_distinct(41)
thf(fact_29_fo__fmla_Orel__distinct_I42_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y71: nat,Y72: fo_fmla_a_b,X4: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Exists_a_b @ Y71 @ Y72 ) @ ( fo_Neg_a_b @ X4 ) ) ).
% fo_fmla.rel_distinct(42)
thf(fact_30_fo__fmla_Orel__inject_I4_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X4: fo_fmla_a_b,Y4: fo_fmla_a_b] :
( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Neg_a_b @ X4 ) @ ( fo_Neg_a_b @ Y4 ) )
= ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X4 @ Y4 ) ) ).
% fo_fmla.rel_inject(4)
thf(fact_31_fo__fmla_Orel__intros_I4_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X4: fo_fmla_a_b,Y4: fo_fmla_a_b] :
( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X4 @ Y4 )
=> ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Neg_a_b @ X4 ) @ ( fo_Neg_a_b @ Y4 ) ) ) ).
% fo_fmla.rel_intros(4)
thf(fact_32_fo__fmla_Orel__inject_I7_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X71: nat,X72: fo_fmla_a_b,Y71: nat,Y72: fo_fmla_a_b] :
( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Exists_a_b @ X71 @ X72 ) @ ( fo_Exists_a_b @ Y71 @ Y72 ) )
= ( ( X71 = Y71 )
& ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X72 @ Y72 ) ) ) ).
% fo_fmla.rel_inject(7)
thf(fact_33_fo__fmla_Orel__intros_I7_J,axiom,
! [X71: nat,Y71: nat,R1: a > a > $o,R2: b > b > $o,X72: fo_fmla_a_b,Y72: fo_fmla_a_b] :
( ( X71 = Y71 )
=> ( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X72 @ Y72 )
=> ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Exists_a_b @ X71 @ X72 ) @ ( fo_Exists_a_b @ Y71 @ Y72 ) ) ) ) ).
% fo_fmla.rel_intros(7)
thf(fact_34_fo__fmla_Orel__intros_I8_J,axiom,
! [X81: nat,Y81: nat,R1: a > a > $o,R2: b > b > $o,X82: fo_fmla_a_b,Y82: fo_fmla_a_b] :
( ( X81 = Y81 )
=> ( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X82 @ Y82 )
=> ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Forall_a_b @ X81 @ X82 ) @ ( fo_Forall_a_b @ Y81 @ Y82 ) ) ) ) ).
% fo_fmla.rel_intros(8)
thf(fact_35_fo__fmla_Orel__inject_I8_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X81: nat,X82: fo_fmla_a_b,Y81: nat,Y82: fo_fmla_a_b] :
( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Forall_a_b @ X81 @ X82 ) @ ( fo_Forall_a_b @ Y81 @ Y82 ) )
= ( ( X81 = Y81 )
& ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X82 @ Y82 ) ) ) ).
% fo_fmla.rel_inject(8)
thf(fact_36_mem__Collect__eq,axiom,
! [A: $o,P: $o > $o] :
( ( member_o @ A @ ( collect_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_37_Collect__mem__eq,axiom,
! [A2: set_o] :
( ( collect_o
@ ^ [X: $o] : ( member_o @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_38_subset__antisym,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_39_subset__antisym,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B )
=> ( ( ord_le1325389633284124927_a_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_40_subsetI,axiom,
! [A2: set_o,B: set_o] :
( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( member_o @ X2 @ B ) )
=> ( ord_less_eq_set_o @ A2 @ B ) ) ).
% subsetI
thf(fact_41_subsetI,axiom,
! [A2: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B ) )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_42_subsetI,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ A2 )
=> ( member_Sum_sum_a_nat @ X2 @ B ) )
=> ( ord_le1325389633284124927_a_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_43_image__eqI,axiom,
! [B2: sum_sum_a_nat,F: nat > sum_sum_a_nat,X3: nat,A2: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A2 )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7293268710728258664_a_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_44_image__eqI,axiom,
! [B2: sum_sum_a_nat,F: a > sum_sum_a_nat,X3: a,A2: set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_a @ X3 @ A2 )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7873763678140191238_a_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_45_image__eqI,axiom,
! [B2: set_nat,F: nat > set_nat,X3: nat,A2: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A2 )
=> ( member_set_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_46_image__eqI,axiom,
! [B2: $o,F: $o > $o,X3: $o,A2: set_o] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_o @ X3 @ A2 )
=> ( member_o @ B2 @ ( image_o_o @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_47_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_48_dual__order_Orefl,axiom,
! [A: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ A @ A ) ).
% dual_order.refl
thf(fact_49_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_50_order__refl,axiom,
! [X3: set_a] : ( ord_less_eq_set_a @ X3 @ X3 ) ).
% order_refl
thf(fact_51_order__refl,axiom,
! [X3: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_52_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_53_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > set_nat,B: set_set_nat] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( member_set_nat @ ( F @ X2 ) @ B ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_54_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( member_Sum_sum_a_nat @ ( F @ X2 ) @ B ) )
=> ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_55_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ! [X2: a] :
( ( P @ X2 )
=> ( member_Sum_sum_a_nat @ ( F @ X2 ) @ B ) )
=> ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ ( collect_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_56_all__subset__image,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_set_nat > $o] :
( ( ! [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ ( image_nat_set_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( P @ ( image_nat_set_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_57_all__subset__image,axiom,
! [F: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( P @ ( image_a_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_58_all__subset__image,axiom,
! [F: sum_sum_a_nat > a,A2: set_Sum_sum_a_nat,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_6322530041254294468_nat_a @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B3 @ A2 )
=> ( P @ ( image_6322530041254294468_nat_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_59_all__subset__image,axiom,
! [F: nat > sum_sum_a_nat,A2: set_nat,P: set_Sum_sum_a_nat > $o] :
( ( ! [B3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B3 @ ( image_7293268710728258664_a_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( P @ ( image_7293268710728258664_a_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_60_all__subset__image,axiom,
! [F: a > sum_sum_a_nat,A2: set_a,P: set_Sum_sum_a_nat > $o] :
( ( ! [B3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B3 @ ( image_7873763678140191238_a_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( P @ ( image_7873763678140191238_a_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_61_all__subset__image,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A2: set_Sum_sum_a_nat,P: set_Sum_sum_a_nat > $o] :
( ( ! [B3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B3 @ ( image_7142520692256960453_a_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B3 @ A2 )
=> ( P @ ( image_7142520692256960453_a_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_62_subset__image__iff,axiom,
! [B: set_set_nat,F: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ ( image_nat_set_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B
= ( image_nat_set_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_63_subset__image__iff,axiom,
! [B: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_64_subset__image__iff,axiom,
! [B: set_a,F: sum_sum_a_nat > a,A2: set_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ B @ ( image_6322530041254294468_nat_a @ F @ A2 ) )
= ( ? [AA: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ AA @ A2 )
& ( B
= ( image_6322530041254294468_nat_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_65_subset__image__iff,axiom,
! [B: set_Sum_sum_a_nat,F: nat > sum_sum_a_nat,A2: set_nat] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_7293268710728258664_a_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B
= ( image_7293268710728258664_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_66_subset__image__iff,axiom,
! [B: set_Sum_sum_a_nat,F: a > sum_sum_a_nat,A2: set_a] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_7873763678140191238_a_nat @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B
= ( image_7873763678140191238_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_67_subset__image__iff,axiom,
! [B: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_7142520692256960453_a_nat @ F @ A2 ) )
= ( ? [AA: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ AA @ A2 )
& ( B
= ( image_7142520692256960453_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_68_image__subset__iff,axiom,
! [F: nat > set_nat,A2: set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_69_image__subset__iff,axiom,
! [F: nat > sum_sum_a_nat,A2: set_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ A2 ) @ B )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_Sum_sum_a_nat @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_70_image__subset__iff,axiom,
! [F: a > sum_sum_a_nat,A2: set_a,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ A2 ) @ B )
= ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( member_Sum_sum_a_nat @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_71_less__eq__set__def,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
( ord_less_eq_o_o
@ ^ [X: $o] : ( member_o @ X @ A3 )
@ ^ [X: $o] : ( member_o @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_72_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ord_less_eq_a_o
@ ^ [X: a] : ( member_a @ X @ A3 )
@ ^ [X: a] : ( member_a @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_73_less__eq__set__def,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A3: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
( ord_le1477630214076318366_nat_o
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ A3 )
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_74_nle__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_75_le__cases3,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_76_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 ) )
= ( ^ [X: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X @ Y3 )
& ( ord_less_eq_set_a @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_77_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] : ( Y2 = Z2 ) )
= ( ^ [X: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X @ Y3 )
& ( ord_le1325389633284124927_a_nat @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_78_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
= ( ^ [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_79_ord__eq__le__trans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( A = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_80_ord__eq__le__trans,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( A = B2 )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ord_le1325389633284124927_a_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_81_ord__eq__le__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_82_ord__le__eq__trans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_83_ord__le__eq__trans,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le1325389633284124927_a_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_84_ord__le__eq__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_85_order__antisym,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_86_order__antisym,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ Y )
=> ( ( ord_le1325389633284124927_a_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_87_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_88_order_Otrans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_89_order_Otrans,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ord_le1325389633284124927_a_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_90_order_Otrans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_91_order__trans,axiom,
! [X3: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_eq_set_a @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_92_order__trans,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ Y )
=> ( ( ord_le1325389633284124927_a_nat @ Y @ Z )
=> ( ord_le1325389633284124927_a_nat @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_93_order__trans,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_94_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_95_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A5 )
& ( ord_less_eq_set_a @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_96_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] : ( Y2 = Z2 ) )
= ( ^ [A5: set_Sum_sum_a_nat,B5: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B5 @ A5 )
& ( ord_le1325389633284124927_a_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_97_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_98_dual__order_Oantisym,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_99_dual__order_Oantisym,axiom,
! [B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A )
=> ( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_100_dual__order_Oantisym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_101_dual__order_Otrans,axiom,
! [B2: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_102_dual__order_Otrans,axiom,
! [B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A )
=> ( ( ord_le1325389633284124927_a_nat @ C @ B2 )
=> ( ord_le1325389633284124927_a_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_103_dual__order_Otrans,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_104_antisym,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_105_antisym,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_106_antisym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_107_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_108_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] : ( Y2 = Z2 ) )
= ( ^ [A5: set_Sum_sum_a_nat,B5: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A5 @ B5 )
& ( ord_le1325389633284124927_a_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_109_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_110_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_111_order__subst1,axiom,
! [A: set_a,F: set_Sum_sum_a_nat > set_a,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ! [X2: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_112_order__subst1,axiom,
! [A: set_a,F: nat > set_a,B2: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_113_order__subst1,axiom,
! [A: set_Sum_sum_a_nat,F: set_a > set_Sum_sum_a_nat,B2: set_a,C: set_a] :
( ( ord_le1325389633284124927_a_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_le1325389633284124927_a_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_114_order__subst1,axiom,
! [A: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ ( F @ B2 ) )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ! [X2: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y5 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_le1325389633284124927_a_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_115_order__subst1,axiom,
! [A: set_Sum_sum_a_nat,F: nat > set_Sum_sum_a_nat,B2: nat,C: nat] :
( ( ord_le1325389633284124927_a_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_le1325389633284124927_a_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_116_order__subst1,axiom,
! [A: nat,F: set_a > nat,B2: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_117_order__subst1,axiom,
! [A: nat,F: set_Sum_sum_a_nat > nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ! [X2: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_118_order__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_119_order__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_120_order__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_le1325389633284124927_a_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_121_order__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_122_order__subst2,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > set_a,C: set_a] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_123_order__subst2,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( ( ord_le1325389633284124927_a_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y5 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_124_order__subst2,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > nat,C: nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_125_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_126_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_le1325389633284124927_a_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_127_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_128_order__eq__refl,axiom,
! [X3: set_a,Y: set_a] :
( ( X3 = Y )
=> ( ord_less_eq_set_a @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_129_order__eq__refl,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( X3 = Y )
=> ( ord_le1325389633284124927_a_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_130_order__eq__refl,axiom,
! [X3: nat,Y: nat] :
( ( X3 = Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_131_linorder__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_132_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_133_ord__eq__le__subst,axiom,
! [A: set_Sum_sum_a_nat,F: set_a > set_Sum_sum_a_nat,B2: set_a,C: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_le1325389633284124927_a_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_134_ord__eq__le__subst,axiom,
! [A: nat,F: set_a > nat,B2: set_a,C: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_135_ord__eq__le__subst,axiom,
! [A: set_a,F: set_Sum_sum_a_nat > set_a,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ! [X2: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_136_ord__eq__le__subst,axiom,
! [A: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ! [X2: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y5 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_le1325389633284124927_a_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_137_ord__eq__le__subst,axiom,
! [A: nat,F: set_Sum_sum_a_nat > nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ! [X2: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_138_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_139_ord__eq__le__subst,axiom,
! [A: set_Sum_sum_a_nat,F: nat > set_Sum_sum_a_nat,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_le1325389633284124927_a_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_140_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_141_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_142_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_143_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_144_ord__le__eq__subst,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > set_a,C: set_a] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_145_ord__le__eq__subst,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y5 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_146_ord__le__eq__subst,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,F: set_Sum_sum_a_nat > nat,C: nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_147_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_148_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_149_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_150_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_151_order__antisym__conv,axiom,
! [Y: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y @ X3 )
=> ( ( ord_less_eq_set_a @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_152_order__antisym__conv,axiom,
! [Y: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ Y @ X3 )
=> ( ( ord_le1325389633284124927_a_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_153_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_154_imageI,axiom,
! [X3: nat,A2: set_nat,F: nat > sum_sum_a_nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_Sum_sum_a_nat @ ( F @ X3 ) @ ( image_7293268710728258664_a_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_155_imageI,axiom,
! [X3: a,A2: set_a,F: a > sum_sum_a_nat] :
( ( member_a @ X3 @ A2 )
=> ( member_Sum_sum_a_nat @ ( F @ X3 ) @ ( image_7873763678140191238_a_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_156_imageI,axiom,
! [X3: nat,A2: set_nat,F: nat > set_nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_set_nat @ ( F @ X3 ) @ ( image_nat_set_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_157_imageI,axiom,
! [X3: $o,A2: set_o,F: $o > $o] :
( ( member_o @ X3 @ A2 )
=> ( member_o @ ( F @ X3 ) @ ( image_o_o @ F @ A2 ) ) ) ).
% imageI
thf(fact_158_image__iff,axiom,
! [Z: sum_sum_a_nat,F: nat > sum_sum_a_nat,A2: set_nat] :
( ( member_Sum_sum_a_nat @ Z @ ( image_7293268710728258664_a_nat @ F @ A2 ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_159_image__iff,axiom,
! [Z: sum_sum_a_nat,F: a > sum_sum_a_nat,A2: set_a] :
( ( member_Sum_sum_a_nat @ Z @ ( image_7873763678140191238_a_nat @ F @ A2 ) )
= ( ? [X: a] :
( ( member_a @ X @ A2 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_160_image__iff,axiom,
! [Z: set_nat,F: nat > set_nat,A2: set_nat] :
( ( member_set_nat @ Z @ ( image_nat_set_nat @ F @ A2 ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_161_bex__imageD,axiom,
! [F: nat > sum_sum_a_nat,A2: set_nat,P: sum_sum_a_nat > $o] :
( ? [X5: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X5 @ ( image_7293268710728258664_a_nat @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_162_bex__imageD,axiom,
! [F: a > sum_sum_a_nat,A2: set_a,P: sum_sum_a_nat > $o] :
( ? [X5: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X5 @ ( image_7873763678140191238_a_nat @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_163_bex__imageD,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_nat > $o] :
( ? [X5: set_nat] :
( ( member_set_nat @ X5 @ ( image_nat_set_nat @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_164_image__cong,axiom,
! [M2: set_nat,N2: set_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ( M2 = N2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_7293268710728258664_a_nat @ F @ M2 )
= ( image_7293268710728258664_a_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_165_image__cong,axiom,
! [M2: set_a,N2: set_a,F: a > sum_sum_a_nat,G: a > sum_sum_a_nat] :
( ( M2 = N2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_7873763678140191238_a_nat @ F @ M2 )
= ( image_7873763678140191238_a_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_166_image__cong,axiom,
! [M2: set_nat,N2: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( M2 = N2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_set_nat @ F @ M2 )
= ( image_nat_set_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_167_ball__imageD,axiom,
! [F: nat > sum_sum_a_nat,A2: set_nat,P: sum_sum_a_nat > $o] :
( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ ( image_7293268710728258664_a_nat @ F @ A2 ) )
=> ( P @ X2 ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_168_ball__imageD,axiom,
! [F: a > sum_sum_a_nat,A2: set_a,P: sum_sum_a_nat > $o] :
( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ ( image_7873763678140191238_a_nat @ F @ A2 ) )
=> ( P @ X2 ) )
=> ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_169_ball__imageD,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_nat > $o] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( image_nat_set_nat @ F @ A2 ) )
=> ( P @ X2 ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_170_rev__image__eqI,axiom,
! [X3: nat,A2: set_nat,B2: sum_sum_a_nat,F: nat > sum_sum_a_nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7293268710728258664_a_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_171_rev__image__eqI,axiom,
! [X3: a,A2: set_a,B2: sum_sum_a_nat,F: a > sum_sum_a_nat] :
( ( member_a @ X3 @ A2 )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7873763678140191238_a_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_172_rev__image__eqI,axiom,
! [X3: nat,A2: set_nat,B2: set_nat,F: nat > set_nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_set_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_173_rev__image__eqI,axiom,
! [X3: $o,A2: set_o,B2: $o,F: $o > $o] :
( ( member_o @ X3 @ A2 )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_o @ B2 @ ( image_o_o @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_174_in__mono,axiom,
! [A2: set_o,B: set_o,X3: $o] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ( member_o @ X3 @ A2 )
=> ( member_o @ X3 @ B ) ) ) ).
% in_mono
thf(fact_175_in__mono,axiom,
! [A2: set_a,B: set_a,X3: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_176_in__mono,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,X3: sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B )
=> ( ( member_Sum_sum_a_nat @ X3 @ A2 )
=> ( member_Sum_sum_a_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_177_subsetD,axiom,
! [A2: set_o,B: set_o,C: $o] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ( member_o @ C @ A2 )
=> ( member_o @ C @ B ) ) ) ).
% subsetD
thf(fact_178_subsetD,axiom,
! [A2: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_179_subsetD,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C: sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B )
=> ( ( member_Sum_sum_a_nat @ C @ A2 )
=> ( member_Sum_sum_a_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_180_equalityE,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B )
=> ~ ( ord_less_eq_set_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_181_equalityE,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( A2 = B )
=> ~ ( ( ord_le1325389633284124927_a_nat @ A2 @ B )
=> ~ ( ord_le1325389633284124927_a_nat @ B @ A2 ) ) ) ).
% equalityE
thf(fact_182_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
! [X: $o] :
( ( member_o @ X @ A3 )
=> ( member_o @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_183_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [X: a] :
( ( member_a @ X @ A3 )
=> ( member_a @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_184_subset__eq,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A3: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A3 )
=> ( member_Sum_sum_a_nat @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_185_equalityD1,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_186_equalityD1,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( A2 = B )
=> ( ord_le1325389633284124927_a_nat @ A2 @ B ) ) ).
% equalityD1
thf(fact_187_Set_OequalityD2,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ).
% Set.equalityD2
thf(fact_188_Set_OequalityD2,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( A2 = B )
=> ( ord_le1325389633284124927_a_nat @ B @ A2 ) ) ).
% Set.equalityD2
thf(fact_189_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
! [T: $o] :
( ( member_o @ T @ A3 )
=> ( member_o @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_190_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [T: a] :
( ( member_a @ T @ A3 )
=> ( member_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_191_subset__iff,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A3: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
! [T: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ T @ A3 )
=> ( member_Sum_sum_a_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_192_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_193_subset__refl,axiom,
! [A2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_194_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_195_Collect__mono,axiom,
! [P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
( ! [X2: sum_sum_a_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le1325389633284124927_a_nat @ ( collec7073057861543223018_a_nat @ P ) @ ( collec7073057861543223018_a_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_196_subset__trans,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_197_subset__trans,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B )
=> ( ( ord_le1325389633284124927_a_nat @ B @ C2 )
=> ( ord_le1325389633284124927_a_nat @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_198_set__eq__subset,axiom,
( ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_199_set__eq__subset,axiom,
( ( ^ [Y2: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] : ( Y2 = Z2 ) )
= ( ^ [A3: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A3 @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_200_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_201_Collect__mono__iff,axiom,
! [P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
( ( ord_le1325389633284124927_a_nat @ ( collec7073057861543223018_a_nat @ P ) @ ( collec7073057861543223018_a_nat @ Q ) )
= ( ! [X: sum_sum_a_nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_202_Compr__image__eq,axiom,
! [F: nat > sum_sum_a_nat,A2: set_nat,P: sum_sum_a_nat > $o] :
( ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( image_7293268710728258664_a_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_7293268710728258664_a_nat @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_203_Compr__image__eq,axiom,
! [F: a > sum_sum_a_nat,A2: set_a,P: sum_sum_a_nat > $o] :
( ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( image_7873763678140191238_a_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_7873763678140191238_a_nat @ F
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_204_Compr__image__eq,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_nat > $o] :
( ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ ( image_nat_set_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_nat_set_nat @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_205_Compr__image__eq,axiom,
! [F: $o > $o,A2: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_o_o @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_o_o @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_206_image__image,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: nat > sum_sum_a_nat,A2: set_nat] :
( ( image_7142520692256960453_a_nat @ F @ ( image_7293268710728258664_a_nat @ G @ A2 ) )
= ( image_7293268710728258664_a_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_207_image__image,axiom,
! [F: sum_sum_a_nat > set_nat,G: nat > sum_sum_a_nat,A2: set_nat] :
( ( image_4589483402070311232et_nat @ F @ ( image_7293268710728258664_a_nat @ G @ A2 ) )
= ( image_nat_set_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_208_image__image,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: a > sum_sum_a_nat,A2: set_a] :
( ( image_7142520692256960453_a_nat @ F @ ( image_7873763678140191238_a_nat @ G @ A2 ) )
= ( image_7873763678140191238_a_nat
@ ^ [X: a] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_209_image__image,axiom,
! [F: set_nat > sum_sum_a_nat,G: nat > set_nat,A2: set_nat] :
( ( image_6589623134797232946_a_nat @ F @ ( image_nat_set_nat @ G @ A2 ) )
= ( image_7293268710728258664_a_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_210_image__image,axiom,
! [F: set_nat > set_nat,G: nat > set_nat,A2: set_nat] :
( ( image_7916887816326733075et_nat @ F @ ( image_nat_set_nat @ G @ A2 ) )
= ( image_nat_set_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_211_image__image,axiom,
! [F: nat > sum_sum_a_nat,G: nat > nat,A2: set_nat] :
( ( image_7293268710728258664_a_nat @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_7293268710728258664_a_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_212_image__image,axiom,
! [F: nat > sum_sum_a_nat,G: a > nat,A2: set_a] :
( ( image_7293268710728258664_a_nat @ F @ ( image_a_nat @ G @ A2 ) )
= ( image_7873763678140191238_a_nat
@ ^ [X: a] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_213_image__image,axiom,
! [F: a > sum_sum_a_nat,G: nat > a,A2: set_nat] :
( ( image_7873763678140191238_a_nat @ F @ ( image_nat_a @ G @ A2 ) )
= ( image_7293268710728258664_a_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_214_image__image,axiom,
! [F: a > sum_sum_a_nat,G: a > a,A2: set_a] :
( ( image_7873763678140191238_a_nat @ F @ ( image_a_a @ G @ A2 ) )
= ( image_7873763678140191238_a_nat
@ ^ [X: a] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_215_image__image,axiom,
! [F: nat > set_nat,G: nat > nat,A2: set_nat] :
( ( image_nat_set_nat @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_set_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_216_imageE,axiom,
! [B2: sum_sum_a_nat,F: nat > sum_sum_a_nat,A2: set_nat] :
( ( member_Sum_sum_a_nat @ B2 @ ( image_7293268710728258664_a_nat @ F @ A2 ) )
=> ~ ! [X2: nat] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member_nat @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_217_imageE,axiom,
! [B2: sum_sum_a_nat,F: a > sum_sum_a_nat,A2: set_a] :
( ( member_Sum_sum_a_nat @ B2 @ ( image_7873763678140191238_a_nat @ F @ A2 ) )
=> ~ ! [X2: a] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member_a @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_218_imageE,axiom,
! [B2: set_nat,F: nat > set_nat,A2: set_nat] :
( ( member_set_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) )
=> ~ ! [X2: nat] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member_nat @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_219_imageE,axiom,
! [B2: $o,F: $o > $o,A2: set_o] :
( ( member_o @ B2 @ ( image_o_o @ F @ A2 ) )
=> ~ ! [X2: $o] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member_o @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_220_Collect__restrict,axiom,
! [X6: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_221_Collect__restrict,axiom,
! [X6: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_222_Collect__restrict,axiom,
! [X6: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ord_le1325389633284124927_a_nat
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_223_prop__restrict,axiom,
! [X3: $o,Z3: set_o,X6: set_o,P: $o > $o] :
( ( member_o @ X3 @ Z3 )
=> ( ( ord_less_eq_set_o @ Z3
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_224_prop__restrict,axiom,
! [X3: a,Z3: set_a,X6: set_a,P: a > $o] :
( ( member_a @ X3 @ Z3 )
=> ( ( ord_less_eq_set_a @ Z3
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_225_prop__restrict,axiom,
! [X3: sum_sum_a_nat,Z3: set_Sum_sum_a_nat,X6: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( member_Sum_sum_a_nat @ X3 @ Z3 )
=> ( ( ord_le1325389633284124927_a_nat @ Z3
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X3 ) ) ) ).
% prop_restrict
thf(fact_226_Collect__subset,axiom,
! [A2: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_227_Collect__subset,axiom,
! [A2: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_228_Collect__subset,axiom,
! [A2: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ord_le1325389633284124927_a_nat
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_229_image__mono,axiom,
! [A2: set_nat,B: set_nat,F: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ ( image_nat_set_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_230_image__mono,axiom,
! [A2: set_nat,B: set_nat,F: nat > sum_sum_a_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ A2 ) @ ( image_7293268710728258664_a_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_231_image__mono,axiom,
! [A2: set_a,B: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_232_image__mono,axiom,
! [A2: set_a,B: set_a,F: a > sum_sum_a_nat] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ A2 ) @ ( image_7873763678140191238_a_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_233_image__mono,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,F: sum_sum_a_nat > a] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B )
=> ( ord_less_eq_set_a @ ( image_6322530041254294468_nat_a @ F @ A2 ) @ ( image_6322530041254294468_nat_a @ F @ B ) ) ) ).
% image_mono
thf(fact_234_image__mono,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B )
=> ( ord_le1325389633284124927_a_nat @ ( image_7142520692256960453_a_nat @ F @ A2 ) @ ( image_7142520692256960453_a_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_235_image__subsetI,axiom,
! [A2: set_nat,F: nat > set_nat,B: set_set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ B ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_236_image__subsetI,axiom,
! [A2: set_o,F: $o > $o,B: set_o] :
( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( member_o @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_237_image__subsetI,axiom,
! [A2: set_o,F: $o > a,B: set_a] :
( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_o_a @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_238_image__subsetI,axiom,
! [A2: set_nat,F: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_Sum_sum_a_nat @ ( F @ X2 ) @ B ) )
=> ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_239_image__subsetI,axiom,
! [A2: set_a,F: a > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_Sum_sum_a_nat @ ( F @ X2 ) @ B ) )
=> ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_240_image__subsetI,axiom,
! [A2: set_o,F: $o > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( member_Sum_sum_a_nat @ ( F @ X2 ) @ B ) )
=> ( ord_le1325389633284124927_a_nat @ ( image_4139480514073730540_a_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_241_subset__imageE,axiom,
! [B: set_set_nat,F: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ ( image_nat_set_nat @ F @ A2 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( B
!= ( image_nat_set_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_242_subset__imageE,axiom,
! [B: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A2 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( B
!= ( image_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_243_subset__imageE,axiom,
! [B: set_a,F: sum_sum_a_nat > a,A2: set_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ B @ ( image_6322530041254294468_nat_a @ F @ A2 ) )
=> ~ ! [C3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C3 @ A2 )
=> ( B
!= ( image_6322530041254294468_nat_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_244_subset__imageE,axiom,
! [B: set_Sum_sum_a_nat,F: nat > sum_sum_a_nat,A2: set_nat] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_7293268710728258664_a_nat @ F @ A2 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( B
!= ( image_7293268710728258664_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_245_subset__imageE,axiom,
! [B: set_Sum_sum_a_nat,F: a > sum_sum_a_nat,A2: set_a] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_7873763678140191238_a_nat @ F @ A2 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( B
!= ( image_7873763678140191238_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_246_subset__imageE,axiom,
! [B: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_7142520692256960453_a_nat @ F @ A2 ) )
=> ~ ! [C3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C3 @ A2 )
=> ( B
!= ( image_7142520692256960453_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_247_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_248_pred__subset__eq,axiom,
! [R: set_a,S: set_a] :
( ( ord_less_eq_a_o
@ ^ [X: a] : ( member_a @ X @ R )
@ ^ [X: a] : ( member_a @ X @ S ) )
= ( ord_less_eq_set_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_249_pred__subset__eq,axiom,
! [R: set_Sum_sum_a_nat,S: set_Sum_sum_a_nat] :
( ( ord_le1477630214076318366_nat_o
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ R )
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ S ) )
= ( ord_le1325389633284124927_a_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_250_subset__Collect__iff,axiom,
! [B: set_o,A2: set_o,P: $o > $o] :
( ( ord_less_eq_set_o @ B @ A2 )
=> ( ( ord_less_eq_set_o @ B
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A2 )
& ( P @ X ) ) ) )
= ( ! [X: $o] :
( ( member_o @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_251_subset__Collect__iff,axiom,
! [B: set_a,A2: set_a,P: a > $o] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( ord_less_eq_set_a @ B
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
& ( P @ X ) ) ) )
= ( ! [X: a] :
( ( member_a @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_252_subset__Collect__iff,axiom,
! [B: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( ord_le1325389633284124927_a_nat @ B @ A2 )
=> ( ( ord_le1325389633284124927_a_nat @ B
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A2 )
& ( P @ X ) ) ) )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_253_subset__CollectI,axiom,
! [B: set_o,A2: set_o,Q: $o > $o,P: $o > $o] :
( ( ord_less_eq_set_o @ B @ A2 )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_less_eq_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ B )
& ( Q @ X ) ) )
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A2 )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_254_subset__CollectI,axiom,
! [B: set_a,A2: set_a,Q: a > $o,P: a > $o] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ B )
& ( Q @ X ) ) )
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_255_subset__CollectI,axiom,
! [B: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,Q: sum_sum_a_nat > $o,P: sum_sum_a_nat > $o] :
( ( ord_le1325389633284124927_a_nat @ B @ A2 )
=> ( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_le1325389633284124927_a_nat
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ B )
& ( Q @ X ) ) )
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A2 )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_256_conj__subset__def,axiom,
! [A2: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2
@ ( collect_a
@ ^ [X: a] :
( ( P @ X )
& ( Q @ X ) ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( collect_a @ P ) )
& ( ord_less_eq_set_a @ A2 @ ( collect_a @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_257_conj__subset__def,axiom,
! [A2: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
( ( ord_le1325389633284124927_a_nat @ A2
@ ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( P @ X )
& ( Q @ X ) ) ) )
= ( ( ord_le1325389633284124927_a_nat @ A2 @ ( collec7073057861543223018_a_nat @ P ) )
& ( ord_le1325389633284124927_a_nat @ A2 @ ( collec7073057861543223018_a_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_258_Forall_Oprems_I3_J,axiom,
ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ ad ) @ ( image_7293268710728258664_a_nat @ sum_Inr_nat_a @ ( set_ord_lessThan_nat @ ( d_a_b @ ( fo_Forall_a_b @ n @ phi ) ) ) ) ) @ x ).
% Forall.prems(3)
thf(fact_259_Forall_OIH,axiom,
! [Sigma: nat > sum_sum_a_nat,Tau: nat > sum_sum_a_nat] :
( ( ad_agr_a_b_nat @ ( fo_Exists_a_b @ n @ ( fo_Neg_a_b @ phi ) ) @ ad @ Sigma @ Tau )
=> ( ( ord_less_eq_set_a @ ( act_edom_a_b @ ( fo_Exists_a_b @ n @ ( fo_Neg_a_b @ phi ) ) @ i ) @ ad )
=> ( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ ad ) @ ( image_7293268710728258664_a_nat @ sum_Inr_nat_a @ ( set_ord_lessThan_nat @ ( d_a_b @ ( fo_Exists_a_b @ n @ ( fo_Neg_a_b @ phi ) ) ) ) ) ) @ x )
=> ( ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ Tau @ ( fv_fo_fmla_a_b @ ( fo_Exists_a_b @ n @ ( fo_Neg_a_b @ phi ) ) ) ) @ x )
=> ( ( esat_a_b @ ( fo_Exists_a_b @ n @ ( fo_Neg_a_b @ phi ) ) @ i @ Sigma @ top_to795618464972521135_a_nat )
= ( esat_a_b @ ( fo_Exists_a_b @ n @ ( fo_Neg_a_b @ phi ) ) @ i @ Tau @ x ) ) ) ) ) ) ).
% Forall.IH
thf(fact_260_image__Fpow__mono,axiom,
! [F: nat > set_nat,A2: set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B )
=> ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F ) @ ( finite_Fpow_nat @ A2 ) ) @ ( finite_Fpow_set_nat @ B ) ) ) ).
% image_Fpow_mono
thf(fact_261_image__Fpow__mono,axiom,
! [F: nat > sum_sum_a_nat,A2: set_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ A2 ) @ B )
=> ( ord_le7974500612278410847_a_nat @ ( image_3578472599065059474_a_nat @ ( image_7293268710728258664_a_nat @ F ) @ ( finite_Fpow_nat @ A2 ) ) @ ( finite171985660919647589_a_nat @ B ) ) ) ).
% image_Fpow_mono
thf(fact_262_image__Fpow__mono,axiom,
! [F: a > sum_sum_a_nat,A2: set_a,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ A2 ) @ B )
=> ( ord_le7974500612278410847_a_nat @ ( image_6715601112060939782_a_nat @ ( image_7873763678140191238_a_nat @ F ) @ ( finite_Fpow_a @ A2 ) ) @ ( finite171985660919647589_a_nat @ B ) ) ) ).
% image_Fpow_mono
thf(fact_263_fv__fo__fmla__list__rec_Osimps_I8_J,axiom,
! [N: nat,Phi: fo_fmla_a_b] :
( ( fv_fo_5581231672024362409ec_a_b @ ( fo_Forall_a_b @ N @ Phi ) )
= ( filter_nat
@ ^ [M: nat] : ( N != M )
@ ( fv_fo_5581231672024362409ec_a_b @ Phi ) ) ) ).
% fv_fo_fmla_list_rec.simps(8)
thf(fact_264_UNIV__I,axiom,
! [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% UNIV_I
thf(fact_265_UNIV__I,axiom,
! [X3: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X3 @ top_to795618464972521135_a_nat ) ).
% UNIV_I
thf(fact_266_UNIV__I,axiom,
! [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_I
thf(fact_267_UnCI,axiom,
! [C: $o,B: set_o,A2: set_o] :
( ( ~ ( member_o @ C @ B )
=> ( member_o @ C @ A2 ) )
=> ( member_o @ C @ ( sup_sup_set_o @ A2 @ B ) ) ) ).
% UnCI
thf(fact_268_UnCI,axiom,
! [C: sum_sum_a_nat,B: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( ~ ( member_Sum_sum_a_nat @ C @ B )
=> ( member_Sum_sum_a_nat @ C @ A2 ) )
=> ( member_Sum_sum_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_269_Un__iff,axiom,
! [C: $o,A2: set_o,B: set_o] :
( ( member_o @ C @ ( sup_sup_set_o @ A2 @ B ) )
= ( ( member_o @ C @ A2 )
| ( member_o @ C @ B ) ) ) ).
% Un_iff
thf(fact_270_Un__iff,axiom,
! [C: sum_sum_a_nat,A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) )
= ( ( member_Sum_sum_a_nat @ C @ A2 )
| ( member_Sum_sum_a_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_271_Un__subset__iff,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C2 )
= ( ( ord_less_eq_set_a @ A2 @ C2 )
& ( ord_less_eq_set_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_272_Un__subset__iff,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) @ C2 )
= ( ( ord_le1325389633284124927_a_nat @ A2 @ C2 )
& ( ord_le1325389633284124927_a_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_273_local_Ounfold_I2_J,axiom,
( ( sup_su6804446743777130803_a_nat @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ ad ) @ ( image_7293268710728258664_a_nat @ sum_Inr_nat_a @ ( set_ord_lessThan_nat @ ( d_a_b @ ( fo_Forall_a_b @ n @ phi ) ) ) ) )
= ( sup_su6804446743777130803_a_nat @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ ad ) @ ( image_7293268710728258664_a_nat @ sum_Inr_nat_a @ ( set_ord_lessThan_nat @ ( d_a_b @ ( fo_Exists_a_b @ n @ ( fo_Neg_a_b @ phi ) ) ) ) ) ) ) ).
% local.unfold(2)
thf(fact_274_UnE,axiom,
! [C: $o,A2: set_o,B: set_o] :
( ( member_o @ C @ ( sup_sup_set_o @ A2 @ B ) )
=> ( ~ ( member_o @ C @ A2 )
=> ( member_o @ C @ B ) ) ) ).
% UnE
thf(fact_275_UnE,axiom,
! [C: sum_sum_a_nat,A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) )
=> ( ~ ( member_Sum_sum_a_nat @ C @ A2 )
=> ( member_Sum_sum_a_nat @ C @ B ) ) ) ).
% UnE
thf(fact_276_UnI1,axiom,
! [C: $o,A2: set_o,B: set_o] :
( ( member_o @ C @ A2 )
=> ( member_o @ C @ ( sup_sup_set_o @ A2 @ B ) ) ) ).
% UnI1
thf(fact_277_UnI1,axiom,
! [C: sum_sum_a_nat,A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ A2 )
=> ( member_Sum_sum_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_278_UnI2,axiom,
! [C: $o,B: set_o,A2: set_o] :
( ( member_o @ C @ B )
=> ( member_o @ C @ ( sup_sup_set_o @ A2 @ B ) ) ) ).
% UnI2
thf(fact_279_UnI2,axiom,
! [C: sum_sum_a_nat,B: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ B )
=> ( member_Sum_sum_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_280_bex__Un,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( ? [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) )
& ( P @ X ) ) )
= ( ? [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A2 )
& ( P @ X ) )
| ? [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ B )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_281_ball__Un,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) )
=> ( P @ X ) ) )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A2 )
=> ( P @ X ) )
& ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_282_Un__assoc,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) @ C2 )
= ( sup_su6804446743777130803_a_nat @ A2 @ ( sup_su6804446743777130803_a_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_283_UNIV__eq__I,axiom,
! [A2: set_o] :
( ! [X2: $o] : ( member_o @ X2 @ A2 )
=> ( top_top_set_o = A2 ) ) ).
% UNIV_eq_I
thf(fact_284_UNIV__eq__I,axiom,
! [A2: set_Sum_sum_a_nat] :
( ! [X2: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X2 @ A2 )
=> ( top_to795618464972521135_a_nat = A2 ) ) ).
% UNIV_eq_I
thf(fact_285_UNIV__eq__I,axiom,
! [A2: set_nat] :
( ! [X2: nat] : ( member_nat @ X2 @ A2 )
=> ( top_top_set_nat = A2 ) ) ).
% UNIV_eq_I
thf(fact_286_Un__absorb,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_287_Un__commute,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [A3: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] : ( sup_su6804446743777130803_a_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_288_UNIV__witness,axiom,
? [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).
% UNIV_witness
thf(fact_289_UNIV__witness,axiom,
? [X2: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X2 @ top_to795618464972521135_a_nat ) ).
% UNIV_witness
thf(fact_290_UNIV__witness,axiom,
? [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_291_Un__UNIV__left,axiom,
! [B: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ top_to795618464972521135_a_nat @ B )
= top_to795618464972521135_a_nat ) ).
% Un_UNIV_left
thf(fact_292_Un__UNIV__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ B )
= top_top_set_nat ) ).
% Un_UNIV_left
thf(fact_293_Un__UNIV__right,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A2 @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat ) ).
% Un_UNIV_right
thf(fact_294_Un__UNIV__right,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ top_top_set_nat )
= top_top_set_nat ) ).
% Un_UNIV_right
thf(fact_295_Un__left__absorb,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) )
= ( sup_su6804446743777130803_a_nat @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_296_Un__left__commute,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A2 @ ( sup_su6804446743777130803_a_nat @ B @ C2 ) )
= ( sup_su6804446743777130803_a_nat @ B @ ( sup_su6804446743777130803_a_nat @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_297_not__arg__cong__Inr,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ( sum_Inr_nat_a @ X3 )
!= ( sum_Inr_nat_a @ Y ) ) ) ).
% not_arg_cong_Inr
thf(fact_298_obj__sumE,axiom,
! [S2: sum_sum_a_nat] :
( ! [X2: a] :
( S2
!= ( sum_Inl_a_nat @ X2 ) )
=> ~ ! [X2: nat] :
( S2
!= ( sum_Inr_nat_a @ X2 ) ) ) ).
% obj_sumE
thf(fact_299_Un__def,axiom,
( sup_sup_set_o
= ( ^ [A3: set_o,B3: set_o] :
( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A3 )
| ( member_o @ X @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_300_Un__def,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [A3: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ A3 )
| ( member_Sum_sum_a_nat @ X @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_301_Collect__disj__eq,axiom,
! [P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
( ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_su6804446743777130803_a_nat @ ( collec7073057861543223018_a_nat @ P ) @ ( collec7073057861543223018_a_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_302_UNIV__def,axiom,
( top_to795618464972521135_a_nat
= ( collec7073057861543223018_a_nat
@ ^ [X: sum_sum_a_nat] : $true ) ) ).
% UNIV_def
thf(fact_303_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X: nat] : $true ) ) ).
% UNIV_def
thf(fact_304_top_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
=> ( A = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_305_top_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
=> ( A = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_306_top_Oextremum__uniqueI,axiom,
! [A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ top_to795618464972521135_a_nat @ A )
=> ( A = top_to795618464972521135_a_nat ) ) ).
% top.extremum_uniqueI
thf(fact_307_top_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
= ( A = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_308_top_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A )
= ( A = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_309_top_Oextremum__unique,axiom,
! [A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ top_to795618464972521135_a_nat @ A )
= ( A = top_to795618464972521135_a_nat ) ) ).
% top.extremum_unique
thf(fact_310_top__greatest,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% top_greatest
thf(fact_311_top__greatest,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% top_greatest
thf(fact_312_top__greatest,axiom,
! [A: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ A @ top_to795618464972521135_a_nat ) ).
% top_greatest
thf(fact_313_image__Un,axiom,
! [F: nat > set_nat,A2: set_nat,B: set_nat] :
( ( image_nat_set_nat @ F @ ( sup_sup_set_nat @ A2 @ B ) )
= ( sup_sup_set_set_nat @ ( image_nat_set_nat @ F @ A2 ) @ ( image_nat_set_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_314_image__Un,axiom,
! [F: nat > sum_sum_a_nat,A2: set_nat,B: set_nat] :
( ( image_7293268710728258664_a_nat @ F @ ( sup_sup_set_nat @ A2 @ B ) )
= ( sup_su6804446743777130803_a_nat @ ( image_7293268710728258664_a_nat @ F @ A2 ) @ ( image_7293268710728258664_a_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_315_image__Un,axiom,
! [F: a > sum_sum_a_nat,A2: set_a,B: set_a] :
( ( image_7873763678140191238_a_nat @ F @ ( sup_sup_set_a @ A2 @ B ) )
= ( sup_su6804446743777130803_a_nat @ ( image_7873763678140191238_a_nat @ F @ A2 ) @ ( image_7873763678140191238_a_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_316_image__Un,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( image_7142520692256960453_a_nat @ F @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) )
= ( sup_su6804446743777130803_a_nat @ ( image_7142520692256960453_a_nat @ F @ A2 ) @ ( image_7142520692256960453_a_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_317_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( sup_sup_set_a @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_318_subset__Un__eq,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A3: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_319_subset__UnE,axiom,
! [C2: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) )
=> ~ ! [A6: set_a] :
( ( ord_less_eq_set_a @ A6 @ A2 )
=> ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ B )
=> ( C2
!= ( sup_sup_set_a @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_320_subset__UnE,axiom,
! [C2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) )
=> ~ ! [A6: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A6 @ A2 )
=> ! [B6: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B6 @ B )
=> ( C2
!= ( sup_su6804446743777130803_a_nat @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_321_Un__absorb2,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_322_Un__absorb2,axiom,
! [B: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B @ A2 )
=> ( ( sup_su6804446743777130803_a_nat @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_323_Un__absorb1,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( sup_sup_set_a @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_324_Un__absorb1,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B )
=> ( ( sup_su6804446743777130803_a_nat @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_325_Un__upper2,axiom,
! [B: set_a,A2: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_upper2
thf(fact_326_Un__upper2,axiom,
! [B: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ B @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) ) ).
% Un_upper2
thf(fact_327_Un__upper1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_upper1
thf(fact_328_Un__upper1,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ A2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) ) ).
% Un_upper1
thf(fact_329_Un__least,axiom,
! [A2: set_a,C2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_330_Un__least,axiom,
! [A2: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ C2 )
=> ( ( ord_le1325389633284124927_a_nat @ B @ C2 )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_331_Un__mono,axiom,
! [A2: set_a,C2: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_332_Un__mono,axiom,
! [A2: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,D: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ C2 )
=> ( ( ord_le1325389633284124927_a_nat @ B @ D )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) @ ( sup_su6804446743777130803_a_nat @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_333_esat_Osimps_I4_J,axiom,
! [Phi: fo_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > sum_sum_a_nat,X6: set_Sum_sum_a_nat] :
( ( esat_a_b @ ( fo_Neg_a_b @ Phi ) @ I @ Sigma @ X6 )
= ( ~ ( esat_a_b @ Phi @ I @ Sigma @ X6 ) ) ) ).
% esat.simps(4)
thf(fact_334_range__eqI,axiom,
! [B2: sum_sum_a_nat,F: a > sum_sum_a_nat,X3: a] :
( ( B2
= ( F @ X3 ) )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7873763678140191238_a_nat @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_335_range__eqI,axiom,
! [B2: $o,F: sum_sum_a_nat > $o,X3: sum_sum_a_nat] :
( ( B2
= ( F @ X3 ) )
=> ( member_o @ B2 @ ( image_6095136190293192542_nat_o @ F @ top_to795618464972521135_a_nat ) ) ) ).
% range_eqI
thf(fact_336_range__eqI,axiom,
! [B2: sum_sum_a_nat,F: nat > sum_sum_a_nat,X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7293268710728258664_a_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_337_range__eqI,axiom,
! [B2: set_nat,F: nat > set_nat,X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ( member_set_nat @ B2 @ ( image_nat_set_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_338_range__eqI,axiom,
! [B2: $o,F: nat > $o,X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ( member_o @ B2 @ ( image_nat_o @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_339_rangeI,axiom,
! [F: a > sum_sum_a_nat,X3: a] : ( member_Sum_sum_a_nat @ ( F @ X3 ) @ ( image_7873763678140191238_a_nat @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_340_rangeI,axiom,
! [F: sum_sum_a_nat > $o,X3: sum_sum_a_nat] : ( member_o @ ( F @ X3 ) @ ( image_6095136190293192542_nat_o @ F @ top_to795618464972521135_a_nat ) ) ).
% rangeI
thf(fact_341_rangeI,axiom,
! [F: nat > sum_sum_a_nat,X3: nat] : ( member_Sum_sum_a_nat @ ( F @ X3 ) @ ( image_7293268710728258664_a_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_342_rangeI,axiom,
! [F: nat > set_nat,X3: nat] : ( member_set_nat @ ( F @ X3 ) @ ( image_nat_set_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_343_rangeI,axiom,
! [F: nat > $o,X3: nat] : ( member_o @ ( F @ X3 ) @ ( image_nat_o @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_344_subset__UNIV,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_345_subset__UNIV,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ top_top_set_a ) ).
% subset_UNIV
thf(fact_346_subset__UNIV,axiom,
! [A2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ A2 @ top_to795618464972521135_a_nat ) ).
% subset_UNIV
thf(fact_347_esat__fv__cong,axiom,
! [Phi: fo_fmla_a_b,Sigma: nat > sum_sum_a_nat,Sigma2: nat > sum_sum_a_nat,I: product_prod_b_nat > set_list_a,X6: set_Sum_sum_a_nat] :
( ! [N3: nat] :
( ( member_nat @ N3 @ ( fv_fo_fmla_a_b @ Phi ) )
=> ( ( Sigma @ N3 )
= ( Sigma2 @ N3 ) ) )
=> ( ( esat_a_b @ Phi @ I @ Sigma @ X6 )
= ( esat_a_b @ Phi @ I @ Sigma2 @ X6 ) ) ) ).
% esat_fv_cong
thf(fact_348_fv__fo__fmla__list__rec_Osimps_I4_J,axiom,
! [Phi: fo_fmla_a_b] :
( ( fv_fo_5581231672024362409ec_a_b @ ( fo_Neg_a_b @ Phi ) )
= ( fv_fo_5581231672024362409ec_a_b @ Phi ) ) ).
% fv_fo_fmla_list_rec.simps(4)
thf(fact_349_rangeE,axiom,
! [B2: sum_sum_a_nat,F: a > sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ B2 @ ( image_7873763678140191238_a_nat @ F @ top_top_set_a ) )
=> ~ ! [X2: a] :
( B2
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_350_rangeE,axiom,
! [B2: $o,F: sum_sum_a_nat > $o] :
( ( member_o @ B2 @ ( image_6095136190293192542_nat_o @ F @ top_to795618464972521135_a_nat ) )
=> ~ ! [X2: sum_sum_a_nat] :
( B2
= ( ~ ( F @ X2 ) ) ) ) ).
% rangeE
thf(fact_351_rangeE,axiom,
! [B2: sum_sum_a_nat,F: nat > sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ B2 @ ( image_7293268710728258664_a_nat @ F @ top_top_set_nat ) )
=> ~ ! [X2: nat] :
( B2
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_352_rangeE,axiom,
! [B2: set_nat,F: nat > set_nat] :
( ( member_set_nat @ B2 @ ( image_nat_set_nat @ F @ top_top_set_nat ) )
=> ~ ! [X2: nat] :
( B2
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_353_rangeE,axiom,
! [B2: $o,F: nat > $o] :
( ( member_o @ B2 @ ( image_nat_o @ F @ top_top_set_nat ) )
=> ~ ! [X2: nat] :
( B2
= ( ~ ( F @ X2 ) ) ) ) ).
% rangeE
thf(fact_354_range__composition,axiom,
! [F: nat > set_nat,G: nat > nat] :
( ( image_nat_set_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_nat_set_nat @ F @ ( image_nat_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_355_range__composition,axiom,
! [F: nat > sum_sum_a_nat,G: a > nat] :
( ( image_7873763678140191238_a_nat
@ ^ [X: a] : ( F @ ( G @ X ) )
@ top_top_set_a )
= ( image_7293268710728258664_a_nat @ F @ ( image_a_nat @ G @ top_top_set_a ) ) ) ).
% range_composition
thf(fact_356_range__composition,axiom,
! [F: a > sum_sum_a_nat,G: a > a] :
( ( image_7873763678140191238_a_nat
@ ^ [X: a] : ( F @ ( G @ X ) )
@ top_top_set_a )
= ( image_7873763678140191238_a_nat @ F @ ( image_a_a @ G @ top_top_set_a ) ) ) ).
% range_composition
thf(fact_357_range__composition,axiom,
! [F: nat > sum_sum_a_nat,G: nat > nat] :
( ( image_7293268710728258664_a_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_7293268710728258664_a_nat @ F @ ( image_nat_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_358_range__composition,axiom,
! [F: a > sum_sum_a_nat,G: nat > a] :
( ( image_7293268710728258664_a_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_7873763678140191238_a_nat @ F @ ( image_nat_a @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_359_range__composition,axiom,
! [F: set_nat > set_nat,G: nat > set_nat] :
( ( image_nat_set_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_7916887816326733075et_nat @ F @ ( image_nat_set_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_360_range__composition,axiom,
! [F: nat > set_nat,G: sum_sum_a_nat > nat] :
( ( image_4589483402070311232et_nat
@ ^ [X: sum_sum_a_nat] : ( F @ ( G @ X ) )
@ top_to795618464972521135_a_nat )
= ( image_nat_set_nat @ F @ ( image_2473878607534554506at_nat @ G @ top_to795618464972521135_a_nat ) ) ) ).
% range_composition
thf(fact_361_range__composition,axiom,
! [F: set_nat > sum_sum_a_nat,G: nat > set_nat] :
( ( image_7293268710728258664_a_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_6589623134797232946_a_nat @ F @ ( image_nat_set_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_362_range__composition,axiom,
! [F: sum_sum_a_nat > set_nat,G: nat > sum_sum_a_nat] :
( ( image_nat_set_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_4589483402070311232et_nat @ F @ ( image_7293268710728258664_a_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_363_range__composition,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: a > sum_sum_a_nat] :
( ( image_7873763678140191238_a_nat
@ ^ [X: a] : ( F @ ( G @ X ) )
@ top_top_set_a )
= ( image_7142520692256960453_a_nat @ F @ ( image_7873763678140191238_a_nat @ G @ top_top_set_a ) ) ) ).
% range_composition
thf(fact_364_Fpow__mono,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ ( finite_Fpow_a @ A2 ) @ ( finite_Fpow_a @ B ) ) ) ).
% Fpow_mono
thf(fact_365_Fpow__mono,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B )
=> ( ord_le7974500612278410847_a_nat @ ( finite171985660919647589_a_nat @ A2 ) @ ( finite171985660919647589_a_nat @ B ) ) ) ).
% Fpow_mono
thf(fact_366_range__subsetD,axiom,
! [F: sum_sum_a_nat > $o,B: set_o,I2: sum_sum_a_nat] :
( ( ord_less_eq_set_o @ ( image_6095136190293192542_nat_o @ F @ top_to795618464972521135_a_nat ) @ B )
=> ( member_o @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_367_range__subsetD,axiom,
! [F: nat > set_nat,B: set_set_nat,I2: nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ top_top_set_nat ) @ B )
=> ( member_set_nat @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_368_range__subsetD,axiom,
! [F: nat > $o,B: set_o,I2: nat] :
( ( ord_less_eq_set_o @ ( image_nat_o @ F @ top_top_set_nat ) @ B )
=> ( member_o @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_369_range__subsetD,axiom,
! [F: sum_sum_a_nat > a,B: set_a,I2: sum_sum_a_nat] :
( ( ord_less_eq_set_a @ ( image_6322530041254294468_nat_a @ F @ top_to795618464972521135_a_nat ) @ B )
=> ( member_a @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_370_range__subsetD,axiom,
! [F: nat > a,B: set_a,I2: nat] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F @ top_top_set_nat ) @ B )
=> ( member_a @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_371_range__subsetD,axiom,
! [F: a > sum_sum_a_nat,B: set_Sum_sum_a_nat,I2: a] :
( ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ top_top_set_a ) @ B )
=> ( member_Sum_sum_a_nat @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_372_range__subsetD,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,B: set_Sum_sum_a_nat,I2: sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7142520692256960453_a_nat @ F @ top_to795618464972521135_a_nat ) @ B )
=> ( member_Sum_sum_a_nat @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_373_range__subsetD,axiom,
! [F: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat,I2: nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ top_top_set_nat ) @ B )
=> ( member_Sum_sum_a_nat @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_374_Sup_OSUP__cong,axiom,
! [A2: set_nat,B: set_nat,C2: nat > sum_sum_a_nat,D: nat > sum_sum_a_nat,Sup: set_Sum_sum_a_nat > sum_sum_a_nat] :
( ( A2 = B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_7293268710728258664_a_nat @ C2 @ A2 ) )
= ( Sup @ ( image_7293268710728258664_a_nat @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_375_Sup_OSUP__cong,axiom,
! [A2: set_a,B: set_a,C2: a > sum_sum_a_nat,D: a > sum_sum_a_nat,Sup: set_Sum_sum_a_nat > sum_sum_a_nat] :
( ( A2 = B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_7873763678140191238_a_nat @ C2 @ A2 ) )
= ( Sup @ ( image_7873763678140191238_a_nat @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_376_Sup_OSUP__cong,axiom,
! [A2: set_nat,B: set_nat,C2: nat > set_nat,D: nat > set_nat,Sup: set_set_nat > set_nat] :
( ( A2 = B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_nat_set_nat @ C2 @ A2 ) )
= ( Sup @ ( image_nat_set_nat @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_377_Inf_OINF__cong,axiom,
! [A2: set_nat,B: set_nat,C2: nat > sum_sum_a_nat,D: nat > sum_sum_a_nat,Inf: set_Sum_sum_a_nat > sum_sum_a_nat] :
( ( A2 = B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_7293268710728258664_a_nat @ C2 @ A2 ) )
= ( Inf @ ( image_7293268710728258664_a_nat @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_378_Inf_OINF__cong,axiom,
! [A2: set_a,B: set_a,C2: a > sum_sum_a_nat,D: a > sum_sum_a_nat,Inf: set_Sum_sum_a_nat > sum_sum_a_nat] :
( ( A2 = B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_7873763678140191238_a_nat @ C2 @ A2 ) )
= ( Inf @ ( image_7873763678140191238_a_nat @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_379_Inf_OINF__cong,axiom,
! [A2: set_nat,B: set_nat,C2: nat > set_nat,D: nat > set_nat,Inf: set_set_nat > set_nat] :
( ( A2 = B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_nat_set_nat @ C2 @ A2 ) )
= ( Inf @ ( image_nat_set_nat @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_380_fv__fo__fmla__list__rec_Osimps_I7_J,axiom,
! [N: nat,Phi: fo_fmla_a_b] :
( ( fv_fo_5581231672024362409ec_a_b @ ( fo_Exists_a_b @ N @ Phi ) )
= ( filter_nat
@ ^ [M: nat] : ( N != M )
@ ( fv_fo_5581231672024362409ec_a_b @ Phi ) ) ) ).
% fv_fo_fmla_list_rec.simps(7)
thf(fact_381_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_382_UNIV__sum,axiom,
( top_to599037537065133003_a_nat
= ( sup_su5691376275764465223_a_nat @ ( image_1339813895292368543_a_nat @ sum_In4896129900671530542_a_nat @ top_to795618464972521135_a_nat ) @ ( image_1339813895292368543_a_nat @ sum_In5020205417008236852_a_nat @ top_to795618464972521135_a_nat ) ) ) ).
% UNIV_sum
thf(fact_383_UNIV__sum,axiom,
( top_to8676068415865862704at_nat
= ( sup_su4098136462271712948at_nat @ ( image_4678444565015437894at_nat @ sum_In1625650605130369377at_nat @ top_to795618464972521135_a_nat ) @ ( image_8912669507014183529at_nat @ sum_In1013769454558682809_a_nat @ top_top_set_nat ) ) ) ).
% UNIV_sum
thf(fact_384_UNIV__sum,axiom,
( top_to7568329678976531286_a_nat
= ( sup_su2990397725382381530_a_nat @ ( image_7108941391349317263_a_nat @ sum_In6445040708324073535_a_nat @ top_top_set_nat ) @ ( image_2874716449350571628_a_nat @ sum_In5417751388219754459at_nat @ top_to795618464972521135_a_nat ) ) ) ).
% UNIV_sum
thf(fact_385_UNIV__sum,axiom,
( top_to6661820994512907621at_nat
= ( sup_su3567568935942035937at_nat @ ( image_678696785212003926at_nat @ sum_Inl_nat_nat @ top_top_set_nat ) @ ( image_678696785212003926at_nat @ sum_Inr_nat_nat @ top_top_set_nat ) ) ) ).
% UNIV_sum
thf(fact_386_UNIV__sum,axiom,
( top_to795618464972521135_a_nat
= ( sup_su6804446743777130803_a_nat @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ top_top_set_a ) @ ( image_7293268710728258664_a_nat @ sum_Inr_nat_a @ top_top_set_nat ) ) ) ).
% UNIV_sum
thf(fact_387_Inr__Inl__False,axiom,
! [X3: nat,Y: a] :
( ( sum_Inr_nat_a @ X3 )
!= ( sum_Inl_a_nat @ Y ) ) ).
% Inr_Inl_False
thf(fact_388_Inl__Inr__False,axiom,
! [X3: a,Y: nat] :
( ( sum_Inl_a_nat @ X3 )
!= ( sum_Inr_nat_a @ Y ) ) ).
% Inl_Inr_False
thf(fact_389_boolean__algebra_Odisj__one__right,axiom,
! [X3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X3 @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat ) ).
% boolean_algebra.disj_one_right
thf(fact_390_boolean__algebra_Odisj__one__right,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ top_top_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_right
thf(fact_391_boolean__algebra_Odisj__one__left,axiom,
! [X3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ top_to795618464972521135_a_nat @ X3 )
= top_to795618464972521135_a_nat ) ).
% boolean_algebra.disj_one_left
thf(fact_392_boolean__algebra_Odisj__one__left,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X3 )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_left
thf(fact_393_sup__top__right,axiom,
! [X3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X3 @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat ) ).
% sup_top_right
thf(fact_394_sup__top__right,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ top_top_set_nat )
= top_top_set_nat ) ).
% sup_top_right
thf(fact_395_sup__top__left,axiom,
! [X3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ top_to795618464972521135_a_nat @ X3 )
= top_to795618464972521135_a_nat ) ).
% sup_top_left
thf(fact_396_sup__top__left,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X3 )
= top_top_set_nat ) ).
% sup_top_left
thf(fact_397_sup_Obounded__iff,axiom,
! [B2: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A )
= ( ( ord_less_eq_set_a @ B2 @ A )
& ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_398_sup_Obounded__iff,axiom,
! [B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ B2 @ C ) @ A )
= ( ( ord_le1325389633284124927_a_nat @ B2 @ A )
& ( ord_le1325389633284124927_a_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_399_sup_Obounded__iff,axiom,
! [B2: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_400_sup_Oidem,axiom,
! [A: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_401_sup__idem,axiom,
! [X3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_402_sup_Oleft__idem,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) )
= ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_403_sup__left__idem,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ X3 @ Y ) )
= ( sup_su6804446743777130803_a_nat @ X3 @ Y ) ) ).
% sup_left_idem
thf(fact_404_sup_Oright__idem,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) @ B2 )
= ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_405_sum_Oinject_I1_J,axiom,
! [X1: a,Y1: a] :
( ( ( sum_Inl_a_nat @ X1 )
= ( sum_Inl_a_nat @ Y1 ) )
= ( X1 = Y1 ) ) ).
% sum.inject(1)
thf(fact_406_old_Osum_Oinject_I1_J,axiom,
! [A: a,A7: a] :
( ( ( sum_Inl_a_nat @ A )
= ( sum_Inl_a_nat @ A7 ) )
= ( A = A7 ) ) ).
% old.sum.inject(1)
thf(fact_407_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_408_sum_Oinject_I2_J,axiom,
! [X22: nat,Y22: nat] :
( ( ( sum_Inr_nat_a @ X22 )
= ( sum_Inr_nat_a @ Y22 ) )
= ( X22 = Y22 ) ) ).
% sum.inject(2)
thf(fact_409_old_Osum_Oinject_I2_J,axiom,
! [B2: nat,B7: nat] :
( ( ( sum_Inr_nat_a @ B2 )
= ( sum_Inr_nat_a @ B7 ) )
= ( B2 = B7 ) ) ).
% old.sum.inject(2)
thf(fact_410_le__sup__iff,axiom,
! [X3: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X3 @ Y ) @ Z )
= ( ( ord_less_eq_set_a @ X3 @ Z )
& ( ord_less_eq_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_411_le__sup__iff,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ X3 @ Y ) @ Z )
= ( ( ord_le1325389633284124927_a_nat @ X3 @ Z )
& ( ord_le1325389633284124927_a_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_412_le__sup__iff,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X3 @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_413_top__empty__eq,axiom,
( top_top_o_o
= ( ^ [X: $o] : ( member_o @ X @ top_top_set_o ) ) ) ).
% top_empty_eq
thf(fact_414_top__empty__eq,axiom,
( top_to1565196397637005550_nat_o
= ( ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ top_to795618464972521135_a_nat ) ) ) ).
% top_empty_eq
thf(fact_415_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_416_top__set__def,axiom,
( top_to795618464972521135_a_nat
= ( collec7073057861543223018_a_nat @ top_to1565196397637005550_nat_o ) ) ).
% top_set_def
thf(fact_417_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_418_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_419_sup__Un__eq,axiom,
! [R: set_Sum_sum_a_nat,S: set_Sum_sum_a_nat] :
( ( sup_su491480579010597738_nat_o
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ R )
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ S ) )
= ( ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_420_sup__set__def,axiom,
( sup_sup_set_o
= ( ^ [A3: set_o,B3: set_o] :
( collect_o
@ ( sup_sup_o_o
@ ^ [X: $o] : ( member_o @ X @ A3 )
@ ^ [X: $o] : ( member_o @ X @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_421_sup__set__def,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [A3: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
( collec7073057861543223018_a_nat
@ ( sup_su491480579010597738_nat_o
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ A3 )
@ ^ [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_422_inf__sup__aci_I8_J,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ X3 @ Y ) )
= ( sup_su6804446743777130803_a_nat @ X3 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_423_inf__sup__aci_I7_J,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ Y @ Z ) )
= ( sup_su6804446743777130803_a_nat @ Y @ ( sup_su6804446743777130803_a_nat @ X3 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_424_inf__sup__aci_I6_J,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ X3 @ Y ) @ Z )
= ( sup_su6804446743777130803_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_425_inf__sup__aci_I5_J,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [X: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] : ( sup_su6804446743777130803_a_nat @ Y3 @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_426_sup_Oassoc,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) @ C )
= ( sup_su6804446743777130803_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_427_sup__assoc,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ X3 @ Y ) @ Z )
= ( sup_su6804446743777130803_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_428_sup_Ocommute,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [A5: set_Sum_sum_a_nat,B5: set_Sum_sum_a_nat] : ( sup_su6804446743777130803_a_nat @ B5 @ A5 ) ) ) ).
% sup.commute
thf(fact_429_sup__commute,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [X: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] : ( sup_su6804446743777130803_a_nat @ Y3 @ X ) ) ) ).
% sup_commute
thf(fact_430_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_Sum_sum_a_nat,K: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( A2
= ( sup_su6804446743777130803_a_nat @ K @ A ) )
=> ( ( sup_su6804446743777130803_a_nat @ A2 @ B2 )
= ( sup_su6804446743777130803_a_nat @ K @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_431_boolean__algebra__cancel_Osup2,axiom,
! [B: set_Sum_sum_a_nat,K: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( B
= ( sup_su6804446743777130803_a_nat @ K @ B2 ) )
=> ( ( sup_su6804446743777130803_a_nat @ A @ B )
= ( sup_su6804446743777130803_a_nat @ K @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_432_sup_Oleft__commute,axiom,
! [B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ B2 @ ( sup_su6804446743777130803_a_nat @ A @ C ) )
= ( sup_su6804446743777130803_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_433_sup__left__commute,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ Y @ Z ) )
= ( sup_su6804446743777130803_a_nat @ Y @ ( sup_su6804446743777130803_a_nat @ X3 @ Z ) ) ) ).
% sup_left_commute
thf(fact_434_Inl__inject,axiom,
! [X3: a,Y: a] :
( ( ( sum_Inl_a_nat @ X3 )
= ( sum_Inl_a_nat @ Y ) )
=> ( X3 = Y ) ) ).
% Inl_inject
thf(fact_435_Inr__inject,axiom,
! [X3: nat,Y: nat] :
( ( ( sum_Inr_nat_a @ X3 )
= ( sum_Inr_nat_a @ Y ) )
=> ( X3 = Y ) ) ).
% Inr_inject
thf(fact_436_inf__sup__ord_I4_J,axiom,
! [Y: set_a,X3: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_437_inf__sup__ord_I4_J,axiom,
! [Y: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ Y @ ( sup_su6804446743777130803_a_nat @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_438_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_439_inf__sup__ord_I3_J,axiom,
! [X3: set_a,Y: set_a] : ( ord_less_eq_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_440_inf__sup__ord_I3_J,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_441_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_442_le__supE,axiom,
! [A: set_a,B2: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ X3 )
=> ~ ( ( ord_less_eq_set_a @ A @ X3 )
=> ~ ( ord_less_eq_set_a @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_443_le__supE,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) @ X3 )
=> ~ ( ( ord_le1325389633284124927_a_nat @ A @ X3 )
=> ~ ( ord_le1325389633284124927_a_nat @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_444_le__supE,axiom,
! [A: nat,B2: nat,X3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ X3 )
=> ~ ( ( ord_less_eq_nat @ A @ X3 )
=> ~ ( ord_less_eq_nat @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_445_le__supI,axiom,
! [A: set_a,X3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ X3 )
=> ( ( ord_less_eq_set_a @ B2 @ X3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_446_le__supI,axiom,
! [A: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ X3 )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ X3 )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_447_le__supI,axiom,
! [A: nat,X3: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ X3 )
=> ( ( ord_less_eq_nat @ B2 @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_448_sup__ge1,axiom,
! [X3: set_a,Y: set_a] : ( ord_less_eq_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_449_sup__ge1,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_450_sup__ge1,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_451_sup__ge2,axiom,
! [Y: set_a,X3: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_452_sup__ge2,axiom,
! [Y: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ Y @ ( sup_su6804446743777130803_a_nat @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_453_sup__ge2,axiom,
! [Y: nat,X3: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_454_le__supI1,axiom,
! [X3: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X3 @ A )
=> ( ord_less_eq_set_a @ X3 @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_455_le__supI1,axiom,
! [X3: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ A )
=> ( ord_le1325389633284124927_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_456_le__supI1,axiom,
! [X3: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ X3 @ A )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_457_le__supI2,axiom,
! [X3: set_a,B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ X3 @ B2 )
=> ( ord_less_eq_set_a @ X3 @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_458_le__supI2,axiom,
! [X3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ B2 )
=> ( ord_le1325389633284124927_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_459_le__supI2,axiom,
! [X3: nat,B2: nat,A: nat] :
( ( ord_less_eq_nat @ X3 @ B2 )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_460_sup_Omono,axiom,
! [C: set_a,A: set_a,D2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ( ord_less_eq_set_a @ D2 @ B2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D2 ) @ ( sup_sup_set_a @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_461_sup_Omono,axiom,
! [C: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,D2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C @ A )
=> ( ( ord_le1325389633284124927_a_nat @ D2 @ B2 )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ C @ D2 ) @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_462_sup_Omono,axiom,
! [C: nat,A: nat,D2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D2 @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_463_sup__mono,axiom,
! [A: set_a,C: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ ( sup_sup_set_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_464_sup__mono,axiom,
! [A: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,D2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ C )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ D2 )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) @ ( sup_su6804446743777130803_a_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_465_sup__mono,axiom,
! [A: nat,C: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_466_sup__least,axiom,
! [Y: set_a,X3: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ Y @ X3 )
=> ( ( ord_less_eq_set_a @ Z @ X3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_467_sup__least,axiom,
! [Y: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ Y @ X3 )
=> ( ( ord_le1325389633284124927_a_nat @ Z @ X3 )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ Y @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_468_sup__least,axiom,
! [Y: nat,X3: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ Z @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X3 ) ) ) ).
% sup_least
thf(fact_469_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X: set_a,Y3: set_a] :
( ( sup_sup_set_a @ X @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_470_le__iff__sup,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [X: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_471_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y3: nat] :
( ( sup_sup_nat @ X @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_472_sup_OorderE,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( A
= ( sup_sup_set_a @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_473_sup_OorderE,axiom,
! [B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A )
=> ( A
= ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_474_sup_OorderE,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( A
= ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_475_sup_OorderI,axiom,
! [A: set_a,B2: set_a] :
( ( A
= ( sup_sup_set_a @ A @ B2 ) )
=> ( ord_less_eq_set_a @ B2 @ A ) ) ).
% sup.orderI
thf(fact_476_sup_OorderI,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( A
= ( sup_su6804446743777130803_a_nat @ A @ B2 ) )
=> ( ord_le1325389633284124927_a_nat @ B2 @ A ) ) ).
% sup.orderI
thf(fact_477_sup_OorderI,axiom,
! [A: nat,B2: nat] :
( ( A
= ( sup_sup_nat @ A @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A ) ) ).
% sup.orderI
thf(fact_478_sup__unique,axiom,
! [F: set_a > set_a > set_a,X3: set_a,Y: set_a] :
( ! [X2: set_a,Y5: set_a] : ( ord_less_eq_set_a @ X2 @ ( F @ X2 @ Y5 ) )
=> ( ! [X2: set_a,Y5: set_a] : ( ord_less_eq_set_a @ Y5 @ ( F @ X2 @ Y5 ) )
=> ( ! [X2: set_a,Y5: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ Y5 @ X2 )
=> ( ( ord_less_eq_set_a @ Z4 @ X2 )
=> ( ord_less_eq_set_a @ ( F @ Y5 @ Z4 ) @ X2 ) ) )
=> ( ( sup_sup_set_a @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_479_sup__unique,axiom,
! [F: set_Sum_sum_a_nat > set_Sum_sum_a_nat > set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ! [X2: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ X2 @ ( F @ X2 @ Y5 ) )
=> ( ! [X2: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ Y5 @ ( F @ X2 @ Y5 ) )
=> ( ! [X2: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat,Z4: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ Y5 @ X2 )
=> ( ( ord_le1325389633284124927_a_nat @ Z4 @ X2 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ Y5 @ Z4 ) @ X2 ) ) )
=> ( ( sup_su6804446743777130803_a_nat @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_480_sup__unique,axiom,
! [F: nat > nat > nat,X3: nat,Y: nat] :
( ! [X2: nat,Y5: nat] : ( ord_less_eq_nat @ X2 @ ( F @ X2 @ Y5 ) )
=> ( ! [X2: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ ( F @ X2 @ Y5 ) )
=> ( ! [X2: nat,Y5: nat,Z4: nat] :
( ( ord_less_eq_nat @ Y5 @ X2 )
=> ( ( ord_less_eq_nat @ Z4 @ X2 )
=> ( ord_less_eq_nat @ ( F @ Y5 @ Z4 ) @ X2 ) ) )
=> ( ( sup_sup_nat @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_481_sup_Oabsorb1,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( sup_sup_set_a @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_482_sup_Oabsorb1,axiom,
! [B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A )
=> ( ( sup_su6804446743777130803_a_nat @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_483_sup_Oabsorb1,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( sup_sup_nat @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_484_sup_Oabsorb2,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( sup_sup_set_a @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_485_sup_Oabsorb2,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( ( sup_su6804446743777130803_a_nat @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_486_sup_Oabsorb2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( sup_sup_nat @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_487_sup__absorb1,axiom,
! [Y: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y @ X3 )
=> ( ( sup_sup_set_a @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_488_sup__absorb1,axiom,
! [Y: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ Y @ X3 )
=> ( ( sup_su6804446743777130803_a_nat @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_489_sup__absorb1,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( sup_sup_nat @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_490_sup__absorb2,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y )
=> ( ( sup_sup_set_a @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_491_sup__absorb2,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ Y )
=> ( ( sup_su6804446743777130803_a_nat @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_492_sup__absorb2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( sup_sup_nat @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_493_sup_OboundedE,axiom,
! [B2: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A )
=> ~ ( ( ord_less_eq_set_a @ B2 @ A )
=> ~ ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_494_sup_OboundedE,axiom,
! [B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ B2 @ C ) @ A )
=> ~ ( ( ord_le1325389633284124927_a_nat @ B2 @ A )
=> ~ ( ord_le1325389633284124927_a_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_495_sup_OboundedE,axiom,
! [B2: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B2 @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_496_sup_OboundedI,axiom,
! [B2: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_497_sup_OboundedI,axiom,
! [B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A )
=> ( ( ord_le1325389633284124927_a_nat @ C @ A )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ B2 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_498_sup_OboundedI,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_499_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B5: set_a,A5: set_a] :
( A5
= ( sup_sup_set_a @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_500_sup_Oorder__iff,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [B5: set_Sum_sum_a_nat,A5: set_Sum_sum_a_nat] :
( A5
= ( sup_su6804446743777130803_a_nat @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_501_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( A5
= ( sup_sup_nat @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_502_sup_Ocobounded1,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_503_sup_Ocobounded1,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_504_sup_Ocobounded1,axiom,
! [A: nat,B2: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_505_sup_Ocobounded2,axiom,
! [B2: set_a,A: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_506_sup_Ocobounded2,axiom,
! [B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ B2 @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_507_sup_Ocobounded2,axiom,
! [B2: nat,A: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_508_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B5: set_a,A5: set_a] :
( ( sup_sup_set_a @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_509_sup_Oabsorb__iff1,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [B5: set_Sum_sum_a_nat,A5: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_510_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_511_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( sup_sup_set_a @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_512_sup_Oabsorb__iff2,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A5: set_Sum_sum_a_nat,B5: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_513_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_514_sup_OcoboundedI1,axiom,
! [C: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_515_sup_OcoboundedI1,axiom,
! [C: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C @ A )
=> ( ord_le1325389633284124927_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_516_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_517_sup_OcoboundedI2,axiom,
! [C: set_a,B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_518_sup_OcoboundedI2,axiom,
! [C: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C @ B2 )
=> ( ord_le1325389633284124927_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_519_sup_OcoboundedI2,axiom,
! [C: nat,B2: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_520_sum_Odistinct_I1_J,axiom,
! [X1: a,X22: nat] :
( ( sum_Inl_a_nat @ X1 )
!= ( sum_Inr_nat_a @ X22 ) ) ).
% sum.distinct(1)
thf(fact_521_old_Osum_Odistinct_I2_J,axiom,
! [B7: nat,A: a] :
( ( sum_Inr_nat_a @ B7 )
!= ( sum_Inl_a_nat @ A ) ) ).
% old.sum.distinct(2)
thf(fact_522_old_Osum_Odistinct_I1_J,axiom,
! [A: a,B7: nat] :
( ( sum_Inl_a_nat @ A )
!= ( sum_Inr_nat_a @ B7 ) ) ).
% old.sum.distinct(1)
thf(fact_523_old_Osum_Oexhaust,axiom,
! [Y: sum_sum_a_nat] :
( ! [A4: a] :
( Y
!= ( sum_Inl_a_nat @ A4 ) )
=> ~ ! [B4: nat] :
( Y
!= ( sum_Inr_nat_a @ B4 ) ) ) ).
% old.sum.exhaust
thf(fact_524_sumE,axiom,
! [S2: sum_sum_a_nat] :
( ! [X2: a] :
( S2
!= ( sum_Inl_a_nat @ X2 ) )
=> ~ ! [Y5: nat] :
( S2
!= ( sum_Inr_nat_a @ Y5 ) ) ) ).
% sumE
thf(fact_525_Inr__not__Inl,axiom,
! [B2: nat,A: a] :
( ( sum_Inr_nat_a @ B2 )
!= ( sum_Inl_a_nat @ A ) ) ).
% Inr_not_Inl
thf(fact_526_split__sum__ex,axiom,
( ( ^ [P2: sum_sum_a_nat > $o] :
? [X7: sum_sum_a_nat] : ( P2 @ X7 ) )
= ( ^ [P3: sum_sum_a_nat > $o] :
( ? [X: a] : ( P3 @ ( sum_Inl_a_nat @ X ) )
| ? [X: nat] : ( P3 @ ( sum_Inr_nat_a @ X ) ) ) ) ) ).
% split_sum_ex
thf(fact_527_split__sum__all,axiom,
( ( ^ [P2: sum_sum_a_nat > $o] :
! [X7: sum_sum_a_nat] : ( P2 @ X7 ) )
= ( ^ [P3: sum_sum_a_nat > $o] :
( ! [X: a] : ( P3 @ ( sum_Inl_a_nat @ X ) )
& ! [X: nat] : ( P3 @ ( sum_Inr_nat_a @ X ) ) ) ) ) ).
% split_sum_all
thf(fact_528_iso__tuple__UNIV__I,axiom,
! [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% iso_tuple_UNIV_I
thf(fact_529_iso__tuple__UNIV__I,axiom,
! [X3: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X3 @ top_to795618464972521135_a_nat ) ).
% iso_tuple_UNIV_I
thf(fact_530_iso__tuple__UNIV__I,axiom,
! [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_531_surjD,axiom,
! [F: a > sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( ( image_7873763678140191238_a_nat @ F @ top_top_set_a )
= top_to795618464972521135_a_nat )
=> ? [X2: a] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_532_surjD,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( ( image_7142520692256960453_a_nat @ F @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat )
=> ? [X2: sum_sum_a_nat] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_533_surjD,axiom,
! [F: sum_sum_a_nat > nat,Y: nat] :
( ( ( image_2473878607534554506at_nat @ F @ top_to795618464972521135_a_nat )
= top_top_set_nat )
=> ? [X2: sum_sum_a_nat] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_534_surjD,axiom,
! [F: nat > set_nat,Y: set_nat] :
( ( ( image_nat_set_nat @ F @ top_top_set_nat )
= top_top_set_set_nat )
=> ? [X2: nat] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_535_surjD,axiom,
! [F: nat > sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( ( image_7293268710728258664_a_nat @ F @ top_top_set_nat )
= top_to795618464972521135_a_nat )
=> ? [X2: nat] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_536_surjD,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X2: nat] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_537_surjE,axiom,
! [F: a > sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( ( image_7873763678140191238_a_nat @ F @ top_top_set_a )
= top_to795618464972521135_a_nat )
=> ~ ! [X2: a] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_538_surjE,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( ( image_7142520692256960453_a_nat @ F @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat )
=> ~ ! [X2: sum_sum_a_nat] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_539_surjE,axiom,
! [F: sum_sum_a_nat > nat,Y: nat] :
( ( ( image_2473878607534554506at_nat @ F @ top_to795618464972521135_a_nat )
= top_top_set_nat )
=> ~ ! [X2: sum_sum_a_nat] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_540_surjE,axiom,
! [F: nat > set_nat,Y: set_nat] :
( ( ( image_nat_set_nat @ F @ top_top_set_nat )
= top_top_set_set_nat )
=> ~ ! [X2: nat] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_541_surjE,axiom,
! [F: nat > sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( ( image_7293268710728258664_a_nat @ F @ top_top_set_nat )
= top_to795618464972521135_a_nat )
=> ~ ! [X2: nat] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_542_surjE,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X2: nat] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_543_surjI,axiom,
! [G: a > sum_sum_a_nat,F: sum_sum_a_nat > a] :
( ! [X2: sum_sum_a_nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_7873763678140191238_a_nat @ G @ top_top_set_a )
= top_to795618464972521135_a_nat ) ) ).
% surjI
thf(fact_544_surjI,axiom,
! [G: sum_sum_a_nat > sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ! [X2: sum_sum_a_nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_7142520692256960453_a_nat @ G @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat ) ) ).
% surjI
thf(fact_545_surjI,axiom,
! [G: sum_sum_a_nat > nat,F: nat > sum_sum_a_nat] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_2473878607534554506at_nat @ G @ top_to795618464972521135_a_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_546_surjI,axiom,
! [G: nat > set_nat,F: set_nat > nat] :
( ! [X2: set_nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_nat_set_nat @ G @ top_top_set_nat )
= top_top_set_set_nat ) ) ).
% surjI
thf(fact_547_surjI,axiom,
! [G: nat > sum_sum_a_nat,F: sum_sum_a_nat > nat] :
( ! [X2: sum_sum_a_nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_7293268710728258664_a_nat @ G @ top_top_set_nat )
= top_to795618464972521135_a_nat ) ) ).
% surjI
thf(fact_548_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_549_surj__def,axiom,
! [F: a > sum_sum_a_nat] :
( ( ( image_7873763678140191238_a_nat @ F @ top_top_set_a )
= top_to795618464972521135_a_nat )
= ( ! [Y3: sum_sum_a_nat] :
? [X: a] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_550_surj__def,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat] :
( ( ( image_7142520692256960453_a_nat @ F @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat )
= ( ! [Y3: sum_sum_a_nat] :
? [X: sum_sum_a_nat] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_551_surj__def,axiom,
! [F: sum_sum_a_nat > nat] :
( ( ( image_2473878607534554506at_nat @ F @ top_to795618464972521135_a_nat )
= top_top_set_nat )
= ( ! [Y3: nat] :
? [X: sum_sum_a_nat] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_552_surj__def,axiom,
! [F: nat > set_nat] :
( ( ( image_nat_set_nat @ F @ top_top_set_nat )
= top_top_set_set_nat )
= ( ! [Y3: set_nat] :
? [X: nat] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_553_surj__def,axiom,
! [F: nat > sum_sum_a_nat] :
( ( ( image_7293268710728258664_a_nat @ F @ top_top_set_nat )
= top_to795618464972521135_a_nat )
= ( ! [Y3: sum_sum_a_nat] :
? [X: nat] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_554_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y3: nat] :
? [X: nat] :
( Y3
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_555_Plus__def,axiom,
( sum_Plus_a_nat
= ( ^ [A3: set_a,B3: set_nat] : ( sup_su6804446743777130803_a_nat @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ A3 ) @ ( image_7293268710728258664_a_nat @ sum_Inr_nat_a @ B3 ) ) ) ) ).
% Plus_def
thf(fact_556_InlI,axiom,
! [A: a,A2: set_a,B: set_nat] :
( ( member_a @ A @ A2 )
=> ( member_Sum_sum_a_nat @ ( sum_Inl_a_nat @ A ) @ ( sum_Plus_a_nat @ A2 @ B ) ) ) ).
% InlI
thf(fact_557_InrI,axiom,
! [B2: nat,B: set_nat,A2: set_a] :
( ( member_nat @ B2 @ B )
=> ( member_Sum_sum_a_nat @ ( sum_Inr_nat_a @ B2 ) @ ( sum_Plus_a_nat @ A2 @ B ) ) ) ).
% InrI
thf(fact_558_UNIV__Plus__UNIV,axiom,
( ( sum_Plus_a_nat @ top_top_set_a @ top_top_set_nat )
= top_to795618464972521135_a_nat ) ).
% UNIV_Plus_UNIV
thf(fact_559_UNIV__Plus__UNIV,axiom,
( ( sum_Pl1179408338895902643_a_nat @ top_to795618464972521135_a_nat @ top_to795618464972521135_a_nat )
= top_to599037537065133003_a_nat ) ).
% UNIV_Plus_UNIV
thf(fact_560_UNIV__Plus__UNIV,axiom,
( ( sum_Pl9138067327779332380at_nat @ top_to795618464972521135_a_nat @ top_top_set_nat )
= top_to8676068415865862704at_nat ) ).
% UNIV_Plus_UNIV
thf(fact_561_UNIV__Plus__UNIV,axiom,
( ( sum_Pl4734085394118260730_a_nat @ top_top_set_nat @ top_to795618464972521135_a_nat )
= top_to7568329678976531286_a_nat ) ).
% UNIV_Plus_UNIV
thf(fact_562_UNIV__Plus__UNIV,axiom,
( ( sum_Plus_nat_nat @ top_top_set_nat @ top_top_set_nat )
= top_to6661820994512907621at_nat ) ).
% UNIV_Plus_UNIV
thf(fact_563_PlusE,axiom,
! [U: sum_sum_o_o,A2: set_o,B: set_o] :
( ( member_Sum_sum_o_o @ U @ ( sum_Plus_o_o @ A2 @ B ) )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( U
!= ( sum_Inl_o_o @ X2 ) ) )
=> ~ ! [Y5: $o] :
( ( member_o @ Y5 @ B )
=> ( U
!= ( sum_Inr_o_o @ Y5 ) ) ) ) ) ).
% PlusE
thf(fact_564_PlusE,axiom,
! [U: sum_sum_a_nat,A2: set_a,B: set_nat] :
( ( member_Sum_sum_a_nat @ U @ ( sum_Plus_a_nat @ A2 @ B ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( U
!= ( sum_Inl_a_nat @ X2 ) ) )
=> ~ ! [Y5: nat] :
( ( member_nat @ Y5 @ B )
=> ( U
!= ( sum_Inr_nat_a @ Y5 ) ) ) ) ) ).
% PlusE
thf(fact_565_image__Pow__mono,axiom,
! [F: nat > set_nat,A2: set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B )
=> ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F ) @ ( pow_nat @ A2 ) ) @ ( pow_set_nat @ B ) ) ) ).
% image_Pow_mono
thf(fact_566_image__Pow__mono,axiom,
! [F: nat > sum_sum_a_nat,A2: set_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ A2 ) @ B )
=> ( ord_le7974500612278410847_a_nat @ ( image_3578472599065059474_a_nat @ ( image_7293268710728258664_a_nat @ F ) @ ( pow_nat @ A2 ) ) @ ( pow_Sum_sum_a_nat @ B ) ) ) ).
% image_Pow_mono
thf(fact_567_image__Pow__mono,axiom,
! [F: a > sum_sum_a_nat,A2: set_a,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ A2 ) @ B )
=> ( ord_le7974500612278410847_a_nat @ ( image_6715601112060939782_a_nat @ ( image_7873763678140191238_a_nat @ F ) @ ( pow_a @ A2 ) ) @ ( pow_Sum_sum_a_nat @ B ) ) ) ).
% image_Pow_mono
thf(fact_568_union__coset__filter,axiom,
! [Xs: list_o,A2: set_o] :
( ( sup_sup_set_o @ ( coset_o @ Xs ) @ A2 )
= ( coset_o
@ ( filter_o
@ ^ [X: $o] :
~ ( member_o @ X @ A2 )
@ Xs ) ) ) ).
% union_coset_filter
thf(fact_569_union__coset__filter,axiom,
! [Xs: list_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( coset_Sum_sum_a_nat @ Xs ) @ A2 )
= ( coset_Sum_sum_a_nat
@ ( filter_Sum_sum_a_nat
@ ^ [X: sum_sum_a_nat] :
~ ( member_Sum_sum_a_nat @ X @ A2 )
@ Xs ) ) ) ).
% union_coset_filter
thf(fact_570_Pow__UNIV,axiom,
( ( pow_Sum_sum_a_nat @ top_to795618464972521135_a_nat )
= top_to9085961846241471503_a_nat ) ).
% Pow_UNIV
thf(fact_571_Pow__UNIV,axiom,
( ( pow_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% Pow_UNIV
thf(fact_572_PowI,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( member_set_a @ A2 @ ( pow_a @ B ) ) ) ).
% PowI
thf(fact_573_PowI,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B )
=> ( member8098812455498974984_a_nat @ A2 @ ( pow_Sum_sum_a_nat @ B ) ) ) ).
% PowI
thf(fact_574_Pow__iff,axiom,
! [A2: set_a,B: set_a] :
( ( member_set_a @ A2 @ ( pow_a @ B ) )
= ( ord_less_eq_set_a @ A2 @ B ) ) ).
% Pow_iff
thf(fact_575_Pow__iff,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ A2 @ ( pow_Sum_sum_a_nat @ B ) )
= ( ord_le1325389633284124927_a_nat @ A2 @ B ) ) ).
% Pow_iff
thf(fact_576_Powp__Pow__eq,axiom,
! [A2: set_o] :
( ( powp_o
@ ^ [X: $o] : ( member_o @ X @ A2 ) )
= ( ^ [X: set_o] : ( member_set_o @ X @ ( pow_o @ A2 ) ) ) ) ).
% Powp_Pow_eq
thf(fact_577_PowD,axiom,
! [A2: set_a,B: set_a] :
( ( member_set_a @ A2 @ ( pow_a @ B ) )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% PowD
thf(fact_578_PowD,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ A2 @ ( pow_Sum_sum_a_nat @ B ) )
=> ( ord_le1325389633284124927_a_nat @ A2 @ B ) ) ).
% PowD
thf(fact_579_Cantors__paradox,axiom,
! [A2: set_nat] :
~ ? [F2: nat > set_nat] :
( ( image_nat_set_nat @ F2 @ A2 )
= ( pow_nat @ A2 ) ) ).
% Cantors_paradox
thf(fact_580_Pow__def,axiom,
( pow_a
= ( ^ [A3: set_a] :
( collect_set_a
@ ^ [B3: set_a] : ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% Pow_def
thf(fact_581_Pow__def,axiom,
( pow_Sum_sum_a_nat
= ( ^ [A3: set_Sum_sum_a_nat] :
( collec4049389696321283146_a_nat
@ ^ [B3: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ B3 @ A3 ) ) ) ) ).
% Pow_def
thf(fact_582_Pow__mono,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ ( pow_a @ A2 ) @ ( pow_a @ B ) ) ) ).
% Pow_mono
thf(fact_583_Pow__mono,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B )
=> ( ord_le7974500612278410847_a_nat @ ( pow_Sum_sum_a_nat @ A2 ) @ ( pow_Sum_sum_a_nat @ B ) ) ) ).
% Pow_mono
thf(fact_584_Un__Pow__subset,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( ord_le7974500612278410847_a_nat @ ( sup_su2291686591051470483_a_nat @ ( pow_Sum_sum_a_nat @ A2 ) @ ( pow_Sum_sum_a_nat @ B ) ) @ ( pow_Sum_sum_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) ) ) ).
% Un_Pow_subset
thf(fact_585_image__Pow__surj,axiom,
! [F: nat > sum_sum_a_nat,A2: set_nat,B: set_Sum_sum_a_nat] :
( ( ( image_7293268710728258664_a_nat @ F @ A2 )
= B )
=> ( ( image_3578472599065059474_a_nat @ ( image_7293268710728258664_a_nat @ F ) @ ( pow_nat @ A2 ) )
= ( pow_Sum_sum_a_nat @ B ) ) ) ).
% image_Pow_surj
thf(fact_586_image__Pow__surj,axiom,
! [F: a > sum_sum_a_nat,A2: set_a,B: set_Sum_sum_a_nat] :
( ( ( image_7873763678140191238_a_nat @ F @ A2 )
= B )
=> ( ( image_6715601112060939782_a_nat @ ( image_7873763678140191238_a_nat @ F ) @ ( pow_a @ A2 ) )
= ( pow_Sum_sum_a_nat @ B ) ) ) ).
% image_Pow_surj
thf(fact_587_image__Pow__surj,axiom,
! [F: nat > set_nat,A2: set_nat,B: set_set_nat] :
( ( ( image_nat_set_nat @ F @ A2 )
= B )
=> ( ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F ) @ ( pow_nat @ A2 ) )
= ( pow_set_nat @ B ) ) ) ).
% image_Pow_surj
thf(fact_588_GreatestI2__order,axiom,
! [P: set_a > $o,X3: set_a,Q: set_a > $o] :
( ( P @ X3 )
=> ( ! [Y5: set_a] :
( ( P @ Y5 )
=> ( ord_less_eq_set_a @ Y5 @ X3 ) )
=> ( ! [X2: set_a] :
( ( P @ X2 )
=> ( ! [Y6: set_a] :
( ( P @ Y6 )
=> ( ord_less_eq_set_a @ Y6 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_589_GreatestI2__order,axiom,
! [P: set_Sum_sum_a_nat > $o,X3: set_Sum_sum_a_nat,Q: set_Sum_sum_a_nat > $o] :
( ( P @ X3 )
=> ( ! [Y5: set_Sum_sum_a_nat] :
( ( P @ Y5 )
=> ( ord_le1325389633284124927_a_nat @ Y5 @ X3 ) )
=> ( ! [X2: set_Sum_sum_a_nat] :
( ( P @ X2 )
=> ( ! [Y6: set_Sum_sum_a_nat] :
( ( P @ Y6 )
=> ( ord_le1325389633284124927_a_nat @ Y6 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_2294890068632620472_a_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_590_GreatestI2__order,axiom,
! [P: nat > $o,X3: nat,Q: nat > $o] :
( ( P @ X3 )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_591_Greatest__equality,axiom,
! [P: set_a > $o,X3: set_a] :
( ( P @ X3 )
=> ( ! [Y5: set_a] :
( ( P @ Y5 )
=> ( ord_less_eq_set_a @ Y5 @ X3 ) )
=> ( ( order_Greatest_set_a @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_592_Greatest__equality,axiom,
! [P: set_Sum_sum_a_nat > $o,X3: set_Sum_sum_a_nat] :
( ( P @ X3 )
=> ( ! [Y5: set_Sum_sum_a_nat] :
( ( P @ Y5 )
=> ( ord_le1325389633284124927_a_nat @ Y5 @ X3 ) )
=> ( ( order_2294890068632620472_a_nat @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_593_Greatest__equality,axiom,
! [P: nat > $o,X3: nat] :
( ( P @ X3 )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) )
=> ( ( order_Greatest_nat @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_594_SUP__union,axiom,
! [M2: sum_sum_a_nat > set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( comple1247738100258233164_a_nat @ ( image_7877458644602423589_a_nat @ M2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) ) )
= ( sup_su6804446743777130803_a_nat @ ( comple1247738100258233164_a_nat @ ( image_7877458644602423589_a_nat @ M2 @ A2 ) ) @ ( comple1247738100258233164_a_nat @ ( image_7877458644602423589_a_nat @ M2 @ B ) ) ) ) ).
% SUP_union
thf(fact_595_SUP__union,axiom,
! [M2: nat > set_nat,A2: set_nat,B: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M2 @ ( sup_sup_set_nat @ A2 @ B ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M2 @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M2 @ B ) ) ) ) ).
% SUP_union
thf(fact_596_SUP__union,axiom,
! [M2: sum_sum_a_nat > set_nat,A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( comple7399068483239264473et_nat @ ( image_4589483402070311232et_nat @ M2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_4589483402070311232et_nat @ M2 @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_4589483402070311232et_nat @ M2 @ B ) ) ) ) ).
% SUP_union
thf(fact_597_SUP__union,axiom,
! [M2: sum_sum_a_nat > $o,A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( complete_Sup_Sup_o @ ( image_6095136190293192542_nat_o @ M2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) ) )
= ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_6095136190293192542_nat_o @ M2 @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_6095136190293192542_nat_o @ M2 @ B ) ) ) ) ).
% SUP_union
thf(fact_598_SUP__subset__mono,axiom,
! [A2: set_o,B: set_o,F: $o > $o,G: $o > $o] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_o @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_o_o @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_599_SUP__subset__mono,axiom,
! [A2: set_a,B: set_a,F: a > $o,G: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_o @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_a_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_a_o @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_600_SUP__subset__mono,axiom,
! [A2: set_o,B: set_o,F: $o > set_a,G: $o > set_a] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_601_SUP__subset__mono,axiom,
! [A2: set_a,B: set_a,F: a > set_a,G: a > set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_602_SUP__subset__mono,axiom,
! [A2: set_nat,B: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_603_SUP__subset__mono,axiom,
! [A2: set_o,B: set_o,F: $o > set_nat,G: $o > set_nat] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_604_SUP__subset__mono,axiom,
! [A2: set_a,B: set_a,F: a > set_nat,G: a > set_nat] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_605_SUP__subset__mono,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,F: sum_sum_a_nat > $o,G: sum_sum_a_nat > $o] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B )
=> ( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ A2 )
=> ( ord_less_eq_o @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_6095136190293192542_nat_o @ F @ A2 ) ) @ ( complete_Sup_Sup_o @ ( image_6095136190293192542_nat_o @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_606_SUP__subset__mono,axiom,
! [A2: set_o,B: set_o,F: $o > set_Sum_sum_a_nat,G: $o > set_Sum_sum_a_nat] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( comple1247738100258233164_a_nat @ ( image_3365592128754359116_a_nat @ F @ A2 ) ) @ ( comple1247738100258233164_a_nat @ ( image_3365592128754359116_a_nat @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_607_SUP__subset__mono,axiom,
! [A2: set_a,B: set_a,F: a > set_Sum_sum_a_nat,G: a > set_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( comple1247738100258233164_a_nat @ ( image_1201892972954314598_a_nat @ F @ A2 ) ) @ ( comple1247738100258233164_a_nat @ ( image_1201892972954314598_a_nat @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_608_INF__superset__mono,axiom,
! [B: set_nat,A2: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ B ) ) ) ) ) ).
% INF_superset_mono
thf(fact_609_INF__superset__mono,axiom,
! [B: set_o,A2: set_o,F: $o > set_a,G: $o > set_a] :
( ( ord_less_eq_set_o @ B @ A2 )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ B )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_o_set_a @ F @ A2 ) ) @ ( comple6135023378680113637_set_a @ ( image_o_set_a @ G @ B ) ) ) ) ) ).
% INF_superset_mono
thf(fact_610_INF__superset__mono,axiom,
! [B: set_o,A2: set_o,F: $o > set_Sum_sum_a_nat,G: $o > set_Sum_sum_a_nat] :
( ( ord_less_eq_set_o @ B @ A2 )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ B )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( comple1528121977673479270_a_nat @ ( image_3365592128754359116_a_nat @ F @ A2 ) ) @ ( comple1528121977673479270_a_nat @ ( image_3365592128754359116_a_nat @ G @ B ) ) ) ) ) ).
% INF_superset_mono
thf(fact_611_INF__superset__mono,axiom,
! [B: set_a,A2: set_a,F: a > set_a,G: a > set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ A2 ) ) @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ G @ B ) ) ) ) ) ).
% INF_superset_mono
thf(fact_612_INF__superset__mono,axiom,
! [B: set_a,A2: set_a,F: a > set_Sum_sum_a_nat,G: a > set_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( comple1528121977673479270_a_nat @ ( image_1201892972954314598_a_nat @ F @ A2 ) ) @ ( comple1528121977673479270_a_nat @ ( image_1201892972954314598_a_nat @ G @ B ) ) ) ) ) ).
% INF_superset_mono
thf(fact_613_INF__superset__mono,axiom,
! [B: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,F: sum_sum_a_nat > set_a,G: sum_sum_a_nat > set_a] :
( ( ord_le1325389633284124927_a_nat @ B @ A2 )
=> ( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ B )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_8809959208761443620_set_a @ F @ A2 ) ) @ ( comple6135023378680113637_set_a @ ( image_8809959208761443620_set_a @ G @ B ) ) ) ) ) ).
% INF_superset_mono
thf(fact_614_INF__superset__mono,axiom,
! [B: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,F: sum_sum_a_nat > set_Sum_sum_a_nat,G: sum_sum_a_nat > set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B @ A2 )
=> ( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ B )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( comple1528121977673479270_a_nat @ ( image_7877458644602423589_a_nat @ F @ A2 ) ) @ ( comple1528121977673479270_a_nat @ ( image_7877458644602423589_a_nat @ G @ B ) ) ) ) ) ).
% INF_superset_mono
thf(fact_615_INF__superset__mono,axiom,
! [B: set_o,A2: set_o,F: $o > $o,G: $o > $o] :
( ( ord_less_eq_set_o @ B @ A2 )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ B )
=> ( ord_less_eq_o @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) @ ( complete_Inf_Inf_o @ ( image_o_o @ G @ B ) ) ) ) ) ).
% INF_superset_mono
thf(fact_616_INF__superset__mono,axiom,
! [B: set_a,A2: set_a,F: a > $o,G: a > $o] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ord_less_eq_o @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_a_o @ F @ A2 ) ) @ ( complete_Inf_Inf_o @ ( image_a_o @ G @ B ) ) ) ) ) ).
% INF_superset_mono
thf(fact_617_INF__superset__mono,axiom,
! [B: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,F: sum_sum_a_nat > $o,G: sum_sum_a_nat > $o] :
( ( ord_le1325389633284124927_a_nat @ B @ A2 )
=> ( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ B )
=> ( ord_less_eq_o @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_6095136190293192542_nat_o @ F @ A2 ) ) @ ( complete_Inf_Inf_o @ ( image_6095136190293192542_nat_o @ G @ B ) ) ) ) ) ).
% INF_superset_mono
thf(fact_618_Inter__UNIV__conv_I1_J,axiom,
! [A2: set_se4904748513628223167_a_nat] :
( ( ( comple1528121977673479270_a_nat @ A2 )
= top_to795618464972521135_a_nat )
= ( ! [X: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X @ A2 )
=> ( X = top_to795618464972521135_a_nat ) ) ) ) ).
% Inter_UNIV_conv(1)
thf(fact_619_Inter__UNIV__conv_I1_J,axiom,
! [A2: set_set_nat] :
( ( ( comple7806235888213564991et_nat @ A2 )
= top_top_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( X = top_top_set_nat ) ) ) ) ).
% Inter_UNIV_conv(1)
thf(fact_620_Inter__UNIV__conv_I2_J,axiom,
! [A2: set_se4904748513628223167_a_nat] :
( ( top_to795618464972521135_a_nat
= ( comple1528121977673479270_a_nat @ A2 ) )
= ( ! [X: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X @ A2 )
=> ( X = top_to795618464972521135_a_nat ) ) ) ) ).
% Inter_UNIV_conv(2)
thf(fact_621_Inter__UNIV__conv_I2_J,axiom,
! [A2: set_set_nat] :
( ( top_top_set_nat
= ( comple7806235888213564991et_nat @ A2 ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( X = top_top_set_nat ) ) ) ) ).
% Inter_UNIV_conv(2)
thf(fact_622_ball__UN,axiom,
! [B: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A2 ) ) )
=> ( P @ X ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( B @ X ) )
=> ( P @ Y3 ) ) ) ) ) ).
% ball_UN
thf(fact_623_bex__UN,axiom,
! [B: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A2 ) ) )
& ( P @ X ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ? [Y3: nat] :
( ( member_nat @ Y3 @ ( B @ X ) )
& ( P @ Y3 ) ) ) ) ) ).
% bex_UN
thf(fact_624_UN__ball__bex__simps_I2_J,axiom,
! [B: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A2 ) ) )
=> ( P @ X ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( B @ X ) )
=> ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_625_UN__ball__bex__simps_I4_J,axiom,
! [B: nat > set_nat,A2: set_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A2 ) ) )
& ( P @ X ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ? [Y3: nat] :
( ( member_nat @ Y3 @ ( B @ X ) )
& ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_626_Union__Pow__eq,axiom,
! [A2: set_nat] :
( ( comple7399068483239264473et_nat @ ( pow_nat @ A2 ) )
= A2 ) ).
% Union_Pow_eq
thf(fact_627_INT__I,axiom,
! [A2: set_nat,B2: nat,B: nat > set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ B2 @ ( B @ X2 ) ) )
=> ( member_nat @ B2 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B @ A2 ) ) ) ) ).
% INT_I
thf(fact_628_INT__I,axiom,
! [A2: set_o,B2: $o,B: $o > set_o] :
( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( member_o @ B2 @ ( B @ X2 ) ) )
=> ( member_o @ B2 @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B @ A2 ) ) ) ) ).
% INT_I
thf(fact_629_INT__iff,axiom,
! [B2: nat,B: nat > set_nat,A2: set_nat] :
( ( member_nat @ B2 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B @ A2 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ B2 @ ( B @ X ) ) ) ) ) ).
% INT_iff
thf(fact_630_UN__I,axiom,
! [A: $o,A2: set_o,B2: $o,B: $o > set_o] :
( ( member_o @ A @ A2 )
=> ( ( member_o @ B2 @ ( B @ A ) )
=> ( member_o @ B2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_631_UN__I,axiom,
! [A: nat,A2: set_nat,B2: nat,B: nat > set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat @ B2 @ ( B @ A ) )
=> ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_632_UN__I,axiom,
! [A: $o,A2: set_o,B2: nat,B: $o > set_nat] :
( ( member_o @ A @ A2 )
=> ( ( member_nat @ B2 @ ( B @ A ) )
=> ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_633_UN__iff,axiom,
! [B2: nat,B: nat > set_nat,A2: set_nat] :
( ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A2 ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( member_nat @ B2 @ ( B @ X ) ) ) ) ) ).
% UN_iff
thf(fact_634_Inf__top__conv_I1_J,axiom,
! [A2: set_se4904748513628223167_a_nat] :
( ( ( comple1528121977673479270_a_nat @ A2 )
= top_to795618464972521135_a_nat )
= ( ! [X: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X @ A2 )
=> ( X = top_to795618464972521135_a_nat ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_635_Inf__top__conv_I1_J,axiom,
! [A2: set_set_nat] :
( ( ( comple7806235888213564991et_nat @ A2 )
= top_top_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( X = top_top_set_nat ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_636_Inf__top__conv_I1_J,axiom,
! [A2: set_o] :
( ( ( complete_Inf_Inf_o @ A2 )
= top_top_o )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( X = top_top_o ) ) ) ) ).
% Inf_top_conv(1)
thf(fact_637_Inf__top__conv_I2_J,axiom,
! [A2: set_se4904748513628223167_a_nat] :
( ( top_to795618464972521135_a_nat
= ( comple1528121977673479270_a_nat @ A2 ) )
= ( ! [X: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X @ A2 )
=> ( X = top_to795618464972521135_a_nat ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_638_Inf__top__conv_I2_J,axiom,
! [A2: set_set_nat] :
( ( top_top_set_nat
= ( comple7806235888213564991et_nat @ A2 ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( X = top_top_set_nat ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_639_Inf__top__conv_I2_J,axiom,
! [A2: set_o] :
( ( top_top_o
= ( complete_Inf_Inf_o @ A2 ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( X = top_top_o ) ) ) ) ).
% Inf_top_conv(2)
thf(fact_640_Union__Un__distrib,axiom,
! [A2: set_se4904748513628223167_a_nat,B: set_se4904748513628223167_a_nat] :
( ( comple1247738100258233164_a_nat @ ( sup_su2291686591051470483_a_nat @ A2 @ B ) )
= ( sup_su6804446743777130803_a_nat @ ( comple1247738100258233164_a_nat @ A2 ) @ ( comple1247738100258233164_a_nat @ B ) ) ) ).
% Union_Un_distrib
thf(fact_641_Union__Un__distrib,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A2 @ B ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_Un_distrib
thf(fact_642_SUP__identity__eq,axiom,
! [A2: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [X: set_nat] : X
@ A2 ) )
= ( comple7399068483239264473et_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_643_SUP__identity__eq,axiom,
! [A2: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X: $o] : X
@ A2 ) )
= ( complete_Sup_Sup_o @ A2 ) ) ).
% SUP_identity_eq
thf(fact_644_INF__identity__eq,axiom,
! [A2: set_o] :
( ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [X: $o] : X
@ A2 ) )
= ( complete_Inf_Inf_o @ A2 ) ) ).
% INF_identity_eq
thf(fact_645_UN__Un,axiom,
! [M2: sum_sum_a_nat > set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( comple1247738100258233164_a_nat @ ( image_7877458644602423589_a_nat @ M2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) ) )
= ( sup_su6804446743777130803_a_nat @ ( comple1247738100258233164_a_nat @ ( image_7877458644602423589_a_nat @ M2 @ A2 ) ) @ ( comple1247738100258233164_a_nat @ ( image_7877458644602423589_a_nat @ M2 @ B ) ) ) ) ).
% UN_Un
thf(fact_646_UN__Un,axiom,
! [M2: nat > set_nat,A2: set_nat,B: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M2 @ ( sup_sup_set_nat @ A2 @ B ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M2 @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M2 @ B ) ) ) ) ).
% UN_Un
thf(fact_647_UN__Un,axiom,
! [M2: sum_sum_a_nat > set_nat,A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( comple7399068483239264473et_nat @ ( image_4589483402070311232et_nat @ M2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_4589483402070311232et_nat @ M2 @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_4589483402070311232et_nat @ M2 @ B ) ) ) ) ).
% UN_Un
thf(fact_648_Sup__UNIV,axiom,
( ( comple1247738100258233164_a_nat @ top_to9085961846241471503_a_nat )
= top_to795618464972521135_a_nat ) ).
% Sup_UNIV
thf(fact_649_Sup__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Sup_UNIV
thf(fact_650_Sup__UNIV,axiom,
( ( complete_Sup_Sup_o @ top_top_set_o )
= top_top_o ) ).
% Sup_UNIV
thf(fact_651_INF__top__conv_I2_J,axiom,
! [B: nat > set_nat,A2: set_nat] :
( ( top_top_set_nat
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B @ A2 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( B @ X )
= top_top_set_nat ) ) ) ) ).
% INF_top_conv(2)
thf(fact_652_INF__top__conv_I1_J,axiom,
! [B: nat > set_nat,A2: set_nat] :
( ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B @ A2 ) )
= top_top_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( B @ X )
= top_top_set_nat ) ) ) ) ).
% INF_top_conv(1)
thf(fact_653_INF__top,axiom,
! [A2: set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : top_top_set_nat
@ A2 ) )
= top_top_set_nat ) ).
% INF_top
thf(fact_654_INF__INT__eq,axiom,
! [R3: nat > set_nat,S: set_nat] :
( ( comple6214475593288795910_nat_o
@ ( image_nat_nat_o2
@ ^ [I3: nat,X: nat] : ( member_nat @ X @ ( R3 @ I3 ) )
@ S ) )
= ( ^ [X: nat] : ( member_nat @ X @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ R3 @ S ) ) ) ) ) ).
% INF_INT_eq
thf(fact_655_SUP__UN__eq,axiom,
! [R3: nat > set_nat,S: set_nat] :
( ( comple8317665133742190828_nat_o
@ ( image_nat_nat_o2
@ ^ [I3: nat,X: nat] : ( member_nat @ X @ ( R3 @ I3 ) )
@ S ) )
= ( ^ [X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ R3 @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_656_INT__extend__simps_I9_J,axiom,
! [C2: nat > set_nat,B: nat > set_nat,A2: set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ C2 @ ( B @ X ) ) )
@ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ C2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A2 ) ) ) ) ) ).
% INT_extend_simps(9)
thf(fact_657_Inf__greatest,axiom,
! [A2: set_set_a,Z: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ Z @ X2 ) )
=> ( ord_less_eq_set_a @ Z @ ( comple6135023378680113637_set_a @ A2 ) ) ) ).
% Inf_greatest
thf(fact_658_Inf__greatest,axiom,
! [A2: set_se4904748513628223167_a_nat,Z: set_Sum_sum_a_nat] :
( ! [X2: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X2 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ Z @ X2 ) )
=> ( ord_le1325389633284124927_a_nat @ Z @ ( comple1528121977673479270_a_nat @ A2 ) ) ) ).
% Inf_greatest
thf(fact_659_Inf__greatest,axiom,
! [A2: set_o,Z: $o] :
( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_o @ Z @ X2 ) )
=> ( ord_less_eq_o @ Z @ ( complete_Inf_Inf_o @ A2 ) ) ) ).
% Inf_greatest
thf(fact_660_le__Inf__iff,axiom,
! [B2: set_a,A2: set_set_a] :
( ( ord_less_eq_set_a @ B2 @ ( comple6135023378680113637_set_a @ A2 ) )
= ( ! [X: set_a] :
( ( member_set_a @ X @ A2 )
=> ( ord_less_eq_set_a @ B2 @ X ) ) ) ) ).
% le_Inf_iff
thf(fact_661_le__Inf__iff,axiom,
! [B2: set_Sum_sum_a_nat,A2: set_se4904748513628223167_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ ( comple1528121977673479270_a_nat @ A2 ) )
= ( ! [X: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X @ A2 )
=> ( ord_le1325389633284124927_a_nat @ B2 @ X ) ) ) ) ).
% le_Inf_iff
thf(fact_662_le__Inf__iff,axiom,
! [B2: $o,A2: set_o] :
( ( ord_less_eq_o @ B2 @ ( complete_Inf_Inf_o @ A2 ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_o @ B2 @ X ) ) ) ) ).
% le_Inf_iff
thf(fact_663_Inf__lower2,axiom,
! [U: set_a,A2: set_set_a,V: set_a] :
( ( member_set_a @ U @ A2 )
=> ( ( ord_less_eq_set_a @ U @ V )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A2 ) @ V ) ) ) ).
% Inf_lower2
thf(fact_664_Inf__lower2,axiom,
! [U: set_Sum_sum_a_nat,A2: set_se4904748513628223167_a_nat,V: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ U @ A2 )
=> ( ( ord_le1325389633284124927_a_nat @ U @ V )
=> ( ord_le1325389633284124927_a_nat @ ( comple1528121977673479270_a_nat @ A2 ) @ V ) ) ) ).
% Inf_lower2
thf(fact_665_Inf__lower2,axiom,
! [U: $o,A2: set_o,V: $o] :
( ( member_o @ U @ A2 )
=> ( ( ord_less_eq_o @ U @ V )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ V ) ) ) ).
% Inf_lower2
thf(fact_666_Inf__lower,axiom,
! [X3: set_a,A2: set_set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A2 ) @ X3 ) ) ).
% Inf_lower
thf(fact_667_Inf__lower,axiom,
! [X3: set_Sum_sum_a_nat,A2: set_se4904748513628223167_a_nat] :
( ( member8098812455498974984_a_nat @ X3 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ ( comple1528121977673479270_a_nat @ A2 ) @ X3 ) ) ).
% Inf_lower
thf(fact_668_Inf__lower,axiom,
! [X3: $o,A2: set_o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ X3 ) ) ).
% Inf_lower
thf(fact_669_Inf__mono,axiom,
! [B: set_set_a,A2: set_set_a] :
( ! [B4: set_a] :
( ( member_set_a @ B4 @ B )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
& ( ord_less_eq_set_a @ X5 @ B4 ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A2 ) @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inf_mono
thf(fact_670_Inf__mono,axiom,
! [B: set_se4904748513628223167_a_nat,A2: set_se4904748513628223167_a_nat] :
( ! [B4: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ B4 @ B )
=> ? [X5: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X5 @ A2 )
& ( ord_le1325389633284124927_a_nat @ X5 @ B4 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( comple1528121977673479270_a_nat @ A2 ) @ ( comple1528121977673479270_a_nat @ B ) ) ) ).
% Inf_mono
thf(fact_671_Inf__mono,axiom,
! [B: set_o,A2: set_o] :
( ! [B4: $o] :
( ( member_o @ B4 @ B )
=> ? [X5: $o] :
( ( member_o @ X5 @ A2 )
& ( ord_less_eq_o @ X5 @ B4 ) ) )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ ( complete_Inf_Inf_o @ B ) ) ) ).
% Inf_mono
thf(fact_672_Inf__eqI,axiom,
! [A2: set_set_a,X3: set_a] :
( ! [I4: set_a] :
( ( member_set_a @ I4 @ A2 )
=> ( ord_less_eq_set_a @ X3 @ I4 ) )
=> ( ! [Y5: set_a] :
( ! [I5: set_a] :
( ( member_set_a @ I5 @ A2 )
=> ( ord_less_eq_set_a @ Y5 @ I5 ) )
=> ( ord_less_eq_set_a @ Y5 @ X3 ) )
=> ( ( comple6135023378680113637_set_a @ A2 )
= X3 ) ) ) ).
% Inf_eqI
thf(fact_673_Inf__eqI,axiom,
! [A2: set_se4904748513628223167_a_nat,X3: set_Sum_sum_a_nat] :
( ! [I4: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ I4 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ X3 @ I4 ) )
=> ( ! [Y5: set_Sum_sum_a_nat] :
( ! [I5: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ I5 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ Y5 @ I5 ) )
=> ( ord_le1325389633284124927_a_nat @ Y5 @ X3 ) )
=> ( ( comple1528121977673479270_a_nat @ A2 )
= X3 ) ) ) ).
% Inf_eqI
thf(fact_674_Inf__eqI,axiom,
! [A2: set_o,X3: $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_o @ X3 @ I4 ) )
=> ( ! [Y5: $o] :
( ! [I5: $o] :
( ( member_o @ I5 @ A2 )
=> ( ord_less_eq_o @ Y5 @ I5 ) )
=> ( ord_less_eq_o @ Y5 @ X3 ) )
=> ( ( complete_Inf_Inf_o @ A2 )
= X3 ) ) ) ).
% Inf_eqI
thf(fact_675_Sup__upper2,axiom,
! [U: set_a,A2: set_set_a,V: set_a] :
( ( member_set_a @ U @ A2 )
=> ( ( ord_less_eq_set_a @ V @ U )
=> ( ord_less_eq_set_a @ V @ ( comple2307003609928055243_set_a @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_676_Sup__upper2,axiom,
! [U: set_Sum_sum_a_nat,A2: set_se4904748513628223167_a_nat,V: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ U @ A2 )
=> ( ( ord_le1325389633284124927_a_nat @ V @ U )
=> ( ord_le1325389633284124927_a_nat @ V @ ( comple1247738100258233164_a_nat @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_677_Sup__upper2,axiom,
! [U: set_nat,A2: set_set_nat,V: set_nat] :
( ( member_set_nat @ U @ A2 )
=> ( ( ord_less_eq_set_nat @ V @ U )
=> ( ord_less_eq_set_nat @ V @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_678_Sup__upper2,axiom,
! [U: $o,A2: set_o,V: $o] :
( ( member_o @ U @ A2 )
=> ( ( ord_less_eq_o @ V @ U )
=> ( ord_less_eq_o @ V @ ( complete_Sup_Sup_o @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_679_Sup__le__iff,axiom,
! [A2: set_set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ B2 )
= ( ! [X: set_a] :
( ( member_set_a @ X @ A2 )
=> ( ord_less_eq_set_a @ X @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_680_Sup__le__iff,axiom,
! [A2: set_se4904748513628223167_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( comple1247738100258233164_a_nat @ A2 ) @ B2 )
= ( ! [X: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X @ A2 )
=> ( ord_le1325389633284124927_a_nat @ X @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_681_Sup__le__iff,axiom,
! [A2: set_set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ B2 )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ X @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_682_Sup__le__iff,axiom,
! [A2: set_o,B2: $o] :
( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ B2 )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( ord_less_eq_o @ X @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_683_Sup__upper,axiom,
! [X3: set_a,A2: set_set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ X3 @ ( comple2307003609928055243_set_a @ A2 ) ) ) ).
% Sup_upper
thf(fact_684_Sup__upper,axiom,
! [X3: set_Sum_sum_a_nat,A2: set_se4904748513628223167_a_nat] :
( ( member8098812455498974984_a_nat @ X3 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ X3 @ ( comple1247738100258233164_a_nat @ A2 ) ) ) ).
% Sup_upper
thf(fact_685_Sup__upper,axiom,
! [X3: set_nat,A2: set_set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ X3 @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% Sup_upper
thf(fact_686_Sup__upper,axiom,
! [X3: $o,A2: set_o] :
( ( member_o @ X3 @ A2 )
=> ( ord_less_eq_o @ X3 @ ( complete_Sup_Sup_o @ A2 ) ) ) ).
% Sup_upper
thf(fact_687_Sup__least,axiom,
! [A2: set_set_a,Z: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ X2 @ Z ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_688_Sup__least,axiom,
! [A2: set_se4904748513628223167_a_nat,Z: set_Sum_sum_a_nat] :
( ! [X2: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X2 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ X2 @ Z ) )
=> ( ord_le1325389633284124927_a_nat @ ( comple1247738100258233164_a_nat @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_689_Sup__least,axiom,
! [A2: set_set_nat,Z: set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ X2 @ Z ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_690_Sup__least,axiom,
! [A2: set_o,Z: $o] :
( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_o @ X2 @ Z ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_691_Sup__mono,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [A4: set_a] :
( ( member_set_a @ A4 @ A2 )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ B )
& ( ord_less_eq_set_a @ A4 @ X5 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Sup_mono
thf(fact_692_Sup__mono,axiom,
! [A2: set_se4904748513628223167_a_nat,B: set_se4904748513628223167_a_nat] :
( ! [A4: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ A4 @ A2 )
=> ? [X5: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X5 @ B )
& ( ord_le1325389633284124927_a_nat @ A4 @ X5 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( comple1247738100258233164_a_nat @ A2 ) @ ( comple1247738100258233164_a_nat @ B ) ) ) ).
% Sup_mono
thf(fact_693_Sup__mono,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ! [A4: set_nat] :
( ( member_set_nat @ A4 @ A2 )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ B )
& ( ord_less_eq_set_nat @ A4 @ X5 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_mono
thf(fact_694_Sup__mono,axiom,
! [A2: set_o,B: set_o] :
( ! [A4: $o] :
( ( member_o @ A4 @ A2 )
=> ? [X5: $o] :
( ( member_o @ X5 @ B )
& ( ord_less_eq_o @ A4 @ X5 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_mono
thf(fact_695_Sup__eqI,axiom,
! [A2: set_set_a,X3: set_a] :
( ! [Y5: set_a] :
( ( member_set_a @ Y5 @ A2 )
=> ( ord_less_eq_set_a @ Y5 @ X3 ) )
=> ( ! [Y5: set_a] :
( ! [Z5: set_a] :
( ( member_set_a @ Z5 @ A2 )
=> ( ord_less_eq_set_a @ Z5 @ Y5 ) )
=> ( ord_less_eq_set_a @ X3 @ Y5 ) )
=> ( ( comple2307003609928055243_set_a @ A2 )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_696_Sup__eqI,axiom,
! [A2: set_se4904748513628223167_a_nat,X3: set_Sum_sum_a_nat] :
( ! [Y5: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ Y5 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ Y5 @ X3 ) )
=> ( ! [Y5: set_Sum_sum_a_nat] :
( ! [Z5: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ Z5 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ Z5 @ Y5 ) )
=> ( ord_le1325389633284124927_a_nat @ X3 @ Y5 ) )
=> ( ( comple1247738100258233164_a_nat @ A2 )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_697_Sup__eqI,axiom,
! [A2: set_set_nat,X3: set_nat] :
( ! [Y5: set_nat] :
( ( member_set_nat @ Y5 @ A2 )
=> ( ord_less_eq_set_nat @ Y5 @ X3 ) )
=> ( ! [Y5: set_nat] :
( ! [Z5: set_nat] :
( ( member_set_nat @ Z5 @ A2 )
=> ( ord_less_eq_set_nat @ Z5 @ Y5 ) )
=> ( ord_less_eq_set_nat @ X3 @ Y5 ) )
=> ( ( comple7399068483239264473et_nat @ A2 )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_698_Sup__eqI,axiom,
! [A2: set_o,X3: $o] :
( ! [Y5: $o] :
( ( member_o @ Y5 @ A2 )
=> ( ord_less_eq_o @ Y5 @ X3 ) )
=> ( ! [Y5: $o] :
( ! [Z5: $o] :
( ( member_o @ Z5 @ A2 )
=> ( ord_less_eq_o @ Z5 @ Y5 ) )
=> ( ord_less_eq_o @ X3 @ Y5 ) )
=> ( ( complete_Sup_Sup_o @ A2 )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_699_INF__cong,axiom,
! [A2: set_nat,B: set_nat,C2: nat > set_nat,D: nat > set_nat] :
( ( A2 = B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ C2 @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ D @ B ) ) ) ) ) ).
% INF_cong
thf(fact_700_INF__cong,axiom,
! [A2: set_o,B: set_o,C2: $o > $o,D: $o > $o] :
( ( A2 = B )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ C2 @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ D @ B ) ) ) ) ) ).
% INF_cong
thf(fact_701_SUP__cong,axiom,
! [A2: set_nat,B: set_nat,C2: nat > set_nat,D: nat > set_nat] :
( ( A2 = B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_702_SUP__cong,axiom,
! [A2: set_o,B: set_o,C2: $o > set_nat,D: $o > set_nat] :
( ( A2 = B )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ C2 @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_703_SUP__cong,axiom,
! [A2: set_o,B: set_o,C2: $o > $o,D: $o > $o] :
( ( A2 = B )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ C2 @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_704_Inter__greatest,axiom,
! [A2: set_set_a,C2: set_a] :
( ! [X8: set_a] :
( ( member_set_a @ X8 @ A2 )
=> ( ord_less_eq_set_a @ C2 @ X8 ) )
=> ( ord_less_eq_set_a @ C2 @ ( comple6135023378680113637_set_a @ A2 ) ) ) ).
% Inter_greatest
thf(fact_705_Inter__greatest,axiom,
! [A2: set_se4904748513628223167_a_nat,C2: set_Sum_sum_a_nat] :
( ! [X8: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X8 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ C2 @ X8 ) )
=> ( ord_le1325389633284124927_a_nat @ C2 @ ( comple1528121977673479270_a_nat @ A2 ) ) ) ).
% Inter_greatest
thf(fact_706_Inter__lower,axiom,
! [B: set_a,A2: set_set_a] :
( ( member_set_a @ B @ A2 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A2 ) @ B ) ) ).
% Inter_lower
thf(fact_707_Inter__lower,axiom,
! [B: set_Sum_sum_a_nat,A2: set_se4904748513628223167_a_nat] :
( ( member8098812455498974984_a_nat @ B @ A2 )
=> ( ord_le1325389633284124927_a_nat @ ( comple1528121977673479270_a_nat @ A2 ) @ B ) ) ).
% Inter_lower
thf(fact_708_Union__UNIV,axiom,
( ( comple1247738100258233164_a_nat @ top_to9085961846241471503_a_nat )
= top_to795618464972521135_a_nat ) ).
% Union_UNIV
thf(fact_709_Union__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Union_UNIV
thf(fact_710_SUP__UNION,axiom,
! [F: nat > set_nat,G: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y3: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( G @ Y3 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_711_SUP__UNION,axiom,
! [F: nat > $o,G: nat > set_nat,A2: set_nat] :
( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A2 ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y3: nat] : ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( G @ Y3 ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_712_Union__subsetI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ? [Y6: set_a] :
( ( member_set_a @ Y6 @ B )
& ( ord_less_eq_set_a @ X2 @ Y6 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_subsetI
thf(fact_713_Union__subsetI,axiom,
! [A2: set_se4904748513628223167_a_nat,B: set_se4904748513628223167_a_nat] :
( ! [X2: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X2 @ A2 )
=> ? [Y6: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ Y6 @ B )
& ( ord_le1325389633284124927_a_nat @ X2 @ Y6 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( comple1247738100258233164_a_nat @ A2 ) @ ( comple1247738100258233164_a_nat @ B ) ) ) ).
% Union_subsetI
thf(fact_714_Union__subsetI,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ? [Y6: set_nat] :
( ( member_set_nat @ Y6 @ B )
& ( ord_less_eq_set_nat @ X2 @ Y6 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_subsetI
thf(fact_715_Union__upper,axiom,
! [B: set_a,A2: set_set_a] :
( ( member_set_a @ B @ A2 )
=> ( ord_less_eq_set_a @ B @ ( comple2307003609928055243_set_a @ A2 ) ) ) ).
% Union_upper
thf(fact_716_Union__upper,axiom,
! [B: set_Sum_sum_a_nat,A2: set_se4904748513628223167_a_nat] :
( ( member8098812455498974984_a_nat @ B @ A2 )
=> ( ord_le1325389633284124927_a_nat @ B @ ( comple1247738100258233164_a_nat @ A2 ) ) ) ).
% Union_upper
thf(fact_717_Union__upper,axiom,
! [B: set_nat,A2: set_set_nat] :
( ( member_set_nat @ B @ A2 )
=> ( ord_less_eq_set_nat @ B @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% Union_upper
thf(fact_718_Union__least,axiom,
! [A2: set_set_a,C2: set_a] :
( ! [X8: set_a] :
( ( member_set_a @ X8 @ A2 )
=> ( ord_less_eq_set_a @ X8 @ C2 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ C2 ) ) ).
% Union_least
thf(fact_719_Union__least,axiom,
! [A2: set_se4904748513628223167_a_nat,C2: set_Sum_sum_a_nat] :
( ! [X8: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X8 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ X8 @ C2 ) )
=> ( ord_le1325389633284124927_a_nat @ ( comple1247738100258233164_a_nat @ A2 ) @ C2 ) ) ).
% Union_least
thf(fact_720_Union__least,axiom,
! [A2: set_set_nat,C2: set_nat] :
( ! [X8: set_nat] :
( ( member_set_nat @ X8 @ A2 )
=> ( ord_less_eq_set_nat @ X8 @ C2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ C2 ) ) ).
% Union_least
thf(fact_721_Pow__INT__eq,axiom,
! [B: nat > set_nat,A2: set_nat] :
( ( pow_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B @ A2 ) ) )
= ( comple1065008630642458357et_nat
@ ( image_2194112158459175443et_nat
@ ^ [X: nat] : ( pow_nat @ ( B @ X ) )
@ A2 ) ) ) ).
% Pow_INT_eq
thf(fact_722_INT__extend__simps_I8_J,axiom,
! [B: nat > set_nat,A2: set_set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_7916887816326733075et_nat
@ ^ [Y3: set_nat] : ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B @ Y3 ) )
@ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% INT_extend_simps(8)
thf(fact_723_INF__commute,axiom,
! [F: nat > nat > set_nat,B: set_nat,A2: set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [I3: nat] : ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ ( F @ I3 ) @ B ) )
@ A2 ) )
= ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [J: nat] :
( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [I3: nat] : ( F @ I3 @ J )
@ A2 ) )
@ B ) ) ) ).
% INF_commute
thf(fact_724_SUP__commute,axiom,
! [F: nat > nat > set_nat,B: set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I3: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F @ I3 ) @ B ) )
@ A2 ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [J: nat] :
( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I3: nat] : ( F @ I3 @ J )
@ A2 ) )
@ B ) ) ) ).
% SUP_commute
thf(fact_725_SUP__UNIV__bool__expand,axiom,
! [A2: $o > set_Sum_sum_a_nat] :
( ( comple1247738100258233164_a_nat @ ( image_3365592128754359116_a_nat @ A2 @ top_top_set_o ) )
= ( sup_su6804446743777130803_a_nat @ ( A2 @ $true ) @ ( A2 @ $false ) ) ) ).
% SUP_UNIV_bool_expand
thf(fact_726_SUP__UNIV__bool__expand,axiom,
! [A2: $o > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A2 @ top_top_set_o ) )
= ( sup_sup_set_nat @ ( A2 @ $true ) @ ( A2 @ $false ) ) ) ).
% SUP_UNIV_bool_expand
thf(fact_727_SUP__UNIV__bool__expand,axiom,
! [A2: $o > $o] :
( ( complete_Sup_Sup_o @ ( image_o_o @ A2 @ top_top_set_o ) )
= ( sup_sup_o @ ( A2 @ $true ) @ ( A2 @ $false ) ) ) ).
% SUP_UNIV_bool_expand
thf(fact_728_INT__D,axiom,
! [B2: nat,B: nat > set_nat,A2: set_nat,A: nat] :
( ( member_nat @ B2 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B @ A2 ) ) )
=> ( ( member_nat @ A @ A2 )
=> ( member_nat @ B2 @ ( B @ A ) ) ) ) ).
% INT_D
thf(fact_729_INT__D,axiom,
! [B2: $o,B: $o > set_o,A2: set_o,A: $o] :
( ( member_o @ B2 @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B @ A2 ) ) )
=> ( ( member_o @ A @ A2 )
=> ( member_o @ B2 @ ( B @ A ) ) ) ) ).
% INT_D
thf(fact_730_INT__E,axiom,
! [B2: nat,B: nat > set_nat,A2: set_nat,A: nat] :
( ( member_nat @ B2 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B @ A2 ) ) )
=> ( ~ ( member_nat @ B2 @ ( B @ A ) )
=> ~ ( member_nat @ A @ A2 ) ) ) ).
% INT_E
thf(fact_731_INT__E,axiom,
! [B2: $o,B: $o > set_o,A2: set_o,A: $o] :
( ( member_o @ B2 @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B @ A2 ) ) )
=> ( ~ ( member_o @ B2 @ ( B @ A ) )
=> ~ ( member_o @ A @ A2 ) ) ) ).
% INT_E
thf(fact_732_Un__eq__UN,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [A3: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
( comple1247738100258233164_a_nat
@ ( image_3365592128754359116_a_nat
@ ^ [B5: $o] : ( if_set_Sum_sum_a_nat @ B5 @ A3 @ B3 )
@ top_top_set_o ) ) ) ) ).
% Un_eq_UN
thf(fact_733_Un__eq__UN,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [B5: $o] : ( if_set_nat @ B5 @ A3 @ B3 )
@ top_top_set_o ) ) ) ) ).
% Un_eq_UN
thf(fact_734_UN__bool__eq,axiom,
! [A2: $o > set_Sum_sum_a_nat] :
( ( comple1247738100258233164_a_nat @ ( image_3365592128754359116_a_nat @ A2 @ top_top_set_o ) )
= ( sup_su6804446743777130803_a_nat @ ( A2 @ $true ) @ ( A2 @ $false ) ) ) ).
% UN_bool_eq
thf(fact_735_UN__bool__eq,axiom,
! [A2: $o > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A2 @ top_top_set_o ) )
= ( sup_sup_set_nat @ ( A2 @ $true ) @ ( A2 @ $false ) ) ) ).
% UN_bool_eq
thf(fact_736_UN__extend__simps_I9_J,axiom,
! [C2: nat > set_nat,B: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( B @ X ) ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_737_UN__E,axiom,
! [B2: $o,B: $o > set_o,A2: set_o] :
( ( member_o @ B2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A2 ) ) )
=> ~ ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ~ ( member_o @ B2 @ ( B @ X2 ) ) ) ) ).
% UN_E
thf(fact_738_UN__E,axiom,
! [B2: nat,B: nat > set_nat,A2: set_nat] :
( ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A2 ) ) )
=> ~ ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ( member_nat @ B2 @ ( B @ X2 ) ) ) ) ).
% UN_E
thf(fact_739_UN__E,axiom,
! [B2: nat,B: $o > set_nat,A2: set_o] :
( ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ A2 ) ) )
=> ~ ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ~ ( member_nat @ B2 @ ( B @ X2 ) ) ) ) ).
% UN_E
thf(fact_740_UN__UN__flatten,axiom,
! [C2: nat > set_nat,B: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A2 ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y3: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( B @ Y3 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_741_UN__lessThan__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_lessThan_UNIV
thf(fact_742_UN__Pow__subset,axiom,
! [B: nat > set_nat,A2: set_nat] :
( ord_le6893508408891458716et_nat
@ ( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [X: nat] : ( pow_nat @ ( B @ X ) )
@ A2 ) )
@ ( pow_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A2 ) ) ) ) ).
% UN_Pow_subset
thf(fact_743_INF__eq,axiom,
! [A2: set_nat,B: set_nat,G: nat > set_nat,F: nat > set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ( ord_less_eq_set_nat @ ( G @ X5 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ord_less_eq_set_nat @ ( F @ X5 ) @ ( G @ J2 ) ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_744_INF__eq,axiom,
! [A2: set_nat,B: set_o,G: $o > set_nat,F: nat > set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ? [X5: $o] :
( ( member_o @ X5 @ B )
& ( ord_less_eq_set_nat @ ( G @ X5 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ord_less_eq_set_nat @ ( F @ X5 ) @ ( G @ J2 ) ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_o_set_nat @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_745_INF__eq,axiom,
! [A2: set_o,B: set_nat,G: nat > set_nat,F: $o > set_nat] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ( ord_less_eq_set_nat @ ( G @ X5 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B )
=> ? [X5: $o] :
( ( member_o @ X5 @ A2 )
& ( ord_less_eq_set_nat @ ( F @ X5 ) @ ( G @ J2 ) ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_746_INF__eq,axiom,
! [A2: set_o,B: set_o,G: $o > set_a,F: $o > set_a] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ? [X5: $o] :
( ( member_o @ X5 @ B )
& ( ord_less_eq_set_a @ ( G @ X5 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X5: $o] :
( ( member_o @ X5 @ A2 )
& ( ord_less_eq_set_a @ ( F @ X5 ) @ ( G @ J2 ) ) ) )
=> ( ( comple6135023378680113637_set_a @ ( image_o_set_a @ F @ A2 ) )
= ( comple6135023378680113637_set_a @ ( image_o_set_a @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_747_INF__eq,axiom,
! [A2: set_o,B: set_o,G: $o > set_Sum_sum_a_nat,F: $o > set_Sum_sum_a_nat] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ? [X5: $o] :
( ( member_o @ X5 @ B )
& ( ord_le1325389633284124927_a_nat @ ( G @ X5 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X5: $o] :
( ( member_o @ X5 @ A2 )
& ( ord_le1325389633284124927_a_nat @ ( F @ X5 ) @ ( G @ J2 ) ) ) )
=> ( ( comple1528121977673479270_a_nat @ ( image_3365592128754359116_a_nat @ F @ A2 ) )
= ( comple1528121977673479270_a_nat @ ( image_3365592128754359116_a_nat @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_748_INF__eq,axiom,
! [A2: set_o,B: set_o,G: $o > $o,F: $o > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ? [X5: $o] :
( ( member_o @ X5 @ B )
& ( ord_less_eq_o @ ( G @ X5 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X5: $o] :
( ( member_o @ X5 @ A2 )
& ( ord_less_eq_o @ ( F @ X5 ) @ ( G @ J2 ) ) ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ G @ B ) ) ) ) ) ).
% INF_eq
thf(fact_749_SUP__eq,axiom,
! [A2: set_o,B: set_o,F: $o > set_a,G: $o > set_a] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ? [X5: $o] :
( ( member_o @ X5 @ B )
& ( ord_less_eq_set_a @ ( F @ I4 ) @ ( G @ X5 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X5: $o] :
( ( member_o @ X5 @ A2 )
& ( ord_less_eq_set_a @ ( G @ J2 ) @ ( F @ X5 ) ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_o_set_a @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_750_SUP__eq,axiom,
! [A2: set_o,B: set_o,F: $o > set_Sum_sum_a_nat,G: $o > set_Sum_sum_a_nat] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ? [X5: $o] :
( ( member_o @ X5 @ B )
& ( ord_le1325389633284124927_a_nat @ ( F @ I4 ) @ ( G @ X5 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X5: $o] :
( ( member_o @ X5 @ A2 )
& ( ord_le1325389633284124927_a_nat @ ( G @ J2 ) @ ( F @ X5 ) ) ) )
=> ( ( comple1247738100258233164_a_nat @ ( image_3365592128754359116_a_nat @ F @ A2 ) )
= ( comple1247738100258233164_a_nat @ ( image_3365592128754359116_a_nat @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_751_SUP__eq,axiom,
! [A2: set_nat,B: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ( ord_less_eq_set_nat @ ( F @ I4 ) @ ( G @ X5 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ord_less_eq_set_nat @ ( G @ J2 ) @ ( F @ X5 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_752_SUP__eq,axiom,
! [A2: set_nat,B: set_o,F: nat > set_nat,G: $o > set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ? [X5: $o] :
( ( member_o @ X5 @ B )
& ( ord_less_eq_set_nat @ ( F @ I4 ) @ ( G @ X5 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ord_less_eq_set_nat @ ( G @ J2 ) @ ( F @ X5 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_753_SUP__eq,axiom,
! [A2: set_o,B: set_nat,F: $o > set_nat,G: nat > set_nat] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ( ord_less_eq_set_nat @ ( F @ I4 ) @ ( G @ X5 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B )
=> ? [X5: $o] :
( ( member_o @ X5 @ A2 )
& ( ord_less_eq_set_nat @ ( G @ J2 ) @ ( F @ X5 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_754_SUP__eq,axiom,
! [A2: set_o,B: set_o,F: $o > set_nat,G: $o > set_nat] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ? [X5: $o] :
( ( member_o @ X5 @ B )
& ( ord_less_eq_set_nat @ ( F @ I4 ) @ ( G @ X5 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X5: $o] :
( ( member_o @ X5 @ A2 )
& ( ord_less_eq_set_nat @ ( G @ J2 ) @ ( F @ X5 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_755_SUP__eq,axiom,
! [A2: set_o,B: set_o,F: $o > $o,G: $o > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ? [X5: $o] :
( ( member_o @ X5 @ B )
& ( ord_less_eq_o @ ( F @ I4 ) @ ( G @ X5 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X5: $o] :
( ( member_o @ X5 @ A2 )
& ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_756_Inf__superset__mono,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A2 ) @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inf_superset_mono
thf(fact_757_Inf__superset__mono,axiom,
! [B: set_se4904748513628223167_a_nat,A2: set_se4904748513628223167_a_nat] :
( ( ord_le7974500612278410847_a_nat @ B @ A2 )
=> ( ord_le1325389633284124927_a_nat @ ( comple1528121977673479270_a_nat @ A2 ) @ ( comple1528121977673479270_a_nat @ B ) ) ) ).
% Inf_superset_mono
thf(fact_758_Inf__superset__mono,axiom,
! [B: set_o,A2: set_o] :
( ( ord_less_eq_set_o @ B @ A2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ ( complete_Inf_Inf_o @ B ) ) ) ).
% Inf_superset_mono
thf(fact_759_Sup__subset__mono,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Sup_subset_mono
thf(fact_760_Sup__subset__mono,axiom,
! [A2: set_se4904748513628223167_a_nat,B: set_se4904748513628223167_a_nat] :
( ( ord_le7974500612278410847_a_nat @ A2 @ B )
=> ( ord_le1325389633284124927_a_nat @ ( comple1247738100258233164_a_nat @ A2 ) @ ( comple1247738100258233164_a_nat @ B ) ) ) ).
% Sup_subset_mono
thf(fact_761_Sup__subset__mono,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_subset_mono
thf(fact_762_Sup__subset__mono,axiom,
! [A2: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_subset_mono
thf(fact_763_Sup__union__distrib,axiom,
! [A2: set_se4904748513628223167_a_nat,B: set_se4904748513628223167_a_nat] :
( ( comple1247738100258233164_a_nat @ ( sup_su2291686591051470483_a_nat @ A2 @ B ) )
= ( sup_su6804446743777130803_a_nat @ ( comple1247738100258233164_a_nat @ A2 ) @ ( comple1247738100258233164_a_nat @ B ) ) ) ).
% Sup_union_distrib
thf(fact_764_Sup__union__distrib,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A2 @ B ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_union_distrib
thf(fact_765_Sup__union__distrib,axiom,
! [A2: set_o,B: set_o] :
( ( complete_Sup_Sup_o @ ( sup_sup_set_o @ A2 @ B ) )
= ( sup_sup_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_union_distrib
thf(fact_766_Inter__anti__mono,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A2 ) @ ( comple6135023378680113637_set_a @ B ) ) ) ).
% Inter_anti_mono
thf(fact_767_Inter__anti__mono,axiom,
! [B: set_se4904748513628223167_a_nat,A2: set_se4904748513628223167_a_nat] :
( ( ord_le7974500612278410847_a_nat @ B @ A2 )
=> ( ord_le1325389633284124927_a_nat @ ( comple1528121977673479270_a_nat @ A2 ) @ ( comple1528121977673479270_a_nat @ B ) ) ) ).
% Inter_anti_mono
thf(fact_768_Union__mono,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_mono
thf(fact_769_Union__mono,axiom,
! [A2: set_se4904748513628223167_a_nat,B: set_se4904748513628223167_a_nat] :
( ( ord_le7974500612278410847_a_nat @ A2 @ B )
=> ( ord_le1325389633284124927_a_nat @ ( comple1247738100258233164_a_nat @ A2 ) @ ( comple1247738100258233164_a_nat @ B ) ) ) ).
% Union_mono
thf(fact_770_Union__mono,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_mono
thf(fact_771_INF__eqI,axiom,
! [A2: set_nat,X3: set_nat,F: nat > set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ X3 @ ( F @ I4 ) ) )
=> ( ! [Y5: set_nat] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A2 )
=> ( ord_less_eq_set_nat @ Y5 @ ( F @ I5 ) ) )
=> ( ord_less_eq_set_nat @ Y5 @ X3 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= X3 ) ) ) ).
% INF_eqI
thf(fact_772_INF__eqI,axiom,
! [A2: set_o,X3: set_a,F: $o > set_a] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_set_a @ X3 @ ( F @ I4 ) ) )
=> ( ! [Y5: set_a] :
( ! [I5: $o] :
( ( member_o @ I5 @ A2 )
=> ( ord_less_eq_set_a @ Y5 @ ( F @ I5 ) ) )
=> ( ord_less_eq_set_a @ Y5 @ X3 ) )
=> ( ( comple6135023378680113637_set_a @ ( image_o_set_a @ F @ A2 ) )
= X3 ) ) ) ).
% INF_eqI
thf(fact_773_INF__eqI,axiom,
! [A2: set_o,X3: set_Sum_sum_a_nat,F: $o > set_Sum_sum_a_nat] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ X3 @ ( F @ I4 ) ) )
=> ( ! [Y5: set_Sum_sum_a_nat] :
( ! [I5: $o] :
( ( member_o @ I5 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ Y5 @ ( F @ I5 ) ) )
=> ( ord_le1325389633284124927_a_nat @ Y5 @ X3 ) )
=> ( ( comple1528121977673479270_a_nat @ ( image_3365592128754359116_a_nat @ F @ A2 ) )
= X3 ) ) ) ).
% INF_eqI
thf(fact_774_INF__eqI,axiom,
! [A2: set_o,X3: $o,F: $o > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_o @ X3 @ ( F @ I4 ) ) )
=> ( ! [Y5: $o] :
( ! [I5: $o] :
( ( member_o @ I5 @ A2 )
=> ( ord_less_eq_o @ Y5 @ ( F @ I5 ) ) )
=> ( ord_less_eq_o @ Y5 @ X3 ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) )
= X3 ) ) ) ).
% INF_eqI
thf(fact_775_INF__mono,axiom,
! [B: set_nat,A2: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ! [M3: nat] :
( ( member_nat @ M3 @ B )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ord_less_eq_set_nat @ ( F @ X5 ) @ ( G @ M3 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ B ) ) ) ) ).
% INF_mono
thf(fact_776_INF__mono,axiom,
! [B: set_o,A2: set_nat,F: nat > set_nat,G: $o > set_nat] :
( ! [M3: $o] :
( ( member_o @ M3 @ B )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ord_less_eq_set_nat @ ( F @ X5 ) @ ( G @ M3 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ G @ B ) ) ) ) ).
% INF_mono
thf(fact_777_INF__lower,axiom,
! [I2: nat,A2: set_nat,F: nat > set_nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( F @ I2 ) ) ) ).
% INF_lower
thf(fact_778_INF__lower,axiom,
! [I2: $o,A2: set_o,F: $o > set_a] :
( ( member_o @ I2 @ A2 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_o_set_a @ F @ A2 ) ) @ ( F @ I2 ) ) ) ).
% INF_lower
thf(fact_779_INF__lower,axiom,
! [I2: $o,A2: set_o,F: $o > set_Sum_sum_a_nat] :
( ( member_o @ I2 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ ( comple1528121977673479270_a_nat @ ( image_3365592128754359116_a_nat @ F @ A2 ) ) @ ( F @ I2 ) ) ) ).
% INF_lower
thf(fact_780_INF__lower,axiom,
! [I2: $o,A2: set_o,F: $o > $o] :
( ( member_o @ I2 @ A2 )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) @ ( F @ I2 ) ) ) ).
% INF_lower
thf(fact_781_INF__mono_H,axiom,
! [F: nat > set_nat,G: nat > set_nat,A2: set_nat] :
( ! [X2: nat] : ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ A2 ) ) ) ) ).
% INF_mono'
thf(fact_782_INF__lower2,axiom,
! [I2: nat,A2: set_nat,F: nat > set_nat,U: set_nat] :
( ( member_nat @ I2 @ A2 )
=> ( ( ord_less_eq_set_nat @ ( F @ I2 ) @ U )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_783_INF__lower2,axiom,
! [I2: $o,A2: set_o,F: $o > set_a,U: set_a] :
( ( member_o @ I2 @ A2 )
=> ( ( ord_less_eq_set_a @ ( F @ I2 ) @ U )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_o_set_a @ F @ A2 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_784_INF__lower2,axiom,
! [I2: $o,A2: set_o,F: $o > set_Sum_sum_a_nat,U: set_Sum_sum_a_nat] :
( ( member_o @ I2 @ A2 )
=> ( ( ord_le1325389633284124927_a_nat @ ( F @ I2 ) @ U )
=> ( ord_le1325389633284124927_a_nat @ ( comple1528121977673479270_a_nat @ ( image_3365592128754359116_a_nat @ F @ A2 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_785_INF__lower2,axiom,
! [I2: $o,A2: set_o,F: $o > $o,U: $o] :
( ( member_o @ I2 @ A2 )
=> ( ( ord_less_eq_o @ ( F @ I2 ) @ U )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_786_le__INF__iff,axiom,
! [U: set_nat,F: nat > set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ U @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A2 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ U @ ( F @ X ) ) ) ) ) ).
% le_INF_iff
thf(fact_787_INF__greatest,axiom,
! [A2: set_nat,U: set_nat,F: nat > set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ U @ ( F @ I4 ) ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% INF_greatest
thf(fact_788_INF__greatest,axiom,
! [A2: set_o,U: set_a,F: $o > set_a] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_set_a @ U @ ( F @ I4 ) ) )
=> ( ord_less_eq_set_a @ U @ ( comple6135023378680113637_set_a @ ( image_o_set_a @ F @ A2 ) ) ) ) ).
% INF_greatest
thf(fact_789_INF__greatest,axiom,
! [A2: set_o,U: set_Sum_sum_a_nat,F: $o > set_Sum_sum_a_nat] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ U @ ( F @ I4 ) ) )
=> ( ord_le1325389633284124927_a_nat @ U @ ( comple1528121977673479270_a_nat @ ( image_3365592128754359116_a_nat @ F @ A2 ) ) ) ) ).
% INF_greatest
thf(fact_790_INF__greatest,axiom,
! [A2: set_o,U: $o,F: $o > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_o @ U @ ( F @ I4 ) ) )
=> ( ord_less_eq_o @ U @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) ) ) ).
% INF_greatest
thf(fact_791_SUP__eqI,axiom,
! [A2: set_o,F: $o > set_a,X3: set_a] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ I4 ) @ X3 ) )
=> ( ! [Y5: set_a] :
( ! [I5: $o] :
( ( member_o @ I5 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ I5 ) @ Y5 ) )
=> ( ord_less_eq_set_a @ X3 @ Y5 ) )
=> ( ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ A2 ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_792_SUP__eqI,axiom,
! [A2: set_o,F: $o > set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ I4 ) @ X3 ) )
=> ( ! [Y5: set_Sum_sum_a_nat] :
( ! [I5: $o] :
( ( member_o @ I5 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ I5 ) @ Y5 ) )
=> ( ord_le1325389633284124927_a_nat @ X3 @ Y5 ) )
=> ( ( comple1247738100258233164_a_nat @ ( image_3365592128754359116_a_nat @ F @ A2 ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_793_SUP__eqI,axiom,
! [A2: set_nat,F: nat > set_nat,X3: set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ X3 ) )
=> ( ! [Y5: set_nat] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I5 ) @ Y5 ) )
=> ( ord_less_eq_set_nat @ X3 @ Y5 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_794_SUP__eqI,axiom,
! [A2: set_o,F: $o > set_nat,X3: set_nat] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ X3 ) )
=> ( ! [Y5: set_nat] :
( ! [I5: $o] :
( ( member_o @ I5 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I5 ) @ Y5 ) )
=> ( ord_less_eq_set_nat @ X3 @ Y5 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_795_SUP__eqI,axiom,
! [A2: set_o,F: $o > $o,X3: $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ X3 ) )
=> ( ! [Y5: $o] :
( ! [I5: $o] :
( ( member_o @ I5 @ A2 )
=> ( ord_less_eq_o @ ( F @ I5 ) @ Y5 ) )
=> ( ord_less_eq_o @ X3 @ Y5 ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_796_SUP__mono,axiom,
! [A2: set_nat,B: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ! [N3: nat] :
( ( member_nat @ N3 @ A2 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( G @ X5 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_797_SUP__mono,axiom,
! [A2: set_o,B: set_nat,F: $o > set_nat,G: nat > set_nat] :
( ! [N3: $o] :
( ( member_o @ N3 @ A2 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( G @ X5 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_798_SUP__least,axiom,
! [A2: set_o,F: $o > set_a,U: set_a] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ I4 ) @ U ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_799_SUP__least,axiom,
! [A2: set_o,F: $o > set_Sum_sum_a_nat,U: set_Sum_sum_a_nat] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ I4 ) @ U ) )
=> ( ord_le1325389633284124927_a_nat @ ( comple1247738100258233164_a_nat @ ( image_3365592128754359116_a_nat @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_800_SUP__least,axiom,
! [A2: set_nat,F: nat > set_nat,U: set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ U ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_801_SUP__least,axiom,
! [A2: set_o,F: $o > set_nat,U: set_nat] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ U ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_802_SUP__least,axiom,
! [A2: set_o,F: $o > $o,U: $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A2 )
=> ( ord_less_eq_o @ ( F @ I4 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_803_SUP__mono_H,axiom,
! [F: nat > set_nat,G: nat > set_nat,A2: set_nat] :
( ! [X2: nat] : ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A2 ) ) ) ) ).
% SUP_mono'
thf(fact_804_SUP__upper,axiom,
! [I2: $o,A2: set_o,F: $o > set_a] :
( ( member_o @ I2 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ I2 ) @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_805_SUP__upper,axiom,
! [I2: $o,A2: set_o,F: $o > set_Sum_sum_a_nat] :
( ( member_o @ I2 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ I2 ) @ ( comple1247738100258233164_a_nat @ ( image_3365592128754359116_a_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_806_SUP__upper,axiom,
! [I2: nat,A2: set_nat,F: nat > set_nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I2 ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_807_SUP__upper,axiom,
! [I2: $o,A2: set_o,F: $o > set_nat] :
( ( member_o @ I2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ I2 ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_808_SUP__upper,axiom,
! [I2: $o,A2: set_o,F: $o > $o] :
( ( member_o @ I2 @ A2 )
=> ( ord_less_eq_o @ ( F @ I2 ) @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_809_SUP__le__iff,axiom,
! [F: nat > set_nat,A2: set_nat,U: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ U )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_810_SUP__upper2,axiom,
! [I2: $o,A2: set_o,U: set_a,F: $o > set_a] :
( ( member_o @ I2 @ A2 )
=> ( ( ord_less_eq_set_a @ U @ ( F @ I2 ) )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_811_SUP__upper2,axiom,
! [I2: $o,A2: set_o,U: set_Sum_sum_a_nat,F: $o > set_Sum_sum_a_nat] :
( ( member_o @ I2 @ A2 )
=> ( ( ord_le1325389633284124927_a_nat @ U @ ( F @ I2 ) )
=> ( ord_le1325389633284124927_a_nat @ U @ ( comple1247738100258233164_a_nat @ ( image_3365592128754359116_a_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_812_SUP__upper2,axiom,
! [I2: nat,A2: set_nat,U: set_nat,F: nat > set_nat] :
( ( member_nat @ I2 @ A2 )
=> ( ( ord_less_eq_set_nat @ U @ ( F @ I2 ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_813_SUP__upper2,axiom,
! [I2: $o,A2: set_o,U: set_nat,F: $o > set_nat] :
( ( member_o @ I2 @ A2 )
=> ( ( ord_less_eq_set_nat @ U @ ( F @ I2 ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_814_SUP__upper2,axiom,
! [I2: $o,A2: set_o,U: $o,F: $o > $o] :
( ( member_o @ I2 @ A2 )
=> ( ( ord_less_eq_o @ U @ ( F @ I2 ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_815_SUP__absorb,axiom,
! [K: $o,I: set_o,A2: $o > set_Sum_sum_a_nat] :
( ( member_o @ K @ I )
=> ( ( sup_su6804446743777130803_a_nat @ ( A2 @ K ) @ ( comple1247738100258233164_a_nat @ ( image_3365592128754359116_a_nat @ A2 @ I ) ) )
= ( comple1247738100258233164_a_nat @ ( image_3365592128754359116_a_nat @ A2 @ I ) ) ) ) ).
% SUP_absorb
thf(fact_816_SUP__absorb,axiom,
! [K: nat,I: set_nat,A2: nat > set_nat] :
( ( member_nat @ K @ I )
=> ( ( sup_sup_set_nat @ ( A2 @ K ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ I ) ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ I ) ) ) ) ).
% SUP_absorb
thf(fact_817_SUP__absorb,axiom,
! [K: $o,I: set_o,A2: $o > set_nat] :
( ( member_o @ K @ I )
=> ( ( sup_sup_set_nat @ ( A2 @ K ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A2 @ I ) ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A2 @ I ) ) ) ) ).
% SUP_absorb
thf(fact_818_SUP__absorb,axiom,
! [K: $o,I: set_o,A2: $o > $o] :
( ( member_o @ K @ I )
=> ( ( sup_sup_o @ ( A2 @ K ) @ ( complete_Sup_Sup_o @ ( image_o_o @ A2 @ I ) ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ A2 @ I ) ) ) ) ).
% SUP_absorb
thf(fact_819_complete__lattice__class_OSUP__sup__distrib,axiom,
! [F: nat > set_nat,A2: set_nat,G: nat > set_nat] :
( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A5: nat] : ( sup_sup_set_nat @ ( F @ A5 ) @ ( G @ A5 ) )
@ A2 ) ) ) ).
% complete_lattice_class.SUP_sup_distrib
thf(fact_820_subset__Pow__Union,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ ( pow_nat @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% subset_Pow_Union
thf(fact_821_INTER__UNIV__conv_I2_J,axiom,
! [B: nat > set_nat,A2: set_nat] :
( ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B @ A2 ) )
= top_top_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( B @ X )
= top_top_set_nat ) ) ) ) ).
% INTER_UNIV_conv(2)
thf(fact_822_INTER__UNIV__conv_I1_J,axiom,
! [B: nat > set_nat,A2: set_nat] :
( ( top_top_set_nat
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B @ A2 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( B @ X )
= top_top_set_nat ) ) ) ) ).
% INTER_UNIV_conv(1)
thf(fact_823_INT__extend__simps_I10_J,axiom,
! [B: sum_sum_a_nat > set_nat,F: nat > sum_sum_a_nat,A2: set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [A5: nat] : ( B @ ( F @ A5 ) )
@ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_4589483402070311232et_nat @ B @ ( image_7293268710728258664_a_nat @ F @ A2 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_824_INT__extend__simps_I10_J,axiom,
! [B: set_nat > set_nat,F: nat > set_nat,A2: set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [A5: nat] : ( B @ ( F @ A5 ) )
@ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ B @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_825_INT__extend__simps_I10_J,axiom,
! [B: nat > set_nat,F: nat > nat,A2: set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [A5: nat] : ( B @ ( F @ A5 ) )
@ A2 ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_826_INT__subset__iff,axiom,
! [B: set_nat,A2: nat > set_nat,I: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A2 @ I ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ I )
=> ( ord_less_eq_set_nat @ B @ ( A2 @ X ) ) ) ) ) ).
% INT_subset_iff
thf(fact_827_INT__anti__mono,axiom,
! [A2: set_nat,B: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ B ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ A2 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_828_INT__anti__mono,axiom,
! [A2: set_o,B: set_o,F: $o > set_a,G: $o > set_a] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_o_set_a @ F @ B ) ) @ ( comple6135023378680113637_set_a @ ( image_o_set_a @ G @ A2 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_829_INT__anti__mono,axiom,
! [A2: set_o,B: set_o,F: $o > set_Sum_sum_a_nat,G: $o > set_Sum_sum_a_nat] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( comple1528121977673479270_a_nat @ ( image_3365592128754359116_a_nat @ F @ B ) ) @ ( comple1528121977673479270_a_nat @ ( image_3365592128754359116_a_nat @ G @ A2 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_830_INT__anti__mono,axiom,
! [A2: set_a,B: set_a,F: a > set_a,G: a > set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ F @ B ) ) @ ( comple6135023378680113637_set_a @ ( image_a_set_a @ G @ A2 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_831_INT__anti__mono,axiom,
! [A2: set_a,B: set_a,F: a > set_Sum_sum_a_nat,G: a > set_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( comple1528121977673479270_a_nat @ ( image_1201892972954314598_a_nat @ F @ B ) ) @ ( comple1528121977673479270_a_nat @ ( image_1201892972954314598_a_nat @ G @ A2 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_832_INT__anti__mono,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,F: sum_sum_a_nat > set_a,G: sum_sum_a_nat > set_a] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B )
=> ( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_8809959208761443620_set_a @ F @ B ) ) @ ( comple6135023378680113637_set_a @ ( image_8809959208761443620_set_a @ G @ A2 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_833_INT__anti__mono,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,F: sum_sum_a_nat > set_Sum_sum_a_nat,G: sum_sum_a_nat > set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B )
=> ( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( comple1528121977673479270_a_nat @ ( image_7877458644602423589_a_nat @ F @ B ) ) @ ( comple1528121977673479270_a_nat @ ( image_7877458644602423589_a_nat @ G @ A2 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_834_INT__greatest,axiom,
! [A2: set_nat,C2: set_nat,B: nat > set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ C2 @ ( B @ X2 ) ) )
=> ( ord_less_eq_set_nat @ C2 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B @ A2 ) ) ) ) ).
% INT_greatest
thf(fact_835_INT__greatest,axiom,
! [A2: set_o,C2: set_a,B: $o > set_a] :
( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_set_a @ C2 @ ( B @ X2 ) ) )
=> ( ord_less_eq_set_a @ C2 @ ( comple6135023378680113637_set_a @ ( image_o_set_a @ B @ A2 ) ) ) ) ).
% INT_greatest
thf(fact_836_INT__greatest,axiom,
! [A2: set_o,C2: set_Sum_sum_a_nat,B: $o > set_Sum_sum_a_nat] :
( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ C2 @ ( B @ X2 ) ) )
=> ( ord_le1325389633284124927_a_nat @ C2 @ ( comple1528121977673479270_a_nat @ ( image_3365592128754359116_a_nat @ B @ A2 ) ) ) ) ).
% INT_greatest
thf(fact_837_INT__lower,axiom,
! [A: nat,A2: set_nat,B: nat > set_nat] :
( ( member_nat @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B @ A2 ) ) @ ( B @ A ) ) ) ).
% INT_lower
thf(fact_838_INT__lower,axiom,
! [A: $o,A2: set_o,B: $o > set_a] :
( ( member_o @ A @ A2 )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ ( image_o_set_a @ B @ A2 ) ) @ ( B @ A ) ) ) ).
% INT_lower
thf(fact_839_INT__lower,axiom,
! [A: $o,A2: set_o,B: $o > set_Sum_sum_a_nat] :
( ( member_o @ A @ A2 )
=> ( ord_le1325389633284124927_a_nat @ ( comple1528121977673479270_a_nat @ ( image_3365592128754359116_a_nat @ B @ A2 ) ) @ ( B @ A ) ) ) ).
% INT_lower
thf(fact_840_INT__extend__simps_I7_J,axiom,
! [A2: set_nat,B: nat > set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B @ C2 ) ) )
= ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( sup_sup_set_nat @ A2 @ ( B @ X ) )
@ C2 ) ) ) ).
% INT_extend_simps(7)
thf(fact_841_INT__extend__simps_I6_J,axiom,
! [A2: nat > set_nat,C2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A2 @ C2 ) ) @ B )
= ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( sup_sup_set_nat @ ( A2 @ X ) @ B )
@ C2 ) ) ) ).
% INT_extend_simps(6)
thf(fact_842_Un__INT__distrib,axiom,
! [B: set_nat,A2: nat > set_nat,I: set_nat] :
( ( sup_sup_set_nat @ B @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A2 @ I ) ) )
= ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [I3: nat] : ( sup_sup_set_nat @ B @ ( A2 @ I3 ) )
@ I ) ) ) ).
% Un_INT_distrib
thf(fact_843_Un__INT__distrib2,axiom,
! [A2: nat > set_nat,I: set_nat,B: nat > set_nat,J3: set_nat] :
( ( sup_sup_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A2 @ I ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B @ J3 ) ) )
= ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [I3: nat] :
( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [J: nat] : ( sup_sup_set_nat @ ( A2 @ I3 ) @ ( B @ J ) )
@ J3 ) )
@ I ) ) ) ).
% Un_INT_distrib2
thf(fact_844_image__UN,axiom,
! [F: nat > sum_sum_a_nat,B: nat > set_nat,A2: set_nat] :
( ( image_7293268710728258664_a_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A2 ) ) )
= ( comple1247738100258233164_a_nat
@ ( image_4085361583586468296_a_nat
@ ^ [X: nat] : ( image_7293268710728258664_a_nat @ F @ ( B @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_845_image__UN,axiom,
! [F: nat > set_nat,B: nat > set_nat,A2: set_nat] :
( ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A2 ) ) )
= ( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [X: nat] : ( image_nat_set_nat @ F @ ( B @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_846_image__UN,axiom,
! [F: nat > nat,B: nat > set_nat,A2: set_nat] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( image_nat_nat @ F @ ( B @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_847_UN__extend__simps_I10_J,axiom,
! [B: sum_sum_a_nat > set_nat,F: a > sum_sum_a_nat,A2: set_a] :
( ( comple7399068483239264473et_nat
@ ( image_a_set_nat
@ ^ [A5: a] : ( B @ ( F @ A5 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_4589483402070311232et_nat @ B @ ( image_7873763678140191238_a_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_848_UN__extend__simps_I10_J,axiom,
! [B: sum_sum_a_nat > set_nat,F: nat > sum_sum_a_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A5: nat] : ( B @ ( F @ A5 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_4589483402070311232et_nat @ B @ ( image_7293268710728258664_a_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_849_UN__extend__simps_I10_J,axiom,
! [B: set_nat > set_nat,F: nat > set_nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A5: nat] : ( B @ ( F @ A5 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_850_UN__extend__simps_I10_J,axiom,
! [B: nat > set_nat,F: nat > nat,A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A5: nat] : ( B @ ( F @ A5 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_851_UN__subset__iff,axiom,
! [A2: nat > set_nat,I: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ I ) ) @ B )
= ( ! [X: nat] :
( ( member_nat @ X @ I )
=> ( ord_less_eq_set_nat @ ( A2 @ X ) @ B ) ) ) ) ).
% UN_subset_iff
thf(fact_852_UN__upper,axiom,
! [A: $o,A2: set_o,B: $o > set_a] :
( ( member_o @ A @ A2 )
=> ( ord_less_eq_set_a @ ( B @ A ) @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ B @ A2 ) ) ) ) ).
% UN_upper
thf(fact_853_UN__upper,axiom,
! [A: $o,A2: set_o,B: $o > set_Sum_sum_a_nat] :
( ( member_o @ A @ A2 )
=> ( ord_le1325389633284124927_a_nat @ ( B @ A ) @ ( comple1247738100258233164_a_nat @ ( image_3365592128754359116_a_nat @ B @ A2 ) ) ) ) ).
% UN_upper
thf(fact_854_UN__upper,axiom,
! [A: nat,A2: set_nat,B: nat > set_nat] :
( ( member_nat @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( B @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A2 ) ) ) ) ).
% UN_upper
thf(fact_855_UN__upper,axiom,
! [A: $o,A2: set_o,B: $o > set_nat] :
( ( member_o @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( B @ A ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ A2 ) ) ) ) ).
% UN_upper
thf(fact_856_UN__least,axiom,
! [A2: set_o,B: $o > set_a,C2: set_a] :
( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( B @ X2 ) @ C2 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ B @ A2 ) ) @ C2 ) ) ).
% UN_least
thf(fact_857_UN__least,axiom,
! [A2: set_o,B: $o > set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ ( B @ X2 ) @ C2 ) )
=> ( ord_le1325389633284124927_a_nat @ ( comple1247738100258233164_a_nat @ ( image_3365592128754359116_a_nat @ B @ A2 ) ) @ C2 ) ) ).
% UN_least
thf(fact_858_UN__least,axiom,
! [A2: set_nat,B: nat > set_nat,C2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( B @ X2 ) @ C2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A2 ) ) @ C2 ) ) ).
% UN_least
thf(fact_859_UN__least,axiom,
! [A2: set_o,B: $o > set_nat,C2: set_nat] :
( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( B @ X2 ) @ C2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ A2 ) ) @ C2 ) ) ).
% UN_least
thf(fact_860_UN__mono,axiom,
! [A2: set_o,B: set_o,F: $o > set_a,G: $o > set_a] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_861_UN__mono,axiom,
! [A2: set_o,B: set_o,F: $o > set_Sum_sum_a_nat,G: $o > set_Sum_sum_a_nat] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( comple1247738100258233164_a_nat @ ( image_3365592128754359116_a_nat @ F @ A2 ) ) @ ( comple1247738100258233164_a_nat @ ( image_3365592128754359116_a_nat @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_862_UN__mono,axiom,
! [A2: set_a,B: set_a,F: a > set_a,G: a > set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_863_UN__mono,axiom,
! [A2: set_a,B: set_a,F: a > set_Sum_sum_a_nat,G: a > set_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( comple1247738100258233164_a_nat @ ( image_1201892972954314598_a_nat @ F @ A2 ) ) @ ( comple1247738100258233164_a_nat @ ( image_1201892972954314598_a_nat @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_864_UN__mono,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,F: sum_sum_a_nat > set_a,G: sum_sum_a_nat > set_a] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B )
=> ( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_8809959208761443620_set_a @ F @ A2 ) ) @ ( comple2307003609928055243_set_a @ ( image_8809959208761443620_set_a @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_865_UN__mono,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,F: sum_sum_a_nat > set_Sum_sum_a_nat,G: sum_sum_a_nat > set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B )
=> ( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( comple1247738100258233164_a_nat @ ( image_7877458644602423589_a_nat @ F @ A2 ) ) @ ( comple1247738100258233164_a_nat @ ( image_7877458644602423589_a_nat @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_866_UN__mono,axiom,
! [A2: set_nat,B: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_867_UN__mono,axiom,
! [A2: set_o,B: set_o,F: $o > set_nat,G: $o > set_nat] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_868_UN__mono,axiom,
! [A2: set_a,B: set_a,F: a > set_nat,G: a > set_nat] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_869_UN__mono,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,F: sum_sum_a_nat > set_nat,G: sum_sum_a_nat > set_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B )
=> ( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_4589483402070311232et_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_4589483402070311232et_nat @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_870_UN__absorb,axiom,
! [K: $o,I: set_o,A2: $o > set_Sum_sum_a_nat] :
( ( member_o @ K @ I )
=> ( ( sup_su6804446743777130803_a_nat @ ( A2 @ K ) @ ( comple1247738100258233164_a_nat @ ( image_3365592128754359116_a_nat @ A2 @ I ) ) )
= ( comple1247738100258233164_a_nat @ ( image_3365592128754359116_a_nat @ A2 @ I ) ) ) ) ).
% UN_absorb
thf(fact_871_UN__absorb,axiom,
! [K: nat,I: set_nat,A2: nat > set_nat] :
( ( member_nat @ K @ I )
=> ( ( sup_sup_set_nat @ ( A2 @ K ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ I ) ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ I ) ) ) ) ).
% UN_absorb
thf(fact_872_UN__absorb,axiom,
! [K: $o,I: set_o,A2: $o > set_nat] :
( ( member_o @ K @ I )
=> ( ( sup_sup_set_nat @ ( A2 @ K ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A2 @ I ) ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A2 @ I ) ) ) ) ).
% UN_absorb
thf(fact_873_UN__Un__distrib,axiom,
! [A2: nat > set_nat,B: nat > set_nat,I: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I3: nat] : ( sup_sup_set_nat @ ( A2 @ I3 ) @ ( B @ I3 ) )
@ I ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ I ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ I ) ) ) ) ).
% UN_Un_distrib
thf(fact_874_Un__Union__image,axiom,
! [A2: nat > set_nat,B: nat > set_nat,C2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( sup_sup_set_nat @ ( A2 @ X ) @ ( B @ X ) )
@ C2 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ C2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ C2 ) ) ) ) ).
% Un_Union_image
thf(fact_875_Un__Inter,axiom,
! [A2: set_Sum_sum_a_nat,B: set_se4904748513628223167_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A2 @ ( comple1528121977673479270_a_nat @ B ) )
= ( comple1528121977673479270_a_nat @ ( image_5599399343138760645_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 ) @ B ) ) ) ).
% Un_Inter
thf(fact_876_image__Union,axiom,
! [F: a > sum_sum_a_nat,S: set_set_a] :
( ( image_7873763678140191238_a_nat @ F @ ( comple2307003609928055243_set_a @ S ) )
= ( comple1247738100258233164_a_nat @ ( image_6715601112060939782_a_nat @ ( image_7873763678140191238_a_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_877_image__Union,axiom,
! [F: nat > sum_sum_a_nat,S: set_set_nat] :
( ( image_7293268710728258664_a_nat @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple1247738100258233164_a_nat @ ( image_3578472599065059474_a_nat @ ( image_7293268710728258664_a_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_878_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_879_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_880_UN__extend__simps_I8_J,axiom,
! [B: nat > set_nat,A2: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [Y3: set_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ Y3 ) )
@ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_881_INF__SUP,axiom,
! [P: sum_sum_a_nat > sum_sum_a_nat > set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_4589483402070311232et_nat
@ ^ [Y3: sum_sum_a_nat] :
( comple7399068483239264473et_nat
@ ( image_4589483402070311232et_nat
@ ^ [X: sum_sum_a_nat] : ( P @ X @ Y3 )
@ top_to795618464972521135_a_nat ) )
@ top_to795618464972521135_a_nat ) )
= ( comple7399068483239264473et_nat
@ ( image_2259655797477813426et_nat
@ ^ [F3: sum_sum_a_nat > sum_sum_a_nat] :
( comple7806235888213564991et_nat
@ ( image_4589483402070311232et_nat
@ ^ [X: sum_sum_a_nat] : ( P @ ( F3 @ X ) @ X )
@ top_to795618464972521135_a_nat ) )
@ top_to7766267419512669707_a_nat ) ) ) ).
% INF_SUP
thf(fact_882_INF__SUP,axiom,
! [P: sum_sum_a_nat > nat > set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [Y3: nat] :
( comple7399068483239264473et_nat
@ ( image_4589483402070311232et_nat
@ ^ [X: sum_sum_a_nat] : ( P @ X @ Y3 )
@ top_to795618464972521135_a_nat ) )
@ top_top_set_nat ) )
= ( comple7399068483239264473et_nat
@ ( image_5790795286017029031et_nat
@ ^ [F3: nat > sum_sum_a_nat] :
( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( P @ ( F3 @ X ) @ X )
@ top_top_set_nat ) )
@ top_to9106040778512017686_a_nat ) ) ) ).
% INF_SUP
thf(fact_883_INF__SUP,axiom,
! [P: nat > sum_sum_a_nat > set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_4589483402070311232et_nat
@ ^ [Y3: sum_sum_a_nat] :
( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( P @ X @ Y3 )
@ top_top_set_nat ) )
@ top_to795618464972521135_a_nat ) )
= ( comple7399068483239264473et_nat
@ ( image_7410052990245744513et_nat
@ ^ [F3: sum_sum_a_nat > nat] :
( comple7806235888213564991et_nat
@ ( image_4589483402070311232et_nat
@ ^ [X: sum_sum_a_nat] : ( P @ ( F3 @ X ) @ X )
@ top_to795618464972521135_a_nat ) )
@ top_to990407478546573296at_nat ) ) ) ).
% INF_SUP
thf(fact_884_INF__SUP,axiom,
! [P: nat > nat > set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [Y3: nat] :
( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( P @ X @ Y3 )
@ top_top_set_nat ) )
@ top_top_set_nat ) )
= ( comple7399068483239264473et_nat
@ ( image_7432509271690132940et_nat
@ ^ [F3: nat > nat] :
( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( P @ ( F3 @ X ) @ X )
@ top_top_set_nat ) )
@ top_top_set_nat_nat ) ) ) ).
% INF_SUP
thf(fact_885_INF__SUP,axiom,
! [P: sum_sum_a_nat > sum_sum_a_nat > $o] :
( ( complete_Inf_Inf_o
@ ( image_6095136190293192542_nat_o
@ ^ [Y3: sum_sum_a_nat] :
( complete_Sup_Sup_o
@ ( image_6095136190293192542_nat_o
@ ^ [X: sum_sum_a_nat] : ( P @ X @ Y3 )
@ top_to795618464972521135_a_nat ) )
@ top_to795618464972521135_a_nat ) )
= ( complete_Sup_Sup_o
@ ( image_1729793951705799532_nat_o
@ ^ [F3: sum_sum_a_nat > sum_sum_a_nat] :
( complete_Inf_Inf_o
@ ( image_6095136190293192542_nat_o
@ ^ [X: sum_sum_a_nat] : ( P @ ( F3 @ X ) @ X )
@ top_to795618464972521135_a_nat ) )
@ top_to7766267419512669707_a_nat ) ) ) ).
% INF_SUP
thf(fact_886_INF__SUP,axiom,
! [P: sum_sum_a_nat > nat > $o] :
( ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [Y3: nat] :
( complete_Sup_Sup_o
@ ( image_6095136190293192542_nat_o
@ ^ [X: sum_sum_a_nat] : ( P @ X @ Y3 )
@ top_to795618464972521135_a_nat ) )
@ top_top_set_nat ) )
= ( complete_Sup_Sup_o
@ ( image_2376713081370839351_nat_o
@ ^ [F3: nat > sum_sum_a_nat] :
( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [X: nat] : ( P @ ( F3 @ X ) @ X )
@ top_top_set_nat ) )
@ top_to9106040778512017686_a_nat ) ) ) ).
% INF_SUP
thf(fact_887_INF__SUP,axiom,
! [P: nat > sum_sum_a_nat > $o] :
( ( complete_Inf_Inf_o
@ ( image_6095136190293192542_nat_o
@ ^ [Y3: sum_sum_a_nat] :
( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [X: nat] : ( P @ X @ Y3 )
@ top_top_set_nat ) )
@ top_to795618464972521135_a_nat ) )
= ( complete_Sup_Sup_o
@ ( image_2859160068955298525_nat_o
@ ^ [F3: sum_sum_a_nat > nat] :
( complete_Inf_Inf_o
@ ( image_6095136190293192542_nat_o
@ ^ [X: sum_sum_a_nat] : ( P @ ( F3 @ X ) @ X )
@ top_to795618464972521135_a_nat ) )
@ top_to990407478546573296at_nat ) ) ) ).
% INF_SUP
thf(fact_888_INF__SUP,axiom,
! [P: nat > nat > $o] :
( ( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [Y3: nat] :
( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [X: nat] : ( P @ X @ Y3 )
@ top_top_set_nat ) )
@ top_top_set_nat ) )
= ( complete_Sup_Sup_o
@ ( image_nat_nat_o
@ ^ [F3: nat > nat] :
( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [X: nat] : ( P @ ( F3 @ X ) @ X )
@ top_top_set_nat ) )
@ top_top_set_nat_nat ) ) ) ).
% INF_SUP
thf(fact_889_SUP__INF,axiom,
! [P: sum_sum_a_nat > sum_sum_a_nat > set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_4589483402070311232et_nat
@ ^ [Y3: sum_sum_a_nat] :
( comple7806235888213564991et_nat
@ ( image_4589483402070311232et_nat
@ ^ [X: sum_sum_a_nat] : ( P @ X @ Y3 )
@ top_to795618464972521135_a_nat ) )
@ top_to795618464972521135_a_nat ) )
= ( comple7806235888213564991et_nat
@ ( image_2259655797477813426et_nat
@ ^ [X: sum_sum_a_nat > sum_sum_a_nat] :
( comple7399068483239264473et_nat
@ ( image_4589483402070311232et_nat
@ ^ [Y3: sum_sum_a_nat] : ( P @ ( X @ Y3 ) @ Y3 )
@ top_to795618464972521135_a_nat ) )
@ top_to7766267419512669707_a_nat ) ) ) ).
% SUP_INF
thf(fact_890_SUP__INF,axiom,
! [P: sum_sum_a_nat > nat > set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y3: nat] :
( comple7806235888213564991et_nat
@ ( image_4589483402070311232et_nat
@ ^ [X: sum_sum_a_nat] : ( P @ X @ Y3 )
@ top_to795618464972521135_a_nat ) )
@ top_top_set_nat ) )
= ( comple7806235888213564991et_nat
@ ( image_5790795286017029031et_nat
@ ^ [X: nat > sum_sum_a_nat] :
( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y3: nat] : ( P @ ( X @ Y3 ) @ Y3 )
@ top_top_set_nat ) )
@ top_to9106040778512017686_a_nat ) ) ) ).
% SUP_INF
thf(fact_891_SUP__INF,axiom,
! [P: nat > sum_sum_a_nat > set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_4589483402070311232et_nat
@ ^ [Y3: sum_sum_a_nat] :
( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( P @ X @ Y3 )
@ top_top_set_nat ) )
@ top_to795618464972521135_a_nat ) )
= ( comple7806235888213564991et_nat
@ ( image_7410052990245744513et_nat
@ ^ [X: sum_sum_a_nat > nat] :
( comple7399068483239264473et_nat
@ ( image_4589483402070311232et_nat
@ ^ [Y3: sum_sum_a_nat] : ( P @ ( X @ Y3 ) @ Y3 )
@ top_to795618464972521135_a_nat ) )
@ top_to990407478546573296at_nat ) ) ) ).
% SUP_INF
thf(fact_892_SUP__INF,axiom,
! [P: nat > nat > set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y3: nat] :
( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( P @ X @ Y3 )
@ top_top_set_nat ) )
@ top_top_set_nat ) )
= ( comple7806235888213564991et_nat
@ ( image_7432509271690132940et_nat
@ ^ [X: nat > nat] :
( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y3: nat] : ( P @ ( X @ Y3 ) @ Y3 )
@ top_top_set_nat ) )
@ top_top_set_nat_nat ) ) ) ).
% SUP_INF
thf(fact_893_SUP__INF,axiom,
! [P: sum_sum_a_nat > sum_sum_a_nat > $o] :
( ( complete_Sup_Sup_o
@ ( image_6095136190293192542_nat_o
@ ^ [Y3: sum_sum_a_nat] :
( complete_Inf_Inf_o
@ ( image_6095136190293192542_nat_o
@ ^ [X: sum_sum_a_nat] : ( P @ X @ Y3 )
@ top_to795618464972521135_a_nat ) )
@ top_to795618464972521135_a_nat ) )
= ( complete_Inf_Inf_o
@ ( image_1729793951705799532_nat_o
@ ^ [X: sum_sum_a_nat > sum_sum_a_nat] :
( complete_Sup_Sup_o
@ ( image_6095136190293192542_nat_o
@ ^ [Y3: sum_sum_a_nat] : ( P @ ( X @ Y3 ) @ Y3 )
@ top_to795618464972521135_a_nat ) )
@ top_to7766267419512669707_a_nat ) ) ) ).
% SUP_INF
thf(fact_894_SUP__INF,axiom,
! [P: sum_sum_a_nat > nat > $o] :
( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y3: nat] :
( complete_Inf_Inf_o
@ ( image_6095136190293192542_nat_o
@ ^ [X: sum_sum_a_nat] : ( P @ X @ Y3 )
@ top_to795618464972521135_a_nat ) )
@ top_top_set_nat ) )
= ( complete_Inf_Inf_o
@ ( image_2376713081370839351_nat_o
@ ^ [X: nat > sum_sum_a_nat] :
( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y3: nat] : ( P @ ( X @ Y3 ) @ Y3 )
@ top_top_set_nat ) )
@ top_to9106040778512017686_a_nat ) ) ) ).
% SUP_INF
thf(fact_895_SUP__INF,axiom,
! [P: nat > sum_sum_a_nat > $o] :
( ( complete_Sup_Sup_o
@ ( image_6095136190293192542_nat_o
@ ^ [Y3: sum_sum_a_nat] :
( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [X: nat] : ( P @ X @ Y3 )
@ top_top_set_nat ) )
@ top_to795618464972521135_a_nat ) )
= ( complete_Inf_Inf_o
@ ( image_2859160068955298525_nat_o
@ ^ [X: sum_sum_a_nat > nat] :
( complete_Sup_Sup_o
@ ( image_6095136190293192542_nat_o
@ ^ [Y3: sum_sum_a_nat] : ( P @ ( X @ Y3 ) @ Y3 )
@ top_to795618464972521135_a_nat ) )
@ top_to990407478546573296at_nat ) ) ) ).
% SUP_INF
thf(fact_896_SUP__INF,axiom,
! [P: nat > nat > $o] :
( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y3: nat] :
( complete_Inf_Inf_o
@ ( image_nat_o
@ ^ [X: nat] : ( P @ X @ Y3 )
@ top_top_set_nat ) )
@ top_top_set_nat ) )
= ( complete_Inf_Inf_o
@ ( image_nat_nat_o
@ ^ [X: nat > nat] :
( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y3: nat] : ( P @ ( X @ Y3 ) @ Y3 )
@ top_top_set_nat ) )
@ top_top_set_nat_nat ) ) ) ).
% SUP_INF
thf(fact_897_INF__sup,axiom,
! [F: nat > set_nat,B: set_nat,A: set_nat] :
( ( sup_sup_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ B ) ) @ A )
= ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [B5: nat] : ( sup_sup_set_nat @ ( F @ B5 ) @ A )
@ B ) ) ) ).
% INF_sup
thf(fact_898_Inf__sup,axiom,
! [B: set_se4904748513628223167_a_nat,A: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( comple1528121977673479270_a_nat @ B ) @ A )
= ( comple1528121977673479270_a_nat
@ ( image_5599399343138760645_a_nat
@ ^ [B5: set_Sum_sum_a_nat] : ( sup_su6804446743777130803_a_nat @ B5 @ A )
@ B ) ) ) ).
% Inf_sup
thf(fact_899_Inf__sup,axiom,
! [B: set_o,A: $o] :
( ( sup_sup_o @ ( complete_Inf_Inf_o @ B ) @ A )
= ( complete_Inf_Inf_o
@ ( image_o_o
@ ^ [B5: $o] : ( sup_sup_o @ B5 @ A )
@ B ) ) ) ).
% Inf_sup
thf(fact_900_sup__INF,axiom,
! [A: set_nat,F: nat > set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ B ) ) )
= ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [B5: nat] : ( sup_sup_set_nat @ A @ ( F @ B5 ) )
@ B ) ) ) ).
% sup_INF
thf(fact_901_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ A2 ) )
& ( P @ X ) ) )
= ( ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ? [Y3: nat] :
( ( member_nat @ Y3 @ X )
& ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_902_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ A2 ) )
=> ( P @ X ) ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ X )
=> ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_903_UnionI,axiom,
! [X6: set_o,C2: set_set_o,A2: $o] :
( ( member_set_o @ X6 @ C2 )
=> ( ( member_o @ A2 @ X6 )
=> ( member_o @ A2 @ ( comple90263536869209701_set_o @ C2 ) ) ) ) ).
% UnionI
thf(fact_904_UnionI,axiom,
! [X6: set_nat,C2: set_set_nat,A2: nat] :
( ( member_set_nat @ X6 @ C2 )
=> ( ( member_nat @ A2 @ X6 )
=> ( member_nat @ A2 @ ( comple7399068483239264473et_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_905_Union__iff,axiom,
! [A2: $o,C2: set_set_o] :
( ( member_o @ A2 @ ( comple90263536869209701_set_o @ C2 ) )
= ( ? [X: set_o] :
( ( member_set_o @ X @ C2 )
& ( member_o @ A2 @ X ) ) ) ) ).
% Union_iff
thf(fact_906_Union__iff,axiom,
! [A2: nat,C2: set_set_nat] :
( ( member_nat @ A2 @ ( comple7399068483239264473et_nat @ C2 ) )
= ( ? [X: set_nat] :
( ( member_set_nat @ X @ C2 )
& ( member_nat @ A2 @ X ) ) ) ) ).
% Union_iff
thf(fact_907_InterI,axiom,
! [C2: set_set_o,A2: $o] :
( ! [X8: set_o] :
( ( member_set_o @ X8 @ C2 )
=> ( member_o @ A2 @ X8 ) )
=> ( member_o @ A2 @ ( comple3063163877087187839_set_o @ C2 ) ) ) ).
% InterI
thf(fact_908_Inter__iff,axiom,
! [A2: $o,C2: set_set_o] :
( ( member_o @ A2 @ ( comple3063163877087187839_set_o @ C2 ) )
= ( ! [X: set_o] :
( ( member_set_o @ X @ C2 )
=> ( member_o @ A2 @ X ) ) ) ) ).
% Inter_iff
thf(fact_909_INF__Int__eq,axiom,
! [S: set_set_o] :
( ( complete_Inf_Inf_o_o
@ ( image_set_o_o_o
@ ^ [I3: set_o,X: $o] : ( member_o @ X @ I3 )
@ S ) )
= ( ^ [X: $o] : ( member_o @ X @ ( comple3063163877087187839_set_o @ S ) ) ) ) ).
% INF_Int_eq
thf(fact_910_SUP__Sup__eq,axiom,
! [S: set_set_o] :
( ( complete_Sup_Sup_o_o
@ ( image_set_o_o_o
@ ^ [I3: set_o,X: $o] : ( member_o @ X @ I3 )
@ S ) )
= ( ^ [X: $o] : ( member_o @ X @ ( comple90263536869209701_set_o @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_911_SUP__Sup__eq,axiom,
! [S: set_set_nat] :
( ( comple8317665133742190828_nat_o
@ ( image_set_nat_nat_o
@ ^ [I3: set_nat,X: nat] : ( member_nat @ X @ I3 )
@ S ) )
= ( ^ [X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_912_Inf__set__def,axiom,
( comple3063163877087187839_set_o
= ( ^ [A3: set_set_o] :
( collect_o
@ ^ [X: $o] : ( complete_Inf_Inf_o @ ( image_set_o_o @ ( member_o @ X ) @ A3 ) ) ) ) ) ).
% Inf_set_def
thf(fact_913_Sup__set__def,axiom,
( comple90263536869209701_set_o
= ( ^ [A3: set_set_o] :
( collect_o
@ ^ [X: $o] : ( complete_Sup_Sup_o @ ( image_set_o_o @ ( member_o @ X ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_914_Sup__set__def,axiom,
( comple7399068483239264473et_nat
= ( ^ [A3: set_set_nat] :
( collect_nat
@ ^ [X: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_915_Sup__SUP__eq,axiom,
( complete_Sup_Sup_o_o
= ( ^ [S3: set_o_o,X: $o] : ( member_o @ X @ ( comple90263536869209701_set_o @ ( image_o_o_set_o @ collect_o @ S3 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_916_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S3: set_nat_o,X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S3 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_917_Inf__INT__eq,axiom,
( complete_Inf_Inf_o_o
= ( ^ [S3: set_o_o,X: $o] : ( member_o @ X @ ( comple3063163877087187839_set_o @ ( image_o_o_set_o @ collect_o @ S3 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_918_InterD,axiom,
! [A2: $o,C2: set_set_o,X6: set_o] :
( ( member_o @ A2 @ ( comple3063163877087187839_set_o @ C2 ) )
=> ( ( member_set_o @ X6 @ C2 )
=> ( member_o @ A2 @ X6 ) ) ) ).
% InterD
thf(fact_919_InterE,axiom,
! [A2: $o,C2: set_set_o,X6: set_o] :
( ( member_o @ A2 @ ( comple3063163877087187839_set_o @ C2 ) )
=> ( ( member_set_o @ X6 @ C2 )
=> ( member_o @ A2 @ X6 ) ) ) ).
% InterE
thf(fact_920_UnionE,axiom,
! [A2: $o,C2: set_set_o] :
( ( member_o @ A2 @ ( comple90263536869209701_set_o @ C2 ) )
=> ~ ! [X8: set_o] :
( ( member_o @ A2 @ X8 )
=> ~ ( member_set_o @ X8 @ C2 ) ) ) ).
% UnionE
thf(fact_921_UnionE,axiom,
! [A2: nat,C2: set_set_nat] :
( ( member_nat @ A2 @ ( comple7399068483239264473et_nat @ C2 ) )
=> ~ ! [X8: set_nat] :
( ( member_nat @ A2 @ X8 )
=> ~ ( member_set_nat @ X8 @ C2 ) ) ) ).
% UnionE
thf(fact_922_cSup__eq__maximum,axiom,
! [Z: set_a,X6: set_set_a] :
( ( member_set_a @ Z @ X6 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X6 )
=> ( ord_less_eq_set_a @ X2 @ Z ) )
=> ( ( comple2307003609928055243_set_a @ X6 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_923_cSup__eq__maximum,axiom,
! [Z: set_Sum_sum_a_nat,X6: set_se4904748513628223167_a_nat] :
( ( member8098812455498974984_a_nat @ Z @ X6 )
=> ( ! [X2: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X2 @ X6 )
=> ( ord_le1325389633284124927_a_nat @ X2 @ Z ) )
=> ( ( comple1247738100258233164_a_nat @ X6 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_924_cSup__eq__maximum,axiom,
! [Z: nat,X6: set_nat] :
( ( member_nat @ Z @ X6 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X6 )
=> ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( complete_Sup_Sup_nat @ X6 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_925_cSup__eq__maximum,axiom,
! [Z: set_nat,X6: set_set_nat] :
( ( member_set_nat @ Z @ X6 )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ X6 )
=> ( ord_less_eq_set_nat @ X2 @ Z ) )
=> ( ( comple7399068483239264473et_nat @ X6 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_926_cSup__eq__maximum,axiom,
! [Z: $o,X6: set_o] :
( ( member_o @ Z @ X6 )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ X6 )
=> ( ord_less_eq_o @ X2 @ Z ) )
=> ( ( complete_Sup_Sup_o @ X6 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_927_cInf__eq__minimum,axiom,
! [Z: set_a,X6: set_set_a] :
( ( member_set_a @ Z @ X6 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X6 )
=> ( ord_less_eq_set_a @ Z @ X2 ) )
=> ( ( comple6135023378680113637_set_a @ X6 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_928_cInf__eq__minimum,axiom,
! [Z: set_Sum_sum_a_nat,X6: set_se4904748513628223167_a_nat] :
( ( member8098812455498974984_a_nat @ Z @ X6 )
=> ( ! [X2: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X2 @ X6 )
=> ( ord_le1325389633284124927_a_nat @ Z @ X2 ) )
=> ( ( comple1528121977673479270_a_nat @ X6 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_929_cInf__eq__minimum,axiom,
! [Z: nat,X6: set_nat] :
( ( member_nat @ Z @ X6 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X6 )
=> ( ord_less_eq_nat @ Z @ X2 ) )
=> ( ( complete_Inf_Inf_nat @ X6 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_930_cInf__eq__minimum,axiom,
! [Z: $o,X6: set_o] :
( ( member_o @ Z @ X6 )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ X6 )
=> ( ord_less_eq_o @ Z @ X2 ) )
=> ( ( complete_Inf_Inf_o @ X6 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_931_cInf__eq,axiom,
! [X6: set_nat,A: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ X6 )
=> ( ord_less_eq_nat @ A @ X2 ) )
=> ( ! [Y5: nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ X6 )
=> ( ord_less_eq_nat @ Y5 @ X5 ) )
=> ( ord_less_eq_nat @ Y5 @ A ) )
=> ( ( complete_Inf_Inf_nat @ X6 )
= A ) ) ) ).
% cInf_eq
thf(fact_932_wellorder__Inf__le1,axiom,
! [K: nat,A2: set_nat] :
( ( member_nat @ K @ A2 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A2 ) @ K ) ) ).
% wellorder_Inf_le1
thf(fact_933_Inf__sup__eq__top__iff,axiom,
! [B: set_se4904748513628223167_a_nat,A: set_Sum_sum_a_nat] :
( ( ( sup_su6804446743777130803_a_nat @ ( comple1528121977673479270_a_nat @ B ) @ A )
= top_to795618464972521135_a_nat )
= ( ! [X: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X @ B )
=> ( ( sup_su6804446743777130803_a_nat @ X @ A )
= top_to795618464972521135_a_nat ) ) ) ) ).
% Inf_sup_eq_top_iff
thf(fact_934_Inf__sup__eq__top__iff,axiom,
! [B: set_set_nat,A: set_nat] :
( ( ( sup_sup_set_nat @ ( comple7806235888213564991et_nat @ B ) @ A )
= top_top_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ B )
=> ( ( sup_sup_set_nat @ X @ A )
= top_top_set_nat ) ) ) ) ).
% Inf_sup_eq_top_iff
thf(fact_935_Inf__sup__eq__top__iff,axiom,
! [B: set_o,A: $o] :
( ( ( sup_sup_o @ ( complete_Inf_Inf_o @ B ) @ A )
= top_top_o )
= ( ! [X: $o] :
( ( member_o @ X @ B )
=> ( ( sup_sup_o @ X @ A )
= top_top_o ) ) ) ) ).
% Inf_sup_eq_top_iff
thf(fact_936_sup__Inf,axiom,
! [A: set_Sum_sum_a_nat,B: set_se4904748513628223167_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A @ ( comple1528121977673479270_a_nat @ B ) )
= ( comple1528121977673479270_a_nat @ ( image_5599399343138760645_a_nat @ ( sup_su6804446743777130803_a_nat @ A ) @ B ) ) ) ).
% sup_Inf
thf(fact_937_sup__Inf,axiom,
! [A: $o,B: set_o] :
( ( sup_sup_o @ A @ ( complete_Inf_Inf_o @ B ) )
= ( complete_Inf_Inf_o @ ( image_o_o @ ( sup_sup_o @ A ) @ B ) ) ) ).
% sup_Inf
thf(fact_938_INF__sup__distrib2,axiom,
! [F: nat > set_nat,A2: set_nat,G: nat > set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ B ) ) )
= ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [A5: nat] :
( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [B5: nat] : ( sup_sup_set_nat @ ( F @ A5 ) @ ( G @ B5 ) )
@ B ) )
@ A2 ) ) ) ).
% INF_sup_distrib2
thf(fact_939_UN__constant__eq,axiom,
! [A: nat,A2: set_nat,F: nat > set_nat,C: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( F @ X2 )
= C ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) )
= C ) ) ) ).
% UN_constant_eq
thf(fact_940_UN__constant__eq,axiom,
! [A: $o,A2: set_o,F: $o > set_nat,C: set_nat] :
( ( member_o @ A @ A2 )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ( F @ X2 )
= C ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) )
= C ) ) ) ).
% UN_constant_eq
thf(fact_941_UN__atLeast__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atLeast_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_atLeast_UNIV
thf(fact_942_INT__constant,axiom,
! [A2: set_nat,C: set_nat] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [Y3: nat] : C
@ A2 ) )
= top_top_set_nat ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [Y3: nat] : C
@ A2 ) )
= C ) ) ) ).
% INT_constant
thf(fact_943_UN__simps_I2_J,axiom,
! [C2: set_nat,A2: nat > set_nat,B: set_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( sup_sup_set_nat @ ( A2 @ X ) @ B )
@ C2 ) )
= bot_bot_set_nat ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( sup_sup_set_nat @ ( A2 @ X ) @ B )
@ C2 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ C2 ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_944_empty__iff,axiom,
! [C: $o] :
~ ( member_o @ C @ bot_bot_set_o ) ).
% empty_iff
thf(fact_945_all__not__in__conv,axiom,
! [A2: set_o] :
( ( ! [X: $o] :
~ ( member_o @ X @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_946_atLeast__eq__iff,axiom,
! [X3: nat,Y: nat] :
( ( ( set_ord_atLeast_nat @ X3 )
= ( set_ord_atLeast_nat @ Y ) )
= ( X3 = Y ) ) ).
% atLeast_eq_iff
thf(fact_947_image__is__empty,axiom,
! [F: nat > sum_sum_a_nat,A2: set_nat] :
( ( ( image_7293268710728258664_a_nat @ F @ A2 )
= bot_bo3438331934148233675_a_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_948_image__is__empty,axiom,
! [F: a > sum_sum_a_nat,A2: set_a] :
( ( ( image_7873763678140191238_a_nat @ F @ A2 )
= bot_bo3438331934148233675_a_nat )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_949_image__is__empty,axiom,
! [F: nat > set_nat,A2: set_nat] :
( ( ( image_nat_set_nat @ F @ A2 )
= bot_bot_set_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_950_empty__is__image,axiom,
! [F: nat > sum_sum_a_nat,A2: set_nat] :
( ( bot_bo3438331934148233675_a_nat
= ( image_7293268710728258664_a_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_951_empty__is__image,axiom,
! [F: a > sum_sum_a_nat,A2: set_a] :
( ( bot_bo3438331934148233675_a_nat
= ( image_7873763678140191238_a_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_952_empty__is__image,axiom,
! [F: nat > set_nat,A2: set_nat] :
( ( bot_bot_set_set_nat
= ( image_nat_set_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_953_image__empty,axiom,
! [F: nat > sum_sum_a_nat] :
( ( image_7293268710728258664_a_nat @ F @ bot_bot_set_nat )
= bot_bo3438331934148233675_a_nat ) ).
% image_empty
thf(fact_954_image__empty,axiom,
! [F: a > sum_sum_a_nat] :
( ( image_7873763678140191238_a_nat @ F @ bot_bot_set_a )
= bot_bo3438331934148233675_a_nat ) ).
% image_empty
thf(fact_955_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_956_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_957_empty__subsetI,axiom,
! [A2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ bot_bo3438331934148233675_a_nat @ A2 ) ).
% empty_subsetI
thf(fact_958_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_959_subset__empty,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ bot_bo3438331934148233675_a_nat )
= ( A2 = bot_bo3438331934148233675_a_nat ) ) ).
% subset_empty
thf(fact_960_sup__bot_Oright__neutral,axiom,
! [A: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A @ bot_bo3438331934148233675_a_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_961_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( bot_bo3438331934148233675_a_nat
= ( sup_su6804446743777130803_a_nat @ A @ B2 ) )
= ( ( A = bot_bo3438331934148233675_a_nat )
& ( B2 = bot_bo3438331934148233675_a_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_962_sup__bot_Oleft__neutral,axiom,
! [A: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ bot_bo3438331934148233675_a_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_963_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ( sup_su6804446743777130803_a_nat @ A @ B2 )
= bot_bo3438331934148233675_a_nat )
= ( ( A = bot_bo3438331934148233675_a_nat )
& ( B2 = bot_bo3438331934148233675_a_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_964_sup__eq__bot__iff,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( ( sup_su6804446743777130803_a_nat @ X3 @ Y )
= bot_bo3438331934148233675_a_nat )
= ( ( X3 = bot_bo3438331934148233675_a_nat )
& ( Y = bot_bo3438331934148233675_a_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_965_bot__eq__sup__iff,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( bot_bo3438331934148233675_a_nat
= ( sup_su6804446743777130803_a_nat @ X3 @ Y ) )
= ( ( X3 = bot_bo3438331934148233675_a_nat )
& ( Y = bot_bo3438331934148233675_a_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_966_sup__bot__right,axiom,
! [X3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X3 @ bot_bo3438331934148233675_a_nat )
= X3 ) ).
% sup_bot_right
thf(fact_967_sup__bot__left,axiom,
! [X3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ bot_bo3438331934148233675_a_nat @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_968_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A2 ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( X = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_969_Sup__bot__conv_I2_J,axiom,
! [A2: set_o] :
( ( bot_bot_o
= ( complete_Sup_Sup_o @ A2 ) )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( X = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_970_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A2 )
= bot_bot_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( X = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_971_Sup__bot__conv_I1_J,axiom,
! [A2: set_o] :
( ( ( complete_Sup_Sup_o @ A2 )
= bot_bot_o )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ( X = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_972_Un__empty,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ( sup_su6804446743777130803_a_nat @ A2 @ B )
= bot_bo3438331934148233675_a_nat )
= ( ( A2 = bot_bo3438331934148233675_a_nat )
& ( B = bot_bo3438331934148233675_a_nat ) ) ) ).
% Un_empty
thf(fact_973_atLeast__iff,axiom,
! [I2: $o,K: $o] :
( ( member_o @ I2 @ ( set_ord_atLeast_o @ K ) )
= ( ord_less_eq_o @ K @ I2 ) ) ).
% atLeast_iff
thf(fact_974_atLeast__iff,axiom,
! [I2: set_a,K: set_a] :
( ( member_set_a @ I2 @ ( set_or8362275514725411625_set_a @ K ) )
= ( ord_less_eq_set_a @ K @ I2 ) ) ).
% atLeast_iff
thf(fact_975_atLeast__iff,axiom,
! [I2: set_Sum_sum_a_nat,K: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ I2 @ ( set_or1144079512921665450_a_nat @ K ) )
= ( ord_le1325389633284124927_a_nat @ K @ I2 ) ) ).
% atLeast_iff
thf(fact_976_atLeast__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat @ I2 @ ( set_ord_atLeast_nat @ K ) )
= ( ord_less_eq_nat @ K @ I2 ) ) ).
% atLeast_iff
thf(fact_977_Inf__atLeast,axiom,
! [X3: $o] :
( ( complete_Inf_Inf_o @ ( set_ord_atLeast_o @ X3 ) )
= X3 ) ).
% Inf_atLeast
thf(fact_978_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collec7073057861543223018_a_nat
@ ^ [S4: sum_sum_a_nat] : P )
= top_to795618464972521135_a_nat ) )
& ( ~ P
=> ( ( collec7073057861543223018_a_nat
@ ^ [S4: sum_sum_a_nat] : P )
= bot_bo3438331934148233675_a_nat ) ) ) ).
% Collect_const
thf(fact_979_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_nat
@ ^ [S4: nat] : P )
= top_top_set_nat ) )
& ( ~ P
=> ( ( collect_nat
@ ^ [S4: nat] : P )
= bot_bot_set_nat ) ) ) ).
% Collect_const
thf(fact_980_Sup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Sup_empty
thf(fact_981_Sup__empty,axiom,
( ( complete_Sup_Sup_o @ bot_bot_set_o )
= bot_bot_o ) ).
% Sup_empty
thf(fact_982_Inf__UNIV,axiom,
( ( complete_Inf_Inf_o @ top_top_set_o )
= bot_bot_o ) ).
% Inf_UNIV
thf(fact_983_Inf__empty,axiom,
( ( comple1528121977673479270_a_nat @ bot_bo2635121477170169643_a_nat )
= top_to795618464972521135_a_nat ) ).
% Inf_empty
thf(fact_984_Inf__empty,axiom,
( ( comple7806235888213564991et_nat @ bot_bot_set_set_nat )
= top_top_set_nat ) ).
% Inf_empty
thf(fact_985_Inf__empty,axiom,
( ( complete_Inf_Inf_o @ bot_bot_set_o )
= top_top_o ) ).
% Inf_empty
thf(fact_986_atLeast__subset__iff,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or8362275514725411625_set_a @ X3 ) @ ( set_or8362275514725411625_set_a @ Y ) )
= ( ord_less_eq_set_a @ Y @ X3 ) ) ).
% atLeast_subset_iff
thf(fact_987_atLeast__subset__iff,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( ord_le7974500612278410847_a_nat @ ( set_or1144079512921665450_a_nat @ X3 ) @ ( set_or1144079512921665450_a_nat @ Y ) )
= ( ord_le1325389633284124927_a_nat @ Y @ X3 ) ) ).
% atLeast_subset_iff
thf(fact_988_atLeast__subset__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atLeast_nat @ X3 ) @ ( set_ord_atLeast_nat @ Y ) )
= ( ord_less_eq_nat @ Y @ X3 ) ) ).
% atLeast_subset_iff
thf(fact_989_Sup__atLeast,axiom,
! [X3: set_Sum_sum_a_nat] :
( ( comple1247738100258233164_a_nat @ ( set_or1144079512921665450_a_nat @ X3 ) )
= top_to795618464972521135_a_nat ) ).
% Sup_atLeast
thf(fact_990_Sup__atLeast,axiom,
! [X3: set_nat] :
( ( comple7399068483239264473et_nat @ ( set_or1731685050470061051et_nat @ X3 ) )
= top_top_set_nat ) ).
% Sup_atLeast
thf(fact_991_Sup__atLeast,axiom,
! [X3: $o] :
( ( complete_Sup_Sup_o @ ( set_ord_atLeast_o @ X3 ) )
= top_top_o ) ).
% Sup_atLeast
thf(fact_992_SUP__bot,axiom,
! [A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : bot_bot_set_nat
@ A2 ) )
= bot_bot_set_nat ) ).
% SUP_bot
thf(fact_993_SUP__bot__conv_I1_J,axiom,
! [B: nat > set_nat,A2: set_nat] :
( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A2 ) )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( B @ X )
= bot_bot_set_nat ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_994_SUP__bot__conv_I2_J,axiom,
! [B: nat > set_nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A2 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( B @ X )
= bot_bot_set_nat ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_995_SUP__const,axiom,
! [A2: set_nat,F: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I3: nat] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_996_cSUP__const,axiom,
! [A2: set_nat,C: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : C
@ A2 ) )
= C ) ) ).
% cSUP_const
thf(fact_997_INF__const,axiom,
! [A2: set_nat,F: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [I3: nat] : F
@ A2 ) )
= F ) ) ).
% INF_const
thf(fact_998_cINF__const,axiom,
! [A2: set_nat,C: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : C
@ A2 ) )
= C ) ) ).
% cINF_const
thf(fact_999_UN__constant,axiom,
! [A2: set_nat,C: set_nat] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y3: nat] : C
@ A2 ) )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y3: nat] : C
@ A2 ) )
= C ) ) ) ).
% UN_constant
thf(fact_1000_UN__simps_I3_J,axiom,
! [C2: set_nat,A2: set_nat,B: nat > set_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( sup_sup_set_nat @ A2 @ ( B @ X ) )
@ C2 ) )
= bot_bot_set_nat ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( sup_sup_set_nat @ A2 @ ( B @ X ) )
@ C2 ) )
= ( sup_sup_set_nat @ A2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ C2 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1001_Union__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Union_empty
thf(fact_1002_Inf__bool__def,axiom,
( complete_Inf_Inf_o
= ( ^ [A3: set_o] :
~ ( member_o @ $false @ A3 ) ) ) ).
% Inf_bool_def
thf(fact_1003_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_1004_atLeast__eq__UNIV__iff,axiom,
! [X3: nat] :
( ( ( set_ord_atLeast_nat @ X3 )
= top_top_set_nat )
= ( X3 = bot_bot_nat ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_1005_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_1006_bot_Oextremum,axiom,
! [A: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ bot_bo3438331934148233675_a_nat @ A ) ).
% bot.extremum
thf(fact_1007_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_1008_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_1009_bot_Oextremum__unique,axiom,
! [A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ bot_bo3438331934148233675_a_nat )
= ( A = bot_bo3438331934148233675_a_nat ) ) ).
% bot.extremum_unique
thf(fact_1010_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_1011_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_1012_bot_Oextremum__uniqueI,axiom,
! [A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ bot_bo3438331934148233675_a_nat )
=> ( A = bot_bo3438331934148233675_a_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1013_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1014_boolean__algebra_Odisj__zero__right,axiom,
! [X3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X3 @ bot_bo3438331934148233675_a_nat )
= X3 ) ).
% boolean_algebra.disj_zero_right
thf(fact_1015_SUP__constant,axiom,
! [A2: set_nat,C: set_nat] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y3: nat] : C
@ A2 ) )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y3: nat] : C
@ A2 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_1016_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_1017_empty__not__UNIV,axiom,
bot_bo3438331934148233675_a_nat != top_to795618464972521135_a_nat ).
% empty_not_UNIV
thf(fact_1018_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_1019_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_1020_not__empty__eq__Ici__eq__empty,axiom,
! [L: nat] :
( bot_bot_set_nat
!= ( set_ord_atLeast_nat @ L ) ) ).
% not_empty_eq_Ici_eq_empty
thf(fact_1021_emptyE,axiom,
! [A: $o] :
~ ( member_o @ A @ bot_bot_set_o ) ).
% emptyE
thf(fact_1022_equals0D,axiom,
! [A2: set_o,A: $o] :
( ( A2 = bot_bot_set_o )
=> ~ ( member_o @ A @ A2 ) ) ).
% equals0D
thf(fact_1023_equals0I,axiom,
! [A2: set_o] :
( ! [Y5: $o] :
~ ( member_o @ Y5 @ A2 )
=> ( A2 = bot_bot_set_o ) ) ).
% equals0I
thf(fact_1024_ex__in__conv,axiom,
! [A2: set_o] :
( ( ? [X: $o] : ( member_o @ X @ A2 ) )
= ( A2 != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_1025_subset__emptyI,axiom,
! [A2: set_o] :
( ! [X2: $o] :
~ ( member_o @ X2 @ A2 )
=> ( ord_less_eq_set_o @ A2 @ bot_bot_set_o ) ) ).
% subset_emptyI
thf(fact_1026_subset__emptyI,axiom,
! [A2: set_a] :
( ! [X2: a] :
~ ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ A2 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_1027_subset__emptyI,axiom,
! [A2: set_Sum_sum_a_nat] :
( ! [X2: sum_sum_a_nat] :
~ ( member_Sum_sum_a_nat @ X2 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ A2 @ bot_bo3438331934148233675_a_nat ) ) ).
% subset_emptyI
thf(fact_1028_Un__empty__left,axiom,
! [B: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ bot_bo3438331934148233675_a_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_1029_Un__empty__right,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A2 @ bot_bo3438331934148233675_a_nat )
= A2 ) ).
% Un_empty_right
thf(fact_1030_empty__Union__conv,axiom,
! [A2: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A2 ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( X = bot_bot_set_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_1031_Union__empty__conv,axiom,
! [A2: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A2 )
= bot_bot_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( X = bot_bot_set_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_1032_atLeast__def,axiom,
( set_or8362275514725411625_set_a
= ( ^ [L2: set_a] : ( collect_set_a @ ( ord_less_eq_set_a @ L2 ) ) ) ) ).
% atLeast_def
thf(fact_1033_atLeast__def,axiom,
( set_or1144079512921665450_a_nat
= ( ^ [L2: set_Sum_sum_a_nat] : ( collec4049389696321283146_a_nat @ ( ord_le1325389633284124927_a_nat @ L2 ) ) ) ) ).
% atLeast_def
thf(fact_1034_atLeast__def,axiom,
( set_ord_atLeast_nat
= ( ^ [L2: nat] : ( collect_nat @ ( ord_less_eq_nat @ L2 ) ) ) ) ).
% atLeast_def
thf(fact_1035_less__eq__Sup,axiom,
! [A2: set_set_a,U: set_a] :
( ! [V2: set_a] :
( ( member_set_a @ V2 @ A2 )
=> ( ord_less_eq_set_a @ U @ V2 ) )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_1036_less__eq__Sup,axiom,
! [A2: set_se4904748513628223167_a_nat,U: set_Sum_sum_a_nat] :
( ! [V2: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ V2 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ U @ V2 ) )
=> ( ( A2 != bot_bo2635121477170169643_a_nat )
=> ( ord_le1325389633284124927_a_nat @ U @ ( comple1247738100258233164_a_nat @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_1037_less__eq__Sup,axiom,
! [A2: set_set_nat,U: set_nat] :
( ! [V2: set_nat] :
( ( member_set_nat @ V2 @ A2 )
=> ( ord_less_eq_set_nat @ U @ V2 ) )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_1038_less__eq__Sup,axiom,
! [A2: set_o,U: $o] :
( ! [V2: $o] :
( ( member_o @ V2 @ A2 )
=> ( ord_less_eq_o @ U @ V2 ) )
=> ( ( A2 != bot_bot_set_o )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_1039_cSup__eq__non__empty,axiom,
! [X6: set_set_a,A: set_a] :
( ( X6 != bot_bot_set_set_a )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X6 )
=> ( ord_less_eq_set_a @ X2 @ A ) )
=> ( ! [Y5: set_a] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ X6 )
=> ( ord_less_eq_set_a @ X5 @ Y5 ) )
=> ( ord_less_eq_set_a @ A @ Y5 ) )
=> ( ( comple2307003609928055243_set_a @ X6 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1040_cSup__eq__non__empty,axiom,
! [X6: set_se4904748513628223167_a_nat,A: set_Sum_sum_a_nat] :
( ( X6 != bot_bo2635121477170169643_a_nat )
=> ( ! [X2: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X2 @ X6 )
=> ( ord_le1325389633284124927_a_nat @ X2 @ A ) )
=> ( ! [Y5: set_Sum_sum_a_nat] :
( ! [X5: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X5 @ X6 )
=> ( ord_le1325389633284124927_a_nat @ X5 @ Y5 ) )
=> ( ord_le1325389633284124927_a_nat @ A @ Y5 ) )
=> ( ( comple1247738100258233164_a_nat @ X6 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1041_cSup__eq__non__empty,axiom,
! [X6: set_nat,A: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X6 )
=> ( ord_less_eq_nat @ X2 @ A ) )
=> ( ! [Y5: nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ X6 )
=> ( ord_less_eq_nat @ X5 @ Y5 ) )
=> ( ord_less_eq_nat @ A @ Y5 ) )
=> ( ( complete_Sup_Sup_nat @ X6 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1042_cSup__eq__non__empty,axiom,
! [X6: set_set_nat,A: set_nat] :
( ( X6 != bot_bot_set_set_nat )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ X6 )
=> ( ord_less_eq_set_nat @ X2 @ A ) )
=> ( ! [Y5: set_nat] :
( ! [X5: set_nat] :
( ( member_set_nat @ X5 @ X6 )
=> ( ord_less_eq_set_nat @ X5 @ Y5 ) )
=> ( ord_less_eq_set_nat @ A @ Y5 ) )
=> ( ( comple7399068483239264473et_nat @ X6 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1043_cSup__eq__non__empty,axiom,
! [X6: set_o,A: $o] :
( ( X6 != bot_bot_set_o )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ X6 )
=> ( ord_less_eq_o @ X2 @ A ) )
=> ( ! [Y5: $o] :
( ! [X5: $o] :
( ( member_o @ X5 @ X6 )
=> ( ord_less_eq_o @ X5 @ Y5 ) )
=> ( ord_less_eq_o @ A @ Y5 ) )
=> ( ( complete_Sup_Sup_o @ X6 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1044_cSup__least,axiom,
! [X6: set_set_a,Z: set_a] :
( ( X6 != bot_bot_set_set_a )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X6 )
=> ( ord_less_eq_set_a @ X2 @ Z ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ X6 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1045_cSup__least,axiom,
! [X6: set_se4904748513628223167_a_nat,Z: set_Sum_sum_a_nat] :
( ( X6 != bot_bo2635121477170169643_a_nat )
=> ( ! [X2: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X2 @ X6 )
=> ( ord_le1325389633284124927_a_nat @ X2 @ Z ) )
=> ( ord_le1325389633284124927_a_nat @ ( comple1247738100258233164_a_nat @ X6 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1046_cSup__least,axiom,
! [X6: set_nat,Z: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X6 )
=> ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X6 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1047_cSup__least,axiom,
! [X6: set_set_nat,Z: set_nat] :
( ( X6 != bot_bot_set_set_nat )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ X6 )
=> ( ord_less_eq_set_nat @ X2 @ Z ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ X6 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1048_cSup__least,axiom,
! [X6: set_o,Z: $o] :
( ( X6 != bot_bot_set_o )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ X6 )
=> ( ord_less_eq_o @ X2 @ Z ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ X6 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1049_Inf__less__eq,axiom,
! [A2: set_set_a,U: set_a] :
( ! [V2: set_a] :
( ( member_set_a @ V2 @ A2 )
=> ( ord_less_eq_set_a @ V2 @ U ) )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A2 ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_1050_Inf__less__eq,axiom,
! [A2: set_se4904748513628223167_a_nat,U: set_Sum_sum_a_nat] :
( ! [V2: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ V2 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ V2 @ U ) )
=> ( ( A2 != bot_bo2635121477170169643_a_nat )
=> ( ord_le1325389633284124927_a_nat @ ( comple1528121977673479270_a_nat @ A2 ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_1051_Inf__less__eq,axiom,
! [A2: set_o,U: $o] :
( ! [V2: $o] :
( ( member_o @ V2 @ A2 )
=> ( ord_less_eq_o @ V2 @ U ) )
=> ( ( A2 != bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ U ) ) ) ).
% Inf_less_eq
thf(fact_1052_cInf__eq__non__empty,axiom,
! [X6: set_set_a,A: set_a] :
( ( X6 != bot_bot_set_set_a )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X6 )
=> ( ord_less_eq_set_a @ A @ X2 ) )
=> ( ! [Y5: set_a] :
( ! [X5: set_a] :
( ( member_set_a @ X5 @ X6 )
=> ( ord_less_eq_set_a @ Y5 @ X5 ) )
=> ( ord_less_eq_set_a @ Y5 @ A ) )
=> ( ( comple6135023378680113637_set_a @ X6 )
= A ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_1053_cInf__eq__non__empty,axiom,
! [X6: set_se4904748513628223167_a_nat,A: set_Sum_sum_a_nat] :
( ( X6 != bot_bo2635121477170169643_a_nat )
=> ( ! [X2: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X2 @ X6 )
=> ( ord_le1325389633284124927_a_nat @ A @ X2 ) )
=> ( ! [Y5: set_Sum_sum_a_nat] :
( ! [X5: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X5 @ X6 )
=> ( ord_le1325389633284124927_a_nat @ Y5 @ X5 ) )
=> ( ord_le1325389633284124927_a_nat @ Y5 @ A ) )
=> ( ( comple1528121977673479270_a_nat @ X6 )
= A ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_1054_cInf__eq__non__empty,axiom,
! [X6: set_nat,A: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X6 )
=> ( ord_less_eq_nat @ A @ X2 ) )
=> ( ! [Y5: nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ X6 )
=> ( ord_less_eq_nat @ Y5 @ X5 ) )
=> ( ord_less_eq_nat @ Y5 @ A ) )
=> ( ( complete_Inf_Inf_nat @ X6 )
= A ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_1055_cInf__eq__non__empty,axiom,
! [X6: set_o,A: $o] :
( ( X6 != bot_bot_set_o )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ X6 )
=> ( ord_less_eq_o @ A @ X2 ) )
=> ( ! [Y5: $o] :
( ! [X5: $o] :
( ( member_o @ X5 @ X6 )
=> ( ord_less_eq_o @ Y5 @ X5 ) )
=> ( ord_less_eq_o @ Y5 @ A ) )
=> ( ( complete_Inf_Inf_o @ X6 )
= A ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_1056_cInf__greatest,axiom,
! [X6: set_set_a,Z: set_a] :
( ( X6 != bot_bot_set_set_a )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X6 )
=> ( ord_less_eq_set_a @ Z @ X2 ) )
=> ( ord_less_eq_set_a @ Z @ ( comple6135023378680113637_set_a @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_1057_cInf__greatest,axiom,
! [X6: set_se4904748513628223167_a_nat,Z: set_Sum_sum_a_nat] :
( ( X6 != bot_bo2635121477170169643_a_nat )
=> ( ! [X2: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X2 @ X6 )
=> ( ord_le1325389633284124927_a_nat @ Z @ X2 ) )
=> ( ord_le1325389633284124927_a_nat @ Z @ ( comple1528121977673479270_a_nat @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_1058_cInf__greatest,axiom,
! [X6: set_nat,Z: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X6 )
=> ( ord_less_eq_nat @ Z @ X2 ) )
=> ( ord_less_eq_nat @ Z @ ( complete_Inf_Inf_nat @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_1059_cInf__greatest,axiom,
! [X6: set_o,Z: $o] :
( ( X6 != bot_bot_set_o )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ X6 )
=> ( ord_less_eq_o @ Z @ X2 ) )
=> ( ord_less_eq_o @ Z @ ( complete_Inf_Inf_o @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_1060_SUP__eq__const,axiom,
! [I: set_nat,F: nat > set_nat,X3: set_nat] :
( ( I != bot_bot_set_nat )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I )
=> ( ( F @ I4 )
= X3 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ I ) )
= X3 ) ) ) ).
% SUP_eq_const
thf(fact_1061_SUP__eq__const,axiom,
! [I: set_o,F: $o > set_nat,X3: set_nat] :
( ( I != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I )
=> ( ( F @ I4 )
= X3 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ I ) )
= X3 ) ) ) ).
% SUP_eq_const
thf(fact_1062_SUP__eq__const,axiom,
! [I: set_o,F: $o > $o,X3: $o] :
( ( I != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I )
=> ( ( F @ I4 )
= X3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ I ) )
= X3 ) ) ) ).
% SUP_eq_const
thf(fact_1063_INF__eq__const,axiom,
! [I: set_nat,F: nat > set_nat,X3: set_nat] :
( ( I != bot_bot_set_nat )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I )
=> ( ( F @ I4 )
= X3 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ I ) )
= X3 ) ) ) ).
% INF_eq_const
thf(fact_1064_INF__eq__const,axiom,
! [I: set_o,F: $o > $o,X3: $o] :
( ( I != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I )
=> ( ( F @ I4 )
= X3 ) )
=> ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ I ) )
= X3 ) ) ) ).
% INF_eq_const
thf(fact_1065_Inter__empty,axiom,
( ( comple1528121977673479270_a_nat @ bot_bo2635121477170169643_a_nat )
= top_to795618464972521135_a_nat ) ).
% Inter_empty
thf(fact_1066_Inter__empty,axiom,
( ( comple7806235888213564991et_nat @ bot_bot_set_set_nat )
= top_top_set_nat ) ).
% Inter_empty
thf(fact_1067_Inter__subset,axiom,
! [A2: set_set_a,B: set_a] :
( ! [X8: set_a] :
( ( member_set_a @ X8 @ A2 )
=> ( ord_less_eq_set_a @ X8 @ B ) )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A2 ) @ B ) ) ) ).
% Inter_subset
thf(fact_1068_Inter__subset,axiom,
! [A2: set_se4904748513628223167_a_nat,B: set_Sum_sum_a_nat] :
( ! [X8: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ X8 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ X8 @ B ) )
=> ( ( A2 != bot_bo2635121477170169643_a_nat )
=> ( ord_le1325389633284124927_a_nat @ ( comple1528121977673479270_a_nat @ A2 ) @ B ) ) ) ).
% Inter_subset
thf(fact_1069_UN__empty2,axiom,
! [A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : bot_bot_set_nat
@ A2 ) )
= bot_bot_set_nat ) ).
% UN_empty2
thf(fact_1070_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_1071_UNION__empty__conv_I1_J,axiom,
! [B: nat > set_nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A2 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( B @ X )
= bot_bot_set_nat ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_1072_UNION__empty__conv_I2_J,axiom,
! [B: nat > set_nat,A2: set_nat] :
( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A2 ) )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( B @ X )
= bot_bot_set_nat ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_1073_SUP__eq__iff,axiom,
! [I: set_o,C: set_a,F: $o > set_a] :
( ( I != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I )
=> ( ord_less_eq_set_a @ C @ ( F @ I4 ) ) )
=> ( ( ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ I ) )
= C )
= ( ! [X: $o] :
( ( member_o @ X @ I )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1074_SUP__eq__iff,axiom,
! [I: set_o,C: set_Sum_sum_a_nat,F: $o > set_Sum_sum_a_nat] :
( ( I != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I )
=> ( ord_le1325389633284124927_a_nat @ C @ ( F @ I4 ) ) )
=> ( ( ( comple1247738100258233164_a_nat @ ( image_3365592128754359116_a_nat @ F @ I ) )
= C )
= ( ! [X: $o] :
( ( member_o @ X @ I )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1075_SUP__eq__iff,axiom,
! [I: set_nat,C: set_nat,F: nat > set_nat] :
( ( I != bot_bot_set_nat )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I )
=> ( ord_less_eq_set_nat @ C @ ( F @ I4 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ I ) )
= C )
= ( ! [X: nat] :
( ( member_nat @ X @ I )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1076_SUP__eq__iff,axiom,
! [I: set_o,C: set_nat,F: $o > set_nat] :
( ( I != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I )
=> ( ord_less_eq_set_nat @ C @ ( F @ I4 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ I ) )
= C )
= ( ! [X: $o] :
( ( member_o @ X @ I )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1077_SUP__eq__iff,axiom,
! [I: set_o,C: $o,F: $o > $o] :
( ( I != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I )
=> ( ord_less_eq_o @ C @ ( F @ I4 ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ I ) )
= C )
= ( ! [X: $o] :
( ( member_o @ X @ I )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1078_cSUP__least,axiom,
! [A2: set_o,F: $o > set_a,M2: set_a] :
( ( A2 != bot_bot_set_o )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ M2 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ A2 ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_1079_cSUP__least,axiom,
! [A2: set_o,F: $o > set_Sum_sum_a_nat,M2: set_Sum_sum_a_nat] :
( ( A2 != bot_bot_set_o )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ X2 ) @ M2 ) )
=> ( ord_le1325389633284124927_a_nat @ ( comple1247738100258233164_a_nat @ ( image_3365592128754359116_a_nat @ F @ A2 ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_1080_cSUP__least,axiom,
! [A2: set_o,F: $o > nat,M2: nat] :
( ( A2 != bot_bot_set_o )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ M2 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ A2 ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_1081_cSUP__least,axiom,
! [A2: set_nat,F: nat > set_nat,M2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ M2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_1082_cSUP__least,axiom,
! [A2: set_o,F: $o > set_nat,M2: set_nat] :
( ( A2 != bot_bot_set_o )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ M2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_1083_cSUP__least,axiom,
! [A2: set_o,F: $o > $o,M2: $o] :
( ( A2 != bot_bot_set_o )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_o @ ( F @ X2 ) @ M2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A2 ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_1084_INF__eq__iff,axiom,
! [I: set_nat,F: nat > set_nat,C: set_nat] :
( ( I != bot_bot_set_nat )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ C ) )
=> ( ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ I ) )
= C )
= ( ! [X: nat] :
( ( member_nat @ X @ I )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1085_INF__eq__iff,axiom,
! [I: set_o,F: $o > set_a,C: set_a] :
( ( I != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I )
=> ( ord_less_eq_set_a @ ( F @ I4 ) @ C ) )
=> ( ( ( comple6135023378680113637_set_a @ ( image_o_set_a @ F @ I ) )
= C )
= ( ! [X: $o] :
( ( member_o @ X @ I )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1086_INF__eq__iff,axiom,
! [I: set_o,F: $o > set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( I != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I )
=> ( ord_le1325389633284124927_a_nat @ ( F @ I4 ) @ C ) )
=> ( ( ( comple1528121977673479270_a_nat @ ( image_3365592128754359116_a_nat @ F @ I ) )
= C )
= ( ! [X: $o] :
( ( member_o @ X @ I )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1087_INF__eq__iff,axiom,
! [I: set_o,F: $o > $o,C: $o] :
( ( I != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I )
=> ( ord_less_eq_o @ ( F @ I4 ) @ C ) )
=> ( ( ( complete_Inf_Inf_o @ ( image_o_o @ F @ I ) )
= C )
= ( ! [X: $o] :
( ( member_o @ X @ I )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_1088_cINF__greatest,axiom,
! [A2: set_nat,M4: set_nat,F: nat > set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ M4 @ ( F @ X2 ) ) )
=> ( ord_less_eq_set_nat @ M4 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_1089_cINF__greatest,axiom,
! [A2: set_o,M4: set_a,F: $o > set_a] :
( ( A2 != bot_bot_set_o )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_set_a @ M4 @ ( F @ X2 ) ) )
=> ( ord_less_eq_set_a @ M4 @ ( comple6135023378680113637_set_a @ ( image_o_set_a @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_1090_cINF__greatest,axiom,
! [A2: set_o,M4: set_Sum_sum_a_nat,F: $o > set_Sum_sum_a_nat] :
( ( A2 != bot_bot_set_o )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_le1325389633284124927_a_nat @ M4 @ ( F @ X2 ) ) )
=> ( ord_le1325389633284124927_a_nat @ M4 @ ( comple1528121977673479270_a_nat @ ( image_3365592128754359116_a_nat @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_1091_cINF__greatest,axiom,
! [A2: set_o,M4: nat,F: $o > nat] :
( ( A2 != bot_bot_set_o )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_nat @ M4 @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ M4 @ ( complete_Inf_Inf_nat @ ( image_o_nat @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_1092_cINF__greatest,axiom,
! [A2: set_o,M4: $o,F: $o > $o] :
( ( A2 != bot_bot_set_o )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ord_less_eq_o @ M4 @ ( F @ X2 ) ) )
=> ( ord_less_eq_o @ M4 @ ( complete_Inf_Inf_o @ ( image_o_o @ F @ A2 ) ) ) ) ) ).
% cINF_greatest
thf(fact_1093_Inf__le__Sup,axiom,
! [A2: set_set_a] :
( ( A2 != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( comple6135023378680113637_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ A2 ) ) ) ).
% Inf_le_Sup
thf(fact_1094_Inf__le__Sup,axiom,
! [A2: set_se4904748513628223167_a_nat] :
( ( A2 != bot_bo2635121477170169643_a_nat )
=> ( ord_le1325389633284124927_a_nat @ ( comple1528121977673479270_a_nat @ A2 ) @ ( comple1247738100258233164_a_nat @ A2 ) ) ) ).
% Inf_le_Sup
thf(fact_1095_Inf__le__Sup,axiom,
! [A2: set_set_nat] :
( ( A2 != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% Inf_le_Sup
thf(fact_1096_Inf__le__Sup,axiom,
! [A2: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ ( complete_Sup_Sup_o @ A2 ) ) ) ).
% Inf_le_Sup
thf(fact_1097_Union__image__empty,axiom,
! [A2: set_nat,F: nat > set_nat] :
( ( sup_sup_set_nat @ A2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ bot_bot_set_nat ) ) )
= A2 ) ).
% Union_image_empty
thf(fact_1098_INF__empty,axiom,
! [F: nat > set_nat] :
( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ bot_bot_set_nat ) )
= top_top_set_nat ) ).
% INF_empty
thf(fact_1099_INF__constant,axiom,
! [A2: set_nat,C: set_nat] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [Y3: nat] : C
@ A2 ) )
= top_top_set_nat ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [Y3: nat] : C
@ A2 ) )
= C ) ) ) ).
% INF_constant
thf(fact_1100_UN__extend__simps_I3_J,axiom,
! [C2: set_nat,A2: set_nat,B: nat > set_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( sup_sup_set_nat @ A2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ C2 ) ) )
= A2 ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( sup_sup_set_nat @ A2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ C2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( sup_sup_set_nat @ A2 @ ( B @ X ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_1101_UN__extend__simps_I2_J,axiom,
! [C2: set_nat,A2: nat > set_nat,B: set_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ C2 ) ) @ B )
= B ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ C2 ) ) @ B )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( sup_sup_set_nat @ ( A2 @ X ) @ B )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_1102_INT__empty,axiom,
! [B: nat > set_nat] :
( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B @ bot_bot_set_nat ) )
= top_top_set_nat ) ).
% INT_empty
thf(fact_1103_INF__le__SUP,axiom,
! [A2: set_nat,F: nat > set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% INF_le_SUP
thf(fact_1104_INT__simps_I4_J,axiom,
! [C2: set_nat,A2: set_nat,B: nat > set_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( minus_minus_set_nat @ A2 @ ( B @ X ) )
@ C2 ) )
= top_top_set_nat ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( minus_minus_set_nat @ A2 @ ( B @ X ) )
@ C2 ) )
= ( minus_minus_set_nat @ A2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ C2 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_1105_INT__simps_I3_J,axiom,
! [C2: set_nat,A2: nat > set_nat,B: set_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( minus_minus_set_nat @ ( A2 @ X ) @ B )
@ C2 ) )
= top_top_set_nat ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( minus_minus_set_nat @ ( A2 @ X ) @ B )
@ C2 ) )
= ( minus_minus_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A2 @ C2 ) ) @ B ) ) ) ) ).
% INT_simps(3)
thf(fact_1106_INT__simps_I1_J,axiom,
! [C2: set_nat,A2: nat > set_nat,B: set_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( inf_inf_set_nat @ ( A2 @ X ) @ B )
@ C2 ) )
= top_top_set_nat ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( inf_inf_set_nat @ ( A2 @ X ) @ B )
@ C2 ) )
= ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A2 @ C2 ) ) @ B ) ) ) ) ).
% INT_simps(1)
thf(fact_1107_INT__simps_I2_J,axiom,
! [C2: set_nat,A2: set_nat,B: nat > set_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( inf_inf_set_nat @ A2 @ ( B @ X ) )
@ C2 ) )
= top_top_set_nat ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( inf_inf_set_nat @ A2 @ ( B @ X ) )
@ C2 ) )
= ( inf_inf_set_nat @ A2 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B @ C2 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_1108_Int__iff,axiom,
! [C: $o,A2: set_o,B: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A2 @ B ) )
= ( ( member_o @ C @ A2 )
& ( member_o @ C @ B ) ) ) ).
% Int_iff
thf(fact_1109_IntI,axiom,
! [C: $o,A2: set_o,B: set_o] :
( ( member_o @ C @ A2 )
=> ( ( member_o @ C @ B )
=> ( member_o @ C @ ( inf_inf_set_o @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_1110_Diff__iff,axiom,
! [C: $o,A2: set_o,B: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A2 @ B ) )
= ( ( member_o @ C @ A2 )
& ~ ( member_o @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1111_DiffI,axiom,
! [C: $o,A2: set_o,B: set_o] :
( ( member_o @ C @ A2 )
=> ( ~ ( member_o @ C @ B )
=> ( member_o @ C @ ( minus_minus_set_o @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_1112_inf_Obounded__iff,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B2 @ C ) )
= ( ( ord_less_eq_set_a @ A @ B2 )
& ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1113_inf_Obounded__iff,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ ( inf_in7084830621192376909_a_nat @ B2 @ C ) )
= ( ( ord_le1325389633284124927_a_nat @ A @ B2 )
& ( ord_le1325389633284124927_a_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1114_inf_Obounded__iff,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B2 @ C ) )
= ( ( ord_less_eq_nat @ A @ B2 )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1115_le__inf__iff,axiom,
! [X3: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X3 @ ( inf_inf_set_a @ Y @ Z ) )
= ( ( ord_less_eq_set_a @ X3 @ Y )
& ( ord_less_eq_set_a @ X3 @ Z ) ) ) ).
% le_inf_iff
thf(fact_1116_le__inf__iff,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ Y @ Z ) )
= ( ( ord_le1325389633284124927_a_nat @ X3 @ Y )
& ( ord_le1325389633284124927_a_nat @ X3 @ Z ) ) ) ).
% le_inf_iff
thf(fact_1117_le__inf__iff,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X3 @ Y )
& ( ord_less_eq_nat @ X3 @ Z ) ) ) ).
% le_inf_iff
thf(fact_1118_inf__top_Oright__neutral,axiom,
! [A: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A @ top_to795618464972521135_a_nat )
= A ) ).
% inf_top.right_neutral
thf(fact_1119_inf__top_Oright__neutral,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ top_top_set_nat )
= A ) ).
% inf_top.right_neutral
thf(fact_1120_inf__top_Oneutr__eq__iff,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( top_to795618464972521135_a_nat
= ( inf_in7084830621192376909_a_nat @ A @ B2 ) )
= ( ( A = top_to795618464972521135_a_nat )
& ( B2 = top_to795618464972521135_a_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1121_inf__top_Oneutr__eq__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ A @ B2 ) )
= ( ( A = top_top_set_nat )
& ( B2 = top_top_set_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1122_inf__top_Oleft__neutral,axiom,
! [A: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ top_to795618464972521135_a_nat @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_1123_inf__top_Oleft__neutral,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_1124_inf__top_Oeq__neutr__iff,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ( inf_in7084830621192376909_a_nat @ A @ B2 )
= top_to795618464972521135_a_nat )
= ( ( A = top_to795618464972521135_a_nat )
& ( B2 = top_to795618464972521135_a_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1125_inf__top_Oeq__neutr__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A @ B2 )
= top_top_set_nat )
= ( ( A = top_top_set_nat )
& ( B2 = top_top_set_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1126_top__eq__inf__iff,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( top_to795618464972521135_a_nat
= ( inf_in7084830621192376909_a_nat @ X3 @ Y ) )
= ( ( X3 = top_to795618464972521135_a_nat )
& ( Y = top_to795618464972521135_a_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_1127_top__eq__inf__iff,axiom,
! [X3: set_nat,Y: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ X3 @ Y ) )
= ( ( X3 = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_1128_inf__eq__top__iff,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( ( inf_in7084830621192376909_a_nat @ X3 @ Y )
= top_to795618464972521135_a_nat )
= ( ( X3 = top_to795618464972521135_a_nat )
& ( Y = top_to795618464972521135_a_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_1129_inf__eq__top__iff,axiom,
! [X3: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X3 @ Y )
= top_top_set_nat )
= ( ( X3 = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_1130_inf__top__right,axiom,
! [X3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X3 @ top_to795618464972521135_a_nat )
= X3 ) ).
% inf_top_right
thf(fact_1131_inf__top__right,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ X3 @ top_top_set_nat )
= X3 ) ).
% inf_top_right
thf(fact_1132_inf__top__left,axiom,
! [X3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ top_to795618464972521135_a_nat @ X3 )
= X3 ) ).
% inf_top_left
thf(fact_1133_inf__top__left,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ X3 )
= X3 ) ).
% inf_top_left
thf(fact_1134_Int__UNIV,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ( inf_in7084830621192376909_a_nat @ A2 @ B )
= top_to795618464972521135_a_nat )
= ( ( A2 = top_to795618464972521135_a_nat )
& ( B = top_to795618464972521135_a_nat ) ) ) ).
% Int_UNIV
thf(fact_1135_Int__UNIV,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B )
= top_top_set_nat )
= ( ( A2 = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_1136_inf__sup__absorb,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ X3 @ Y ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_1137_sup__inf__absorb,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ X3 @ Y ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_1138_Int__subset__iff,axiom,
! [C2: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( ord_less_eq_set_a @ C2 @ A2 )
& ( ord_less_eq_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_1139_Int__subset__iff,axiom,
! [C2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C2 @ ( inf_in7084830621192376909_a_nat @ A2 @ B ) )
= ( ( ord_le1325389633284124927_a_nat @ C2 @ A2 )
& ( ord_le1325389633284124927_a_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_1140_Un__Int__eq_I1_J,axiom,
! [S: set_Sum_sum_a_nat,T2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_1141_Un__Int__eq_I2_J,axiom,
! [S: set_Sum_sum_a_nat,T2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_1142_Un__Int__eq_I3_J,axiom,
! [S: set_Sum_sum_a_nat,T2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ S @ ( sup_su6804446743777130803_a_nat @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_1143_Un__Int__eq_I4_J,axiom,
! [T2: set_Sum_sum_a_nat,S: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ T2 @ ( sup_su6804446743777130803_a_nat @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_1144_Int__Un__eq_I1_J,axiom,
! [S: set_Sum_sum_a_nat,T2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_1145_Int__Un__eq_I2_J,axiom,
! [S: set_Sum_sum_a_nat,T2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_1146_Int__Un__eq_I3_J,axiom,
! [S: set_Sum_sum_a_nat,T2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ S @ ( inf_in7084830621192376909_a_nat @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_1147_Int__Un__eq_I4_J,axiom,
! [T2: set_Sum_sum_a_nat,S: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ T2 @ ( inf_in7084830621192376909_a_nat @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_1148_Un__Diff__cancel2,axiom,
! [B: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( minus_1134630996077396038_a_nat @ B @ A2 ) @ A2 )
= ( sup_su6804446743777130803_a_nat @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_1149_Un__Diff__cancel,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A2 @ ( minus_1134630996077396038_a_nat @ B @ A2 ) )
= ( sup_su6804446743777130803_a_nat @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_1150_Diff__UNIV,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( minus_1134630996077396038_a_nat @ A2 @ top_to795618464972521135_a_nat )
= bot_bo3438331934148233675_a_nat ) ).
% Diff_UNIV
thf(fact_1151_Diff__UNIV,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Diff_UNIV
thf(fact_1152_Diff__eq__empty__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1153_Diff__eq__empty__iff,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ( minus_1134630996077396038_a_nat @ A2 @ B )
= bot_bo3438331934148233675_a_nat )
= ( ord_le1325389633284124927_a_nat @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1154_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_1155_if__image__distrib,axiom,
! [P: nat > $o,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat,S: set_nat] :
( ( image_7293268710728258664_a_nat
@ ^ [X: nat] : ( if_Sum_sum_a_nat @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ S )
= ( sup_su6804446743777130803_a_nat @ ( image_7293268710728258664_a_nat @ F @ ( inf_inf_set_nat @ S @ ( collect_nat @ P ) ) )
@ ( image_7293268710728258664_a_nat @ G
@ ( inf_inf_set_nat @ S
@ ( collect_nat
@ ^ [X: nat] :
~ ( P @ X ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1156_if__image__distrib,axiom,
! [P: a > $o,F: a > sum_sum_a_nat,G: a > sum_sum_a_nat,S: set_a] :
( ( image_7873763678140191238_a_nat
@ ^ [X: a] : ( if_Sum_sum_a_nat @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ S )
= ( sup_su6804446743777130803_a_nat @ ( image_7873763678140191238_a_nat @ F @ ( inf_inf_set_a @ S @ ( collect_a @ P ) ) )
@ ( image_7873763678140191238_a_nat @ G
@ ( inf_inf_set_a @ S
@ ( collect_a
@ ^ [X: a] :
~ ( P @ X ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1157_Union__Int__subset,axiom,
! [A2: set_set_a,B: set_set_a] : ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A2 @ B ) ) @ ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_Int_subset
thf(fact_1158_Union__Int__subset,axiom,
! [A2: set_se4904748513628223167_a_nat,B: set_se4904748513628223167_a_nat] : ( ord_le1325389633284124927_a_nat @ ( comple1247738100258233164_a_nat @ ( inf_in3446711685962385325_a_nat @ A2 @ B ) ) @ ( inf_in7084830621192376909_a_nat @ ( comple1247738100258233164_a_nat @ A2 ) @ ( comple1247738100258233164_a_nat @ B ) ) ) ).
% Union_Int_subset
thf(fact_1159_Union__Int__subset,axiom,
! [A2: set_set_nat,B: set_set_nat] : ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( inf_inf_set_set_nat @ A2 @ B ) ) @ ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_Int_subset
thf(fact_1160_Sup__inter__less__eq,axiom,
! [A2: set_set_a,B: set_set_a] : ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A2 @ B ) ) @ ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_1161_Sup__inter__less__eq,axiom,
! [A2: set_se4904748513628223167_a_nat,B: set_se4904748513628223167_a_nat] : ( ord_le1325389633284124927_a_nat @ ( comple1247738100258233164_a_nat @ ( inf_in3446711685962385325_a_nat @ A2 @ B ) ) @ ( inf_in7084830621192376909_a_nat @ ( comple1247738100258233164_a_nat @ A2 ) @ ( comple1247738100258233164_a_nat @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_1162_Sup__inter__less__eq,axiom,
! [A2: set_set_nat,B: set_set_nat] : ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( inf_inf_set_set_nat @ A2 @ B ) ) @ ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_1163_Sup__inter__less__eq,axiom,
! [A2: set_o,B: set_o] : ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( inf_inf_set_o @ A2 @ B ) ) @ ( inf_inf_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_1164_Int__emptyI,axiom,
! [A2: set_o,B: set_o] :
( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ~ ( member_o @ X2 @ B ) )
=> ( ( inf_inf_set_o @ A2 @ B )
= bot_bot_set_o ) ) ).
% Int_emptyI
thf(fact_1165_disjoint__iff,axiom,
! [A2: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A2 @ B )
= bot_bot_set_o )
= ( ! [X: $o] :
( ( member_o @ X @ A2 )
=> ~ ( member_o @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_1166_bot__empty__eq,axiom,
( bot_bot_o_o
= ( ^ [X: $o] : ( member_o @ X @ bot_bot_set_o ) ) ) ).
% bot_empty_eq
thf(fact_1167_Un__Diff__Int,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( minus_1134630996077396038_a_nat @ A2 @ B ) @ ( inf_in7084830621192376909_a_nat @ A2 @ B ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_1168_Int__Diff__Un,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ A2 @ B ) @ ( minus_1134630996077396038_a_nat @ A2 @ B ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_1169_Diff__Int,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( minus_1134630996077396038_a_nat @ A2 @ ( inf_in7084830621192376909_a_nat @ B @ C2 ) )
= ( sup_su6804446743777130803_a_nat @ ( minus_1134630996077396038_a_nat @ A2 @ B ) @ ( minus_1134630996077396038_a_nat @ A2 @ C2 ) ) ) ).
% Diff_Int
thf(fact_1170_Diff__Un,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( minus_1134630996077396038_a_nat @ A2 @ ( sup_su6804446743777130803_a_nat @ B @ C2 ) )
= ( inf_in7084830621192376909_a_nat @ ( minus_1134630996077396038_a_nat @ A2 @ B ) @ ( minus_1134630996077396038_a_nat @ A2 @ C2 ) ) ) ).
% Diff_Un
thf(fact_1171_DiffD2,axiom,
! [C: $o,A2: set_o,B: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A2 @ B ) )
=> ~ ( member_o @ C @ B ) ) ).
% DiffD2
thf(fact_1172_DiffD1,axiom,
! [C: $o,A2: set_o,B: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A2 @ B ) )
=> ( member_o @ C @ A2 ) ) ).
% DiffD1
thf(fact_1173_IntD2,axiom,
! [C: $o,A2: set_o,B: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A2 @ B ) )
=> ( member_o @ C @ B ) ) ).
% IntD2
thf(fact_1174_IntD1,axiom,
! [C: $o,A2: set_o,B: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A2 @ B ) )
=> ( member_o @ C @ A2 ) ) ).
% IntD1
thf(fact_1175_DiffE,axiom,
! [C: $o,A2: set_o,B: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A2 @ B ) )
=> ~ ( ( member_o @ C @ A2 )
=> ( member_o @ C @ B ) ) ) ).
% DiffE
thf(fact_1176_IntE,axiom,
! [C: $o,A2: set_o,B: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A2 @ B ) )
=> ~ ( ( member_o @ C @ A2 )
=> ~ ( member_o @ C @ B ) ) ) ).
% IntE
thf(fact_1177_Un__Diff,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( minus_1134630996077396038_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) @ C2 )
= ( sup_su6804446743777130803_a_nat @ ( minus_1134630996077396038_a_nat @ A2 @ C2 ) @ ( minus_1134630996077396038_a_nat @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_1178_Un__Int__crazy,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ A2 @ B ) @ ( inf_in7084830621192376909_a_nat @ B @ C2 ) ) @ ( inf_in7084830621192376909_a_nat @ C2 @ A2 ) )
= ( inf_in7084830621192376909_a_nat @ ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) @ ( sup_su6804446743777130803_a_nat @ B @ C2 ) ) @ ( sup_su6804446743777130803_a_nat @ C2 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_1179_Int__Un__distrib,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A2 @ ( sup_su6804446743777130803_a_nat @ B @ C2 ) )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ A2 @ B ) @ ( inf_in7084830621192376909_a_nat @ A2 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1180_Un__Int__distrib,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A2 @ ( inf_in7084830621192376909_a_nat @ B @ C2 ) )
= ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) @ ( sup_su6804446743777130803_a_nat @ A2 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1181_Int__Un__distrib2,axiom,
! [B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ B @ C2 ) @ A2 )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ B @ A2 ) @ ( inf_in7084830621192376909_a_nat @ C2 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_1182_Un__Int__distrib2,axiom,
! [B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ B @ C2 ) @ A2 )
= ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ B @ A2 ) @ ( sup_su6804446743777130803_a_nat @ C2 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_1183_double__diff,axiom,
! [A2: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_1184_double__diff,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B )
=> ( ( ord_le1325389633284124927_a_nat @ B @ C2 )
=> ( ( minus_1134630996077396038_a_nat @ B @ ( minus_1134630996077396038_a_nat @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_1185_Diff__subset,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_1186_Diff__subset,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( minus_1134630996077396038_a_nat @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_1187_Diff__mono,axiom,
! [A2: set_a,C2: set_a,D: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ D @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( minus_minus_set_a @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_1188_Diff__mono,axiom,
! [A2: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat,D: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ C2 )
=> ( ( ord_le1325389633284124927_a_nat @ D @ B )
=> ( ord_le1325389633284124927_a_nat @ ( minus_1134630996077396038_a_nat @ A2 @ B ) @ ( minus_1134630996077396038_a_nat @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_1189_Int__Collect__mono,axiom,
! [A2: set_o,B: set_o,P: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A2 @ ( collect_o @ P ) ) @ ( inf_inf_set_o @ B @ ( collect_o @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1190_Int__Collect__mono,axiom,
! [A2: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1191_Int__Collect__mono,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B )
=> ( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A2 @ ( collec7073057861543223018_a_nat @ P ) ) @ ( inf_in7084830621192376909_a_nat @ B @ ( collec7073057861543223018_a_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1192_Int__greatest,axiom,
! [C2: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ B )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_1193_Int__greatest,axiom,
! [C2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C2 @ A2 )
=> ( ( ord_le1325389633284124927_a_nat @ C2 @ B )
=> ( ord_le1325389633284124927_a_nat @ C2 @ ( inf_in7084830621192376909_a_nat @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_1194_Int__absorb2,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( inf_inf_set_a @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_1195_Int__absorb2,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B )
=> ( ( inf_in7084830621192376909_a_nat @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_1196_Int__absorb1,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_1197_Int__absorb1,axiom,
! [B: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B @ A2 )
=> ( ( inf_in7084830621192376909_a_nat @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_1198_Int__lower2,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_1199_Int__lower2,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_1200_Int__lower1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_1201_Int__lower1,axiom,
! [A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_1202_Int__mono,axiom,
! [A2: set_a,C2: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_1203_Int__mono,axiom,
! [A2: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,D: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ C2 )
=> ( ( ord_le1325389633284124927_a_nat @ B @ D )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A2 @ B ) @ ( inf_in7084830621192376909_a_nat @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_1204_Int__UNIV__left,axiom,
! [B: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ top_to795618464972521135_a_nat @ B )
= B ) ).
% Int_UNIV_left
thf(fact_1205_Int__UNIV__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ B )
= B ) ).
% Int_UNIV_left
thf(fact_1206_Int__UNIV__right,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A2 @ top_to795618464972521135_a_nat )
= A2 ) ).
% Int_UNIV_right
thf(fact_1207_Int__UNIV__right,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ top_top_set_nat )
= A2 ) ).
% Int_UNIV_right
thf(fact_1208_distrib__imp1,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ! [X2: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat,Z4: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X2 @ ( sup_su6804446743777130803_a_nat @ Y5 @ Z4 ) )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ X2 @ Y5 ) @ ( inf_in7084830621192376909_a_nat @ X2 @ Z4 ) ) )
=> ( ( sup_su6804446743777130803_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ Y @ Z ) )
= ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ X3 @ Y ) @ ( sup_su6804446743777130803_a_nat @ X3 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_1209_distrib__imp2,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ! [X2: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat,Z4: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X2 @ ( inf_in7084830621192376909_a_nat @ Y5 @ Z4 ) )
= ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ X2 @ Y5 ) @ ( sup_su6804446743777130803_a_nat @ X2 @ Z4 ) ) )
=> ( ( inf_in7084830621192376909_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ Y @ Z ) )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ X3 @ Y ) @ ( inf_in7084830621192376909_a_nat @ X3 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_1210_inf__sup__distrib1,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ Y @ Z ) )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ X3 @ Y ) @ ( inf_in7084830621192376909_a_nat @ X3 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_1211_inf__sup__distrib2,axiom,
! [Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ Y @ Z ) @ X3 )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ Y @ X3 ) @ ( inf_in7084830621192376909_a_nat @ Z @ X3 ) ) ) ).
% inf_sup_distrib2
thf(fact_1212_sup__inf__distrib1,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ Y @ Z ) )
= ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ X3 @ Y ) @ ( sup_su6804446743777130803_a_nat @ X3 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_1213_sup__inf__distrib2,axiom,
! [Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ Y @ Z ) @ X3 )
= ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ Y @ X3 ) @ ( sup_su6804446743777130803_a_nat @ Z @ X3 ) ) ) ).
% sup_inf_distrib2
thf(fact_1214_boolean__algebra_Oconj__disj__distrib,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X3 @ ( sup_su6804446743777130803_a_nat @ Y @ Z ) )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ X3 @ Y ) @ ( inf_in7084830621192376909_a_nat @ X3 @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1215_boolean__algebra_Odisj__conj__distrib,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ Y @ Z ) )
= ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ X3 @ Y ) @ ( sup_su6804446743777130803_a_nat @ X3 @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_1216_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ Y @ Z ) @ X3 )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ Y @ X3 ) @ ( inf_in7084830621192376909_a_nat @ Z @ X3 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_1217_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ Y @ Z ) @ X3 )
= ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ Y @ X3 ) @ ( sup_su6804446743777130803_a_nat @ Z @ X3 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_1218_boolean__algebra_Oconj__one__right,axiom,
! [X3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X3 @ top_to795618464972521135_a_nat )
= X3 ) ).
% boolean_algebra.conj_one_right
thf(fact_1219_boolean__algebra_Oconj__one__right,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ X3 @ top_top_set_nat )
= X3 ) ).
% boolean_algebra.conj_one_right
thf(fact_1220_inf__sup__ord_I2_J,axiom,
! [X3: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X3 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1221_inf__sup__ord_I2_J,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ X3 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1222_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_1223_inf__sup__ord_I1_J,axiom,
! [X3: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X3 @ Y ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_1224_inf__sup__ord_I1_J,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ X3 @ Y ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_1225_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_1226_inf__le1,axiom,
! [X3: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X3 @ Y ) @ X3 ) ).
% inf_le1
thf(fact_1227_inf__le1,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ X3 @ Y ) @ X3 ) ).
% inf_le1
thf(fact_1228_inf__le1,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ X3 ) ).
% inf_le1
thf(fact_1229_inf__le2,axiom,
! [X3: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_1230_inf__le2,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_1231_inf__le2,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ Y ) ).
% inf_le2
thf(fact_1232_le__infE,axiom,
! [X3: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X3 @ ( inf_inf_set_a @ A @ B2 ) )
=> ~ ( ( ord_less_eq_set_a @ X3 @ A )
=> ~ ( ord_less_eq_set_a @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_1233_le__infE,axiom,
! [X3: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ A @ B2 ) )
=> ~ ( ( ord_le1325389633284124927_a_nat @ X3 @ A )
=> ~ ( ord_le1325389633284124927_a_nat @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_1234_le__infE,axiom,
! [X3: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X3 @ A )
=> ~ ( ord_less_eq_nat @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_1235_le__infI,axiom,
! [X3: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X3 @ A )
=> ( ( ord_less_eq_set_a @ X3 @ B2 )
=> ( ord_less_eq_set_a @ X3 @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_1236_le__infI,axiom,
! [X3: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ A )
=> ( ( ord_le1325389633284124927_a_nat @ X3 @ B2 )
=> ( ord_le1325389633284124927_a_nat @ X3 @ ( inf_in7084830621192376909_a_nat @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_1237_le__infI,axiom,
! [X3: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ X3 @ A )
=> ( ( ord_less_eq_nat @ X3 @ B2 )
=> ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_1238_inf__mono,axiom,
! [A: set_a,C: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1239_inf__mono,axiom,
! [A: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,D2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ C )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ D2 )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B2 ) @ ( inf_in7084830621192376909_a_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1240_inf__mono,axiom,
! [A: nat,C: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1241_le__infI1,axiom,
! [A: set_a,X3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ X3 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_1242_le__infI1,axiom,
! [A: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ X3 )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_1243_le__infI1,axiom,
! [A: nat,X3: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ X3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_1244_le__infI2,axiom,
! [B2: set_a,X3: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ X3 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_1245_le__infI2,axiom,
! [B2: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ X3 )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_1246_le__infI2,axiom,
! [B2: nat,X3: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ X3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_1247_inf_OorderE,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( A
= ( inf_inf_set_a @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_1248_inf_OorderE,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( A
= ( inf_in7084830621192376909_a_nat @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_1249_inf_OorderE,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( A
= ( inf_inf_nat @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_1250_inf_OorderI,axiom,
! [A: set_a,B2: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B2 ) )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% inf.orderI
thf(fact_1251_inf_OorderI,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( A
= ( inf_in7084830621192376909_a_nat @ A @ B2 ) )
=> ( ord_le1325389633284124927_a_nat @ A @ B2 ) ) ).
% inf.orderI
thf(fact_1252_inf_OorderI,axiom,
! [A: nat,B2: nat] :
( ( A
= ( inf_inf_nat @ A @ B2 ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% inf.orderI
thf(fact_1253_inf__unique,axiom,
! [F: set_a > set_a > set_a,X3: set_a,Y: set_a] :
( ! [X2: set_a,Y5: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y5 ) @ X2 )
=> ( ! [X2: set_a,Y5: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y5 ) @ Y5 )
=> ( ! [X2: set_a,Y5: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y5 )
=> ( ( ord_less_eq_set_a @ X2 @ Z4 )
=> ( ord_less_eq_set_a @ X2 @ ( F @ Y5 @ Z4 ) ) ) )
=> ( ( inf_inf_set_a @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1254_inf__unique,axiom,
! [F: set_Sum_sum_a_nat > set_Sum_sum_a_nat > set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ! [X2: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( F @ X2 @ Y5 ) @ X2 )
=> ( ! [X2: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( F @ X2 @ Y5 ) @ Y5 )
=> ( ! [X2: set_Sum_sum_a_nat,Y5: set_Sum_sum_a_nat,Z4: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y5 )
=> ( ( ord_le1325389633284124927_a_nat @ X2 @ Z4 )
=> ( ord_le1325389633284124927_a_nat @ X2 @ ( F @ Y5 @ Z4 ) ) ) )
=> ( ( inf_in7084830621192376909_a_nat @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1255_inf__unique,axiom,
! [F: nat > nat > nat,X3: nat,Y: nat] :
( ! [X2: nat,Y5: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y5 ) @ X2 )
=> ( ! [X2: nat,Y5: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y5 ) @ Y5 )
=> ( ! [X2: nat,Y5: nat,Z4: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
=> ( ( ord_less_eq_nat @ X2 @ Z4 )
=> ( ord_less_eq_nat @ X2 @ ( F @ Y5 @ Z4 ) ) ) )
=> ( ( inf_inf_nat @ X3 @ Y )
= ( F @ X3 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1256_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X @ Y3 )
= X ) ) ) ).
% le_iff_inf
thf(fact_1257_le__iff__inf,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [X: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X @ Y3 )
= X ) ) ) ).
% le_iff_inf
thf(fact_1258_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y3: nat] :
( ( inf_inf_nat @ X @ Y3 )
= X ) ) ) ).
% le_iff_inf
thf(fact_1259_inf_Oabsorb1,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( inf_inf_set_a @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_1260_inf_Oabsorb1,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( ( inf_in7084830621192376909_a_nat @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_1261_inf_Oabsorb1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( inf_inf_nat @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_1262_inf_Oabsorb2,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( inf_inf_set_a @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_1263_inf_Oabsorb2,axiom,
! [B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A )
=> ( ( inf_in7084830621192376909_a_nat @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_1264_inf_Oabsorb2,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( inf_inf_nat @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_1265_inf__absorb1,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y )
=> ( ( inf_inf_set_a @ X3 @ Y )
= X3 ) ) ).
% inf_absorb1
thf(fact_1266_inf__absorb1,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ Y )
=> ( ( inf_in7084830621192376909_a_nat @ X3 @ Y )
= X3 ) ) ).
% inf_absorb1
thf(fact_1267_inf__absorb1,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( inf_inf_nat @ X3 @ Y )
= X3 ) ) ).
% inf_absorb1
thf(fact_1268_inf__absorb2,axiom,
! [Y: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y @ X3 )
=> ( ( inf_inf_set_a @ X3 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1269_inf__absorb2,axiom,
! [Y: set_Sum_sum_a_nat,X3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ Y @ X3 )
=> ( ( inf_in7084830621192376909_a_nat @ X3 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1270_inf__absorb2,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( inf_inf_nat @ X3 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1271_inf_OboundedE,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B2 )
=> ~ ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_1272_inf_OboundedE,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ ( inf_in7084830621192376909_a_nat @ B2 @ C ) )
=> ~ ( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ~ ( ord_le1325389633284124927_a_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_1273_inf_OboundedE,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B2 )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_1274_inf_OboundedI,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1275_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M2: nat] :
( ( P @ X3 )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M2 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
% Helper facts (7)
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 ) ).
thf(help_If_2_1_If_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_T,axiom,
! [X3: sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( if_Sum_sum_a_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_T,axiom,
! [X3: sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( if_Sum_sum_a_nat @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_T,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( if_set_Sum_sum_a_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_T,axiom,
! [X3: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( if_set_Sum_sum_a_nat @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ad_agr_a_b_nat @ ( fo_Exists_a_b @ n @ ( fo_Neg_a_b @ phi ) ) @ ad @ sigma @ tau ).
%------------------------------------------------------------------------------