TPTP Problem File: SLH0040^1.p
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- 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 : Median_Method/0000_Median/prob_00073_002547__14638942_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1448 ( 662 unt; 176 typ; 0 def)
% Number of atoms : 3511 (1225 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 9893 ( 360 ~; 60 |; 241 &;7994 @)
% ( 0 <=>;1238 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 6 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 643 ( 643 >; 0 *; 0 +; 0 <<)
% Number of symbols : 162 ( 161 usr; 20 con; 0-5 aty)
% Number of variables : 3172 ( 137 ^;2934 !; 101 ?;3172 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:43:12.772
%------------------------------------------------------------------------------
% Could-be-implicit typings (15)
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_Itf__a_J_Mt__Extended____Nonnegative____Real__Oennreal_J_J,type,
set_se9209621484078883815nnreal: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Set__Oset_Itf__a_J_J,type,
sigma_measure_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_Itf__a_J_M_Eo_J_J,type,
set_set_a_o: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Nat__Onat_J,type,
sigma_measure_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__Extended____Nonnegative____Real__Oennreal,type,
extend8495563244428889912nnreal: $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__Sigma____Algebra__Omeasure_Itf__a_J,type,
sigma_measure_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_I_062_Itf__a_M_Eo_J_J,type,
set_a_o: $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__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (161)
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra_001t__Set__Oset_It__Nat__Onat_J,type,
boolea778851993438741648et_nat: ( set_nat > set_nat > set_nat ) > ( set_nat > set_nat > set_nat ) > ( set_nat > set_nat ) > set_nat > set_nat > $o ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
boolea3433950929776517940_set_a: ( set_set_a > set_set_a > set_set_a ) > ( set_set_a > set_set_a > set_set_a ) > ( set_set_a > set_set_a ) > set_set_a > set_set_a > $o ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra_001t__Set__Oset_Itf__a_J,type,
boolea6678413348699952596_set_a: ( set_a > set_a > set_a ) > ( set_a > set_a > set_a ) > ( set_a > set_a ) > set_a > set_a > $o ).
thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Nat__Onat,type,
borel_8449730974584783410el_nat: sigma_measure_nat ).
thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001tf__a,type,
borel_5459123734250506524orel_a: sigma_measure_a ).
thf(sy_c_Disjoint__Sets_Odisjointed_001t__Nat__Onat,type,
disjoi6656021742987073733ed_nat: ( nat > set_nat ) > nat > set_nat ).
thf(sy_c_Disjoint__Sets_Odisjointed_001tf__a,type,
disjoi660815876502745417nted_a: ( nat > set_a ) > nat > set_a ).
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Nat__Onat,type,
monotone_on_nat_nat: set_nat > ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_Itf__a_J_001t__Nat__Onat,type,
monoto4790297507788910087_a_nat: set_set_a > ( set_a > set_a > $o ) > ( nat > nat > $o ) > ( set_a > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001tf__a_001t__Nat__Onat,type,
monotone_on_a_nat: set_a > ( a > a > $o ) > ( nat > nat > $o ) > ( a > nat ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
minus_5736297505244876581_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
uminus613421341184616069et_nat: set_set_nat > set_set_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
uminus1263758147157775637_set_a: set_set_set_a > set_set_set_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
uminus6103902357914783669_set_a: set_set_a > set_set_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
uminus_uminus_set_a: set_a > set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__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_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_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Set__Oset_It__Nat__Onat_J,type,
semila1667268886620078168et_nat: ( set_nat > set_nat > set_nat ) > set_nat > ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $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_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_Itf__a_J_J,type,
sup_sup_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Sigma____Algebra__Omeasure_Itf__a_J,type,
sup_su27664952386392231sure_a: sigma_measure_a > sigma_measure_a > sigma_measure_a ).
thf(sy_c_Measure__Space_Oincreasing_001tf__a_001t__Nat__Onat,type,
measur8151441426001876059_a_nat: set_set_a > ( set_a > nat ) > $o ).
thf(sy_c_Measure__Space_Osup__lexord_001t__Nat__Onat_001t__Nat__Onat,type,
measur4601247141005857854at_nat: nat > nat > ( nat > nat ) > nat > nat > nat ).
thf(sy_c_Measure__Space_Osup__measure_H_001tf__a,type,
measur3004909623614618064sure_a: sigma_measure_a > sigma_measure_a > sigma_measure_a ).
thf(sy_c_Median_Odown__ray_001t__Nat__Onat,type,
down_ray_nat: set_nat > $o ).
thf(sy_c_Median_Odown__ray_001tf__a,type,
down_ray_a: set_a > $o ).
thf(sy_c_Median_Ointerval_001t__Nat__Onat,type,
interval_nat: set_nat > $o ).
thf(sy_c_Median_Ointerval_001tf__a,type,
interval_a: set_a > $o ).
thf(sy_c_Median_Oup__ray_001t__Nat__Onat,type,
up_ray_nat: set_nat > $o ).
thf(sy_c_Median_Oup__ray_001tf__a,type,
up_ray_a: set_a > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nonnegative__Lebesgue__Integration_Ouniform__count__measure_001t__Nat__Onat,type,
nonneg7031465154080143958re_nat: set_nat > sigma_measure_nat ).
thf(sy_c_Nonnegative__Lebesgue__Integration_Ouniform__count__measure_001tf__a,type,
nonneg7367794086797660664sure_a: set_a > sigma_measure_a ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
bot_bot_set_a_o: set_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
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_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Sigma____Algebra__Omeasure_It__Nat__Onat_J,type,
bot_bo6718502177978453909re_nat: sigma_measure_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Sigma____Algebra__Omeasure_Itf__a_J,type,
bot_bo2108912051383640591sure_a: sigma_measure_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_less_set_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001tf__a,type,
ord_less_a: a > a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_Itf__a_J_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
ord_le6700572704167691815nnreal: ( set_a > extend8495563244428889912nnreal ) > ( set_a > extend8495563244428889912nnreal ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nonnegative____Real__Oennreal,type,
ord_le3935885782089961368nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Set__Oset_Itf__a_J_Mt__Extended____Nonnegative____Real__Oennreal_J_J,type,
ord_le2718372912061172615nnreal: set_se9209621484078883815nnreal > set_se9209621484078883815nnreal > $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_Itf__a_J_J_J,type,
ord_le5722252365846178494_set_a: set_set_set_a > set_set_set_a > $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_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Sigma____Algebra__Omeasure_Itf__a_J,type,
ord_le254669795585780187sure_a: sigma_measure_a > sigma_measure_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
ord_less_eq_a: a > a > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Nat__Onat_J,type,
ordering_top_set_nat: ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > set_nat > $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__Set__Oset_Itf__a_J_M_Eo_J,type,
top_top_set_a_o: set_a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_Itf__a_M_Eo_J,type,
top_top_a_o: a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_Eo,type,
top_top_o: $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Extended____Nonnegative____Real__Oennreal,type,
top_to1496364449551166952nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
top_top_set_nat_o: set_nat_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Set__Oset_Itf__a_J_M_Eo_J_J,type,
top_top_set_set_a_o: set_set_a_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
top_top_set_a_o2: set_a_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__Set__Oset_Itf__a_J_J_J,type,
top_to4027821306633060462_set_a: set_set_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
top_top_set_set_a: set_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
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__Set__Oset_Itf__a_J_J,type,
collect_set_set_a: ( set_set_a > $o ) > set_set_set_a ).
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_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
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_Itf__a_J,type,
image_nat_set_a: ( nat > set_a ) > set_nat > set_set_a ).
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_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__Set__Oset_Itf__a_J_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_1042221919965026181_set_a: ( set_set_a > set_set_a ) > set_set_set_a > set_set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Nat__Onat,type,
image_set_a_nat: ( set_a > nat ) > set_set_a > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001tf__a,type,
image_set_a_a: ( set_a > a ) > set_set_a > set_a ).
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_Itf__a_J,type,
image_a_set_a: ( a > set_a ) > set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
insert_set_set_a: set_set_a > set_set_set_a > set_set_set_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_Itf__a_J,type,
is_singleton_set_a: set_set_a > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oremove_001t__Set__Oset_Itf__a_J,type,
remove_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oremove_001tf__a,type,
remove_a: a > set_a > set_a ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_Itf__a_J,type,
set_or6288561110385358355_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001tf__a,type,
set_or672772299803893939Most_a: a > a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_Itf__a_J,type,
set_or2348907005316661231_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001tf__a,type,
set_or5139330845457685135Than_a: a > a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001_062_It__Set__Oset_Itf__a_J_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
set_or3682978270108489938nnreal: ( set_a > extend8495563244428889912nnreal ) > set_se9209621484078883815nnreal ).
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__Set__Oset_Itf__a_J_J,type,
set_or3904034815786525833_set_a: set_set_a > set_set_set_a ).
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_OatLeast_001tf__a,type,
set_ord_atLeast_a: a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_It__Nat__Onat_M_Eo_J,type,
set_ord_atMost_nat_o: ( nat > $o ) > set_nat_o ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
set_or4722645096382476952et_a_o: ( set_a > $o ) > set_set_a_o ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_It__Set__Oset_Itf__a_J_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
set_or7674582181447345870nnreal: ( set_a > extend8495563244428889912nnreal ) > set_se9209621484078883815nnreal ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_Itf__a_M_Eo_J,type,
set_ord_atMost_a_o: ( a > $o ) > set_a_o ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_or4016371710855203973_set_a: set_set_a > set_set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_Itf__a_J,type,
set_ord_atMost_set_a: set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001tf__a,type,
set_ord_atMost_a: a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Set__Oset_Itf__a_J,type,
set_or2503527069484367278_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001tf__a,type,
set_or4472690218693186638Most_a: a > a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Set__Oset_Itf__a_J,type,
set_or6017932776736107018_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001tf__a,type,
set_or5939364468397584554Than_a: a > a > set_a ).
thf(sy_c_Sigma__Algebra_Oalgebra_001t__Nat__Onat,type,
sigma_algebra_nat: set_nat > set_set_nat > $o ).
thf(sy_c_Sigma__Algebra_Oalgebra_001tf__a,type,
sigma_algebra_a: set_a > set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Obinary_001t__Nat__Onat,type,
sigma_binary_nat: nat > nat > nat > nat ).
thf(sy_c_Sigma__Algebra_Oemeasure_001t__Nat__Onat,type,
sigma_emeasure_nat: sigma_measure_nat > set_nat > extend8495563244428889912nnreal ).
thf(sy_c_Sigma__Algebra_Oemeasure_001tf__a,type,
sigma_emeasure_a: sigma_measure_a > set_a > extend8495563244428889912nnreal ).
thf(sy_c_Sigma__Algebra_Osets_001t__Nat__Onat,type,
sigma_sets_nat: sigma_measure_nat > set_set_nat ).
thf(sy_c_Sigma__Algebra_Osets_001t__Set__Oset_Itf__a_J,type,
sigma_sets_set_a: sigma_measure_set_a > set_set_set_a ).
thf(sy_c_Sigma__Algebra_Osets_001tf__a,type,
sigma_sets_a: sigma_measure_a > set_set_a ).
thf(sy_c_Sigma__Algebra_Osmallest__ccdi__sets_001t__Nat__Onat,type,
sigma_5553761350045521333ts_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Sigma__Algebra_Osmallest__ccdi__sets_001tf__a,type,
sigma_5648178489087971417sets_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Sigma__Algebra_Ospace_001t__Nat__Onat,type,
sigma_space_nat: sigma_measure_nat > set_nat ).
thf(sy_c_Sigma__Algebra_Ospace_001tf__a,type,
sigma_space_a: sigma_measure_a > set_a ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
topolo4902158794631467389eq_nat: ( nat > nat ) > $o ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Nat__Onat,type,
topolo4328251076210115529en_nat: set_nat > $o ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001tf__a,type,
topolo8477419352202985285open_a: set_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_Itf__a_J_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
member4180043592386426928nnreal: ( set_a > extend8495563244428889912nnreal ) > set_se9209621484078883815nnreal > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $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_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_I,type,
i: set_a ).
% Relevant facts (1271)
thf(fact_0_assms,axiom,
down_ray_a @ i ).
% assms
thf(fact_1__092_060open_062up__ray_A_I_N_AI_J_092_060close_062,axiom,
up_ray_a @ ( uminus_uminus_set_a @ i ) ).
% \<open>up_ray (- I)\<close>
thf(fact_2_up__ray__borel,axiom,
! [I: set_nat] :
( ( up_ray_nat @ I )
=> ( member_set_nat @ I @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ) ).
% up_ray_borel
thf(fact_3_up__ray__borel,axiom,
! [I: set_a] :
( ( up_ray_a @ I )
=> ( member_set_a @ I @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ) ) ).
% up_ray_borel
thf(fact_4_borel__comp,axiom,
! [A: set_nat] :
( ( member_set_nat @ A @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) )
=> ( member_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ) ).
% borel_comp
thf(fact_5_borel__comp,axiom,
! [A: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) )
=> ( member_set_a @ ( uminus_uminus_set_a @ A ) @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ) ) ).
% borel_comp
thf(fact_6_ComplI,axiom,
! [C: set_nat,A: set_set_nat] :
( ~ ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A ) ) ) ).
% ComplI
thf(fact_7_ComplI,axiom,
! [C: set_set_a,A: set_set_set_a] :
( ~ ( member_set_set_a @ C @ A )
=> ( member_set_set_a @ C @ ( uminus1263758147157775637_set_a @ A ) ) ) ).
% ComplI
thf(fact_8_ComplI,axiom,
! [C: nat,A: set_nat] :
( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% ComplI
thf(fact_9_ComplI,axiom,
! [C: set_a,A: set_set_a] :
( ~ ( member_set_a @ C @ A )
=> ( member_set_a @ C @ ( uminus6103902357914783669_set_a @ A ) ) ) ).
% ComplI
thf(fact_10_ComplI,axiom,
! [C: a,A: set_a] :
( ~ ( member_a @ C @ A )
=> ( member_a @ C @ ( uminus_uminus_set_a @ A ) ) ) ).
% ComplI
thf(fact_11_Compl__iff,axiom,
! [C: set_nat,A: set_set_nat] :
( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A ) )
= ( ~ ( member_set_nat @ C @ A ) ) ) ).
% Compl_iff
thf(fact_12_Compl__iff,axiom,
! [C: set_set_a,A: set_set_set_a] :
( ( member_set_set_a @ C @ ( uminus1263758147157775637_set_a @ A ) )
= ( ~ ( member_set_set_a @ C @ A ) ) ) ).
% Compl_iff
thf(fact_13_Compl__iff,axiom,
! [C: nat,A: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) )
= ( ~ ( member_nat @ C @ A ) ) ) ).
% Compl_iff
thf(fact_14_Compl__iff,axiom,
! [C: set_a,A: set_set_a] :
( ( member_set_a @ C @ ( uminus6103902357914783669_set_a @ A ) )
= ( ~ ( member_set_a @ C @ A ) ) ) ).
% Compl_iff
thf(fact_15_Compl__iff,axiom,
! [C: a,A: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A ) )
= ( ~ ( member_a @ C @ A ) ) ) ).
% Compl_iff
thf(fact_16_Compl__eq__Compl__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( uminus5710092332889474511et_nat @ A )
= ( uminus5710092332889474511et_nat @ B ) )
= ( A = B ) ) ).
% Compl_eq_Compl_iff
thf(fact_17_Compl__eq__Compl__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( uminus6103902357914783669_set_a @ A )
= ( uminus6103902357914783669_set_a @ B ) )
= ( A = B ) ) ).
% Compl_eq_Compl_iff
thf(fact_18_Compl__eq__Compl__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( uminus_uminus_set_a @ A )
= ( uminus_uminus_set_a @ B ) )
= ( A = B ) ) ).
% Compl_eq_Compl_iff
thf(fact_19_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [X: set_nat] :
( ( uminus5710092332889474511et_nat @ ( uminus5710092332889474511et_nat @ X ) )
= X ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_20_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [X: set_a] :
( ( uminus_uminus_set_a @ ( uminus_uminus_set_a @ X ) )
= X ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_21_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [X: set_set_a] :
( ( uminus6103902357914783669_set_a @ ( uminus6103902357914783669_set_a @ X ) )
= X ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_22_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( uminus5710092332889474511et_nat @ X )
= ( uminus5710092332889474511et_nat @ Y ) )
= ( X = Y ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_23_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( uminus_uminus_set_a @ X )
= ( uminus_uminus_set_a @ Y ) )
= ( X = Y ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_24_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ( uminus6103902357914783669_set_a @ X )
= ( uminus6103902357914783669_set_a @ Y ) )
= ( X = Y ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_25_ComplD,axiom,
! [C: set_nat,A: set_set_nat] :
( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A ) )
=> ~ ( member_set_nat @ C @ A ) ) ).
% ComplD
thf(fact_26_ComplD,axiom,
! [C: set_set_a,A: set_set_set_a] :
( ( member_set_set_a @ C @ ( uminus1263758147157775637_set_a @ A ) )
=> ~ ( member_set_set_a @ C @ A ) ) ).
% ComplD
thf(fact_27_ComplD,axiom,
! [C: nat,A: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) )
=> ~ ( member_nat @ C @ A ) ) ).
% ComplD
thf(fact_28_ComplD,axiom,
! [C: a,A: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A ) )
=> ~ ( member_a @ C @ A ) ) ).
% ComplD
thf(fact_29_ComplD,axiom,
! [C: set_a,A: set_set_a] :
( ( member_set_a @ C @ ( uminus6103902357914783669_set_a @ A ) )
=> ~ ( member_set_a @ C @ A ) ) ).
% ComplD
thf(fact_30_double__complement,axiom,
! [A: set_nat] :
( ( uminus5710092332889474511et_nat @ ( uminus5710092332889474511et_nat @ A ) )
= A ) ).
% double_complement
thf(fact_31_double__complement,axiom,
! [A: set_a] :
( ( uminus_uminus_set_a @ ( uminus_uminus_set_a @ A ) )
= A ) ).
% double_complement
thf(fact_32_double__complement,axiom,
! [A: set_set_a] :
( ( uminus6103902357914783669_set_a @ ( uminus6103902357914783669_set_a @ A ) )
= A ) ).
% double_complement
thf(fact_33_exists__diff,axiom,
! [P: set_nat > $o] :
( ( ? [S: set_nat] : ( P @ ( uminus5710092332889474511et_nat @ S ) ) )
= ( ? [X2: set_nat] : ( P @ X2 ) ) ) ).
% exists_diff
thf(fact_34_exists__diff,axiom,
! [P: set_a > $o] :
( ( ? [S: set_a] : ( P @ ( uminus_uminus_set_a @ S ) ) )
= ( ? [X2: set_a] : ( P @ X2 ) ) ) ).
% exists_diff
thf(fact_35_exists__diff,axiom,
! [P: set_set_a > $o] :
( ( ? [S: set_set_a] : ( P @ ( uminus6103902357914783669_set_a @ S ) ) )
= ( ? [X2: set_set_a] : ( P @ X2 ) ) ) ).
% exists_diff
thf(fact_36_sets__Ball,axiom,
! [I: set_a,A: a > set_a,M: a > sigma_measure_a,I2: a] :
( ! [X3: a] :
( ( member_a @ X3 @ I )
=> ( member_set_a @ ( A @ X3 ) @ ( sigma_sets_a @ ( M @ X3 ) ) ) )
=> ( ( member_a @ I2 @ I )
=> ( member_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_37_sets__Ball,axiom,
! [I: set_set_a,A: set_a > set_a,M: set_a > sigma_measure_a,I2: set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ I )
=> ( member_set_a @ ( A @ X3 ) @ ( sigma_sets_a @ ( M @ X3 ) ) ) )
=> ( ( member_set_a @ I2 @ I )
=> ( member_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_38_sets__Ball,axiom,
! [I: set_nat,A: nat > set_a,M: nat > sigma_measure_a,I2: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ I )
=> ( member_set_a @ ( A @ X3 ) @ ( sigma_sets_a @ ( M @ X3 ) ) ) )
=> ( ( member_nat @ I2 @ I )
=> ( member_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_39_sets__Ball,axiom,
! [I: set_a,A: a > set_nat,M: a > sigma_measure_nat,I2: a] :
( ! [X3: a] :
( ( member_a @ X3 @ I )
=> ( member_set_nat @ ( A @ X3 ) @ ( sigma_sets_nat @ ( M @ X3 ) ) ) )
=> ( ( member_a @ I2 @ I )
=> ( member_set_nat @ ( A @ I2 ) @ ( sigma_sets_nat @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_40_sets__Ball,axiom,
! [I: set_nat,A: nat > set_nat,M: nat > sigma_measure_nat,I2: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ I )
=> ( member_set_nat @ ( A @ X3 ) @ ( sigma_sets_nat @ ( M @ X3 ) ) ) )
=> ( ( member_nat @ I2 @ I )
=> ( member_set_nat @ ( A @ I2 ) @ ( sigma_sets_nat @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_41_sets__Ball,axiom,
! [I: set_a,A: a > set_set_a,M: a > sigma_measure_set_a,I2: a] :
( ! [X3: a] :
( ( member_a @ X3 @ I )
=> ( member_set_set_a @ ( A @ X3 ) @ ( sigma_sets_set_a @ ( M @ X3 ) ) ) )
=> ( ( member_a @ I2 @ I )
=> ( member_set_set_a @ ( A @ I2 ) @ ( sigma_sets_set_a @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_42_sets__Ball,axiom,
! [I: set_nat,A: nat > set_set_a,M: nat > sigma_measure_set_a,I2: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ I )
=> ( member_set_set_a @ ( A @ X3 ) @ ( sigma_sets_set_a @ ( M @ X3 ) ) ) )
=> ( ( member_nat @ I2 @ I )
=> ( member_set_set_a @ ( A @ I2 ) @ ( sigma_sets_set_a @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_43_sets__Ball,axiom,
! [I: set_set_nat,A: set_nat > set_a,M: set_nat > sigma_measure_a,I2: set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ I )
=> ( member_set_a @ ( A @ X3 ) @ ( sigma_sets_a @ ( M @ X3 ) ) ) )
=> ( ( member_set_nat @ I2 @ I )
=> ( member_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_44_sets__Ball,axiom,
! [I: set_set_a,A: set_a > set_nat,M: set_a > sigma_measure_nat,I2: set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ I )
=> ( member_set_nat @ ( A @ X3 ) @ ( sigma_sets_nat @ ( M @ X3 ) ) ) )
=> ( ( member_set_a @ I2 @ I )
=> ( member_set_nat @ ( A @ I2 ) @ ( sigma_sets_nat @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_45_sets__Ball,axiom,
! [I: set_set_nat,A: set_nat > set_nat,M: set_nat > sigma_measure_nat,I2: set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ I )
=> ( member_set_nat @ ( A @ X3 ) @ ( sigma_sets_nat @ ( M @ X3 ) ) ) )
=> ( ( member_set_nat @ I2 @ I )
=> ( member_set_nat @ ( A @ I2 ) @ ( sigma_sets_nat @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_46_down__ray__def,axiom,
( down_ray_a
= ( ^ [I3: set_a] :
! [X4: a,Y2: a] :
( ( member_a @ Y2 @ I3 )
=> ( ( ord_less_eq_a @ X4 @ Y2 )
=> ( member_a @ X4 @ I3 ) ) ) ) ) ).
% down_ray_def
thf(fact_47_down__ray__def,axiom,
( down_ray_nat
= ( ^ [I3: set_nat] :
! [X4: nat,Y2: nat] :
( ( member_nat @ Y2 @ I3 )
=> ( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( member_nat @ X4 @ I3 ) ) ) ) ) ).
% down_ray_def
thf(fact_48_space__in__borel,axiom,
member_set_nat @ top_top_set_nat @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ).
% space_in_borel
thf(fact_49_space__in__borel,axiom,
member_set_a @ top_top_set_a @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ).
% space_in_borel
thf(fact_50_greaterThanLessThan__borel,axiom,
! [A2: a,B2: a] : ( member_set_a @ ( set_or5939364468397584554Than_a @ A2 @ B2 ) @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ) ).
% greaterThanLessThan_borel
thf(fact_51_greaterThanLessThan__borel,axiom,
! [A2: nat,B2: nat] : ( member_set_nat @ ( set_or5834768355832116004an_nat @ A2 @ B2 ) @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ).
% greaterThanLessThan_borel
thf(fact_52_greaterThanAtMost__borel,axiom,
! [A2: nat,B2: nat] : ( member_set_nat @ ( set_or6659071591806873216st_nat @ A2 @ B2 ) @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ).
% greaterThanAtMost_borel
thf(fact_53_greaterThanAtMost__borel,axiom,
! [A2: a,B2: a] : ( member_set_a @ ( set_or4472690218693186638Most_a @ A2 @ B2 ) @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ) ).
% greaterThanAtMost_borel
thf(fact_54_borel__singleton,axiom,
! [A: set_nat,X: nat] :
( ( member_set_nat @ A @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) )
=> ( member_set_nat @ ( insert_nat @ X @ A ) @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ) ).
% borel_singleton
thf(fact_55_borel__singleton,axiom,
! [A: set_a,X: a] :
( ( member_set_a @ A @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) )
=> ( member_set_a @ ( insert_a @ X @ A ) @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ) ) ).
% borel_singleton
thf(fact_56_borel__open,axiom,
! [A: set_nat] :
( ( topolo4328251076210115529en_nat @ A )
=> ( member_set_nat @ A @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ) ).
% borel_open
thf(fact_57_borel__open,axiom,
! [A: set_a] :
( ( topolo8477419352202985285open_a @ A )
=> ( member_set_a @ A @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ) ) ).
% borel_open
thf(fact_58_atLeast__borel,axiom,
! [A2: nat] : ( member_set_nat @ ( set_ord_atLeast_nat @ A2 ) @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ).
% atLeast_borel
thf(fact_59_atLeast__borel,axiom,
! [A2: a] : ( member_set_a @ ( set_ord_atLeast_a @ A2 ) @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ) ).
% atLeast_borel
thf(fact_60_atMost__borel,axiom,
! [A2: nat] : ( member_set_nat @ ( set_ord_atMost_nat @ A2 ) @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ).
% atMost_borel
thf(fact_61_atMost__borel,axiom,
! [A2: a] : ( member_set_a @ ( set_ord_atMost_a @ A2 ) @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ) ).
% atMost_borel
thf(fact_62_UNIV__I,axiom,
! [X: set_nat] : ( member_set_nat @ X @ top_top_set_set_nat ) ).
% UNIV_I
thf(fact_63_UNIV__I,axiom,
! [X: set_set_a] : ( member_set_set_a @ X @ top_to4027821306633060462_set_a ) ).
% UNIV_I
thf(fact_64_UNIV__I,axiom,
! [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ).
% UNIV_I
thf(fact_65_UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_I
thf(fact_66_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_67_insertCI,axiom,
! [A2: set_nat,B: set_set_nat,B2: set_nat] :
( ( ~ ( member_set_nat @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_set_nat @ A2 @ ( insert_set_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_68_insertCI,axiom,
! [A2: set_set_a,B: set_set_set_a,B2: set_set_a] :
( ( ~ ( member_set_set_a @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_set_set_a @ A2 @ ( insert_set_set_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_69_insertCI,axiom,
! [A2: nat,B: set_nat,B2: nat] :
( ( ~ ( member_nat @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_nat @ A2 @ ( insert_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_70_insertCI,axiom,
! [A2: set_a,B: set_set_a,B2: set_a] :
( ( ~ ( member_set_a @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_set_a @ A2 @ ( insert_set_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_71_insertCI,axiom,
! [A2: a,B: set_a,B2: a] :
( ( ~ ( member_a @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_72_insert__iff,axiom,
! [A2: set_nat,B2: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ ( insert_set_nat @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_set_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_73_insert__iff,axiom,
! [A2: set_set_a,B2: set_set_a,A: set_set_set_a] :
( ( member_set_set_a @ A2 @ ( insert_set_set_a @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_set_set_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_74_insert__iff,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_75_insert__iff,axiom,
! [A2: set_a,B2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ ( insert_set_a @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_set_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_76_insert__iff,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_77_insert__subset,axiom,
! [X: set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X @ A ) @ B )
= ( ( member_set_nat @ X @ B )
& ( ord_le6893508408891458716et_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_78_insert__subset,axiom,
! [X: set_set_a,A: set_set_set_a,B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( insert_set_set_a @ X @ A ) @ B )
= ( ( member_set_set_a @ X @ B )
& ( ord_le5722252365846178494_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_79_insert__subset,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A ) @ B )
= ( ( member_nat @ X @ B )
& ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_80_insert__subset,axiom,
! [X: set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A ) @ B )
= ( ( member_set_a @ X @ B )
& ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_81_insert__subset,axiom,
! [X: a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A ) @ B )
= ( ( member_a @ X @ B )
& ( ord_less_eq_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_82_insert__absorb2,axiom,
! [X: a,A: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A ) )
= ( insert_a @ X @ A ) ) ).
% insert_absorb2
thf(fact_83_insert__absorb2,axiom,
! [X: set_a,A: set_set_a] :
( ( insert_set_a @ X @ ( insert_set_a @ X @ A ) )
= ( insert_set_a @ X @ A ) ) ).
% insert_absorb2
thf(fact_84_insert__absorb2,axiom,
! [X: set_nat,A: set_set_nat] :
( ( insert_set_nat @ X @ ( insert_set_nat @ X @ A ) )
= ( insert_set_nat @ X @ A ) ) ).
% insert_absorb2
thf(fact_85_insert__absorb2,axiom,
! [X: nat,A: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ X @ A ) )
= ( insert_nat @ X @ A ) ) ).
% insert_absorb2
thf(fact_86_Compl__anti__mono,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B ) @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% Compl_anti_mono
thf(fact_87_Compl__anti__mono,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B ) @ ( uminus_uminus_set_a @ A ) ) ) ).
% Compl_anti_mono
thf(fact_88_Compl__anti__mono,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ B ) @ ( uminus6103902357914783669_set_a @ A ) ) ) ).
% Compl_anti_mono
thf(fact_89_Compl__subset__Compl__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( uminus5710092332889474511et_nat @ B ) )
= ( ord_less_eq_set_nat @ B @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_90_Compl__subset__Compl__iff,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A ) @ ( uminus_uminus_set_a @ B ) )
= ( ord_less_eq_set_a @ B @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_91_Compl__subset__Compl__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ A ) @ ( uminus6103902357914783669_set_a @ B ) )
= ( ord_le3724670747650509150_set_a @ B @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_92_compl__le__compl__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) )
= ( ord_less_eq_set_nat @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_93_compl__le__compl__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ ( uminus_uminus_set_a @ Y ) )
= ( ord_less_eq_set_a @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_94_compl__le__compl__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ X ) @ ( uminus6103902357914783669_set_a @ Y ) )
= ( ord_le3724670747650509150_set_a @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_95_verit__comp__simplify1_I2_J,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_96_verit__comp__simplify1_I2_J,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_97_verit__comp__simplify1_I2_J,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_98_verit__comp__simplify1_I2_J,axiom,
! [A2: set_a > extend8495563244428889912nnreal] : ( ord_le6700572704167691815nnreal @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_99_verit__comp__simplify1_I2_J,axiom,
! [A2: a] : ( ord_less_eq_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_100_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_101_insertE,axiom,
! [A2: set_nat,B2: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ ( insert_set_nat @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_set_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_102_insertE,axiom,
! [A2: set_set_a,B2: set_set_a,A: set_set_set_a] :
( ( member_set_set_a @ A2 @ ( insert_set_set_a @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_set_set_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_103_insertE,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_104_insertE,axiom,
! [A2: set_a,B2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ ( insert_set_a @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_set_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_105_insertE,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_106_insertI1,axiom,
! [A2: set_nat,B: set_set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_107_insertI1,axiom,
! [A2: set_set_a,B: set_set_set_a] : ( member_set_set_a @ A2 @ ( insert_set_set_a @ A2 @ B ) ) ).
% insertI1
thf(fact_108_insertI1,axiom,
! [A2: nat,B: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_109_insertI1,axiom,
! [A2: set_a,B: set_set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ B ) ) ).
% insertI1
thf(fact_110_insertI1,axiom,
! [A2: a,B: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B ) ) ).
% insertI1
thf(fact_111_insertI2,axiom,
! [A2: set_nat,B: set_set_nat,B2: set_nat] :
( ( member_set_nat @ A2 @ B )
=> ( member_set_nat @ A2 @ ( insert_set_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_112_insertI2,axiom,
! [A2: set_set_a,B: set_set_set_a,B2: set_set_a] :
( ( member_set_set_a @ A2 @ B )
=> ( member_set_set_a @ A2 @ ( insert_set_set_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_113_insertI2,axiom,
! [A2: nat,B: set_nat,B2: nat] :
( ( member_nat @ A2 @ B )
=> ( member_nat @ A2 @ ( insert_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_114_insertI2,axiom,
! [A2: set_a,B: set_set_a,B2: set_a] :
( ( member_set_a @ A2 @ B )
=> ( member_set_a @ A2 @ ( insert_set_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_115_insertI2,axiom,
! [A2: a,B: set_a,B2: a] :
( ( member_a @ A2 @ B )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_116_UNIV__eq__I,axiom,
! [A: set_set_nat] :
( ! [X3: set_nat] : ( member_set_nat @ X3 @ A )
=> ( top_top_set_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_117_UNIV__eq__I,axiom,
! [A: set_set_set_a] :
( ! [X3: set_set_a] : ( member_set_set_a @ X3 @ A )
=> ( top_to4027821306633060462_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_118_UNIV__eq__I,axiom,
! [A: set_set_a] :
( ! [X3: set_a] : ( member_set_a @ X3 @ A )
=> ( top_top_set_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_119_UNIV__eq__I,axiom,
! [A: set_a] :
( ! [X3: a] : ( member_a @ X3 @ A )
=> ( top_top_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_120_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X3: nat] : ( member_nat @ X3 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_121_mem__Collect__eq,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A2 @ ( collect_set_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_122_mem__Collect__eq,axiom,
! [A2: set_set_a,P: set_set_a > $o] :
( ( member_set_set_a @ A2 @ ( collect_set_set_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_123_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_124_mem__Collect__eq,axiom,
! [A2: set_a,P: set_a > $o] :
( ( member_set_a @ A2 @ ( collect_set_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_125_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_126_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_127_Collect__mem__eq,axiom,
! [A: set_set_set_a] :
( ( collect_set_set_a
@ ^ [X4: set_set_a] : ( member_set_set_a @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_128_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_129_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X4: set_a] : ( member_set_a @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_130_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_131_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_132_Collect__cong,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X3: set_a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_set_a @ P )
= ( collect_set_a @ Q ) ) ) ).
% Collect_cong
thf(fact_133_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_134_Set_Oset__insert,axiom,
! [X: set_nat,A: set_set_nat] :
( ( member_set_nat @ X @ A )
=> ~ ! [B3: set_set_nat] :
( ( A
= ( insert_set_nat @ X @ B3 ) )
=> ( member_set_nat @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_135_Set_Oset__insert,axiom,
! [X: set_set_a,A: set_set_set_a] :
( ( member_set_set_a @ X @ A )
=> ~ ! [B3: set_set_set_a] :
( ( A
= ( insert_set_set_a @ X @ B3 ) )
=> ( member_set_set_a @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_136_Set_Oset__insert,axiom,
! [X: nat,A: set_nat] :
( ( member_nat @ X @ A )
=> ~ ! [B3: set_nat] :
( ( A
= ( insert_nat @ X @ B3 ) )
=> ( member_nat @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_137_Set_Oset__insert,axiom,
! [X: set_a,A: set_set_a] :
( ( member_set_a @ X @ A )
=> ~ ! [B3: set_set_a] :
( ( A
= ( insert_set_a @ X @ B3 ) )
=> ( member_set_a @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_138_Set_Oset__insert,axiom,
! [X: a,A: set_a] :
( ( member_a @ X @ A )
=> ~ ! [B3: set_a] :
( ( A
= ( insert_a @ X @ B3 ) )
=> ( member_a @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_139_insert__UNIV,axiom,
! [X: set_nat] :
( ( insert_set_nat @ X @ top_top_set_set_nat )
= top_top_set_set_nat ) ).
% insert_UNIV
thf(fact_140_insert__UNIV,axiom,
! [X: set_a] :
( ( insert_set_a @ X @ top_top_set_set_a )
= top_top_set_set_a ) ).
% insert_UNIV
thf(fact_141_insert__UNIV,axiom,
! [X: a] :
( ( insert_a @ X @ top_top_set_a )
= top_top_set_a ) ).
% insert_UNIV
thf(fact_142_insert__UNIV,axiom,
! [X: nat] :
( ( insert_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% insert_UNIV
thf(fact_143_insert__mono,axiom,
! [C2: set_set_nat,D: set_set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ D )
=> ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ A2 @ C2 ) @ ( insert_set_nat @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_144_insert__mono,axiom,
! [C2: set_set_a,D: set_set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ D )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ A2 @ C2 ) @ ( insert_set_a @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_145_insert__mono,axiom,
! [C2: set_a,D: set_a,A2: a] :
( ( ord_less_eq_set_a @ C2 @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A2 @ C2 ) @ ( insert_a @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_146_insert__mono,axiom,
! [C2: set_nat,D: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ C2 @ D )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ C2 ) @ ( insert_nat @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_147_subset__UNIV,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ top_top_set_set_a ) ).
% subset_UNIV
thf(fact_148_subset__UNIV,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% subset_UNIV
thf(fact_149_subset__UNIV,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_150_UNIV__witness,axiom,
? [X3: set_nat] : ( member_set_nat @ X3 @ top_top_set_set_nat ) ).
% UNIV_witness
thf(fact_151_UNIV__witness,axiom,
? [X3: set_set_a] : ( member_set_set_a @ X3 @ top_to4027821306633060462_set_a ) ).
% UNIV_witness
thf(fact_152_UNIV__witness,axiom,
? [X3: set_a] : ( member_set_a @ X3 @ top_top_set_set_a ) ).
% UNIV_witness
thf(fact_153_UNIV__witness,axiom,
? [X3: a] : ( member_a @ X3 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_154_UNIV__witness,axiom,
? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_155_insert__ident,axiom,
! [X: set_nat,A: set_set_nat,B: set_set_nat] :
( ~ ( member_set_nat @ X @ A )
=> ( ~ ( member_set_nat @ X @ B )
=> ( ( ( insert_set_nat @ X @ A )
= ( insert_set_nat @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_156_insert__ident,axiom,
! [X: set_set_a,A: set_set_set_a,B: set_set_set_a] :
( ~ ( member_set_set_a @ X @ A )
=> ( ~ ( member_set_set_a @ X @ B )
=> ( ( ( insert_set_set_a @ X @ A )
= ( insert_set_set_a @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_157_insert__ident,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ~ ( member_nat @ X @ B )
=> ( ( ( insert_nat @ X @ A )
= ( insert_nat @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_158_insert__ident,axiom,
! [X: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X @ A )
=> ( ~ ( member_set_a @ X @ B )
=> ( ( ( insert_set_a @ X @ A )
= ( insert_set_a @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_159_insert__ident,axiom,
! [X: a,A: set_a,B: set_a] :
( ~ ( member_a @ X @ A )
=> ( ~ ( member_a @ X @ B )
=> ( ( ( insert_a @ X @ A )
= ( insert_a @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_160_insert__absorb,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ A )
=> ( ( insert_set_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_161_insert__absorb,axiom,
! [A2: set_set_a,A: set_set_set_a] :
( ( member_set_set_a @ A2 @ A )
=> ( ( insert_set_set_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_162_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_163_insert__absorb,axiom,
! [A2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ( ( insert_set_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_164_insert__absorb,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_165_insert__eq__iff,axiom,
! [A2: set_nat,A: set_set_nat,B2: set_nat,B: set_set_nat] :
( ~ ( member_set_nat @ A2 @ A )
=> ( ~ ( member_set_nat @ B2 @ B )
=> ( ( ( insert_set_nat @ A2 @ A )
= ( insert_set_nat @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_set_nat] :
( ( A
= ( insert_set_nat @ B2 @ C3 ) )
& ~ ( member_set_nat @ B2 @ C3 )
& ( B
= ( insert_set_nat @ A2 @ C3 ) )
& ~ ( member_set_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_166_insert__eq__iff,axiom,
! [A2: set_set_a,A: set_set_set_a,B2: set_set_a,B: set_set_set_a] :
( ~ ( member_set_set_a @ A2 @ A )
=> ( ~ ( member_set_set_a @ B2 @ B )
=> ( ( ( insert_set_set_a @ A2 @ A )
= ( insert_set_set_a @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_set_set_a] :
( ( A
= ( insert_set_set_a @ B2 @ C3 ) )
& ~ ( member_set_set_a @ B2 @ C3 )
& ( B
= ( insert_set_set_a @ A2 @ C3 ) )
& ~ ( member_set_set_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_167_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B2: nat,B: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ B2 @ B )
=> ( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_nat] :
( ( A
= ( insert_nat @ B2 @ C3 ) )
& ~ ( member_nat @ B2 @ C3 )
& ( B
= ( insert_nat @ A2 @ C3 ) )
& ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_168_insert__eq__iff,axiom,
! [A2: set_a,A: set_set_a,B2: set_a,B: set_set_a] :
( ~ ( member_set_a @ A2 @ A )
=> ( ~ ( member_set_a @ B2 @ B )
=> ( ( ( insert_set_a @ A2 @ A )
= ( insert_set_a @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_set_a] :
( ( A
= ( insert_set_a @ B2 @ C3 ) )
& ~ ( member_set_a @ B2 @ C3 )
& ( B
= ( insert_set_a @ A2 @ C3 ) )
& ~ ( member_set_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_169_insert__eq__iff,axiom,
! [A2: a,A: set_a,B2: a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ~ ( member_a @ B2 @ B )
=> ( ( ( insert_a @ A2 @ A )
= ( insert_a @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_a] :
( ( A
= ( insert_a @ B2 @ C3 ) )
& ~ ( member_a @ B2 @ C3 )
& ( B
= ( insert_a @ A2 @ C3 ) )
& ~ ( member_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_170_subset__insert,axiom,
! [X: set_nat,A: set_set_nat,B: set_set_nat] :
( ~ ( member_set_nat @ X @ A )
=> ( ( ord_le6893508408891458716et_nat @ A @ ( insert_set_nat @ X @ B ) )
= ( ord_le6893508408891458716et_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_171_subset__insert,axiom,
! [X: set_set_a,A: set_set_set_a,B: set_set_set_a] :
( ~ ( member_set_set_a @ X @ A )
=> ( ( ord_le5722252365846178494_set_a @ A @ ( insert_set_set_a @ X @ B ) )
= ( ord_le5722252365846178494_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_172_subset__insert,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_173_subset__insert,axiom,
! [X: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X @ B ) )
= ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_174_subset__insert,axiom,
! [X: a,A: set_a,B: set_a] :
( ~ ( member_a @ X @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_175_insert__commute,axiom,
! [X: a,Y: a,A: set_a] :
( ( insert_a @ X @ ( insert_a @ Y @ A ) )
= ( insert_a @ Y @ ( insert_a @ X @ A ) ) ) ).
% insert_commute
thf(fact_176_insert__commute,axiom,
! [X: set_a,Y: set_a,A: set_set_a] :
( ( insert_set_a @ X @ ( insert_set_a @ Y @ A ) )
= ( insert_set_a @ Y @ ( insert_set_a @ X @ A ) ) ) ).
% insert_commute
thf(fact_177_insert__commute,axiom,
! [X: set_nat,Y: set_nat,A: set_set_nat] :
( ( insert_set_nat @ X @ ( insert_set_nat @ Y @ A ) )
= ( insert_set_nat @ Y @ ( insert_set_nat @ X @ A ) ) ) ).
% insert_commute
thf(fact_178_insert__commute,axiom,
! [X: nat,Y: nat,A: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ Y @ A ) )
= ( insert_nat @ Y @ ( insert_nat @ X @ A ) ) ) ).
% insert_commute
thf(fact_179_subset__insertI,axiom,
! [B: set_set_nat,A2: set_nat] : ( ord_le6893508408891458716et_nat @ B @ ( insert_set_nat @ A2 @ B ) ) ).
% subset_insertI
thf(fact_180_subset__insertI,axiom,
! [B: set_set_a,A2: set_a] : ( ord_le3724670747650509150_set_a @ B @ ( insert_set_a @ A2 @ B ) ) ).
% subset_insertI
thf(fact_181_subset__insertI,axiom,
! [B: set_a,A2: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A2 @ B ) ) ).
% subset_insertI
thf(fact_182_subset__insertI,axiom,
! [B: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat @ A2 @ B ) ) ).
% subset_insertI
thf(fact_183_subset__insertI2,axiom,
! [A: set_set_nat,B: set_set_nat,B2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ A @ ( insert_set_nat @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_184_subset__insertI2,axiom,
! [A: set_set_a,B: set_set_a,B2: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_185_subset__insertI2,axiom,
! [A: set_a,B: set_a,B2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_186_subset__insertI2,axiom,
! [A: set_nat,B: set_nat,B2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_187_mk__disjoint__insert,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ A )
=> ? [B3: set_set_nat] :
( ( A
= ( insert_set_nat @ A2 @ B3 ) )
& ~ ( member_set_nat @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_188_mk__disjoint__insert,axiom,
! [A2: set_set_a,A: set_set_set_a] :
( ( member_set_set_a @ A2 @ A )
=> ? [B3: set_set_set_a] :
( ( A
= ( insert_set_set_a @ A2 @ B3 ) )
& ~ ( member_set_set_a @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_189_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ? [B3: set_nat] :
( ( A
= ( insert_nat @ A2 @ B3 ) )
& ~ ( member_nat @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_190_mk__disjoint__insert,axiom,
! [A2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ? [B3: set_set_a] :
( ( A
= ( insert_set_a @ A2 @ B3 ) )
& ~ ( member_set_a @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_191_mk__disjoint__insert,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ? [B3: set_a] :
( ( A
= ( insert_a @ A2 @ B3 ) )
& ~ ( member_a @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_192_verit__la__disequality,axiom,
! [A2: a,B2: a] :
( ( A2 = B2 )
| ~ ( ord_less_eq_a @ A2 @ B2 )
| ~ ( ord_less_eq_a @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_193_verit__la__disequality,axiom,
! [A2: nat,B2: nat] :
( ( A2 = B2 )
| ~ ( ord_less_eq_nat @ A2 @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_194_compl__mono,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ ( uminus5710092332889474511et_nat @ X ) ) ) ).
% compl_mono
thf(fact_195_compl__mono,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ ( uminus_uminus_set_a @ X ) ) ) ).
% compl_mono
thf(fact_196_compl__mono,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ Y ) @ ( uminus6103902357914783669_set_a @ X ) ) ) ).
% compl_mono
thf(fact_197_compl__le__swap1,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) )
=> ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% compl_le_swap1
thf(fact_198_compl__le__swap1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ ( uminus_uminus_set_a @ X ) )
=> ( ord_less_eq_set_a @ X @ ( uminus_uminus_set_a @ Y ) ) ) ).
% compl_le_swap1
thf(fact_199_compl__le__swap1,axiom,
! [Y: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y @ ( uminus6103902357914783669_set_a @ X ) )
=> ( ord_le3724670747650509150_set_a @ X @ ( uminus6103902357914783669_set_a @ Y ) ) ) ).
% compl_le_swap1
thf(fact_200_compl__le__swap2,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ X )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_201_compl__le__swap2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ X )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_202_compl__le__swap2,axiom,
! [Y: set_set_a,X: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ Y ) @ X )
=> ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_203_up__ray__def,axiom,
( up_ray_a
= ( ^ [I3: set_a] :
! [X4: a,Y2: a] :
( ( member_a @ X4 @ I3 )
=> ( ( ord_less_eq_a @ X4 @ Y2 )
=> ( member_a @ Y2 @ I3 ) ) ) ) ) ).
% up_ray_def
thf(fact_204_up__ray__def,axiom,
( up_ray_nat
= ( ^ [I3: set_nat] :
! [X4: nat,Y2: nat] :
( ( member_nat @ X4 @ I3 )
=> ( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( member_nat @ Y2 @ I3 ) ) ) ) ) ).
% up_ray_def
thf(fact_205_atLeast__subset__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_or3904034815786525833_set_a @ X ) @ ( set_or3904034815786525833_set_a @ Y ) )
= ( ord_le3724670747650509150_set_a @ Y @ X ) ) ).
% atLeast_subset_iff
thf(fact_206_atLeast__subset__iff,axiom,
! [X: set_a > extend8495563244428889912nnreal,Y: set_a > extend8495563244428889912nnreal] :
( ( ord_le2718372912061172615nnreal @ ( set_or3682978270108489938nnreal @ X ) @ ( set_or3682978270108489938nnreal @ Y ) )
= ( ord_le6700572704167691815nnreal @ Y @ X ) ) ).
% atLeast_subset_iff
thf(fact_207_atLeast__subset__iff,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_set_a @ ( set_ord_atLeast_a @ X ) @ ( set_ord_atLeast_a @ Y ) )
= ( ord_less_eq_a @ Y @ X ) ) ).
% atLeast_subset_iff
thf(fact_208_atLeast__subset__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or8362275514725411625_set_a @ X ) @ ( set_or8362275514725411625_set_a @ Y ) )
= ( ord_less_eq_set_a @ Y @ X ) ) ).
% atLeast_subset_iff
thf(fact_209_atLeast__subset__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or1731685050470061051et_nat @ X ) @ ( set_or1731685050470061051et_nat @ Y ) )
= ( ord_less_eq_set_nat @ Y @ X ) ) ).
% atLeast_subset_iff
thf(fact_210_atLeast__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atLeast_nat @ X ) @ ( set_ord_atLeast_nat @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% atLeast_subset_iff
thf(fact_211_atLeast__iff,axiom,
! [I2: set_set_a,K: set_set_a] :
( ( member_set_set_a @ I2 @ ( set_or3904034815786525833_set_a @ K ) )
= ( ord_le3724670747650509150_set_a @ K @ I2 ) ) ).
% atLeast_iff
thf(fact_212_atLeast__iff,axiom,
! [I2: set_a > extend8495563244428889912nnreal,K: set_a > extend8495563244428889912nnreal] :
( ( member4180043592386426928nnreal @ I2 @ ( set_or3682978270108489938nnreal @ K ) )
= ( ord_le6700572704167691815nnreal @ K @ I2 ) ) ).
% atLeast_iff
thf(fact_213_atLeast__iff,axiom,
! [I2: set_nat,K: set_nat] :
( ( member_set_nat @ I2 @ ( set_or1731685050470061051et_nat @ K ) )
= ( ord_less_eq_set_nat @ K @ I2 ) ) ).
% atLeast_iff
thf(fact_214_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_215_atLeast__iff,axiom,
! [I2: a,K: a] :
( ( member_a @ I2 @ ( set_ord_atLeast_a @ K ) )
= ( ord_less_eq_a @ K @ I2 ) ) ).
% atLeast_iff
thf(fact_216_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_217_atMost__subset__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_or4016371710855203973_set_a @ X ) @ ( set_or4016371710855203973_set_a @ Y ) )
= ( ord_le3724670747650509150_set_a @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_218_atMost__subset__iff,axiom,
! [X: set_a > extend8495563244428889912nnreal,Y: set_a > extend8495563244428889912nnreal] :
( ( ord_le2718372912061172615nnreal @ ( set_or7674582181447345870nnreal @ X ) @ ( set_or7674582181447345870nnreal @ Y ) )
= ( ord_le6700572704167691815nnreal @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_219_atMost__subset__iff,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_set_a @ ( set_ord_atMost_a @ X ) @ ( set_ord_atMost_a @ Y ) )
= ( ord_less_eq_a @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_220_atMost__subset__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_ord_atMost_set_a @ X ) @ ( set_ord_atMost_set_a @ Y ) )
= ( ord_less_eq_set_a @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_221_atMost__subset__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4236626031148496127et_nat @ X ) @ ( set_or4236626031148496127et_nat @ Y ) )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_222_atMost__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_223_atMost__iff,axiom,
! [I2: set_set_a,K: set_set_a] :
( ( member_set_set_a @ I2 @ ( set_or4016371710855203973_set_a @ K ) )
= ( ord_le3724670747650509150_set_a @ I2 @ K ) ) ).
% atMost_iff
thf(fact_224_atMost__iff,axiom,
! [I2: set_a > extend8495563244428889912nnreal,K: set_a > extend8495563244428889912nnreal] :
( ( member4180043592386426928nnreal @ I2 @ ( set_or7674582181447345870nnreal @ K ) )
= ( ord_le6700572704167691815nnreal @ I2 @ K ) ) ).
% atMost_iff
thf(fact_225_atMost__iff,axiom,
! [I2: set_nat,K: set_nat] :
( ( member_set_nat @ I2 @ ( set_or4236626031148496127et_nat @ K ) )
= ( ord_less_eq_set_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_226_atMost__iff,axiom,
! [I2: set_a,K: set_a] :
( ( member_set_a @ I2 @ ( set_ord_atMost_set_a @ K ) )
= ( ord_less_eq_set_a @ I2 @ K ) ) ).
% atMost_iff
thf(fact_227_atMost__iff,axiom,
! [I2: a,K: a] :
( ( member_a @ I2 @ ( set_ord_atMost_a @ K ) )
= ( ord_less_eq_a @ I2 @ K ) ) ).
% atMost_iff
thf(fact_228_atMost__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat @ I2 @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_229_open__UNIV,axiom,
topolo8477419352202985285open_a @ top_top_set_a ).
% open_UNIV
thf(fact_230_open__UNIV,axiom,
topolo4328251076210115529en_nat @ top_top_set_nat ).
% open_UNIV
thf(fact_231_atMost__eq__UNIV__iff,axiom,
! [X: set_set_a] :
( ( ( set_or4016371710855203973_set_a @ X )
= top_to4027821306633060462_set_a )
= ( X = top_top_set_set_a ) ) ).
% atMost_eq_UNIV_iff
thf(fact_232_atMost__eq__UNIV__iff,axiom,
! [X: set_a > $o] :
( ( ( set_or4722645096382476952et_a_o @ X )
= top_top_set_set_a_o )
= ( X = top_top_set_a_o ) ) ).
% atMost_eq_UNIV_iff
thf(fact_233_atMost__eq__UNIV__iff,axiom,
! [X: nat > $o] :
( ( ( set_ord_atMost_nat_o @ X )
= top_top_set_nat_o )
= ( X = top_top_nat_o ) ) ).
% atMost_eq_UNIV_iff
thf(fact_234_atMost__eq__UNIV__iff,axiom,
! [X: a > $o] :
( ( ( set_ord_atMost_a_o @ X )
= top_top_set_a_o2 )
= ( X = top_top_a_o ) ) ).
% atMost_eq_UNIV_iff
thf(fact_235_atMost__eq__UNIV__iff,axiom,
! [X: set_a] :
( ( ( set_ord_atMost_set_a @ X )
= top_top_set_set_a )
= ( X = top_top_set_a ) ) ).
% atMost_eq_UNIV_iff
thf(fact_236_atMost__eq__UNIV__iff,axiom,
! [X: set_nat] :
( ( ( set_or4236626031148496127et_nat @ X )
= top_top_set_set_nat )
= ( X = top_top_set_nat ) ) ).
% atMost_eq_UNIV_iff
thf(fact_237_atLeast__eq__iff,axiom,
! [X: a,Y: a] :
( ( ( set_ord_atLeast_a @ X )
= ( set_ord_atLeast_a @ Y ) )
= ( X = Y ) ) ).
% atLeast_eq_iff
thf(fact_238_atLeast__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_atLeast_nat @ X )
= ( set_ord_atLeast_nat @ Y ) )
= ( X = Y ) ) ).
% atLeast_eq_iff
thf(fact_239_atLeast__eq__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( set_or8362275514725411625_set_a @ X )
= ( set_or8362275514725411625_set_a @ Y ) )
= ( X = Y ) ) ).
% atLeast_eq_iff
thf(fact_240_atLeast__eq__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( set_or1731685050470061051et_nat @ X )
= ( set_or1731685050470061051et_nat @ Y ) )
= ( X = Y ) ) ).
% atLeast_eq_iff
thf(fact_241_atMost__eq__iff,axiom,
! [X: a,Y: a] :
( ( ( set_ord_atMost_a @ X )
= ( set_ord_atMost_a @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_242_atMost__eq__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( set_ord_atMost_set_a @ X )
= ( set_ord_atMost_set_a @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_243_atMost__eq__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( set_or4236626031148496127et_nat @ X )
= ( set_or4236626031148496127et_nat @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_244_atMost__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_atMost_nat @ X )
= ( set_ord_atMost_nat @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_245_ball__insert,axiom,
! [A2: a,B: set_a,P: a > $o] :
( ( ! [X4: a] :
( ( member_a @ X4 @ ( insert_a @ A2 @ B ) )
=> ( P @ X4 ) ) )
= ( ( P @ A2 )
& ! [X4: a] :
( ( member_a @ X4 @ B )
=> ( P @ X4 ) ) ) ) ).
% ball_insert
thf(fact_246_ball__insert,axiom,
! [A2: set_a,B: set_set_a,P: set_a > $o] :
( ( ! [X4: set_a] :
( ( member_set_a @ X4 @ ( insert_set_a @ A2 @ B ) )
=> ( P @ X4 ) ) )
= ( ( P @ A2 )
& ! [X4: set_a] :
( ( member_set_a @ X4 @ B )
=> ( P @ X4 ) ) ) ) ).
% ball_insert
thf(fact_247_ball__insert,axiom,
! [A2: set_nat,B: set_set_nat,P: set_nat > $o] :
( ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ ( insert_set_nat @ A2 @ B ) )
=> ( P @ X4 ) ) )
= ( ( P @ A2 )
& ! [X4: set_nat] :
( ( member_set_nat @ X4 @ B )
=> ( P @ X4 ) ) ) ) ).
% ball_insert
thf(fact_248_ball__insert,axiom,
! [A2: nat,B: set_nat,P: nat > $o] :
( ( ! [X4: nat] :
( ( member_nat @ X4 @ ( insert_nat @ A2 @ B ) )
=> ( P @ X4 ) ) )
= ( ( P @ A2 )
& ! [X4: nat] :
( ( member_nat @ X4 @ B )
=> ( P @ X4 ) ) ) ) ).
% ball_insert
thf(fact_249_atMost__UNIV__triv,axiom,
( ( set_or4016371710855203973_set_a @ top_top_set_set_a )
= top_to4027821306633060462_set_a ) ).
% atMost_UNIV_triv
thf(fact_250_atMost__UNIV__triv,axiom,
( ( set_ord_atMost_set_a @ top_top_set_a )
= top_top_set_set_a ) ).
% atMost_UNIV_triv
thf(fact_251_atMost__UNIV__triv,axiom,
( ( set_or4236626031148496127et_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% atMost_UNIV_triv
thf(fact_252_iso__tuple__UNIV__I,axiom,
! [X: set_nat] : ( member_set_nat @ X @ top_top_set_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_253_iso__tuple__UNIV__I,axiom,
! [X: set_set_a] : ( member_set_set_a @ X @ top_to4027821306633060462_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_254_iso__tuple__UNIV__I,axiom,
! [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_255_iso__tuple__UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_256_iso__tuple__UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_257_subsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( member_set_nat @ X3 @ B ) )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% subsetI
thf(fact_258_subsetI,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A )
=> ( member_set_set_a @ X3 @ B ) )
=> ( ord_le5722252365846178494_set_a @ A @ B ) ) ).
% subsetI
thf(fact_259_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_260_subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( member_set_a @ X3 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% subsetI
thf(fact_261_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_262_subset__antisym,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_263_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_264_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_265_in__mono,axiom,
! [A: set_set_nat,B: set_set_nat,X: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ X @ A )
=> ( member_set_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_266_in__mono,axiom,
! [A: set_set_set_a,B: set_set_set_a,X: set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( member_set_set_a @ X @ A )
=> ( member_set_set_a @ X @ B ) ) ) ).
% in_mono
thf(fact_267_in__mono,axiom,
! [A: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_268_in__mono,axiom,
! [A: set_set_a,B: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ X @ A )
=> ( member_set_a @ X @ B ) ) ) ).
% in_mono
thf(fact_269_in__mono,axiom,
! [A: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X @ A )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_270_subsetD,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_271_subsetD,axiom,
! [A: set_set_set_a,B: set_set_set_a,C: set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ( member_set_set_a @ C @ A )
=> ( member_set_set_a @ C @ B ) ) ) ).
% subsetD
thf(fact_272_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_273_subsetD,axiom,
! [A: set_set_a,B: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% subsetD
thf(fact_274_subsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_275_equalityE,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ~ ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ~ ( ord_le3724670747650509150_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_276_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_277_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_278_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B4: set_set_nat] :
! [X4: set_nat] :
( ( member_set_nat @ X4 @ A3 )
=> ( member_set_nat @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_279_subset__eq,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [A3: set_set_set_a,B4: set_set_set_a] :
! [X4: set_set_a] :
( ( member_set_set_a @ X4 @ A3 )
=> ( member_set_set_a @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_280_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ( member_nat @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_281_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B4: set_set_a] :
! [X4: set_a] :
( ( member_set_a @ X4 @ A3 )
=> ( member_set_a @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_282_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B4: set_a] :
! [X4: a] :
( ( member_a @ X4 @ A3 )
=> ( member_a @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_283_equalityD1,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_284_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_285_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_286_equalityD2,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le3724670747650509150_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_287_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_288_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_289_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B4: set_set_nat] :
! [T: set_nat] :
( ( member_set_nat @ T @ A3 )
=> ( member_set_nat @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_290_subset__iff,axiom,
( ord_le5722252365846178494_set_a
= ( ^ [A3: set_set_set_a,B4: set_set_set_a] :
! [T: set_set_a] :
( ( member_set_set_a @ T @ A3 )
=> ( member_set_set_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_291_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_292_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B4: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A3 )
=> ( member_set_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_293_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B4: set_a] :
! [T: a] :
( ( member_a @ T @ A3 )
=> ( member_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_294_subset__refl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% subset_refl
thf(fact_295_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_296_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_297_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X3: set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_298_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_299_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_300_subset__trans,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ord_le3724670747650509150_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_301_subset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_302_subset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_303_set__eq__subset,axiom,
( ( ^ [Y3: set_set_a,Z: set_set_a] : ( Y3 = Z ) )
= ( ^ [A3: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_304_set__eq__subset,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_305_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_306_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X4: set_a] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_307_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_308_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_309_top__set__def,axiom,
( top_top_set_set_a
= ( collect_set_a @ top_top_set_a_o ) ) ).
% top_set_def
thf(fact_310_top__set__def,axiom,
( top_top_set_a
= ( collect_a @ top_top_a_o ) ) ).
% top_set_def
thf(fact_311_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_312_t0__space,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ? [U: set_nat] :
( ( topolo4328251076210115529en_nat @ U )
& ( ( member_nat @ X @ U )
!= ( member_nat @ Y @ U ) ) ) ) ).
% t0_space
thf(fact_313_t0__space,axiom,
! [X: a,Y: a] :
( ( X != Y )
=> ? [U: set_a] :
( ( topolo8477419352202985285open_a @ U )
& ( ( member_a @ X @ U )
!= ( member_a @ Y @ U ) ) ) ) ).
% t0_space
thf(fact_314_t1__space,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ? [U: set_nat] :
( ( topolo4328251076210115529en_nat @ U )
& ( member_nat @ X @ U )
& ~ ( member_nat @ Y @ U ) ) ) ).
% t1_space
thf(fact_315_t1__space,axiom,
! [X: a,Y: a] :
( ( X != Y )
=> ? [U: set_a] :
( ( topolo8477419352202985285open_a @ U )
& ( member_a @ X @ U )
& ~ ( member_a @ Y @ U ) ) ) ).
% t1_space
thf(fact_316_separation__t0,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ? [U2: set_nat] :
( ( topolo4328251076210115529en_nat @ U2 )
& ( ( member_nat @ X @ U2 )
!= ( member_nat @ Y @ U2 ) ) ) ) ) ).
% separation_t0
thf(fact_317_separation__t0,axiom,
! [X: a,Y: a] :
( ( X != Y )
= ( ? [U2: set_a] :
( ( topolo8477419352202985285open_a @ U2 )
& ( ( member_a @ X @ U2 )
!= ( member_a @ Y @ U2 ) ) ) ) ) ).
% separation_t0
thf(fact_318_separation__t1,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ? [U2: set_nat] :
( ( topolo4328251076210115529en_nat @ U2 )
& ( member_nat @ X @ U2 )
& ~ ( member_nat @ Y @ U2 ) ) ) ) ).
% separation_t1
thf(fact_319_separation__t1,axiom,
! [X: a,Y: a] :
( ( X != Y )
= ( ? [U2: set_a] :
( ( topolo8477419352202985285open_a @ U2 )
& ( member_a @ X @ U2 )
& ~ ( member_a @ Y @ U2 ) ) ) ) ).
% separation_t1
thf(fact_320_topological__space__class_OopenI,axiom,
! [S2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ S2 )
=> ? [T2: set_nat] :
( ( topolo4328251076210115529en_nat @ T2 )
& ( member_nat @ X3 @ T2 )
& ( ord_less_eq_set_nat @ T2 @ S2 ) ) )
=> ( topolo4328251076210115529en_nat @ S2 ) ) ).
% topological_space_class.openI
thf(fact_321_topological__space__class_OopenI,axiom,
! [S2: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ S2 )
=> ? [T2: set_a] :
( ( topolo8477419352202985285open_a @ T2 )
& ( member_a @ X3 @ T2 )
& ( ord_less_eq_set_a @ T2 @ S2 ) ) )
=> ( topolo8477419352202985285open_a @ S2 ) ) ).
% topological_space_class.openI
thf(fact_322_open__subopen,axiom,
( topolo4328251076210115529en_nat
= ( ^ [S: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ S )
=> ? [T3: set_nat] :
( ( topolo4328251076210115529en_nat @ T3 )
& ( member_nat @ X4 @ T3 )
& ( ord_less_eq_set_nat @ T3 @ S ) ) ) ) ) ).
% open_subopen
thf(fact_323_open__subopen,axiom,
( topolo8477419352202985285open_a
= ( ^ [S: set_a] :
! [X4: a] :
( ( member_a @ X4 @ S )
=> ? [T3: set_a] :
( ( topolo8477419352202985285open_a @ T3 )
& ( member_a @ X4 @ T3 )
& ( ord_less_eq_set_a @ T3 @ S ) ) ) ) ) ).
% open_subopen
thf(fact_324_open__discrete,axiom,
! [A: set_nat] : ( topolo4328251076210115529en_nat @ A ) ).
% open_discrete
thf(fact_325_first__countable__basis,axiom,
! [X: nat] :
? [A4: nat > set_nat] :
( ! [I4: nat] :
( ( member_nat @ X @ ( A4 @ I4 ) )
& ( topolo4328251076210115529en_nat @ ( A4 @ I4 ) ) )
& ! [S3: set_nat] :
( ( ( topolo4328251076210115529en_nat @ S3 )
& ( member_nat @ X @ S3 ) )
=> ? [I5: nat] : ( ord_less_eq_set_nat @ ( A4 @ I5 ) @ S3 ) ) ) ).
% first_countable_basis
thf(fact_326_not__UNIV__eq__Iic,axiom,
! [H: nat] :
( top_top_set_nat
!= ( set_ord_atMost_nat @ H ) ) ).
% not_UNIV_eq_Iic
thf(fact_327_not__UNIV__le__Iic,axiom,
! [H2: nat] :
~ ( ord_less_eq_set_nat @ top_top_set_nat @ ( set_ord_atMost_nat @ H2 ) ) ).
% not_UNIV_le_Iic
thf(fact_328_Ioc__inj,axiom,
! [A2: a,B2: a,C: a,D2: a] :
( ( ( set_or4472690218693186638Most_a @ A2 @ B2 )
= ( set_or4472690218693186638Most_a @ C @ D2 ) )
= ( ( ( ord_less_eq_a @ B2 @ A2 )
& ( ord_less_eq_a @ D2 @ C ) )
| ( ( A2 = C )
& ( B2 = D2 ) ) ) ) ).
% Ioc_inj
thf(fact_329_Ioc__inj,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ( set_or6659071591806873216st_nat @ A2 @ B2 )
= ( set_or6659071591806873216st_nat @ C @ D2 ) )
= ( ( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ D2 @ C ) )
| ( ( A2 = C )
& ( B2 = D2 ) ) ) ) ).
% Ioc_inj
thf(fact_330_Ioc__subset__iff,axiom,
! [A2: a,B2: a,C: a,D2: a] :
( ( ord_less_eq_set_a @ ( set_or4472690218693186638Most_a @ A2 @ B2 ) @ ( set_or4472690218693186638Most_a @ C @ D2 ) )
= ( ( ord_less_eq_a @ B2 @ A2 )
| ( ( ord_less_eq_a @ C @ A2 )
& ( ord_less_eq_a @ B2 @ D2 ) ) ) ) ).
% Ioc_subset_iff
thf(fact_331_Ioc__subset__iff,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_set_nat @ ( set_or6659071591806873216st_nat @ A2 @ B2 ) @ ( set_or6659071591806873216st_nat @ C @ D2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
| ( ( ord_less_eq_nat @ C @ A2 )
& ( ord_less_eq_nat @ B2 @ D2 ) ) ) ) ).
% Ioc_subset_iff
thf(fact_332_open__greaterThanLessThan,axiom,
! [A2: a,B2: a] : ( topolo8477419352202985285open_a @ ( set_or5939364468397584554Than_a @ A2 @ B2 ) ) ).
% open_greaterThanLessThan
thf(fact_333_open__greaterThanLessThan,axiom,
! [A2: nat,B2: nat] : ( topolo4328251076210115529en_nat @ ( set_or5834768355832116004an_nat @ A2 @ B2 ) ) ).
% open_greaterThanLessThan
thf(fact_334_not__Ici__le__Iic,axiom,
! [L: nat,H: nat] :
~ ( ord_less_eq_set_nat @ ( set_ord_atLeast_nat @ L ) @ ( set_ord_atMost_nat @ H ) ) ).
% not_Ici_le_Iic
thf(fact_335_not__Iic__eq__Ici,axiom,
! [H2: nat,L2: nat] :
( ( set_ord_atMost_nat @ H2 )
!= ( set_ord_atLeast_nat @ L2 ) ) ).
% not_Iic_eq_Ici
thf(fact_336_top__apply,axiom,
( top_top_set_a_o
= ( ^ [X4: set_a] : top_top_o ) ) ).
% top_apply
thf(fact_337_top__apply,axiom,
( top_top_nat_o
= ( ^ [X4: nat] : top_top_o ) ) ).
% top_apply
thf(fact_338_top__apply,axiom,
( top_top_a_o
= ( ^ [X4: a] : top_top_o ) ) ).
% top_apply
thf(fact_339_order__refl,axiom,
! [X: set_set_a] : ( ord_le3724670747650509150_set_a @ X @ X ) ).
% order_refl
thf(fact_340_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_341_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_342_order__refl,axiom,
! [X: set_a > extend8495563244428889912nnreal] : ( ord_le6700572704167691815nnreal @ X @ X ) ).
% order_refl
thf(fact_343_order__refl,axiom,
! [X: a] : ( ord_less_eq_a @ X @ X ) ).
% order_refl
thf(fact_344_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_345_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_346_dual__order_Orefl,axiom,
! [A2: set_a > extend8495563244428889912nnreal] : ( ord_le6700572704167691815nnreal @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_347_dual__order_Orefl,axiom,
! [A2: a] : ( ord_less_eq_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_348_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_349_interval__def,axiom,
( interval_a
= ( ^ [I3: set_a] :
! [X4: a,Y2: a,Z2: a] :
( ( member_a @ X4 @ I3 )
=> ( ( member_a @ Z2 @ I3 )
=> ( ( ord_less_eq_a @ X4 @ Y2 )
=> ( ( ord_less_eq_a @ Y2 @ Z2 )
=> ( member_a @ Y2 @ I3 ) ) ) ) ) ) ) ).
% interval_def
thf(fact_350_interval__def,axiom,
( interval_nat
= ( ^ [I3: set_nat] :
! [X4: nat,Y2: nat,Z2: nat] :
( ( member_nat @ X4 @ I3 )
=> ( ( member_nat @ Z2 @ I3 )
=> ( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( member_nat @ Y2 @ I3 ) ) ) ) ) ) ) ).
% interval_def
thf(fact_351_insert__subsetI,axiom,
! [X: nat,A: set_nat,X5: set_nat] :
( ( member_nat @ X @ A )
=> ( ( ord_less_eq_set_nat @ X5 @ A )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_352_insert__subsetI,axiom,
! [X: set_a,A: set_set_a,X5: set_set_a] :
( ( member_set_a @ X @ A )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ A )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_353_insert__subsetI,axiom,
! [X: a,A: set_a,X5: set_a] :
( ( member_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X5 @ A )
=> ( ord_less_eq_set_a @ ( insert_a @ X @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_354_top__greatest,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% top_greatest
thf(fact_355_top_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
= ( A2 = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_356_top_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
=> ( A2 = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_357_sets__uniform__count__measure__eq__UNIV_I1_J,axiom,
( ( sigma_sets_a @ ( nonneg7367794086797660664sure_a @ top_top_set_a ) )
= top_top_set_set_a ) ).
% sets_uniform_count_measure_eq_UNIV(1)
thf(fact_358_sets__uniform__count__measure__eq__UNIV_I1_J,axiom,
( ( sigma_sets_nat @ ( nonneg7031465154080143958re_nat @ top_top_set_nat ) )
= top_top_set_set_nat ) ).
% sets_uniform_count_measure_eq_UNIV(1)
thf(fact_359_compl__less__compl__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ ( uminus_uminus_set_a @ X ) @ ( uminus_uminus_set_a @ Y ) )
= ( ord_less_set_a @ Y @ X ) ) ).
% compl_less_compl_iff
thf(fact_360_compl__less__compl__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ord_less_set_set_a @ ( uminus6103902357914783669_set_a @ X ) @ ( uminus6103902357914783669_set_a @ Y ) )
= ( ord_less_set_set_a @ Y @ X ) ) ).
% compl_less_compl_iff
thf(fact_361_greaterThanLessThan__iff,axiom,
! [I2: set_a,L: set_a,U3: set_a] :
( ( member_set_a @ I2 @ ( set_or6017932776736107018_set_a @ L @ U3 ) )
= ( ( ord_less_set_a @ L @ I2 )
& ( ord_less_set_a @ I2 @ U3 ) ) ) ).
% greaterThanLessThan_iff
thf(fact_362_greaterThanLessThan__iff,axiom,
! [I2: a,L: a,U3: a] :
( ( member_a @ I2 @ ( set_or5939364468397584554Than_a @ L @ U3 ) )
= ( ( ord_less_a @ L @ I2 )
& ( ord_less_a @ I2 @ U3 ) ) ) ).
% greaterThanLessThan_iff
thf(fact_363_greaterThanLessThan__iff,axiom,
! [I2: nat,L: nat,U3: nat] :
( ( member_nat @ I2 @ ( set_or5834768355832116004an_nat @ L @ U3 ) )
= ( ( ord_less_nat @ L @ I2 )
& ( ord_less_nat @ I2 @ U3 ) ) ) ).
% greaterThanLessThan_iff
thf(fact_364_greaterThanAtMost__iff,axiom,
! [I2: set_a,L: set_a,U3: set_a] :
( ( member_set_a @ I2 @ ( set_or2503527069484367278_set_a @ L @ U3 ) )
= ( ( ord_less_set_a @ L @ I2 )
& ( ord_less_eq_set_a @ I2 @ U3 ) ) ) ).
% greaterThanAtMost_iff
thf(fact_365_greaterThanAtMost__iff,axiom,
! [I2: a,L: a,U3: a] :
( ( member_a @ I2 @ ( set_or4472690218693186638Most_a @ L @ U3 ) )
= ( ( ord_less_a @ L @ I2 )
& ( ord_less_eq_a @ I2 @ U3 ) ) ) ).
% greaterThanAtMost_iff
thf(fact_366_greaterThanAtMost__iff,axiom,
! [I2: nat,L: nat,U3: nat] :
( ( member_nat @ I2 @ ( set_or6659071591806873216st_nat @ L @ U3 ) )
= ( ( ord_less_nat @ L @ I2 )
& ( ord_less_eq_nat @ I2 @ U3 ) ) ) ).
% greaterThanAtMost_iff
thf(fact_367_sets__uniform__count__measure__eq__UNIV_I2_J,axiom,
( top_top_set_set_a
= ( sigma_sets_a @ ( nonneg7367794086797660664sure_a @ top_top_set_a ) ) ) ).
% sets_uniform_count_measure_eq_UNIV(2)
thf(fact_368_sets__uniform__count__measure__eq__UNIV_I2_J,axiom,
( top_top_set_set_nat
= ( sigma_sets_nat @ ( nonneg7031465154080143958re_nat @ top_top_set_nat ) ) ) ).
% sets_uniform_count_measure_eq_UNIV(2)
thf(fact_369_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_370_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_371_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_372_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_373_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_374_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_375_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_376_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_377_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_378_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_379_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_380_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_381_order__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_382_order__less__asym_H,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_383_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_384_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_385_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_386_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_387_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_388_dual__order_Ostrict__trans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_389_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_390_order_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_391_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_392_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [N: nat] :
( ( P3 @ N )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_393_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_394_dual__order_Oasym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_395_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_396_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_397_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_398_ord__less__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_399_ord__eq__less__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_400_order_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_401_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_402_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_403_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_404_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_405_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_406_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_407_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_408_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_409_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_410_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_411_order__neq__le__trans,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_412_order__le__neq__trans,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_413_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_414_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_415_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_416_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
& ( X4 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_417_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
| ( X4 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_418_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_419_order_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_420_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B6: nat,A6: nat] :
( ( ord_less_eq_nat @ B6 @ A6 )
& ~ ( ord_less_eq_nat @ A6 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_421_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_422_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_423_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B6: nat,A6: nat] :
( ( ord_less_eq_nat @ B6 @ A6 )
& ( A6 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_424_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B6: nat,A6: nat] :
( ( ord_less_nat @ B6 @ A6 )
| ( A6 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_425_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
& ~ ( ord_less_eq_nat @ B6 @ A6 ) ) ) ) ).
% order.strict_iff_not
thf(fact_426_order_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_427_order_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_428_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_429_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B6: nat] :
( ( ord_less_nat @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_430_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_431_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_432_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_433_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_434_nless__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_435_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_436_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_437_top_Onot__eq__extremum,axiom,
! [A2: set_nat] :
( ( A2 != top_top_set_nat )
= ( ord_less_set_nat @ A2 @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_438_top_Oextremum__strict,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A2 ) ).
% top.extremum_strict
thf(fact_439_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A7: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A7 ) )
= ( ord_less_nat @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_440_compl__less__swap1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_set_a @ Y @ ( uminus_uminus_set_a @ X ) )
=> ( ord_less_set_a @ X @ ( uminus_uminus_set_a @ Y ) ) ) ).
% compl_less_swap1
thf(fact_441_compl__less__swap1,axiom,
! [Y: set_set_a,X: set_set_a] :
( ( ord_less_set_set_a @ Y @ ( uminus6103902357914783669_set_a @ X ) )
=> ( ord_less_set_set_a @ X @ ( uminus6103902357914783669_set_a @ Y ) ) ) ).
% compl_less_swap1
thf(fact_442_compl__less__swap2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_set_a @ ( uminus_uminus_set_a @ Y ) @ X )
=> ( ord_less_set_a @ ( uminus_uminus_set_a @ X ) @ Y ) ) ).
% compl_less_swap2
thf(fact_443_compl__less__swap2,axiom,
! [Y: set_set_a,X: set_set_a] :
( ( ord_less_set_set_a @ ( uminus6103902357914783669_set_a @ Y ) @ X )
=> ( ord_less_set_set_a @ ( uminus6103902357914783669_set_a @ X ) @ Y ) ) ).
% compl_less_swap2
thf(fact_444_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_445_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_446_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_447_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_448_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_449_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_450_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_451_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_452_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
& ( ord_less_eq_nat @ B6 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_453_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_454_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_455_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_456_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ B6 @ A6 )
& ( ord_less_eq_nat @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_457_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_458_order__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_459_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_460_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_461_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_462_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_463_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_464_le__cases3,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_465_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_466_open__left,axiom,
! [S2: set_a,X: a,Y: a] :
( ( topolo8477419352202985285open_a @ S2 )
=> ( ( member_a @ X @ S2 )
=> ( ( ord_less_a @ Y @ X )
=> ? [B5: a] :
( ( ord_less_a @ B5 @ X )
& ( ord_less_eq_set_a @ ( set_or4472690218693186638Most_a @ B5 @ X ) @ S2 ) ) ) ) ) ).
% open_left
thf(fact_467_open__left,axiom,
! [S2: set_nat,X: nat,Y: nat] :
( ( topolo4328251076210115529en_nat @ S2 )
=> ( ( member_nat @ X @ S2 )
=> ( ( ord_less_nat @ Y @ X )
=> ? [B5: nat] :
( ( ord_less_nat @ B5 @ X )
& ( ord_less_eq_set_nat @ ( set_or6659071591806873216st_nat @ B5 @ X ) @ S2 ) ) ) ) ) ).
% open_left
thf(fact_468_top__empty__eq,axiom,
( top_top_set_a_o
= ( ^ [X4: set_a] : ( member_set_a @ X4 @ top_top_set_set_a ) ) ) ).
% top_empty_eq
thf(fact_469_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X4: a] : ( member_a @ X4 @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_470_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X4: nat] : ( member_nat @ X4 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_471_complete__interval,axiom,
! [A2: nat,B2: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C4: nat] :
( ( ord_less_eq_nat @ A2 @ C4 )
& ( ord_less_eq_nat @ C4 @ B2 )
& ! [X7: nat] :
( ( ( ord_less_eq_nat @ A2 @ X7 )
& ( ord_less_nat @ X7 @ C4 ) )
=> ( P @ X7 ) )
& ! [D3: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A2 @ X3 )
& ( ord_less_nat @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D3 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_472_pinf_I6_J,axiom,
! [T4: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ~ ( ord_less_eq_nat @ X7 @ T4 ) ) ).
% pinf(6)
thf(fact_473_pinf_I8_J,axiom,
! [T4: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ( ord_less_eq_nat @ T4 @ X7 ) ) ).
% pinf(8)
thf(fact_474_minf_I6_J,axiom,
! [T4: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ( ord_less_eq_nat @ X7 @ T4 ) ) ).
% minf(6)
thf(fact_475_minf_I8_J,axiom,
! [T4: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ~ ( ord_less_eq_nat @ T4 @ X7 ) ) ).
% minf(8)
thf(fact_476_psubsetD,axiom,
! [A: set_set_a,B: set_set_a,C: set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% psubsetD
thf(fact_477_psubsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% psubsetD
thf(fact_478_minf_I7_J,axiom,
! [T4: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ~ ( ord_less_nat @ T4 @ X7 ) ) ).
% minf(7)
thf(fact_479_minf_I5_J,axiom,
! [T4: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ( ord_less_nat @ X7 @ T4 ) ) ).
% minf(5)
thf(fact_480_minf_I4_J,axiom,
! [T4: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ( X7 != T4 ) ) ).
% minf(4)
thf(fact_481_minf_I3_J,axiom,
! [T4: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ( X7 != T4 ) ) ).
% minf(3)
thf(fact_482_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P4 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(2)
thf(fact_483_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P4 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(1)
thf(fact_484_pinf_I7_J,axiom,
! [T4: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ( ord_less_nat @ T4 @ X7 ) ) ).
% pinf(7)
thf(fact_485_pinf_I5_J,axiom,
! [T4: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ~ ( ord_less_nat @ X7 @ T4 ) ) ).
% pinf(5)
thf(fact_486_pinf_I4_J,axiom,
! [T4: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ( X7 != T4 ) ) ).
% pinf(4)
thf(fact_487_pinf_I3_J,axiom,
! [T4: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ( X7 != T4 ) ) ).
% pinf(3)
thf(fact_488_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P4 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_489_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P4 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_490_top_Oordering__top__axioms,axiom,
ordering_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat @ top_top_set_nat ).
% top.ordering_top_axioms
thf(fact_491_subset__Compl__singleton,axiom,
! [A: set_nat,B2: nat] :
( ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) )
= ( ~ ( member_nat @ B2 @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_492_subset__Compl__singleton,axiom,
! [A: set_a,B2: a] :
( ( ord_less_eq_set_a @ A @ ( uminus_uminus_set_a @ ( insert_a @ B2 @ bot_bot_set_a ) ) )
= ( ~ ( member_a @ B2 @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_493_subset__Compl__singleton,axiom,
! [A: set_set_a,B2: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( uminus6103902357914783669_set_a @ ( insert_set_a @ B2 @ bot_bot_set_set_a ) ) )
= ( ~ ( member_set_a @ B2 @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_494_ivl__disj__un__two_I2_J,axiom,
! [L: nat,M3: nat,U3: nat] :
( ( ord_less_eq_nat @ L @ M3 )
=> ( ( ord_less_nat @ M3 @ U3 )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M3 ) @ ( set_or5834768355832116004an_nat @ M3 @ U3 ) )
= ( set_or5834768355832116004an_nat @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_495_image__eqI,axiom,
! [B2: nat,F: nat > nat,X: nat,A: set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_496_image__eqI,axiom,
! [B2: set_a,F: set_a > set_a,X: set_a,A: set_set_a] :
( ( B2
= ( F @ X ) )
=> ( ( member_set_a @ X @ A )
=> ( member_set_a @ B2 @ ( image_set_a_set_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_497_image__eqI,axiom,
! [B2: a,F: set_a > a,X: set_a,A: set_set_a] :
( ( B2
= ( F @ X ) )
=> ( ( member_set_a @ X @ A )
=> ( member_a @ B2 @ ( image_set_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_498_image__eqI,axiom,
! [B2: set_a,F: a > set_a,X: a,A: set_a] :
( ( B2
= ( F @ X ) )
=> ( ( member_a @ X @ A )
=> ( member_set_a @ B2 @ ( image_a_set_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_499_image__eqI,axiom,
! [B2: a,F: a > a,X: a,A: set_a] :
( ( B2
= ( F @ X ) )
=> ( ( member_a @ X @ A )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_500_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_501_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_502_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_503_all__not__in__conv,axiom,
! [A: set_set_a] :
( ( ! [X4: set_a] :
~ ( member_set_a @ X4 @ A ) )
= ( A = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_504_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X4: a] :
~ ( member_a @ X4 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_505_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X4: nat] :
~ ( member_nat @ X4 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_506_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X4: nat] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_507_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X4: nat] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_508_UnCI,axiom,
! [C: set_a,B: set_set_a,A: set_set_a] :
( ( ~ ( member_set_a @ C @ B )
=> ( member_set_a @ C @ A ) )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_509_UnCI,axiom,
! [C: a,B: set_a,A: set_a] :
( ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ A ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_510_Un__iff,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) )
= ( ( member_set_a @ C @ A )
| ( member_set_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_511_Un__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
| ( member_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_512_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_513_empty__is__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_514_image__is__empty,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_515_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_516_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_517_le__sup__iff,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z3 )
= ( ( ord_less_eq_nat @ X @ Z3 )
& ( ord_less_eq_nat @ Y @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_518_sup_Obounded__iff,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_519_atLeastAtMost__iff,axiom,
! [I2: set_a,L: set_a,U3: set_a] :
( ( member_set_a @ I2 @ ( set_or6288561110385358355_set_a @ L @ U3 ) )
= ( ( ord_less_eq_set_a @ L @ I2 )
& ( ord_less_eq_set_a @ I2 @ U3 ) ) ) ).
% atLeastAtMost_iff
thf(fact_520_atLeastAtMost__iff,axiom,
! [I2: a,L: a,U3: a] :
( ( member_a @ I2 @ ( set_or672772299803893939Most_a @ L @ U3 ) )
= ( ( ord_less_eq_a @ L @ I2 )
& ( ord_less_eq_a @ I2 @ U3 ) ) ) ).
% atLeastAtMost_iff
thf(fact_521_atLeastAtMost__iff,axiom,
! [I2: nat,L: nat,U3: nat] :
( ( member_nat @ I2 @ ( set_or1269000886237332187st_nat @ L @ U3 ) )
= ( ( ord_less_eq_nat @ L @ I2 )
& ( ord_less_eq_nat @ I2 @ U3 ) ) ) ).
% atLeastAtMost_iff
thf(fact_522_Icc__eq__Icc,axiom,
! [L: nat,H2: nat,L2: nat,H: nat] :
( ( ( set_or1269000886237332187st_nat @ L @ H2 )
= ( set_or1269000886237332187st_nat @ L2 @ H ) )
= ( ( ( L = L2 )
& ( H2 = H ) )
| ( ~ ( ord_less_eq_nat @ L @ H2 )
& ~ ( ord_less_eq_nat @ L2 @ H ) ) ) ) ).
% Icc_eq_Icc
thf(fact_523_image__insert,axiom,
! [F: nat > nat,A2: nat,B: set_nat] :
( ( image_nat_nat @ F @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ ( F @ A2 ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_524_insert__image,axiom,
! [X: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( insert_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A ) )
= ( image_nat_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_525_insert__image,axiom,
! [X: set_a,A: set_set_a,F: set_a > nat] :
( ( member_set_a @ X @ A )
=> ( ( insert_nat @ ( F @ X ) @ ( image_set_a_nat @ F @ A ) )
= ( image_set_a_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_526_insert__image,axiom,
! [X: a,A: set_a,F: a > nat] :
( ( member_a @ X @ A )
=> ( ( insert_nat @ ( F @ X ) @ ( image_a_nat @ F @ A ) )
= ( image_a_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_527_singletonI,axiom,
! [A2: set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_528_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_529_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_530_sup__top__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X )
= top_top_set_nat ) ).
% sup_top_left
thf(fact_531_sup__top__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% sup_top_right
thf(fact_532_boolean__algebra_Odisj__one__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_left
thf(fact_533_boolean__algebra_Odisj__one__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_right
thf(fact_534_open__empty,axiom,
topolo4328251076210115529en_nat @ bot_bot_set_nat ).
% open_empty
thf(fact_535_sup__bot__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_536_sup__bot__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% sup_bot_right
thf(fact_537_bot__eq__sup__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X @ Y ) )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_538_sup__eq__bot__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_539_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_540_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_541_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_542_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_543_Un__empty,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_544_Un__insert__left,axiom,
! [A2: nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( insert_nat @ A2 @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_545_Un__insert__right,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_546_atLeast__empty__triv,axiom,
( ( set_or1731685050470061051et_nat @ bot_bot_set_nat )
= top_top_set_set_nat ) ).
% atLeast_empty_triv
thf(fact_547_boolean__algebra_Ocompl__zero,axiom,
( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.compl_zero
thf(fact_548_boolean__algebra_Ocompl__zero,axiom,
( ( uminus_uminus_set_a @ bot_bot_set_a )
= top_top_set_a ) ).
% boolean_algebra.compl_zero
thf(fact_549_boolean__algebra_Ocompl__zero,axiom,
( ( uminus6103902357914783669_set_a @ bot_bot_set_set_a )
= top_top_set_set_a ) ).
% boolean_algebra.compl_zero
thf(fact_550_boolean__algebra_Ocompl__one,axiom,
( ( uminus5710092332889474511et_nat @ top_top_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.compl_one
thf(fact_551_boolean__algebra_Ocompl__one,axiom,
( ( uminus_uminus_set_a @ top_top_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.compl_one
thf(fact_552_boolean__algebra_Ocompl__one,axiom,
( ( uminus6103902357914783669_set_a @ top_top_set_set_a )
= bot_bot_set_set_a ) ).
% boolean_algebra.compl_one
thf(fact_553_atLeastatMost__subset__iff,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) @ ( set_or1269000886237332187st_nat @ C @ D2 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( ( ord_less_eq_nat @ C @ A2 )
& ( ord_less_eq_nat @ B2 @ D2 ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_554_atLeastatMost__empty__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( set_or1269000886237332187st_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_555_atLeastatMost__empty__iff2,axiom,
! [A2: nat,B2: nat] :
( ( bot_bot_set_nat
= ( set_or1269000886237332187st_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_556_atLeastatMost__empty_H,axiom,
! [A2: nat,B2: nat] :
( ~ ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( set_or1269000886237332187st_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ).
% atLeastatMost_empty'
thf(fact_557_singleton__insert__inj__eq,axiom,
! [B2: nat,A2: nat,A: set_nat] :
( ( ( insert_nat @ B2 @ bot_bot_set_nat )
= ( insert_nat @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_558_singleton__insert__inj__eq_H,axiom,
! [A2: nat,A: set_nat,B2: nat] :
( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_559_atLeastatMost__empty,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( set_or1269000886237332187st_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ).
% atLeastatMost_empty
thf(fact_560_boolean__algebra_Odisj__cancel__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ ( uminus5710092332889474511et_nat @ X ) )
= top_top_set_nat ) ).
% boolean_algebra.disj_cancel_right
thf(fact_561_boolean__algebra_Odisj__cancel__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ ( uminus_uminus_set_a @ X ) )
= top_top_set_a ) ).
% boolean_algebra.disj_cancel_right
thf(fact_562_boolean__algebra_Odisj__cancel__right,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ X @ ( uminus6103902357914783669_set_a @ X ) )
= top_top_set_set_a ) ).
% boolean_algebra.disj_cancel_right
thf(fact_563_boolean__algebra_Odisj__cancel__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ X )
= top_top_set_nat ) ).
% boolean_algebra.disj_cancel_left
thf(fact_564_boolean__algebra_Odisj__cancel__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ ( uminus_uminus_set_a @ X ) @ X )
= top_top_set_a ) ).
% boolean_algebra.disj_cancel_left
thf(fact_565_boolean__algebra_Odisj__cancel__left,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ ( uminus6103902357914783669_set_a @ X ) @ X )
= top_top_set_set_a ) ).
% boolean_algebra.disj_cancel_left
thf(fact_566_sup__compl__top__left2,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) )
= top_top_set_nat ) ).
% sup_compl_top_left2
thf(fact_567_sup__compl__top__left2,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ ( uminus_uminus_set_a @ X ) @ Y ) )
= top_top_set_a ) ).
% sup_compl_top_left2
thf(fact_568_sup__compl__top__left2,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( sup_sup_set_set_a @ X @ ( sup_sup_set_set_a @ ( uminus6103902357914783669_set_a @ X ) @ Y ) )
= top_top_set_set_a ) ).
% sup_compl_top_left2
thf(fact_569_sup__compl__top__left1,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( sup_sup_set_nat @ X @ Y ) )
= top_top_set_nat ) ).
% sup_compl_top_left1
thf(fact_570_sup__compl__top__left1,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ ( uminus_uminus_set_a @ X ) @ ( sup_sup_set_a @ X @ Y ) )
= top_top_set_a ) ).
% sup_compl_top_left1
thf(fact_571_sup__compl__top__left1,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( sup_sup_set_set_a @ ( uminus6103902357914783669_set_a @ X ) @ ( sup_sup_set_set_a @ X @ Y ) )
= top_top_set_set_a ) ).
% sup_compl_top_left1
thf(fact_572_atLeastLessThan__iff,axiom,
! [I2: set_a,L: set_a,U3: set_a] :
( ( member_set_a @ I2 @ ( set_or2348907005316661231_set_a @ L @ U3 ) )
= ( ( ord_less_eq_set_a @ L @ I2 )
& ( ord_less_set_a @ I2 @ U3 ) ) ) ).
% atLeastLessThan_iff
thf(fact_573_atLeastLessThan__iff,axiom,
! [I2: a,L: a,U3: a] :
( ( member_a @ I2 @ ( set_or5139330845457685135Than_a @ L @ U3 ) )
= ( ( ord_less_eq_a @ L @ I2 )
& ( ord_less_a @ I2 @ U3 ) ) ) ).
% atLeastLessThan_iff
thf(fact_574_atLeastLessThan__iff,axiom,
! [I2: nat,L: nat,U3: nat] :
( ( member_nat @ I2 @ ( set_or4665077453230672383an_nat @ L @ U3 ) )
= ( ( ord_less_eq_nat @ L @ I2 )
& ( ord_less_nat @ I2 @ U3 ) ) ) ).
% atLeastLessThan_iff
thf(fact_575_atLeastAtMost__singleton__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ( set_or1269000886237332187st_nat @ A2 @ B2 )
= ( insert_nat @ C @ bot_bot_set_nat ) )
= ( ( A2 = B2 )
& ( B2 = C ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_576_atLeastAtMost__singleton,axiom,
! [A2: nat] :
( ( set_or1269000886237332187st_nat @ A2 @ A2 )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% atLeastAtMost_singleton
thf(fact_577_ivl__subset,axiom,
! [I2: nat,J: nat,M3: nat,N2: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ J ) @ ( set_or4665077453230672383an_nat @ M3 @ N2 ) )
= ( ( ord_less_eq_nat @ J @ I2 )
| ( ( ord_less_eq_nat @ M3 @ I2 )
& ( ord_less_eq_nat @ J @ N2 ) ) ) ) ).
% ivl_subset
thf(fact_578_atLeastLessThan__empty,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( set_or4665077453230672383an_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ).
% atLeastLessThan_empty
thf(fact_579_atLeastLessThan__empty__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( set_or4665077453230672383an_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ~ ( ord_less_nat @ A2 @ B2 ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_580_atLeastLessThan__empty__iff2,axiom,
! [A2: nat,B2: nat] :
( ( bot_bot_set_nat
= ( set_or4665077453230672383an_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_nat @ A2 @ B2 ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_581_greaterThanLessThan__empty,axiom,
! [L: nat,K: nat] :
( ( ord_less_eq_nat @ L @ K )
=> ( ( set_or5834768355832116004an_nat @ K @ L )
= bot_bot_set_nat ) ) ).
% greaterThanLessThan_empty
thf(fact_582_greaterThanAtMost__empty,axiom,
! [L: nat,K: nat] :
( ( ord_less_eq_nat @ L @ K )
=> ( ( set_or6659071591806873216st_nat @ K @ L )
= bot_bot_set_nat ) ) ).
% greaterThanAtMost_empty
thf(fact_583_greaterThanAtMost__empty__iff,axiom,
! [K: nat,L: nat] :
( ( ( set_or6659071591806873216st_nat @ K @ L )
= bot_bot_set_nat )
= ( ~ ( ord_less_nat @ K @ L ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_584_greaterThanAtMost__empty__iff2,axiom,
! [K: nat,L: nat] :
( ( bot_bot_set_nat
= ( set_or6659071591806873216st_nat @ K @ L ) )
= ( ~ ( ord_less_nat @ K @ L ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_585_Icc__subset__Iic__iff,axiom,
! [L: nat,H2: nat,H: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H2 ) @ ( set_ord_atMost_nat @ H ) )
= ( ~ ( ord_less_eq_nat @ L @ H2 )
| ( ord_less_eq_nat @ H2 @ H ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_586_Icc__subset__Ici__iff,axiom,
! [L: nat,H2: nat,L2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H2 ) @ ( set_ord_atLeast_nat @ L2 ) )
= ( ~ ( ord_less_eq_nat @ L @ H2 )
| ( ord_less_eq_nat @ L2 @ L ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_587_ivl__disj__un__two__touch_I2_J,axiom,
! [L: nat,M3: nat,U3: nat] :
( ( ord_less_eq_nat @ L @ M3 )
=> ( ( ord_less_nat @ M3 @ U3 )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M3 ) @ ( set_or4665077453230672383an_nat @ M3 @ U3 ) )
= ( set_or4665077453230672383an_nat @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_588_ivl__disj__un__two_I7_J,axiom,
! [L: nat,M3: nat,U3: nat] :
( ( ord_less_eq_nat @ L @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ U3 )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ M3 ) @ ( set_or1269000886237332187st_nat @ M3 @ U3 ) )
= ( set_or1269000886237332187st_nat @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_589_ivl__disj__un__two_I3_J,axiom,
! [L: nat,M3: nat,U3: nat] :
( ( ord_less_eq_nat @ L @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ U3 )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ M3 ) @ ( set_or4665077453230672383an_nat @ M3 @ U3 ) )
= ( set_or4665077453230672383an_nat @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_590_ivl__disj__un__singleton_I6_J,axiom,
! [L: nat,U3: nat] :
( ( ord_less_eq_nat @ L @ U3 )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ U3 ) @ ( insert_nat @ U3 @ bot_bot_set_nat ) )
= ( set_or1269000886237332187st_nat @ L @ U3 ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_591_ivl__disj__un__two__touch_I4_J,axiom,
! [L: nat,M3: nat,U3: nat] :
( ( ord_less_eq_nat @ L @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ U3 )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M3 ) @ ( set_or1269000886237332187st_nat @ M3 @ U3 ) )
= ( set_or1269000886237332187st_nat @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_592_atLeastAtMost__singleton_H,axiom,
! [A2: nat,B2: nat] :
( ( A2 = B2 )
=> ( ( set_or1269000886237332187st_nat @ A2 @ B2 )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_593_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_594_UnE,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) )
=> ( ~ ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% UnE
thf(fact_595_UnE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
=> ( ~ ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% UnE
thf(fact_596_UnI1,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_597_UnI1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_598_UnI2,axiom,
! [C: set_a,B: set_set_a,A: set_set_a] :
( ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_599_UnI2,axiom,
! [C: a,B: set_a,A: set_a] :
( ( member_a @ C @ B )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_600_emptyE,axiom,
! [A2: set_a] :
~ ( member_set_a @ A2 @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_601_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_602_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_603_imageI,axiom,
! [X: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_604_imageI,axiom,
! [X: set_a,A: set_set_a,F: set_a > set_a] :
( ( member_set_a @ X @ A )
=> ( member_set_a @ ( F @ X ) @ ( image_set_a_set_a @ F @ A ) ) ) ).
% imageI
thf(fact_605_imageI,axiom,
! [X: set_a,A: set_set_a,F: set_a > a] :
( ( member_set_a @ X @ A )
=> ( member_a @ ( F @ X ) @ ( image_set_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_606_imageI,axiom,
! [X: a,A: set_a,F: a > set_a] :
( ( member_a @ X @ A )
=> ( member_set_a @ ( F @ X ) @ ( image_a_set_a @ F @ A ) ) ) ).
% imageI
thf(fact_607_imageI,axiom,
! [X: a,A: set_a,F: a > a] :
( ( member_a @ X @ A )
=> ( member_a @ ( F @ X ) @ ( image_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_608_equals0D,axiom,
! [A: set_set_a,A2: set_a] :
( ( A = bot_bot_set_set_a )
=> ~ ( member_set_a @ A2 @ A ) ) ).
% equals0D
thf(fact_609_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_610_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_611_equals0I,axiom,
! [A: set_set_a] :
( ! [Y4: set_a] :
~ ( member_set_a @ Y4 @ A )
=> ( A = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_612_equals0I,axiom,
! [A: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_613_equals0I,axiom,
! [A: set_nat] :
( ! [Y4: nat] :
~ ( member_nat @ Y4 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_614_image__Un,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_615_image__iff,axiom,
! [Z3: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Z3 @ ( image_nat_nat @ F @ A ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( Z3
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_616_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X7: nat] :
( ( member_nat @ X7 @ ( image_nat_nat @ F @ A ) )
& ( P @ X7 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_617_ex__in__conv,axiom,
! [A: set_set_a] :
( ( ? [X4: set_a] : ( member_set_a @ X4 @ A ) )
= ( A != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_618_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X4: a] : ( member_a @ X4 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_619_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X4: nat] : ( member_nat @ X4 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_620_image__cong,axiom,
! [M: set_nat,N3: set_nat,F: nat > nat,G: nat > nat] :
( ( M = N3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N3 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_nat @ F @ M )
= ( image_nat_nat @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_621_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X7: nat] :
( ( member_nat @ X7 @ A )
=> ( P @ ( F @ X7 ) ) ) ) ).
% ball_imageD
thf(fact_622_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_623_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B2: nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_624_rev__image__eqI,axiom,
! [X: set_a,A: set_set_a,B2: set_a,F: set_a > set_a] :
( ( member_set_a @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_set_a @ B2 @ ( image_set_a_set_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_625_rev__image__eqI,axiom,
! [X: set_a,A: set_set_a,B2: a,F: set_a > a] :
( ( member_set_a @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_a @ B2 @ ( image_set_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_626_rev__image__eqI,axiom,
! [X: a,A: set_a,B2: set_a,F: a > set_a] :
( ( member_a @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_set_a @ B2 @ ( image_a_set_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_627_rev__image__eqI,axiom,
! [X: a,A: set_a,B2: a,F: a > a] :
( ( member_a @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_628_Un__empty__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Un_empty_right
thf(fact_629_singleton__Un__iff,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ( ( insert_nat @ X @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A @ B ) )
= ( ( ( A = bot_bot_set_nat )
& ( B
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_630_Un__singleton__iff,axiom,
! [A: set_nat,B: set_nat,X: nat] :
( ( ( sup_sup_set_nat @ A @ B )
= ( insert_nat @ X @ bot_bot_set_nat ) )
= ( ( ( A = bot_bot_set_nat )
& ( B
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_631_insert__is__Un,axiom,
( insert_nat
= ( ^ [A6: nat] : ( sup_sup_set_nat @ ( insert_nat @ A6 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_632_ivl__disj__un__two_I4_J,axiom,
! [L: nat,M3: nat,U3: nat] :
( ( ord_less_eq_nat @ L @ M3 )
=> ( ( ord_less_nat @ M3 @ U3 )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M3 ) @ ( set_or5834768355832116004an_nat @ M3 @ U3 ) )
= ( set_or4665077453230672383an_nat @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_633_atLeastAtMost__eq__UNIV__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( set_or4548717258645045905et_nat @ X @ Y )
= top_top_set_set_nat )
= ( ( X = bot_bot_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_634_ivl__disj__un__singleton_I3_J,axiom,
! [L: nat,U3: nat] :
( ( ord_less_nat @ L @ U3 )
=> ( ( sup_sup_set_nat @ ( insert_nat @ L @ bot_bot_set_nat ) @ ( set_or5834768355832116004an_nat @ L @ U3 ) )
= ( set_or4665077453230672383an_nat @ L @ U3 ) ) ) ).
% ivl_disj_un_singleton(3)
thf(fact_635_ivl__disj__un__singleton_I5_J,axiom,
! [L: nat,U3: nat] :
( ( ord_less_eq_nat @ L @ U3 )
=> ( ( sup_sup_set_nat @ ( insert_nat @ L @ bot_bot_set_nat ) @ ( set_or6659071591806873216st_nat @ L @ U3 ) )
= ( set_or1269000886237332187st_nat @ L @ U3 ) ) ) ).
% ivl_disj_un_singleton(5)
thf(fact_636_ivl__disj__un__two_I8_J,axiom,
! [L: nat,M3: nat,U3: nat] :
( ( ord_less_eq_nat @ L @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ U3 )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M3 ) @ ( set_or6659071591806873216st_nat @ M3 @ U3 ) )
= ( set_or1269000886237332187st_nat @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_637_ivl__disj__un__one_I8_J,axiom,
! [L: nat,U3: nat] :
( ( ord_less_eq_nat @ L @ U3 )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ U3 ) @ ( set_ord_atLeast_nat @ U3 ) )
= ( set_ord_atLeast_nat @ L ) ) ) ).
% ivl_disj_un_one(8)
thf(fact_638_range__eq__singletonD,axiom,
! [F: nat > nat,A2: nat,X: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( ( F @ X )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_639_not__UNIV__eq__Icc,axiom,
! [L2: nat,H: nat] :
( top_top_set_nat
!= ( set_or1269000886237332187st_nat @ L2 @ H ) ) ).
% not_UNIV_eq_Icc
thf(fact_640_atLeastLessThan__inj_I2_J,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ( set_or4665077453230672383an_nat @ A2 @ B2 )
= ( set_or4665077453230672383an_nat @ C @ D2 ) )
=> ( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( B2 = D2 ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_641_atLeastLessThan__inj_I1_J,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ( set_or4665077453230672383an_nat @ A2 @ B2 )
= ( set_or4665077453230672383an_nat @ C @ D2 ) )
=> ( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( A2 = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_642_Ico__eq__Ico,axiom,
! [L: nat,H2: nat,L2: nat,H: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H2 )
= ( set_or4665077453230672383an_nat @ L2 @ H ) )
= ( ( ( L = L2 )
& ( H2 = H ) )
| ( ~ ( ord_less_nat @ L @ H2 )
& ~ ( ord_less_nat @ L2 @ H ) ) ) ) ).
% Ico_eq_Ico
thf(fact_643_atLeastLessThan__eq__iff,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ( set_or4665077453230672383an_nat @ A2 @ B2 )
= ( set_or4665077453230672383an_nat @ C @ D2 ) )
= ( ( A2 = C )
& ( B2 = D2 ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_644_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_645_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_646_le__supE,axiom,
! [A2: nat,B2: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_nat @ A2 @ X )
=> ~ ( ord_less_eq_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_647_le__supI,axiom,
! [A2: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_648_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_649_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_650_le__supI1,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_651_le__supI2,axiom,
! [X: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_652_sup_Omono,axiom,
! [C: nat,A2: nat,D2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D2 @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_653_sup__mono,axiom,
! [A2: nat,C: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_654_sup__least,axiom,
! [Y: nat,X: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z3 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z3 ) @ X ) ) ) ).
% sup_least
thf(fact_655_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y2: nat] :
( ( sup_sup_nat @ X4 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_656_sup_OorderE,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_657_sup_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_658_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ ( F @ X3 @ Y4 ) )
=> ( ! [X3: nat,Y4: nat,Z4: nat] :
( ( ord_less_eq_nat @ Y4 @ X3 )
=> ( ( ord_less_eq_nat @ Z4 @ X3 )
=> ( ord_less_eq_nat @ ( F @ Y4 @ Z4 ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_659_sup_Oabsorb1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_660_sup_Oabsorb2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_661_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_662_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_663_sup_OboundedE,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_664_sup_OboundedI,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_665_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B6: nat,A6: nat] :
( A6
= ( sup_sup_nat @ A6 @ B6 ) ) ) ) ).
% sup.order_iff
thf(fact_666_sup_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_667_sup_Ocobounded2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_668_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B6: nat,A6: nat] :
( ( sup_sup_nat @ A6 @ B6 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_669_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B6: nat] :
( ( sup_sup_nat @ A6 @ B6 )
= B6 ) ) ) ).
% sup.absorb_iff2
thf(fact_670_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_671_sup_OcoboundedI2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_672_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_673_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_674_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_675_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_676_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_677_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_678_less__supI1,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ X @ A2 )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_679_less__supI2,axiom,
! [X: nat,B2: nat,A2: nat] :
( ( ord_less_nat @ X @ B2 )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_680_sup_Oabsorb3,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_681_sup_Oabsorb4,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_682_sup_Ostrict__boundedE,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_683_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B6: nat,A6: nat] :
( ( A6
= ( sup_sup_nat @ A6 @ B6 ) )
& ( A6 != B6 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_684_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ C @ A2 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_685_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_686_bot_Onot__eq__extremum,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_687_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_688_bot_Oextremum__strict,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_689_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_690_rangeI,axiom,
! [F: nat > nat,X: nat] : ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_691_rangeI,axiom,
! [F: nat > set_a,X: nat] : ( member_set_a @ ( F @ X ) @ ( image_nat_set_a @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_692_rangeI,axiom,
! [F: nat > a,X: nat] : ( member_a @ ( F @ X ) @ ( image_nat_a @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_693_range__eqI,axiom,
! [B2: nat,F: nat > nat,X: nat] :
( ( B2
= ( F @ X ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_694_range__eqI,axiom,
! [B2: set_a,F: nat > set_a,X: nat] :
( ( B2
= ( F @ X ) )
=> ( member_set_a @ B2 @ ( image_nat_set_a @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_695_range__eqI,axiom,
! [B2: a,F: nat > a,X: nat] :
( ( B2
= ( F @ X ) )
=> ( member_a @ B2 @ ( image_nat_a @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_696_subset__image__iff,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_697_image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_698_subset__imageE,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A )
=> ( B
!= ( image_nat_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_699_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_700_image__subsetI,axiom,
! [A: set_set_a,F: set_a > set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( member_set_a @ ( F @ X3 ) @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_701_image__subsetI,axiom,
! [A: set_set_a,F: set_a > a,B: set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_702_image__subsetI,axiom,
! [A: set_a,F: a > set_a,B: set_set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_set_a @ ( F @ X3 ) @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_703_image__subsetI,axiom,
! [A: set_a,F: a > a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_704_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_705_sets__range,axiom,
! [A: set_a > set_a,I: set_set_a,M: sigma_measure_a,I2: set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ A @ I ) @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ I2 @ I )
=> ( member_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ M ) ) ) ) ).
% sets_range
thf(fact_706_sets__range,axiom,
! [A: a > set_a,I: set_a,M: sigma_measure_a,I2: a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ A @ I ) @ ( sigma_sets_a @ M ) )
=> ( ( member_a @ I2 @ I )
=> ( member_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ M ) ) ) ) ).
% sets_range
thf(fact_707_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_708_Un__UNIV__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ top_top_set_nat )
= top_top_set_nat ) ).
% Un_UNIV_right
thf(fact_709_not__Ici__eq__Icc,axiom,
! [L2: nat,L: nat,H2: nat] :
( ( set_ord_atLeast_nat @ L2 )
!= ( set_or1269000886237332187st_nat @ L @ H2 ) ) ).
% not_Ici_eq_Icc
thf(fact_710_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_711_subset__emptyI,axiom,
! [A: set_set_a] :
( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A )
=> ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a ) ) ).
% subset_emptyI
thf(fact_712_subset__emptyI,axiom,
! [A: set_a] :
( ! [X3: a] :
~ ( member_a @ X3 @ A )
=> ( ord_less_eq_set_a @ A @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_713_subset__emptyI,axiom,
! [A: set_nat] :
( ! [X3: nat] :
~ ( member_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_714_singletonD,axiom,
! [B2: set_a,A2: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_715_singletonD,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_716_singletonD,axiom,
! [B2: nat,A2: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_717_singleton__iff,axiom,
! [B2: set_a,A2: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_718_singleton__iff,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_719_singleton__iff,axiom,
! [B2: nat,A2: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_720_doubleton__eq__iff,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ( insert_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A2 = C )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_721_insert__not__empty,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ A )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_722_singleton__inject,axiom,
! [A2: nat,B2: nat] :
( ( ( insert_nat @ A2 @ bot_bot_set_nat )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_723_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) @ ( set_or4665077453230672383an_nat @ C @ D2 ) )
= ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
& ( ord_less_nat @ B2 @ D2 ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_724_not__empty__eq__Iic__eq__empty,axiom,
! [H2: nat] :
( bot_bot_set_nat
!= ( set_ord_atMost_nat @ H2 ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_725_not__psubset__empty,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_726_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_727_ivl__disj__un__two_I1_J,axiom,
! [L: nat,M3: nat,U3: nat] :
( ( ord_less_nat @ L @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ U3 )
=> ( ( sup_sup_set_nat @ ( set_or5834768355832116004an_nat @ L @ M3 ) @ ( set_or4665077453230672383an_nat @ M3 @ U3 ) )
= ( set_or5834768355832116004an_nat @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_728_ivl__disj__un__two__touch_I3_J,axiom,
! [L: nat,M3: nat,U3: nat] :
( ( ord_less_nat @ L @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ U3 )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M3 ) @ ( set_or1269000886237332187st_nat @ M3 @ U3 ) )
= ( set_or6659071591806873216st_nat @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_729_ivl__disj__un__two__touch_I1_J,axiom,
! [L: nat,M3: nat,U3: nat] :
( ( ord_less_nat @ L @ M3 )
=> ( ( ord_less_nat @ M3 @ U3 )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M3 ) @ ( set_or4665077453230672383an_nat @ M3 @ U3 ) )
= ( set_or5834768355832116004an_nat @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_730_atLeastLessThan__subset__iff,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A2 @ B2 ) @ ( set_or4665077453230672383an_nat @ C @ D2 ) )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
| ( ( ord_less_eq_nat @ C @ A2 )
& ( ord_less_eq_nat @ B2 @ D2 ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_731_not__UNIV__le__Icc,axiom,
! [L: nat,H2: nat] :
~ ( ord_less_eq_set_nat @ top_top_set_nat @ ( set_or1269000886237332187st_nat @ L @ H2 ) ) ).
% not_UNIV_le_Icc
thf(fact_732_ivl__disj__un__two_I5_J,axiom,
! [L: nat,M3: nat,U3: nat] :
( ( ord_less_nat @ L @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ U3 )
=> ( ( sup_sup_set_nat @ ( set_or5834768355832116004an_nat @ L @ M3 ) @ ( set_or1269000886237332187st_nat @ M3 @ U3 ) )
= ( set_or6659071591806873216st_nat @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_733_atLeastAtMost__borel,axiom,
! [A2: nat,B2: nat] : ( member_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ).
% atLeastAtMost_borel
thf(fact_734_atLeastAtMost__borel,axiom,
! [A2: a,B2: a] : ( member_set_a @ ( set_or672772299803893939Most_a @ A2 @ B2 ) @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ) ).
% atLeastAtMost_borel
thf(fact_735_sup__cancel__left2,axiom,
! [X: set_nat,A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ A2 ) @ ( sup_sup_set_nat @ X @ B2 ) )
= top_top_set_nat ) ).
% sup_cancel_left2
thf(fact_736_sup__cancel__left2,axiom,
! [X: set_a,A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( uminus_uminus_set_a @ X ) @ A2 ) @ ( sup_sup_set_a @ X @ B2 ) )
= top_top_set_a ) ).
% sup_cancel_left2
thf(fact_737_sup__cancel__left2,axiom,
! [X: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ ( uminus6103902357914783669_set_a @ X ) @ A2 ) @ ( sup_sup_set_set_a @ X @ B2 ) )
= top_top_set_set_a ) ).
% sup_cancel_left2
thf(fact_738_sup__cancel__left1,axiom,
! [X: set_nat,A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ A2 ) @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ B2 ) )
= top_top_set_nat ) ).
% sup_cancel_left1
thf(fact_739_sup__cancel__left1,axiom,
! [X: set_a,A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ A2 ) @ ( sup_sup_set_a @ ( uminus_uminus_set_a @ X ) @ B2 ) )
= top_top_set_a ) ).
% sup_cancel_left1
thf(fact_740_sup__cancel__left1,axiom,
! [X: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( sup_sup_set_set_a @ ( sup_sup_set_set_a @ X @ A2 ) @ ( sup_sup_set_set_a @ ( uminus6103902357914783669_set_a @ X ) @ B2 ) )
= top_top_set_set_a ) ).
% sup_cancel_left1
thf(fact_741_atLeastLessThan__borel,axiom,
! [A2: nat,B2: nat] : ( member_set_nat @ ( set_or4665077453230672383an_nat @ A2 @ B2 ) @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ).
% atLeastLessThan_borel
thf(fact_742_atLeastLessThan__borel,axiom,
! [A2: a,B2: a] : ( member_set_a @ ( set_or5139330845457685135Than_a @ A2 @ B2 ) @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ) ).
% atLeastLessThan_borel
thf(fact_743_range__subsetD,axiom,
! [F: nat > nat,B: set_nat,I2: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ top_top_set_nat ) @ B )
=> ( member_nat @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_744_range__subsetD,axiom,
! [F: nat > set_a,B: set_set_a,I2: nat] :
( ( ord_le3724670747650509150_set_a @ ( image_nat_set_a @ F @ top_top_set_nat ) @ B )
=> ( member_set_a @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_745_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_746_not__Ici__le__Icc,axiom,
! [L: nat,L2: nat,H: nat] :
~ ( ord_less_eq_set_nat @ ( set_ord_atLeast_nat @ L ) @ ( set_or1269000886237332187st_nat @ L2 @ H ) ) ).
% not_Ici_le_Icc
thf(fact_747_atLeast__eq__UNIV__iff,axiom,
! [X: set_nat] :
( ( ( set_or1731685050470061051et_nat @ X )
= top_top_set_set_nat )
= ( X = bot_bot_set_nat ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_748_atLeast__eq__UNIV__iff,axiom,
! [X: nat] :
( ( ( set_ord_atLeast_nat @ X )
= top_top_set_nat )
= ( X = bot_bot_nat ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_749_ivl__disj__un__singleton_I4_J,axiom,
! [L: nat,U3: nat] :
( ( ord_less_nat @ L @ U3 )
=> ( ( sup_sup_set_nat @ ( set_or5834768355832116004an_nat @ L @ U3 ) @ ( insert_nat @ U3 @ bot_bot_set_nat ) )
= ( set_or6659071591806873216st_nat @ L @ U3 ) ) ) ).
% ivl_disj_un_singleton(4)
thf(fact_750_Compl__partition2,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ A )
= top_top_set_nat ) ).
% Compl_partition2
thf(fact_751_Compl__partition2,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ ( uminus_uminus_set_a @ A ) @ A )
= top_top_set_a ) ).
% Compl_partition2
thf(fact_752_Compl__partition2,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ ( uminus6103902357914783669_set_a @ A ) @ A )
= top_top_set_set_a ) ).
% Compl_partition2
thf(fact_753_Compl__partition,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ ( uminus5710092332889474511et_nat @ A ) )
= top_top_set_nat ) ).
% Compl_partition
thf(fact_754_Compl__partition,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ ( uminus_uminus_set_a @ A ) )
= top_top_set_a ) ).
% Compl_partition
thf(fact_755_Compl__partition,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ A @ ( uminus6103902357914783669_set_a @ A ) )
= top_top_set_set_a ) ).
% Compl_partition
thf(fact_756_ivl__disj__un__two_I6_J,axiom,
! [L: nat,M3: nat,U3: nat] :
( ( ord_less_eq_nat @ L @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ U3 )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M3 ) @ ( set_or6659071591806873216st_nat @ M3 @ U3 ) )
= ( set_or6659071591806873216st_nat @ L @ U3 ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_757_subset__singletonD,axiom,
! [A: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( A = bot_bot_set_nat )
| ( A
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_758_subset__singleton__iff,axiom,
! [X5: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ X5 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
| ( X5
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_759_Compl__empty__eq,axiom,
( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
= top_top_set_nat ) ).
% Compl_empty_eq
thf(fact_760_Compl__empty__eq,axiom,
( ( uminus_uminus_set_a @ bot_bot_set_a )
= top_top_set_a ) ).
% Compl_empty_eq
thf(fact_761_Compl__empty__eq,axiom,
( ( uminus6103902357914783669_set_a @ bot_bot_set_set_a )
= top_top_set_set_a ) ).
% Compl_empty_eq
thf(fact_762_Compl__UNIV__eq,axiom,
( ( uminus5710092332889474511et_nat @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Compl_UNIV_eq
thf(fact_763_Compl__UNIV__eq,axiom,
( ( uminus_uminus_set_a @ top_top_set_a )
= bot_bot_set_a ) ).
% Compl_UNIV_eq
thf(fact_764_Compl__UNIV__eq,axiom,
( ( uminus6103902357914783669_set_a @ top_top_set_set_a )
= bot_bot_set_set_a ) ).
% Compl_UNIV_eq
thf(fact_765_subset__Compl__self__eq,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% subset_Compl_self_eq
thf(fact_766_subset__Compl__self__eq,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ ( uminus_uminus_set_a @ A ) )
= ( A = bot_bot_set_a ) ) ).
% subset_Compl_self_eq
thf(fact_767_subset__Compl__self__eq,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( uminus6103902357914783669_set_a @ A ) )
= ( A = bot_bot_set_set_a ) ) ).
% subset_Compl_self_eq
thf(fact_768_open__right,axiom,
! [S2: set_a,X: a,Y: a] :
( ( topolo8477419352202985285open_a @ S2 )
=> ( ( member_a @ X @ S2 )
=> ( ( ord_less_a @ X @ Y )
=> ? [B5: a] :
( ( ord_less_a @ X @ B5 )
& ( ord_less_eq_set_a @ ( set_or5139330845457685135Than_a @ X @ B5 ) @ S2 ) ) ) ) ) ).
% open_right
thf(fact_769_open__right,axiom,
! [S2: set_nat,X: nat,Y: nat] :
( ( topolo4328251076210115529en_nat @ S2 )
=> ( ( member_nat @ X @ S2 )
=> ( ( ord_less_nat @ X @ Y )
=> ? [B5: nat] :
( ( ord_less_nat @ X @ B5 )
& ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ X @ B5 ) @ S2 ) ) ) ) ) ).
% open_right
thf(fact_770_sup__shunt,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y )
= top_top_set_nat )
= ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) ) ).
% sup_shunt
thf(fact_771_sup__shunt,axiom,
! [X: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X @ Y )
= top_top_set_a )
= ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ Y ) ) ).
% sup_shunt
thf(fact_772_sup__shunt,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ( sup_sup_set_set_a @ X @ Y )
= top_top_set_set_a )
= ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ X ) @ Y ) ) ).
% sup_shunt
thf(fact_773_atLeastatMost__psubset__iff,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) @ ( set_or1269000886237332187st_nat @ C @ D2 ) )
= ( ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( ( ord_less_eq_nat @ C @ A2 )
& ( ord_less_eq_nat @ B2 @ D2 )
& ( ( ord_less_nat @ C @ A2 )
| ( ord_less_nat @ B2 @ D2 ) ) ) )
& ( ord_less_eq_nat @ C @ D2 ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_774_ivl__disj__un__one_I3_J,axiom,
! [L: nat,U3: nat] :
( ( ord_less_eq_nat @ L @ U3 )
=> ( ( sup_sup_set_nat @ ( set_ord_atMost_nat @ L ) @ ( set_or6659071591806873216st_nat @ L @ U3 ) )
= ( set_ord_atMost_nat @ U3 ) ) ) ).
% ivl_disj_un_one(3)
thf(fact_775_sets_OUn,axiom,
! [A2: set_a,M: sigma_measure_a,B2: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ B2 @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sigma_sets_a @ M ) ) ) ) ).
% sets.Un
thf(fact_776_sets_Oempty__sets,axiom,
! [M: sigma_measure_a] : ( member_set_a @ bot_bot_set_a @ ( sigma_sets_a @ M ) ) ).
% sets.empty_sets
thf(fact_777_sets_Oempty__sets,axiom,
! [M: sigma_measure_nat] : ( member_set_nat @ bot_bot_set_nat @ ( sigma_sets_nat @ M ) ) ).
% sets.empty_sets
thf(fact_778_surj__Compl__image__subset,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_nat_nat @ F @ A ) ) @ ( image_nat_nat @ F @ ( uminus5710092332889474511et_nat @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_779_surj__Compl__image__subset,axiom,
! [F: a > nat,A: set_a] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_a_nat @ F @ A ) ) @ ( image_a_nat @ F @ ( uminus_uminus_set_a @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_780_surj__Compl__image__subset,axiom,
! [F: set_a > nat,A: set_set_a] :
( ( ( image_set_a_nat @ F @ top_top_set_set_a )
= top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_set_a_nat @ F @ A ) ) @ ( image_set_a_nat @ F @ ( uminus6103902357914783669_set_a @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_781_surj__Compl__image__subset,axiom,
! [F: nat > a,A: set_nat] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= top_top_set_a )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ ( image_nat_a @ F @ A ) ) @ ( image_nat_a @ F @ ( uminus5710092332889474511et_nat @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_782_surj__Compl__image__subset,axiom,
! [F: a > a,A: set_a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ ( image_a_a @ F @ A ) ) @ ( image_a_a @ F @ ( uminus_uminus_set_a @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_783_surj__Compl__image__subset,axiom,
! [F: set_a > a,A: set_set_a] :
( ( ( image_set_a_a @ F @ top_top_set_set_a )
= top_top_set_a )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ ( image_set_a_a @ F @ A ) ) @ ( image_set_a_a @ F @ ( uminus6103902357914783669_set_a @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_784_surj__Compl__image__subset,axiom,
! [F: nat > set_a,A: set_nat] :
( ( ( image_nat_set_a @ F @ top_top_set_nat )
= top_top_set_set_a )
=> ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ ( image_nat_set_a @ F @ A ) ) @ ( image_nat_set_a @ F @ ( uminus5710092332889474511et_nat @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_785_surj__Compl__image__subset,axiom,
! [F: a > set_a,A: set_a] :
( ( ( image_a_set_a @ F @ top_top_set_a )
= top_top_set_set_a )
=> ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ ( image_a_set_a @ F @ A ) ) @ ( image_a_set_a @ F @ ( uminus_uminus_set_a @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_786_surj__Compl__image__subset,axiom,
! [F: set_a > set_a,A: set_set_a] :
( ( ( image_set_a_set_a @ F @ top_top_set_set_a )
= top_top_set_set_a )
=> ( ord_le3724670747650509150_set_a @ ( uminus6103902357914783669_set_a @ ( image_set_a_set_a @ F @ A ) ) @ ( image_set_a_set_a @ F @ ( uminus6103902357914783669_set_a @ A ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_787_sets_Oinsert__in__sets,axiom,
! [X: a,M: sigma_measure_a,A: set_a] :
( ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( insert_a @ X @ A ) @ ( sigma_sets_a @ M ) ) ) ) ).
% sets.insert_in_sets
thf(fact_788_sets_Oinsert__in__sets,axiom,
! [X: nat,M: sigma_measure_nat,A: set_nat] :
( ( member_set_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( sigma_sets_nat @ M ) )
=> ( ( member_set_nat @ A @ ( sigma_sets_nat @ M ) )
=> ( member_set_nat @ ( insert_nat @ X @ A ) @ ( sigma_sets_nat @ M ) ) ) ) ).
% sets.insert_in_sets
thf(fact_789_the__elem__eq,axiom,
! [X: nat] :
( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% the_elem_eq
thf(fact_790_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_791_bot__empty__eq,axiom,
( bot_bot_set_a_o
= ( ^ [X4: set_a] : ( member_set_a @ X4 @ bot_bot_set_set_a ) ) ) ).
% bot_empty_eq
thf(fact_792_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : ( member_a @ X4 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_793_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X4: nat] : ( member_nat @ X4 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_794_the__elem__image__unique,axiom,
! [A: set_nat,F: nat > nat,X: nat] :
( ( A != bot_bot_set_nat )
=> ( ! [Y4: nat] :
( ( member_nat @ Y4 @ A )
=> ( ( F @ Y4 )
= ( F @ X ) ) )
=> ( ( the_elem_nat @ ( image_nat_nat @ F @ A ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_795_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y2: nat] :
? [X4: nat] :
( Y2
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_796_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_797_surjE,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X3: nat] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_798_surjD,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X3: nat] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_799_sets__bot,axiom,
( ( sigma_sets_a @ bot_bo2108912051383640591sure_a )
= ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) ) ).
% sets_bot
thf(fact_800_sets__bot,axiom,
( ( sigma_sets_nat @ bot_bo6718502177978453909re_nat )
= ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).
% sets_bot
thf(fact_801_range__binary__eq,axiom,
! [A2: nat,B2: nat] :
( ( image_nat_nat @ ( sigma_binary_nat @ A2 @ B2 ) @ top_top_set_nat )
= ( insert_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ).
% range_binary_eq
thf(fact_802_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
( A3
= ( insert_nat @ ( the_elem_nat @ A3 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_803_is__singletonI,axiom,
! [X: nat] : ( is_singleton_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_804_is__singletonI_H,axiom,
! [A: set_set_a] :
( ( A != bot_bot_set_set_a )
=> ( ! [X3: set_a,Y4: set_a] :
( ( member_set_a @ X3 @ A )
=> ( ( member_set_a @ Y4 @ A )
=> ( X3 = Y4 ) ) )
=> ( is_singleton_set_a @ A ) ) ) ).
% is_singletonI'
thf(fact_805_is__singletonI_H,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
=> ( ! [X3: a,Y4: a] :
( ( member_a @ X3 @ A )
=> ( ( member_a @ Y4 @ A )
=> ( X3 = Y4 ) ) )
=> ( is_singleton_a @ A ) ) ) ).
% is_singletonI'
thf(fact_806_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X3: nat,Y4: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y4 @ A )
=> ( X3 = Y4 ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_807_sets__eq__iff__bounded,axiom,
! [A: sigma_measure_a,B: sigma_measure_a,C2: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A @ B )
=> ( ( ord_le254669795585780187sure_a @ B @ C2 )
=> ( ( ( sigma_sets_a @ A )
= ( sigma_sets_a @ C2 ) )
=> ( ( sigma_sets_a @ B )
= ( sigma_sets_a @ A ) ) ) ) ) ).
% sets_eq_iff_bounded
thf(fact_808_sets__sup,axiom,
! [A: sigma_measure_a,M: sigma_measure_a,B: sigma_measure_a] :
( ( ( sigma_sets_a @ A )
= ( sigma_sets_a @ M ) )
=> ( ( ( sigma_sets_a @ B )
= ( sigma_sets_a @ M ) )
=> ( ( sigma_sets_a @ ( sup_su27664952386392231sure_a @ A @ B ) )
= ( sigma_sets_a @ M ) ) ) ) ).
% sets_sup
thf(fact_809_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
? [X4: nat] :
( A3
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_810_is__singletonE,axiom,
! [A: set_nat] :
( ( is_singleton_nat @ A )
=> ~ ! [X3: nat] :
( A
!= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_811_sets__eq__bot2,axiom,
! [M: sigma_measure_a] :
( ( ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a )
= ( sigma_sets_a @ M ) )
= ( M = bot_bo2108912051383640591sure_a ) ) ).
% sets_eq_bot2
thf(fact_812_sets__eq__bot2,axiom,
! [M: sigma_measure_nat] :
( ( ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat )
= ( sigma_sets_nat @ M ) )
= ( M = bot_bo6718502177978453909re_nat ) ) ).
% sets_eq_bot2
thf(fact_813_sets__eq__bot,axiom,
! [M: sigma_measure_a] :
( ( ( sigma_sets_a @ M )
= ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) )
= ( M = bot_bo2108912051383640591sure_a ) ) ).
% sets_eq_bot
thf(fact_814_sets__eq__bot,axiom,
! [M: sigma_measure_nat] :
( ( ( sigma_sets_nat @ M )
= ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) )
= ( M = bot_bo6718502177978453909re_nat ) ) ).
% sets_eq_bot
thf(fact_815_increasingD,axiom,
! [M: set_set_a,F: set_a > nat,X: set_a,Y: set_a] :
( ( measur8151441426001876059_a_nat @ M @ F )
=> ( ( ord_less_eq_set_a @ X @ Y )
=> ( ( member_set_a @ X @ M )
=> ( ( member_set_a @ Y @ M )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% increasingD
thf(fact_816_all__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( P @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_817_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A3: set_nat] : ( A3 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_818_image__Fpow__mono,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A ) ) @ ( finite_Fpow_nat @ B ) ) ) ).
% image_Fpow_mono
thf(fact_819_le__sup__lexord,axiom,
! [K: nat > nat,A: nat,B: nat,Ca: nat,C: nat,S4: nat] :
( ( ( ord_less_nat @ ( K @ A ) @ ( K @ B ) )
=> ( ord_less_eq_nat @ Ca @ B ) )
=> ( ( ( ord_less_nat @ ( K @ B ) @ ( K @ A ) )
=> ( ord_less_eq_nat @ Ca @ A ) )
=> ( ( ( ( K @ A )
= ( K @ B ) )
=> ( ord_less_eq_nat @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_nat @ ( K @ B ) @ ( K @ A ) )
=> ( ~ ( ord_less_eq_nat @ ( K @ A ) @ ( K @ B ) )
=> ( ord_less_eq_nat @ Ca @ S4 ) ) )
=> ( ord_less_eq_nat @ Ca @ ( measur4601247141005857854at_nat @ A @ B @ K @ S4 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_820_mono__image__least,axiom,
! [F: nat > nat,M3: nat,N2: nat,M4: nat,N4: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ( image_nat_nat @ F @ ( set_or4665077453230672383an_nat @ M3 @ N2 ) )
= ( set_or4665077453230672383an_nat @ M4 @ N4 ) )
=> ( ( ord_less_nat @ M3 @ N2 )
=> ( ( F @ M3 )
= M4 ) ) ) ) ).
% mono_image_least
thf(fact_821_mono__onI,axiom,
! [A: set_set_a,F: set_a > nat] :
( ! [R: set_a,S5: set_a] :
( ( member_set_a @ R @ A )
=> ( ( member_set_a @ S5 @ A )
=> ( ( ord_less_eq_set_a @ R @ S5 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S5 ) ) ) ) )
=> ( monoto4790297507788910087_a_nat @ A @ ord_less_eq_set_a @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_822_mono__onI,axiom,
! [A: set_a,F: a > nat] :
( ! [R: a,S5: a] :
( ( member_a @ R @ A )
=> ( ( member_a @ S5 @ A )
=> ( ( ord_less_eq_a @ R @ S5 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S5 ) ) ) ) )
=> ( monotone_on_a_nat @ A @ ord_less_eq_a @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_823_mono__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [R: nat,S5: nat] :
( ( member_nat @ R @ A )
=> ( ( member_nat @ S5 @ A )
=> ( ( ord_less_eq_nat @ R @ S5 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S5 ) ) ) ) )
=> ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_824_mono__onD,axiom,
! [A: set_set_a,F: set_a > nat,R2: set_a,S4: set_a] :
( ( monoto4790297507788910087_a_nat @ A @ ord_less_eq_set_a @ ord_less_eq_nat @ F )
=> ( ( member_set_a @ R2 @ A )
=> ( ( member_set_a @ S4 @ A )
=> ( ( ord_less_eq_set_a @ R2 @ S4 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) ) ) ).
% mono_onD
thf(fact_825_mono__onD,axiom,
! [A: set_a,F: a > nat,R2: a,S4: a] :
( ( monotone_on_a_nat @ A @ ord_less_eq_a @ ord_less_eq_nat @ F )
=> ( ( member_a @ R2 @ A )
=> ( ( member_a @ S4 @ A )
=> ( ( ord_less_eq_a @ R2 @ S4 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) ) ) ).
% mono_onD
thf(fact_826_mono__onD,axiom,
! [A: set_nat,F: nat > nat,R2: nat,S4: nat] :
( ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S4 @ A )
=> ( ( ord_less_eq_nat @ R2 @ S4 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) ) ) ).
% mono_onD
thf(fact_827_ord_Omono__on__def,axiom,
! [A: set_set_a,Less_eq: set_a > set_a > $o,F: set_a > nat] :
( ( monoto4790297507788910087_a_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: set_a,S6: set_a] :
( ( ( member_set_a @ R3 @ A )
& ( member_set_a @ S6 @ A )
& ( Less_eq @ R3 @ S6 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_828_ord_Omono__on__def,axiom,
! [A: set_a,Less_eq: a > a > $o,F: a > nat] :
( ( monotone_on_a_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: a,S6: a] :
( ( ( member_a @ R3 @ A )
& ( member_a @ S6 @ A )
& ( Less_eq @ R3 @ S6 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_829_ord_Omono__on__def,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ( monotone_on_nat_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
= ( ! [R3: nat,S6: nat] :
( ( ( member_nat @ R3 @ A )
& ( member_nat @ S6 @ A )
& ( Less_eq @ R3 @ S6 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_830_ord_Omono__onI,axiom,
! [A: set_set_a,Less_eq: set_a > set_a > $o,F: set_a > nat] :
( ! [R: set_a,S5: set_a] :
( ( member_set_a @ R @ A )
=> ( ( member_set_a @ S5 @ A )
=> ( ( Less_eq @ R @ S5 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S5 ) ) ) ) )
=> ( monoto4790297507788910087_a_nat @ A @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_831_ord_Omono__onI,axiom,
! [A: set_a,Less_eq: a > a > $o,F: a > nat] :
( ! [R: a,S5: a] :
( ( member_a @ R @ A )
=> ( ( member_a @ S5 @ A )
=> ( ( Less_eq @ R @ S5 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S5 ) ) ) ) )
=> ( monotone_on_a_nat @ A @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_832_ord_Omono__onI,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > nat] :
( ! [R: nat,S5: nat] :
( ( member_nat @ R @ A )
=> ( ( member_nat @ S5 @ A )
=> ( ( Less_eq @ R @ S5 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S5 ) ) ) ) )
=> ( monotone_on_nat_nat @ A @ Less_eq @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_833_ord_Omono__onD,axiom,
! [A: set_set_a,Less_eq: set_a > set_a > $o,F: set_a > nat,R2: set_a,S4: set_a] :
( ( monoto4790297507788910087_a_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_set_a @ R2 @ A )
=> ( ( member_set_a @ S4 @ A )
=> ( ( Less_eq @ R2 @ S4 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_834_ord_Omono__onD,axiom,
! [A: set_a,Less_eq: a > a > $o,F: a > nat,R2: a,S4: a] :
( ( monotone_on_a_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_a @ R2 @ A )
=> ( ( member_a @ S4 @ A )
=> ( ( Less_eq @ R2 @ S4 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_835_ord_Omono__onD,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > nat,R2: nat,S4: nat] :
( ( monotone_on_nat_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S4 @ A )
=> ( ( Less_eq @ R2 @ S4 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_836_monotone__on__subset,axiom,
! [A: set_nat,Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat,B: set_nat] :
( ( monotone_on_nat_nat @ A @ Orda @ Ordb @ F )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( monotone_on_nat_nat @ B @ Orda @ Ordb @ F ) ) ) ).
% monotone_on_subset
thf(fact_837_strict__mono__on__eqD,axiom,
! [A: set_a,F: a > nat,X: a,Y: a] :
( ( monotone_on_a_nat @ A @ ord_less_a @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_a @ X @ A )
=> ( ( member_a @ Y @ A )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_838_strict__mono__on__eqD,axiom,
! [A: set_nat,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( Y = X ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_839_strict__mono__onI,axiom,
! [A: set_set_a,F: set_a > nat] :
( ! [R: set_a,S5: set_a] :
( ( member_set_a @ R @ A )
=> ( ( member_set_a @ S5 @ A )
=> ( ( ord_less_set_a @ R @ S5 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S5 ) ) ) ) )
=> ( monoto4790297507788910087_a_nat @ A @ ord_less_set_a @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_840_strict__mono__onI,axiom,
! [A: set_a,F: a > nat] :
( ! [R: a,S5: a] :
( ( member_a @ R @ A )
=> ( ( member_a @ S5 @ A )
=> ( ( ord_less_a @ R @ S5 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S5 ) ) ) ) )
=> ( monotone_on_a_nat @ A @ ord_less_a @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_841_strict__mono__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [R: nat,S5: nat] :
( ( member_nat @ R @ A )
=> ( ( member_nat @ S5 @ A )
=> ( ( ord_less_nat @ R @ S5 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S5 ) ) ) ) )
=> ( monotone_on_nat_nat @ A @ ord_less_nat @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_842_strict__mono__onD,axiom,
! [A: set_set_a,F: set_a > nat,R2: set_a,S4: set_a] :
( ( monoto4790297507788910087_a_nat @ A @ ord_less_set_a @ ord_less_nat @ F )
=> ( ( member_set_a @ R2 @ A )
=> ( ( member_set_a @ S4 @ A )
=> ( ( ord_less_set_a @ R2 @ S4 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_843_strict__mono__onD,axiom,
! [A: set_a,F: a > nat,R2: a,S4: a] :
( ( monotone_on_a_nat @ A @ ord_less_a @ ord_less_nat @ F )
=> ( ( member_a @ R2 @ A )
=> ( ( member_a @ S4 @ A )
=> ( ( ord_less_a @ R2 @ S4 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_844_strict__mono__onD,axiom,
! [A: set_nat,F: nat > nat,R2: nat,S4: nat] :
( ( monotone_on_nat_nat @ A @ ord_less_nat @ ord_less_nat @ F )
=> ( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S4 @ A )
=> ( ( ord_less_nat @ R2 @ S4 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_845_ord_Ostrict__mono__on__def,axiom,
! [A: set_set_a,Less: set_a > set_a > $o,F: set_a > nat] :
( ( monoto4790297507788910087_a_nat @ A @ Less @ ord_less_nat @ F )
= ( ! [R3: set_a,S6: set_a] :
( ( ( member_set_a @ R3 @ A )
& ( member_set_a @ S6 @ A )
& ( Less @ R3 @ S6 ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_846_ord_Ostrict__mono__on__def,axiom,
! [A: set_a,Less: a > a > $o,F: a > nat] :
( ( monotone_on_a_nat @ A @ Less @ ord_less_nat @ F )
= ( ! [R3: a,S6: a] :
( ( ( member_a @ R3 @ A )
& ( member_a @ S6 @ A )
& ( Less @ R3 @ S6 ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_847_ord_Ostrict__mono__on__def,axiom,
! [A: set_nat,Less: nat > nat > $o,F: nat > nat] :
( ( monotone_on_nat_nat @ A @ Less @ ord_less_nat @ F )
= ( ! [R3: nat,S6: nat] :
( ( ( member_nat @ R3 @ A )
& ( member_nat @ S6 @ A )
& ( Less @ R3 @ S6 ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S6 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_848_ord_Ostrict__mono__onI,axiom,
! [A: set_set_a,Less: set_a > set_a > $o,F: set_a > nat] :
( ! [R: set_a,S5: set_a] :
( ( member_set_a @ R @ A )
=> ( ( member_set_a @ S5 @ A )
=> ( ( Less @ R @ S5 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S5 ) ) ) ) )
=> ( monoto4790297507788910087_a_nat @ A @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_849_ord_Ostrict__mono__onI,axiom,
! [A: set_a,Less: a > a > $o,F: a > nat] :
( ! [R: a,S5: a] :
( ( member_a @ R @ A )
=> ( ( member_a @ S5 @ A )
=> ( ( Less @ R @ S5 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S5 ) ) ) ) )
=> ( monotone_on_a_nat @ A @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_850_ord_Ostrict__mono__onI,axiom,
! [A: set_nat,Less: nat > nat > $o,F: nat > nat] :
( ! [R: nat,S5: nat] :
( ( member_nat @ R @ A )
=> ( ( member_nat @ S5 @ A )
=> ( ( Less @ R @ S5 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S5 ) ) ) ) )
=> ( monotone_on_nat_nat @ A @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_851_ord_Ostrict__mono__onD,axiom,
! [A: set_set_a,Less: set_a > set_a > $o,F: set_a > nat,R2: set_a,S4: set_a] :
( ( monoto4790297507788910087_a_nat @ A @ Less @ ord_less_nat @ F )
=> ( ( member_set_a @ R2 @ A )
=> ( ( member_set_a @ S4 @ A )
=> ( ( Less @ R2 @ S4 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_852_ord_Ostrict__mono__onD,axiom,
! [A: set_a,Less: a > a > $o,F: a > nat,R2: a,S4: a] :
( ( monotone_on_a_nat @ A @ Less @ ord_less_nat @ F )
=> ( ( member_a @ R2 @ A )
=> ( ( member_a @ S4 @ A )
=> ( ( Less @ R2 @ S4 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_853_ord_Ostrict__mono__onD,axiom,
! [A: set_nat,Less: nat > nat > $o,F: nat > nat,R2: nat,S4: nat] :
( ( monotone_on_nat_nat @ A @ Less @ ord_less_nat @ F )
=> ( ( member_nat @ R2 @ A )
=> ( ( member_nat @ S4 @ A )
=> ( ( Less @ R2 @ S4 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_854_monotoneI,axiom,
! [Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat] :
( ! [X3: nat,Y4: nat] :
( ( Orda @ X3 @ Y4 )
=> ( Ordb @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ Orda @ Ordb @ F ) ) ).
% monotoneI
thf(fact_855_monotoneD,axiom,
! [Orda: nat > nat > $o,Ordb: nat > nat > $o,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ Orda @ Ordb @ F )
=> ( ( Orda @ X @ Y )
=> ( Ordb @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monotoneD
thf(fact_856_strict__mono__on__imp__mono__on,axiom,
! [A: set_nat,F: nat > nat] :
( ( monotone_on_nat_nat @ A @ ord_less_nat @ ord_less_nat @ F )
=> ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_857_strict__mono__on__leD,axiom,
! [A: set_a,F: a > nat,X: a,Y: a] :
( ( monotone_on_a_nat @ A @ ord_less_a @ ord_less_nat @ F )
=> ( ( member_a @ X @ A )
=> ( ( member_a @ Y @ A )
=> ( ( ord_less_eq_a @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_858_strict__mono__on__leD,axiom,
! [A: set_nat,F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A @ ord_less_nat @ ord_less_nat @ F )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_859_mono__on__greaterD,axiom,
! [A: set_a,G: a > nat,X: a,Y: a] :
( ( monotone_on_a_nat @ A @ ord_less_eq_a @ ord_less_eq_nat @ G )
=> ( ( member_a @ X @ A )
=> ( ( member_a @ Y @ A )
=> ( ( ord_less_nat @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_a @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_860_mono__on__greaterD,axiom,
! [A: set_nat,G: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ G )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ord_less_nat @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less_nat @ Y @ X ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_861_mono__imp__mono__on,axiom,
! [F: nat > nat,A: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_862_monoI,axiom,
! [F: nat > nat] :
( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_863_monoE,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoE
thf(fact_864_monoD,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monoD
thf(fact_865_mono__on__subset,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ( monotone_on_nat_nat @ A @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( monotone_on_nat_nat @ B @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_866_ord_Omono__on__subset,axiom,
! [A: set_nat,Less_eq: nat > nat > $o,F: nat > nat,B: set_nat] :
( ( monotone_on_nat_nat @ A @ Less_eq @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( monotone_on_nat_nat @ B @ Less_eq @ ord_less_eq_nat @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_867_strict__mono__less,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% strict_mono_less
thf(fact_868_strict__mono__eq,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% strict_mono_eq
thf(fact_869_strict__monoI,axiom,
! [F: nat > nat] :
( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F ) ) ).
% strict_monoI
thf(fact_870_strict__monoD,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% strict_monoD
thf(fact_871_incseqD,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% incseqD
thf(fact_872_incseq__def,axiom,
! [X5: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ X5 )
= ( ! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( X5 @ M2 ) @ ( X5 @ N ) ) ) ) ) ).
% incseq_def
thf(fact_873_strict__mono__leD,axiom,
! [R2: nat > nat,M3: nat,N2: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ R2 )
=> ( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ ( R2 @ M3 ) @ ( R2 @ N2 ) ) ) ) ).
% strict_mono_leD
thf(fact_874_strict__mono__less__eq,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% strict_mono_less_eq
thf(fact_875_mono__strict__invE,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_nat @ X @ Y ) ) ) ).
% mono_strict_invE
thf(fact_876_strict__mono__mono,axiom,
! [F: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% strict_mono_mono
thf(fact_877_mono__invE,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_eq_nat @ X @ Y ) ) ) ).
% mono_invE
thf(fact_878_strict__mono__inv,axiom,
! [F: nat > nat,G: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ! [X3: nat] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_879_mono__sup,axiom,
! [F: nat > nat,A: nat,B: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ ( F @ A ) @ ( F @ B ) ) @ ( F @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% mono_sup
thf(fact_880_atMost__Int__atLeast,axiom,
! [N2: nat] :
( ( inf_inf_set_nat @ ( set_ord_atMost_nat @ N2 ) @ ( set_ord_atLeast_nat @ N2 ) )
= ( insert_nat @ N2 @ bot_bot_set_nat ) ) ).
% atMost_Int_atLeast
thf(fact_881_space__empty__iff,axiom,
! [N3: sigma_measure_a] :
( ( ( sigma_space_a @ N3 )
= bot_bot_set_a )
= ( ( sigma_sets_a @ N3 )
= ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) ) ) ).
% space_empty_iff
thf(fact_882_space__empty__iff,axiom,
! [N3: sigma_measure_nat] :
( ( ( sigma_space_nat @ N3 )
= bot_bot_set_nat )
= ( ( sigma_sets_nat @ N3 )
= ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ) ).
% space_empty_iff
thf(fact_883_sets_Orange__disjointed__sets_H,axiom,
! [A: nat > set_a,M: sigma_measure_a] :
( ( ord_le3724670747650509150_set_a @ ( image_nat_set_a @ A @ top_top_set_nat ) @ ( sigma_sets_a @ M ) )
=> ( ord_le3724670747650509150_set_a @ ( image_nat_set_a @ ( disjoi660815876502745417nted_a @ A ) @ top_top_set_nat ) @ ( sigma_sets_a @ M ) ) ) ).
% sets.range_disjointed_sets'
thf(fact_884_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
( set_or6659071591806873216st_nat
= ( ^ [A6: nat,B6: nat] : ( minus_minus_set_nat @ ( set_or1269000886237332187st_nat @ A6 @ B6 ) @ ( insert_nat @ A6 @ bot_bot_set_nat ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_885_Int__iff,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
= ( ( member_set_a @ C @ A )
& ( member_set_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_886_Int__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ( member_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_887_IntI,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A )
=> ( ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_888_IntI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ( member_a @ C @ B )
=> ( member_a @ C @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_889_Diff__iff,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) )
= ( ( member_set_a @ C @ A )
& ~ ( member_set_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_890_Diff__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ~ ( member_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_891_DiffI,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A )
=> ( ~ ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_892_DiffI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_893_inf_Obounded__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_894_le__inf__iff,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z3 ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_895_inf__top_Oright__neutral,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ top_top_set_nat )
= A2 ) ).
% inf_top.right_neutral
thf(fact_896_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ A2 @ B2 ) )
= ( ( A2 = top_top_set_nat )
& ( B2 = top_top_set_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_897_inf__top_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_898_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= top_top_set_nat )
= ( ( A2 = top_top_set_nat )
& ( B2 = top_top_set_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_899_top__eq__inf__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ X @ Y ) )
= ( ( X = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_900_inf__eq__top__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X @ Y )
= top_top_set_nat )
= ( ( X = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_901_inf__top__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ top_top_set_nat )
= X ) ).
% inf_top_right
thf(fact_902_inf__top__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ X )
= X ) ).
% inf_top_left
thf(fact_903_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_904_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_905_inf__bot__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_906_inf__bot__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_907_Int__UNIV,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= top_top_set_nat )
= ( ( A = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_908_Int__insert__right__if1,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_909_Int__insert__right__if1,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_910_Int__insert__right__if1,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_911_Int__insert__right__if0,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_912_Int__insert__right__if0,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( inf_inf_set_set_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_913_Int__insert__right__if0,axiom,
! [A2: a,A: set_a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_914_insert__inter__insert,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_915_Int__insert__left__if1,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_916_Int__insert__left__if1,axiom,
! [A2: set_a,C2: set_set_a,B: set_set_a] :
( ( member_set_a @ A2 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C2 )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_917_Int__insert__left__if1,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_918_Int__insert__left__if0,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_nat @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_919_Int__insert__left__if0,axiom,
! [A2: set_a,C2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A2 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_920_Int__insert__left__if0,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_921_sets_OInt,axiom,
! [A2: set_a,M: sigma_measure_a,B2: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ B2 @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( sigma_sets_a @ M ) ) ) ) ).
% sets.Int
thf(fact_922_Diff__cancel,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ A )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_923_empty__Diff,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_924_Diff__empty,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Diff_empty
thf(fact_925_sets_Otop,axiom,
! [M: sigma_measure_a] : ( member_set_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) ).
% sets.top
thf(fact_926_insert__Diff1,axiom,
! [X: nat,B: set_nat,A: set_nat] :
( ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_927_insert__Diff1,axiom,
! [X: set_a,B: set_set_a,A: set_set_a] :
( ( member_set_a @ X @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A ) @ B )
= ( minus_5736297505244876581_set_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_928_insert__Diff1,axiom,
! [X: a,B: set_a,A: set_a] :
( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_929_Diff__insert0,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ B ) )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_930_Diff__insert0,axiom,
! [X: set_a,A: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X @ A )
=> ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X @ B ) )
= ( minus_5736297505244876581_set_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_931_Diff__insert0,axiom,
! [X: a,A: set_a,B: set_a] :
( ~ ( member_a @ X @ A )
=> ( ( minus_minus_set_a @ A @ ( insert_a @ X @ B ) )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_932_sets_ODiff,axiom,
! [A2: set_a,M: sigma_measure_a,B2: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ B2 @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( sigma_sets_a @ M ) ) ) ) ).
% sets.Diff
thf(fact_933_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ ( uminus5710092332889474511et_nat @ X ) )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_cancel_right
thf(fact_934_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ ( uminus_uminus_set_a @ X ) )
= bot_bot_set_a ) ).
% boolean_algebra.conj_cancel_right
thf(fact_935_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( uminus6103902357914783669_set_a @ X ) )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_cancel_right
thf(fact_936_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ X )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_cancel_left
thf(fact_937_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ X )
= bot_bot_set_a ) ).
% boolean_algebra.conj_cancel_left
thf(fact_938_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ ( uminus6103902357914783669_set_a @ X ) @ X )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_cancel_left
thf(fact_939_inf__compl__bot__right,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_right
thf(fact_940_inf__compl__bot__right,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ ( uminus_uminus_set_a @ X ) ) )
= bot_bot_set_a ) ).
% inf_compl_bot_right
thf(fact_941_inf__compl__bot__right,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ Y @ ( uminus6103902357914783669_set_a @ X ) ) )
= bot_bot_set_set_a ) ).
% inf_compl_bot_right
thf(fact_942_inf__compl__bot__left2,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_left2
thf(fact_943_inf__compl__bot__left2,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ Y ) )
= bot_bot_set_a ) ).
% inf_compl_bot_left2
thf(fact_944_inf__compl__bot__left2,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( inf_inf_set_set_a @ X @ ( inf_inf_set_set_a @ ( uminus6103902357914783669_set_a @ X ) @ Y ) )
= bot_bot_set_set_a ) ).
% inf_compl_bot_left2
thf(fact_945_inf__compl__bot__left1,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( inf_inf_set_nat @ X @ Y ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_left1
thf(fact_946_inf__compl__bot__left1,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ ( inf_inf_set_a @ X @ Y ) )
= bot_bot_set_a ) ).
% inf_compl_bot_left1
thf(fact_947_inf__compl__bot__left1,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( inf_inf_set_set_a @ ( uminus6103902357914783669_set_a @ X ) @ ( inf_inf_set_set_a @ X @ Y ) )
= bot_bot_set_set_a ) ).
% inf_compl_bot_left1
thf(fact_948_disjoint__insert_I2_J,axiom,
! [A: set_set_a,B2: set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A @ ( insert_set_a @ B2 @ B ) ) )
= ( ~ ( member_set_a @ B2 @ A )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_949_disjoint__insert_I2_J,axiom,
! [A: set_a,B2: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A @ ( insert_a @ B2 @ B ) ) )
= ( ~ ( member_a @ B2 @ A )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_950_disjoint__insert_I2_J,axiom,
! [A: set_nat,B2: nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ ( insert_nat @ B2 @ B ) ) )
= ( ~ ( member_nat @ B2 @ A )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_951_disjoint__insert_I1_J,axiom,
! [B: set_set_a,A2: set_a,A: set_set_a] :
( ( ( inf_inf_set_set_a @ B @ ( insert_set_a @ A2 @ A ) )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A2 @ B )
& ( ( inf_inf_set_set_a @ B @ A )
= bot_bot_set_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_952_disjoint__insert_I1_J,axiom,
! [B: set_a,A2: a,A: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A2 @ A ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ B @ A )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_953_disjoint__insert_I1_J,axiom,
! [B: set_nat,A2: nat,A: set_nat] :
( ( ( inf_inf_set_nat @ B @ ( insert_nat @ A2 @ A ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B )
& ( ( inf_inf_set_nat @ B @ A )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_954_insert__disjoint_I2_J,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ A ) @ B ) )
= ( ~ ( member_set_a @ A2 @ B )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_955_insert__disjoint_I2_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B ) )
= ( ~ ( member_a @ A2 @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_956_insert__disjoint_I2_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B ) )
= ( ~ ( member_nat @ A2 @ B )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_957_insert__disjoint_I1_J,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ A ) @ B )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A2 @ B )
& ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_958_insert__disjoint_I1_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_959_insert__disjoint_I1_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B )
& ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_960_Diff__UNIV,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Diff_UNIV
thf(fact_961_Diff__eq__empty__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( minus_minus_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_962_boolean__algebra_Ode__Morgan__disj,axiom,
! [X: set_a,Y: set_a] :
( ( uminus_uminus_set_a @ ( sup_sup_set_a @ X @ Y ) )
= ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ ( uminus_uminus_set_a @ Y ) ) ) ).
% boolean_algebra.de_Morgan_disj
thf(fact_963_boolean__algebra_Ode__Morgan__disj,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( uminus6103902357914783669_set_a @ ( sup_sup_set_set_a @ X @ Y ) )
= ( inf_inf_set_set_a @ ( uminus6103902357914783669_set_a @ X ) @ ( uminus6103902357914783669_set_a @ Y ) ) ) ).
% boolean_algebra.de_Morgan_disj
thf(fact_964_boolean__algebra_Ode__Morgan__conj,axiom,
! [X: set_a,Y: set_a] :
( ( uminus_uminus_set_a @ ( inf_inf_set_a @ X @ Y ) )
= ( sup_sup_set_a @ ( uminus_uminus_set_a @ X ) @ ( uminus_uminus_set_a @ Y ) ) ) ).
% boolean_algebra.de_Morgan_conj
thf(fact_965_boolean__algebra_Ode__Morgan__conj,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( uminus6103902357914783669_set_a @ ( inf_inf_set_set_a @ X @ Y ) )
= ( sup_sup_set_set_a @ ( uminus6103902357914783669_set_a @ X ) @ ( uminus6103902357914783669_set_a @ Y ) ) ) ).
% boolean_algebra.de_Morgan_conj
thf(fact_966_insert__Diff__single,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= ( insert_nat @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_967_Diff__disjoint,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_968_space__borel,axiom,
( ( sigma_space_nat @ borel_8449730974584783410el_nat )
= top_top_set_nat ) ).
% space_borel
thf(fact_969_space__borel,axiom,
( ( sigma_space_a @ borel_5459123734250506524orel_a )
= top_top_set_a ) ).
% space_borel
thf(fact_970_sets_OInt__space__eq1,axiom,
! [X: set_a,M: sigma_measure_a] :
( ( member_set_a @ X @ ( sigma_sets_a @ M ) )
=> ( ( inf_inf_set_a @ ( sigma_space_a @ M ) @ X )
= X ) ) ).
% sets.Int_space_eq1
thf(fact_971_sets_OInt__space__eq2,axiom,
! [X: set_a,M: sigma_measure_a] :
( ( member_set_a @ X @ ( sigma_sets_a @ M ) )
=> ( ( inf_inf_set_a @ X @ ( sigma_space_a @ M ) )
= X ) ) ).
% sets.Int_space_eq2
thf(fact_972_ivl__diff,axiom,
! [I2: nat,N2: nat,M3: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ M3 ) @ ( set_or4665077453230672383an_nat @ I2 @ N2 ) )
= ( set_or4665077453230672383an_nat @ N2 @ M3 ) ) ) ).
% ivl_diff
thf(fact_973_sets_Ocompl__sets,axiom,
! [A2: set_a,M: sigma_measure_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( minus_minus_set_a @ ( sigma_space_a @ M ) @ A2 ) @ ( sigma_sets_a @ M ) ) ) ).
% sets.compl_sets
thf(fact_974_Compl__disjoint2,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ A )
= bot_bot_set_nat ) ).
% Compl_disjoint2
thf(fact_975_Compl__disjoint2,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ A ) @ A )
= bot_bot_set_a ) ).
% Compl_disjoint2
thf(fact_976_Compl__disjoint2,axiom,
! [A: set_set_a] :
( ( inf_inf_set_set_a @ ( uminus6103902357914783669_set_a @ A ) @ A )
= bot_bot_set_set_a ) ).
% Compl_disjoint2
thf(fact_977_Compl__disjoint,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ ( uminus5710092332889474511et_nat @ A ) )
= bot_bot_set_nat ) ).
% Compl_disjoint
thf(fact_978_Compl__disjoint,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ ( uminus_uminus_set_a @ A ) )
= bot_bot_set_a ) ).
% Compl_disjoint
thf(fact_979_Compl__disjoint,axiom,
! [A: set_set_a] :
( ( inf_inf_set_set_a @ A @ ( uminus6103902357914783669_set_a @ A ) )
= bot_bot_set_set_a ) ).
% Compl_disjoint
thf(fact_980_Diff__Compl,axiom,
! [A: set_a,B: set_a] :
( ( minus_minus_set_a @ A @ ( uminus_uminus_set_a @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% Diff_Compl
thf(fact_981_Diff__Compl,axiom,
! [A: set_set_a,B: set_set_a] :
( ( minus_5736297505244876581_set_a @ A @ ( uminus6103902357914783669_set_a @ B ) )
= ( inf_inf_set_set_a @ A @ B ) ) ).
% Diff_Compl
thf(fact_982_Compl__Diff__eq,axiom,
! [A: set_a,B: set_a] :
( ( uminus_uminus_set_a @ ( minus_minus_set_a @ A @ B ) )
= ( sup_sup_set_a @ ( uminus_uminus_set_a @ A ) @ B ) ) ).
% Compl_Diff_eq
thf(fact_983_Compl__Diff__eq,axiom,
! [A: set_set_a,B: set_set_a] :
( ( uminus6103902357914783669_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) )
= ( sup_sup_set_set_a @ ( uminus6103902357914783669_set_a @ A ) @ B ) ) ).
% Compl_Diff_eq
thf(fact_984_space__uniform__count__measure__empty__iff,axiom,
! [X5: set_nat] :
( ( ( sigma_space_nat @ ( nonneg7031465154080143958re_nat @ X5 ) )
= bot_bot_set_nat )
= ( X5 = bot_bot_set_nat ) ) ).
% space_uniform_count_measure_empty_iff
thf(fact_985_inf_Ostrict__coboundedI2,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_986_inf_Ostrict__coboundedI1,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_nat @ A2 @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_987_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A6: nat,B6: nat] :
( ( A6
= ( inf_inf_nat @ A6 @ B6 ) )
& ( A6 != B6 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_988_inf_Ostrict__boundedE,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
=> ~ ( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ A2 @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_989_inf_Oabsorb4,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_990_inf_Oabsorb3,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb3
thf(fact_991_less__infI2,axiom,
! [B2: nat,X: nat,A2: nat] :
( ( ord_less_nat @ B2 @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).
% less_infI2
thf(fact_992_less__infI1,axiom,
! [A2: nat,X: nat,B2: nat] :
( ( ord_less_nat @ A2 @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).
% less_infI1
thf(fact_993_psubset__imp__ex__mem,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ? [B5: set_a] : ( member_set_a @ B5 @ ( minus_5736297505244876581_set_a @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_994_psubset__imp__ex__mem,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ? [B5: a] : ( member_a @ B5 @ ( minus_minus_set_a @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_995_Diff__triv,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_996_Int__Diff__disjoint,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ B ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_997_boolean__algebra_Oconj__one__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ top_top_set_nat )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_998_Int__UNIV__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ top_top_set_nat )
= A ) ).
% Int_UNIV_right
thf(fact_999_Int__UNIV__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ B )
= B ) ).
% Int_UNIV_left
thf(fact_1000_disjoint__iff__not__equal,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ B )
=> ( X4 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1001_Int__empty__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_1002_Int__empty__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_1003_disjoint__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A )
=> ~ ( member_set_a @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_1004_disjoint__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ~ ( member_a @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_1005_disjoint__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ~ ( member_nat @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_1006_Int__emptyI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ~ ( member_set_a @ X3 @ B ) )
=> ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_1007_Int__emptyI,axiom,
! [A: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ~ ( member_a @ X3 @ B ) )
=> ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_1008_Int__emptyI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ~ ( member_nat @ X3 @ B ) )
=> ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_1009_Int__insert__right,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) )
& ( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_1010_Int__insert__right,axiom,
! [A2: set_a,A: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ A @ B ) ) ) )
& ( ~ ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B ) )
= ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_1011_Int__insert__right,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) )
& ( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_1012_Int__insert__left,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B @ C2 ) ) ) )
& ( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1013_Int__insert__left,axiom,
! [A2: set_a,C2: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A2 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C2 )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_set_a @ A2 @ C2 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1014_Int__insert__left,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1015_insert__Diff__if,axiom,
! [X: nat,B: set_nat,A: set_nat] :
( ( ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) )
& ( ~ ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B )
= ( insert_nat @ X @ ( minus_minus_set_nat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1016_insert__Diff__if,axiom,
! [X: set_a,B: set_set_a,A: set_set_a] :
( ( ( member_set_a @ X @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A ) @ B )
= ( minus_5736297505244876581_set_a @ A @ B ) ) )
& ( ~ ( member_set_a @ X @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A ) @ B )
= ( insert_set_a @ X @ ( minus_5736297505244876581_set_a @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1017_insert__Diff__if,axiom,
! [X: a,B: set_a,A: set_a] :
( ( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) )
& ( ~ ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ B )
= ( insert_a @ X @ ( minus_minus_set_a @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1018_Diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A3: set_a,B4: set_a] : ( inf_inf_set_a @ A3 @ ( uminus_uminus_set_a @ B4 ) ) ) ) ).
% Diff_eq
thf(fact_1019_Diff__eq,axiom,
( minus_5736297505244876581_set_a
= ( ^ [A3: set_set_a,B4: set_set_a] : ( inf_inf_set_set_a @ A3 @ ( uminus6103902357914783669_set_a @ B4 ) ) ) ) ).
% Diff_eq
thf(fact_1020_DiffD2,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) )
=> ~ ( member_set_a @ C @ B ) ) ).
% DiffD2
thf(fact_1021_DiffD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( member_a @ C @ B ) ) ).
% DiffD2
thf(fact_1022_DiffD1,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) )
=> ( member_set_a @ C @ A ) ) ).
% DiffD1
thf(fact_1023_DiffD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% DiffD1
thf(fact_1024_IntD2,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ( member_set_a @ C @ B ) ) ).
% IntD2
thf(fact_1025_IntD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ B ) ) ).
% IntD2
thf(fact_1026_IntD1,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ( member_set_a @ C @ A ) ) ).
% IntD1
thf(fact_1027_IntD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% IntD1
thf(fact_1028_DiffE,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) )
=> ~ ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% DiffE
thf(fact_1029_DiffE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% DiffE
thf(fact_1030_IntE,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ~ ( ( member_set_a @ C @ A )
=> ~ ( member_set_a @ C @ B ) ) ) ).
% IntE
thf(fact_1031_IntE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ~ ( member_a @ C @ B ) ) ) ).
% IntE
thf(fact_1032_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1033_diff__eq,axiom,
( minus_minus_set_a
= ( ^ [X4: set_a,Y2: set_a] : ( inf_inf_set_a @ X4 @ ( uminus_uminus_set_a @ Y2 ) ) ) ) ).
% diff_eq
thf(fact_1034_diff__eq,axiom,
( minus_5736297505244876581_set_a
= ( ^ [X4: set_set_a,Y2: set_set_a] : ( inf_inf_set_set_a @ X4 @ ( uminus6103902357914783669_set_a @ Y2 ) ) ) ) ).
% diff_eq
thf(fact_1035_sets__eq__imp__space__eq,axiom,
! [M: sigma_measure_a,M5: sigma_measure_a] :
( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ M5 ) )
=> ( ( sigma_space_a @ M )
= ( sigma_space_a @ M5 ) ) ) ).
% sets_eq_imp_space_eq
thf(fact_1036_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M ) )
=> ~ ! [M6: nat] :
( ( P @ M6 )
=> ~ ! [X7: nat] :
( ( P @ X7 )
=> ( ord_less_eq_nat @ X7 @ M6 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1037_inf_OcoboundedI2,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1038_inf_OcoboundedI1,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1039_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B6: nat,A6: nat] :
( ( inf_inf_nat @ A6 @ B6 )
= B6 ) ) ) ).
% inf.absorb_iff2
thf(fact_1040_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B6: nat] :
( ( inf_inf_nat @ A6 @ B6 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_1041_inf_Ocobounded2,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_1042_inf_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_1043_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B6: nat] :
( A6
= ( inf_inf_nat @ A6 @ B6 ) ) ) ) ).
% inf.order_iff
thf(fact_1044_inf__greatest,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z3 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z3 ) ) ) ) ).
% inf_greatest
thf(fact_1045_inf_OboundedI,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1046_inf_OboundedE,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_1047_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1048_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_1049_inf_Oabsorb2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_1050_inf_Oabsorb1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_1051_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y2: nat] :
( ( inf_inf_nat @ X4 @ Y2 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_1052_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y4 ) @ X3 )
=> ( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y4 ) @ Y4 )
=> ( ! [X3: nat,Y4: nat,Z4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ( ord_less_eq_nat @ X3 @ Z4 )
=> ( ord_less_eq_nat @ X3 @ ( F @ Y4 @ Z4 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1053_inf_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_1054_inf_OorderE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_1055_le__infI2,axiom,
! [B2: nat,X: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_1056_le__infI1,axiom,
! [A2: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_1057_inf__mono,axiom,
! [A2: nat,C: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1058_le__infI,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_1059_le__infE,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X @ A2 )
=> ~ ( ord_less_eq_nat @ X @ B2 ) ) ) ).
% le_infE
thf(fact_1060_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_1061_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_1062_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_1063_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1064_Int__Collect__mono,axiom,
! [A: set_set_a,B: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1065_Int__Collect__mono,axiom,
! [A: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1066_inf__cancel__left2,axiom,
! [X: set_nat,A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ A2 ) @ ( inf_inf_set_nat @ X @ B2 ) )
= bot_bot_set_nat ) ).
% inf_cancel_left2
thf(fact_1067_inf__cancel__left2,axiom,
! [X: set_a,A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ A2 ) @ ( inf_inf_set_a @ X @ B2 ) )
= bot_bot_set_a ) ).
% inf_cancel_left2
thf(fact_1068_inf__cancel__left2,axiom,
! [X: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ ( uminus6103902357914783669_set_a @ X ) @ A2 ) @ ( inf_inf_set_set_a @ X @ B2 ) )
= bot_bot_set_set_a ) ).
% inf_cancel_left2
thf(fact_1069_inf__cancel__left1,axiom,
! [X: set_nat,A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ A2 ) @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ B2 ) )
= bot_bot_set_nat ) ).
% inf_cancel_left1
thf(fact_1070_inf__cancel__left1,axiom,
! [X: set_a,A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ A2 ) @ ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ B2 ) )
= bot_bot_set_a ) ).
% inf_cancel_left1
thf(fact_1071_inf__cancel__left1,axiom,
! [X: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ X @ A2 ) @ ( inf_inf_set_set_a @ ( uminus6103902357914783669_set_a @ X ) @ B2 ) )
= bot_bot_set_set_a ) ).
% inf_cancel_left1
thf(fact_1072_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z3: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z3 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z3 ) ) ) ).
% distrib_sup_le
thf(fact_1073_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z3: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z3 ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z3 ) ) ) ).
% distrib_inf_le
thf(fact_1074_diff__shunt__var,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( minus_minus_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_1075_sets_Osets__into__space,axiom,
! [X: set_a,M: sigma_measure_a] :
( ( member_set_a @ X @ ( sigma_sets_a @ M ) )
=> ( ord_less_eq_set_a @ X @ ( sigma_space_a @ M ) ) ) ).
% sets.sets_into_space
thf(fact_1076_image__Int__subset,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( inf_inf_set_nat @ A @ B ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_1077_image__diff__subset,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_1078_subset__Diff__insert,axiom,
! [A: set_nat,B: set_nat,X: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ ( insert_nat @ X @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ C2 ) )
& ~ ( member_nat @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1079_subset__Diff__insert,axiom,
! [A: set_set_a,B: set_set_a,X: set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( minus_5736297505244876581_set_a @ B @ ( insert_set_a @ X @ C2 ) ) )
= ( ( ord_le3724670747650509150_set_a @ A @ ( minus_5736297505244876581_set_a @ B @ C2 ) )
& ~ ( member_set_a @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1080_subset__Diff__insert,axiom,
! [A: set_a,B: set_a,X: a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ ( insert_a @ X @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ C2 ) )
& ~ ( member_a @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1081_Diff__insert,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_1082_insert__Diff,axiom,
! [A2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ( ( insert_set_a @ A2 @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1083_insert__Diff,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1084_insert__Diff,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1085_Diff__insert2,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_1086_Diff__insert__absorb,axiom,
! [X: set_a,A: set_set_a] :
( ~ ( member_set_a @ X @ A )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A ) @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1087_Diff__insert__absorb,axiom,
! [X: a,A: set_a] :
( ~ ( member_a @ X @ A )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1088_Diff__insert__absorb,axiom,
! [X: nat,A: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1089_hausdorff,axiom,
! [X: a,Y: a] :
( ( X != Y )
=> ? [U: set_a,V: set_a] :
( ( topolo8477419352202985285open_a @ U )
& ( topolo8477419352202985285open_a @ V )
& ( member_a @ X @ U )
& ( member_a @ Y @ V )
& ( ( inf_inf_set_a @ U @ V )
= bot_bot_set_a ) ) ) ).
% hausdorff
thf(fact_1090_hausdorff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ? [U: set_nat,V: set_nat] :
( ( topolo4328251076210115529en_nat @ U )
& ( topolo4328251076210115529en_nat @ V )
& ( member_nat @ X @ U )
& ( member_nat @ Y @ V )
& ( ( inf_inf_set_nat @ U @ V )
= bot_bot_set_nat ) ) ) ).
% hausdorff
thf(fact_1091_separation__t2,axiom,
! [X: a,Y: a] :
( ( X != Y )
= ( ? [U2: set_a,V2: set_a] :
( ( topolo8477419352202985285open_a @ U2 )
& ( topolo8477419352202985285open_a @ V2 )
& ( member_a @ X @ U2 )
& ( member_a @ Y @ V2 )
& ( ( inf_inf_set_a @ U2 @ V2 )
= bot_bot_set_a ) ) ) ) ).
% separation_t2
thf(fact_1092_separation__t2,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ? [U2: set_nat,V2: set_nat] :
( ( topolo4328251076210115529en_nat @ U2 )
& ( topolo4328251076210115529en_nat @ V2 )
& ( member_nat @ X @ U2 )
& ( member_nat @ Y @ V2 )
& ( ( inf_inf_set_nat @ U2 @ V2 )
= bot_bot_set_nat ) ) ) ) ).
% separation_t2
thf(fact_1093_ivl__disj__int__two_I3_J,axiom,
! [L: nat,M3: nat,U3: nat] :
( ( inf_inf_set_nat @ ( set_or4665077453230672383an_nat @ L @ M3 ) @ ( set_or4665077453230672383an_nat @ M3 @ U3 ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(3)
thf(fact_1094_Compl__eq__Diff__UNIV,axiom,
( uminus5710092332889474511et_nat
= ( minus_minus_set_nat @ top_top_set_nat ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_1095_Compl__eq__Diff__UNIV,axiom,
( uminus_uminus_set_a
= ( minus_minus_set_a @ top_top_set_a ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_1096_Compl__eq__Diff__UNIV,axiom,
( uminus6103902357914783669_set_a
= ( minus_5736297505244876581_set_a @ top_top_set_set_a ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_1097_Compl__Int,axiom,
! [A: set_a,B: set_a] :
( ( uminus_uminus_set_a @ ( inf_inf_set_a @ A @ B ) )
= ( sup_sup_set_a @ ( uminus_uminus_set_a @ A ) @ ( uminus_uminus_set_a @ B ) ) ) ).
% Compl_Int
thf(fact_1098_Compl__Int,axiom,
! [A: set_set_a,B: set_set_a] :
( ( uminus6103902357914783669_set_a @ ( inf_inf_set_set_a @ A @ B ) )
= ( sup_sup_set_set_a @ ( uminus6103902357914783669_set_a @ A ) @ ( uminus6103902357914783669_set_a @ B ) ) ) ).
% Compl_Int
thf(fact_1099_Compl__Un,axiom,
! [A: set_a,B: set_a] :
( ( uminus_uminus_set_a @ ( sup_sup_set_a @ A @ B ) )
= ( inf_inf_set_a @ ( uminus_uminus_set_a @ A ) @ ( uminus_uminus_set_a @ B ) ) ) ).
% Compl_Un
thf(fact_1100_Compl__Un,axiom,
! [A: set_set_a,B: set_set_a] :
( ( uminus6103902357914783669_set_a @ ( sup_sup_set_set_a @ A @ B ) )
= ( inf_inf_set_set_a @ ( uminus6103902357914783669_set_a @ A ) @ ( uminus6103902357914783669_set_a @ B ) ) ) ).
% Compl_Un
thf(fact_1101_ivl__disj__int__two_I6_J,axiom,
! [L: nat,M3: nat,U3: nat] :
( ( inf_inf_set_nat @ ( set_or6659071591806873216st_nat @ L @ M3 ) @ ( set_or6659071591806873216st_nat @ M3 @ U3 ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(6)
thf(fact_1102_less__eq__measure_Ointros_I2_J,axiom,
! [M: sigma_measure_a,N3: sigma_measure_a] :
( ( ( sigma_space_a @ M )
= ( sigma_space_a @ N3 ) )
=> ( ( ord_less_set_set_a @ ( sigma_sets_a @ M ) @ ( sigma_sets_a @ N3 ) )
=> ( ord_le254669795585780187sure_a @ M @ N3 ) ) ) ).
% less_eq_measure.intros(2)
thf(fact_1103_inf__shunt,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% inf_shunt
thf(fact_1104_inf__shunt,axiom,
! [X: set_a,Y: set_a] :
( ( ( inf_inf_set_a @ X @ Y )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X @ ( uminus_uminus_set_a @ Y ) ) ) ).
% inf_shunt
thf(fact_1105_inf__shunt,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ( inf_inf_set_set_a @ X @ Y )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ X @ ( uminus6103902357914783669_set_a @ Y ) ) ) ).
% inf_shunt
thf(fact_1106_sup__neg__inf,axiom,
! [P5: set_a,Q3: set_a,R2: set_a] :
( ( ord_less_eq_set_a @ P5 @ ( sup_sup_set_a @ Q3 @ R2 ) )
= ( ord_less_eq_set_a @ ( inf_inf_set_a @ P5 @ ( uminus_uminus_set_a @ Q3 ) ) @ R2 ) ) ).
% sup_neg_inf
thf(fact_1107_sup__neg__inf,axiom,
! [P5: set_set_a,Q3: set_set_a,R2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ P5 @ ( sup_sup_set_set_a @ Q3 @ R2 ) )
= ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ P5 @ ( uminus6103902357914783669_set_a @ Q3 ) ) @ R2 ) ) ).
% sup_neg_inf
thf(fact_1108_shunt2,axiom,
! [X: set_a,Y: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ ( uminus_uminus_set_a @ Y ) ) @ Z3 )
= ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ Y @ Z3 ) ) ) ).
% shunt2
thf(fact_1109_shunt2,axiom,
! [X: set_set_a,Y: set_set_a,Z3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ ( uminus6103902357914783669_set_a @ Y ) ) @ Z3 )
= ( ord_le3724670747650509150_set_a @ X @ ( sup_sup_set_set_a @ Y @ Z3 ) ) ) ).
% shunt2
thf(fact_1110_shunt1,axiom,
! [X: set_a,Y: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z3 )
= ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ ( uminus_uminus_set_a @ Y ) @ Z3 ) ) ) ).
% shunt1
thf(fact_1111_shunt1,axiom,
! [X: set_set_a,Y: set_set_a,Z3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ X @ Y ) @ Z3 )
= ( ord_le3724670747650509150_set_a @ X @ ( sup_sup_set_set_a @ ( uminus6103902357914783669_set_a @ Y ) @ Z3 ) ) ) ).
% shunt1
thf(fact_1112_boolean__algebra_Ocomplement__unique,axiom,
! [A2: set_nat,X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ X )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ A2 @ X )
= top_top_set_nat )
=> ( ( ( inf_inf_set_nat @ A2 @ Y )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ A2 @ Y )
= top_top_set_nat )
=> ( X = Y ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1113_mono__inf,axiom,
! [F: nat > nat,A: nat,B: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ord_less_eq_nat @ ( F @ ( inf_inf_nat @ A @ B ) ) @ ( inf_inf_nat @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% mono_inf
thf(fact_1114_sets__le__imp__space__le,axiom,
! [A: sigma_measure_a,B: sigma_measure_a] :
( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ A ) @ ( sigma_sets_a @ B ) )
=> ( ord_less_eq_set_a @ ( sigma_space_a @ A ) @ ( sigma_space_a @ B ) ) ) ).
% sets_le_imp_space_le
thf(fact_1115_Diff__single__insert,axiom,
! [A: set_nat,X: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_1116_subset__insert__iff,axiom,
! [A: set_set_a,X: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X @ B ) )
= ( ( ( member_set_a @ X @ A )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B ) )
& ( ~ ( member_set_a @ X @ A )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1117_subset__insert__iff,axiom,
! [A: set_a,X: a,B: set_a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ B ) )
= ( ( ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1118_subset__insert__iff,axiom,
! [A: set_nat,X: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B ) )
= ( ( ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1119_disjoint__eq__subset__Compl,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ B ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1120_disjoint__eq__subset__Compl,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A @ ( uminus_uminus_set_a @ B ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1121_disjoint__eq__subset__Compl,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ A @ ( uminus6103902357914783669_set_a @ B ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1122_ivl__disj__int__two_I7_J,axiom,
! [L: nat,M3: nat,U3: nat] :
( ( inf_inf_set_nat @ ( set_or4665077453230672383an_nat @ L @ M3 ) @ ( set_or1269000886237332187st_nat @ M3 @ U3 ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(7)
thf(fact_1123_Compl__insert,axiom,
! [X: nat,A: set_nat] :
( ( uminus5710092332889474511et_nat @ ( insert_nat @ X @ A ) )
= ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% Compl_insert
thf(fact_1124_Compl__insert,axiom,
! [X: a,A: set_a] :
( ( uminus_uminus_set_a @ ( insert_a @ X @ A ) )
= ( minus_minus_set_a @ ( uminus_uminus_set_a @ A ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) ).
% Compl_insert
thf(fact_1125_Compl__insert,axiom,
! [X: set_a,A: set_set_a] :
( ( uminus6103902357914783669_set_a @ ( insert_set_a @ X @ A ) )
= ( minus_5736297505244876581_set_a @ ( uminus6103902357914783669_set_a @ A ) @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ).
% Compl_insert
thf(fact_1126_Ioc__disjoint,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ( inf_inf_set_nat @ ( set_or6659071591806873216st_nat @ A2 @ B2 ) @ ( set_or6659071591806873216st_nat @ C @ D2 ) )
= bot_bot_set_nat )
= ( ( ord_less_eq_nat @ B2 @ A2 )
| ( ord_less_eq_nat @ D2 @ C )
| ( ord_less_eq_nat @ B2 @ C )
| ( ord_less_eq_nat @ D2 @ A2 ) ) ) ).
% Ioc_disjoint
thf(fact_1127_ivl__disj__int__two_I5_J,axiom,
! [L: nat,M3: nat,U3: nat] :
( ( inf_inf_set_nat @ ( set_or5834768355832116004an_nat @ L @ M3 ) @ ( set_or1269000886237332187st_nat @ M3 @ U3 ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(5)
thf(fact_1128_ivl__disj__int__two_I4_J,axiom,
! [L: nat,M3: nat,U3: nat] :
( ( inf_inf_set_nat @ ( set_or1269000886237332187st_nat @ L @ M3 ) @ ( set_or5834768355832116004an_nat @ M3 @ U3 ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(4)
thf(fact_1129_ivl__disj__int__two_I8_J,axiom,
! [L: nat,M3: nat,U3: nat] :
( ( inf_inf_set_nat @ ( set_or1269000886237332187st_nat @ L @ M3 ) @ ( set_or6659071591806873216st_nat @ M3 @ U3 ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(8)
thf(fact_1130_ivl__disj__int__two_I1_J,axiom,
! [L: nat,M3: nat,U3: nat] :
( ( inf_inf_set_nat @ ( set_or5834768355832116004an_nat @ L @ M3 ) @ ( set_or4665077453230672383an_nat @ M3 @ U3 ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(1)
thf(fact_1131_ivl__disj__int__one_I1_J,axiom,
! [L: nat,U3: nat] :
( ( inf_inf_set_nat @ ( set_ord_atMost_nat @ L ) @ ( set_or5834768355832116004an_nat @ L @ U3 ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_one(1)
thf(fact_1132_in__image__insert__iff,axiom,
! [B: set_set_set_a,X: set_a,A: set_set_a] :
( ! [C5: set_set_a] :
( ( member_set_set_a @ C5 @ B )
=> ~ ( member_set_a @ X @ C5 ) )
=> ( ( member_set_set_a @ A @ ( image_1042221919965026181_set_a @ ( insert_set_a @ X ) @ B ) )
= ( ( member_set_a @ X @ A )
& ( member_set_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1133_in__image__insert__iff,axiom,
! [B: set_set_a,X: a,A: set_a] :
( ! [C5: set_a] :
( ( member_set_a @ C5 @ B )
=> ~ ( member_a @ X @ C5 ) )
=> ( ( member_set_a @ A @ ( image_set_a_set_a @ ( insert_a @ X ) @ B ) )
= ( ( member_a @ X @ A )
& ( member_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1134_in__image__insert__iff,axiom,
! [B: set_set_nat,X: nat,A: set_nat] :
( ! [C5: set_nat] :
( ( member_set_nat @ C5 @ B )
=> ~ ( member_nat @ X @ C5 ) )
=> ( ( member_set_nat @ A @ ( image_7916887816326733075et_nat @ ( insert_nat @ X ) @ B ) )
= ( ( member_nat @ X @ A )
& ( member_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1135_ivl__disj__int__one_I8_J,axiom,
! [L: nat,U3: nat] :
( ( inf_inf_set_nat @ ( set_or4665077453230672383an_nat @ L @ U3 ) @ ( set_ord_atLeast_nat @ U3 ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_one(8)
thf(fact_1136_ivl__disj__int__one_I3_J,axiom,
! [L: nat,U3: nat] :
( ( inf_inf_set_nat @ ( set_ord_atMost_nat @ L ) @ ( set_or6659071591806873216st_nat @ L @ U3 ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_one(3)
thf(fact_1137_ivl__disj__int__one_I6_J,axiom,
! [L: nat,U3: nat] :
( ( inf_inf_set_nat @ ( set_or5834768355832116004an_nat @ L @ U3 ) @ ( set_ord_atLeast_nat @ U3 ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_one(6)
thf(fact_1138_atLeastAtMost__def,axiom,
( set_or1269000886237332187st_nat
= ( ^ [L3: nat,U4: nat] : ( inf_inf_set_nat @ ( set_ord_atLeast_nat @ L3 ) @ ( set_ord_atMost_nat @ U4 ) ) ) ) ).
% atLeastAtMost_def
thf(fact_1139_ivl__disj__int__two_I2_J,axiom,
! [L: nat,M3: nat,U3: nat] :
( ( inf_inf_set_nat @ ( set_or6659071591806873216st_nat @ L @ M3 ) @ ( set_or5834768355832116004an_nat @ M3 @ U3 ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(2)
thf(fact_1140_le__measureD2,axiom,
! [A: sigma_measure_a,B: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A @ B )
=> ( ( ( sigma_space_a @ A )
= ( sigma_space_a @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ A ) @ ( sigma_sets_a @ B ) ) ) ) ).
% le_measureD2
thf(fact_1141_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X @ Y )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ X @ Y )
= top_top_set_nat )
=> ( ( uminus5710092332889474511et_nat @ X )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_1142_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_a,Y: set_a] :
( ( ( inf_inf_set_a @ X @ Y )
= bot_bot_set_a )
=> ( ( ( sup_sup_set_a @ X @ Y )
= top_top_set_a )
=> ( ( uminus_uminus_set_a @ X )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_1143_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ( inf_inf_set_set_a @ X @ Y )
= bot_bot_set_set_a )
=> ( ( ( sup_sup_set_set_a @ X @ Y )
= top_top_set_set_a )
=> ( ( uminus6103902357914783669_set_a @ X )
= Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_1144_psubset__insert__iff,axiom,
! [A: set_set_a,X: set_a,B: set_set_a] :
( ( ord_less_set_set_a @ A @ ( insert_set_a @ X @ B ) )
= ( ( ( member_set_a @ X @ B )
=> ( ord_less_set_set_a @ A @ B ) )
& ( ~ ( member_set_a @ X @ B )
=> ( ( ( member_set_a @ X @ A )
=> ( ord_less_set_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B ) )
& ( ~ ( member_set_a @ X @ A )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1145_psubset__insert__iff,axiom,
! [A: set_a,X: a,B: set_a] :
( ( ord_less_set_a @ A @ ( insert_a @ X @ B ) )
= ( ( ( member_a @ X @ B )
=> ( ord_less_set_a @ A @ B ) )
& ( ~ ( member_a @ X @ B )
=> ( ( ( member_a @ X @ A )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ A @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1146_psubset__insert__iff,axiom,
! [A: set_nat,X: nat,B: set_nat] :
( ( ord_less_set_nat @ A @ ( insert_nat @ X @ B ) )
= ( ( ( member_nat @ X @ B )
=> ( ord_less_set_nat @ A @ B ) )
& ( ~ ( member_nat @ X @ B )
=> ( ( ( member_nat @ X @ A )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ A @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1147_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
( set_or4665077453230672383an_nat
= ( ^ [A6: nat,B6: nat] : ( minus_minus_set_nat @ ( set_or1269000886237332187st_nat @ A6 @ B6 ) @ ( insert_nat @ B6 @ bot_bot_set_nat ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_1148_atLeastAtMost__diff__ends,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) @ ( insert_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) )
= ( set_or5834768355832116004an_nat @ A2 @ B2 ) ) ).
% atLeastAtMost_diff_ends
thf(fact_1149_open__delete,axiom,
! [S4: set_nat,X: nat] :
( ( topolo4328251076210115529en_nat @ S4 )
=> ( topolo4328251076210115529en_nat @ ( minus_minus_set_nat @ S4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% open_delete
thf(fact_1150_less__disjoint__disjointed,axiom,
! [M3: nat,N2: nat,A: nat > set_nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ( inf_inf_set_nat @ ( disjoi6656021742987073733ed_nat @ A @ M3 ) @ ( disjoi6656021742987073733ed_nat @ A @ N2 ) )
= bot_bot_set_nat ) ) ).
% less_disjoint_disjointed
thf(fact_1151_strict__mono__imp__increasing,axiom,
! [F: nat > nat,N2: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ord_less_eq_nat @ N2 @ ( F @ N2 ) ) ) ).
% strict_mono_imp_increasing
thf(fact_1152_diff__diff__cancel,axiom,
! [I2: nat,N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_1153_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B2 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1154_nat__le__linear,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
| ( ord_less_eq_nat @ N2 @ M3 ) ) ).
% nat_le_linear
thf(fact_1155_le__antisym,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M3 )
=> ( M3 = N2 ) ) ) ).
% le_antisym
thf(fact_1156_eq__imp__le,axiom,
! [M3: nat,N2: nat] :
( ( M3 = N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% eq_imp_le
thf(fact_1157_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_1158_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_1159_eq__diff__iff,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M3 @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M3 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1160_le__diff__iff,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1161_Nat_Odiff__diff__eq,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M3 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1162_diff__le__mono,axiom,
! [M3: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_1163_diff__le__self,axiom,
! [M3: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ N2 ) @ M3 ) ).
% diff_le_self
thf(fact_1164_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1165_diff__le__mono2,axiom,
! [M3: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M3 ) ) ) ).
% diff_le_mono2
thf(fact_1166_nat__neq__iff,axiom,
! [M3: nat,N2: nat] :
( ( M3 != N2 )
= ( ( ord_less_nat @ M3 @ N2 )
| ( ord_less_nat @ N2 @ M3 ) ) ) ).
% nat_neq_iff
thf(fact_1167_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_1168_less__not__refl2,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_nat @ N2 @ M3 )
=> ( M3 != N2 ) ) ).
% less_not_refl2
thf(fact_1169_less__not__refl3,axiom,
! [S4: nat,T4: nat] :
( ( ord_less_nat @ S4 @ T4 )
=> ( S4 != T4 ) ) ).
% less_not_refl3
thf(fact_1170_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_1171_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N5: nat] :
( ! [M7: nat] :
( ( ord_less_nat @ M7 @ N5 )
=> ( P @ M7 ) )
=> ( P @ N5 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_1172_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N5: nat] :
( ~ ( P @ N5 )
=> ? [M7: nat] :
( ( ord_less_nat @ M7 @ N5 )
& ~ ( P @ M7 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_1173_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_1174_less__imp__diff__less,axiom,
! [J: nat,K: nat,N2: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1175_diff__less__mono2,axiom,
! [M3: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ( ord_less_nat @ M3 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M3 ) ) ) ) ).
% diff_less_mono2
thf(fact_1176_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
& ( M2 != N ) ) ) ) ).
% nat_less_le
thf(fact_1177_less__imp__le__nat,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_1178_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1179_less__or__eq__imp__le,axiom,
! [M3: nat,N2: nat] :
( ( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_1180_le__neq__implies__less,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( M3 != N2 )
=> ( ord_less_nat @ M3 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_1181_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I5: nat,J2: nat] :
( ( ord_less_nat @ I5 @ J2 )
=> ( ord_less_nat @ ( F @ I5 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1182_diff__less__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1183_less__diff__iff,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_nat @ M3 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_1184_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M3: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N2 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M3 ) ) ) ).
% nat_descend_induct
thf(fact_1185_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila1667268886620078168et_nat @ inf_inf_set_nat @ top_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_1186_incseq__imp__monoseq,axiom,
! [X5: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ X5 )
=> ( topolo4902158794631467389eq_nat @ X5 ) ) ).
% incseq_imp_monoseq
thf(fact_1187_space__sup__measure_H,axiom,
! [B: sigma_measure_a,A: sigma_measure_a] :
( ( ( sigma_sets_a @ B )
= ( sigma_sets_a @ A ) )
=> ( ( sigma_space_a @ ( measur3004909623614618064sure_a @ A @ B ) )
= ( sigma_space_a @ A ) ) ) ).
% space_sup_measure'
thf(fact_1188_sets__sup__measure_H,axiom,
! [B: sigma_measure_a,A: sigma_measure_a] :
( ( ( sigma_sets_a @ B )
= ( sigma_sets_a @ A ) )
=> ( ( sigma_sets_a @ ( measur3004909623614618064sure_a @ A @ B ) )
= ( sigma_sets_a @ A ) ) ) ).
% sets_sup_measure'
thf(fact_1189_monoseq__def,axiom,
( topolo4902158794631467389eq_nat
= ( ^ [X2: nat > nat] :
( ! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( X2 @ M2 ) @ ( X2 @ N ) ) )
| ! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( X2 @ N ) @ ( X2 @ M2 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_1190_monoI2,axiom,
! [X5: nat > nat] :
( ! [M6: nat,N5: nat] :
( ( ord_less_eq_nat @ M6 @ N5 )
=> ( ord_less_eq_nat @ ( X5 @ N5 ) @ ( X5 @ M6 ) ) )
=> ( topolo4902158794631467389eq_nat @ X5 ) ) ).
% monoI2
thf(fact_1191_monoI1,axiom,
! [X5: nat > nat] :
( ! [M6: nat,N5: nat] :
( ( ord_less_eq_nat @ M6 @ N5 )
=> ( ord_less_eq_nat @ ( X5 @ M6 ) @ ( X5 @ N5 ) ) )
=> ( topolo4902158794631467389eq_nat @ X5 ) ) ).
% monoI1
thf(fact_1192_remove__def,axiom,
( remove_nat
= ( ^ [X4: nat,A3: set_nat] : ( minus_minus_set_nat @ A3 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).
% remove_def
thf(fact_1193_algebra__single__set,axiom,
! [X5: set_nat,S2: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ S2 )
=> ( sigma_algebra_nat @ S2 @ ( insert_set_nat @ bot_bot_set_nat @ ( insert_set_nat @ X5 @ ( insert_set_nat @ ( minus_minus_set_nat @ S2 @ X5 ) @ ( insert_set_nat @ S2 @ bot_bot_set_set_nat ) ) ) ) ) ) ).
% algebra_single_set
thf(fact_1194_member__remove,axiom,
! [X: set_a,Y: set_a,A: set_set_a] :
( ( member_set_a @ X @ ( remove_set_a @ Y @ A ) )
= ( ( member_set_a @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_1195_member__remove,axiom,
! [X: a,Y: a,A: set_a] :
( ( member_a @ X @ ( remove_a @ Y @ A ) )
= ( ( member_a @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_1196_sets_Oalgebra__axioms,axiom,
! [M: sigma_measure_a] : ( sigma_algebra_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) ).
% sets.algebra_axioms
thf(fact_1197_sets_Orestricted__algebra,axiom,
! [A: set_a,M: sigma_measure_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( sigma_algebra_a @ A @ ( image_set_a_set_a @ ( inf_inf_set_a @ A ) @ ( sigma_sets_a @ M ) ) ) ) ).
% sets.restricted_algebra
thf(fact_1198_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
boolea778851993438741648et_nat @ inf_inf_set_nat @ sup_sup_set_nat @ uminus5710092332889474511et_nat @ bot_bot_set_nat @ top_top_set_nat ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_1199_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
boolea6678413348699952596_set_a @ inf_inf_set_a @ sup_sup_set_a @ uminus_uminus_set_a @ bot_bot_set_a @ top_top_set_a ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_1200_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
boolea3433950929776517940_set_a @ inf_inf_set_set_a @ sup_sup_set_set_a @ uminus6103902357914783669_set_a @ bot_bot_set_set_a @ top_top_set_set_a ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_1201_measure__eqI,axiom,
! [M: sigma_measure_a,N3: sigma_measure_a] :
( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ N3 ) )
=> ( ! [A4: set_a] :
( ( member_set_a @ A4 @ ( sigma_sets_a @ M ) )
=> ( ( sigma_emeasure_a @ M @ A4 )
= ( sigma_emeasure_a @ N3 @ A4 ) ) )
=> ( M = N3 ) ) ) ).
% measure_eqI
thf(fact_1202_le__measureD3,axiom,
! [A: sigma_measure_a,B: sigma_measure_a,X5: set_a] :
( ( ord_le254669795585780187sure_a @ A @ B )
=> ( ( ( sigma_sets_a @ A )
= ( sigma_sets_a @ B ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ A @ X5 ) @ ( sigma_emeasure_a @ B @ X5 ) ) ) ) ).
% le_measureD3
thf(fact_1203_le__measure,axiom,
! [M: sigma_measure_a,N3: sigma_measure_a] :
( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ N3 ) )
=> ( ( ord_le254669795585780187sure_a @ M @ N3 )
= ( ! [X4: set_a] :
( ( member_set_a @ X4 @ ( sigma_sets_a @ M ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ M @ X4 ) @ ( sigma_emeasure_a @ N3 @ X4 ) ) ) ) ) ) ).
% le_measure
thf(fact_1204_emeasure__mono,axiom,
! [A2: set_a,B2: set_a,M: sigma_measure_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_set_a @ B2 @ ( sigma_sets_a @ M ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ M @ A2 ) @ ( sigma_emeasure_a @ M @ B2 ) ) ) ) ).
% emeasure_mono
thf(fact_1205_le__emeasure__sup__measure_H1,axiom,
! [B: sigma_measure_a,A: sigma_measure_a,X5: set_a] :
( ( ( sigma_sets_a @ B )
= ( sigma_sets_a @ A ) )
=> ( ( member_set_a @ X5 @ ( sigma_sets_a @ A ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ A @ X5 ) @ ( sigma_emeasure_a @ ( measur3004909623614618064sure_a @ A @ B ) @ X5 ) ) ) ) ).
% le_emeasure_sup_measure'1
thf(fact_1206_le__emeasure__sup__measure_H2,axiom,
! [B: sigma_measure_a,A: sigma_measure_a,X5: set_a] :
( ( ( sigma_sets_a @ B )
= ( sigma_sets_a @ A ) )
=> ( ( member_set_a @ X5 @ ( sigma_sets_a @ A ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ B @ X5 ) @ ( sigma_emeasure_a @ ( measur3004909623614618064sure_a @ A @ B ) @ X5 ) ) ) ) ).
% le_emeasure_sup_measure'2
thf(fact_1207_less__eq__measure_Ointros_I3_J,axiom,
! [M: sigma_measure_a,N3: sigma_measure_a] :
( ( ( sigma_space_a @ M )
= ( sigma_space_a @ N3 ) )
=> ( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ N3 ) )
=> ( ( ord_le6700572704167691815nnreal @ ( sigma_emeasure_a @ M ) @ ( sigma_emeasure_a @ N3 ) )
=> ( ord_le254669795585780187sure_a @ M @ N3 ) ) ) ) ).
% less_eq_measure.intros(3)
thf(fact_1208_emeasure__sup__measure_H__le2,axiom,
! [B: sigma_measure_a,C2: sigma_measure_a,A: sigma_measure_a,X5: set_a] :
( ( ( sigma_sets_a @ B )
= ( sigma_sets_a @ C2 ) )
=> ( ( ( sigma_sets_a @ A )
= ( sigma_sets_a @ C2 ) )
=> ( ( member_set_a @ X5 @ ( sigma_sets_a @ C2 ) )
=> ( ! [Y6: set_a] :
( ( ord_less_eq_set_a @ Y6 @ X5 )
=> ( ( member_set_a @ Y6 @ ( sigma_sets_a @ A ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ A @ Y6 ) @ ( sigma_emeasure_a @ C2 @ Y6 ) ) ) )
=> ( ! [Y6: set_a] :
( ( ord_less_eq_set_a @ Y6 @ X5 )
=> ( ( member_set_a @ Y6 @ ( sigma_sets_a @ A ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ B @ Y6 ) @ ( sigma_emeasure_a @ C2 @ Y6 ) ) ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ ( measur3004909623614618064sure_a @ A @ B ) @ X5 ) @ ( sigma_emeasure_a @ C2 @ X5 ) ) ) ) ) ) ) ).
% emeasure_sup_measure'_le2
thf(fact_1209_unsigned__Hahn__decomposition,axiom,
! [N3: sigma_measure_a,M: sigma_measure_a,A: set_a] :
( ( ( sigma_sets_a @ N3 )
= ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ( ( sigma_emeasure_a @ M @ A )
!= top_to1496364449551166952nnreal )
=> ( ( ( sigma_emeasure_a @ N3 @ A )
!= top_to1496364449551166952nnreal )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ ( sigma_sets_a @ M ) )
& ( ord_less_eq_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( sigma_sets_a @ M ) )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ N3 @ Xa ) @ ( sigma_emeasure_a @ M @ Xa ) ) ) )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( sigma_sets_a @ M ) )
=> ( ( ord_less_eq_set_a @ Xa @ A )
=> ( ( ( inf_inf_set_a @ Xa @ X3 )
= bot_bot_set_a )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ M @ Xa ) @ ( sigma_emeasure_a @ N3 @ Xa ) ) ) ) ) ) ) ) ) ) ).
% unsigned_Hahn_decomposition
thf(fact_1210_unsigned__Hahn__decomposition,axiom,
! [N3: sigma_measure_nat,M: sigma_measure_nat,A: set_nat] :
( ( ( sigma_sets_nat @ N3 )
= ( sigma_sets_nat @ M ) )
=> ( ( member_set_nat @ A @ ( sigma_sets_nat @ M ) )
=> ( ( ( sigma_emeasure_nat @ M @ A )
!= top_to1496364449551166952nnreal )
=> ( ( ( sigma_emeasure_nat @ N3 @ A )
!= top_to1496364449551166952nnreal )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ ( sigma_sets_nat @ M ) )
& ( ord_less_eq_set_nat @ X3 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ ( sigma_sets_nat @ M ) )
=> ( ( ord_less_eq_set_nat @ Xa @ X3 )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_nat @ N3 @ Xa ) @ ( sigma_emeasure_nat @ M @ Xa ) ) ) )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ ( sigma_sets_nat @ M ) )
=> ( ( ord_less_eq_set_nat @ Xa @ A )
=> ( ( ( inf_inf_set_nat @ Xa @ X3 )
= bot_bot_set_nat )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_nat @ M @ Xa ) @ ( sigma_emeasure_nat @ N3 @ Xa ) ) ) ) ) ) ) ) ) ) ).
% unsigned_Hahn_decomposition
thf(fact_1211_le__measure__iff,axiom,
( ord_le254669795585780187sure_a
= ( ^ [M8: sigma_measure_a,N6: sigma_measure_a] :
( ( ( ( sigma_space_a @ M8 )
= ( sigma_space_a @ N6 ) )
=> ( ( ( ( sigma_sets_a @ M8 )
= ( sigma_sets_a @ N6 ) )
=> ( ord_le6700572704167691815nnreal @ ( sigma_emeasure_a @ M8 ) @ ( sigma_emeasure_a @ N6 ) ) )
& ( ( ( sigma_sets_a @ M8 )
!= ( sigma_sets_a @ N6 ) )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ M8 ) @ ( sigma_sets_a @ N6 ) ) ) ) )
& ( ( ( sigma_space_a @ M8 )
!= ( sigma_space_a @ N6 ) )
=> ( ord_less_eq_set_a @ ( sigma_space_a @ M8 ) @ ( sigma_space_a @ N6 ) ) ) ) ) ) ).
% le_measure_iff
thf(fact_1212_less__eq__measure_Ocases,axiom,
! [A1: sigma_measure_a,A22: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A1 @ A22 )
=> ( ~ ( ord_less_set_a @ ( sigma_space_a @ A1 ) @ ( sigma_space_a @ A22 ) )
=> ( ( ( ( sigma_space_a @ A1 )
= ( sigma_space_a @ A22 ) )
=> ~ ( ord_less_set_set_a @ ( sigma_sets_a @ A1 ) @ ( sigma_sets_a @ A22 ) ) )
=> ~ ( ( ( sigma_space_a @ A1 )
= ( sigma_space_a @ A22 ) )
=> ( ( ( sigma_sets_a @ A1 )
= ( sigma_sets_a @ A22 ) )
=> ~ ( ord_le6700572704167691815nnreal @ ( sigma_emeasure_a @ A1 ) @ ( sigma_emeasure_a @ A22 ) ) ) ) ) ) ) ).
% less_eq_measure.cases
thf(fact_1213_less__eq__measure_Osimps,axiom,
( ord_le254669795585780187sure_a
= ( ^ [A12: sigma_measure_a,A23: sigma_measure_a] :
( ? [M8: sigma_measure_a,N6: sigma_measure_a] :
( ( A12 = M8 )
& ( A23 = N6 )
& ( ord_less_set_a @ ( sigma_space_a @ M8 ) @ ( sigma_space_a @ N6 ) ) )
| ? [M8: sigma_measure_a,N6: sigma_measure_a] :
( ( A12 = M8 )
& ( A23 = N6 )
& ( ( sigma_space_a @ M8 )
= ( sigma_space_a @ N6 ) )
& ( ord_less_set_set_a @ ( sigma_sets_a @ M8 ) @ ( sigma_sets_a @ N6 ) ) )
| ? [M8: sigma_measure_a,N6: sigma_measure_a] :
( ( A12 = M8 )
& ( A23 = N6 )
& ( ( sigma_space_a @ M8 )
= ( sigma_space_a @ N6 ) )
& ( ( sigma_sets_a @ M8 )
= ( sigma_sets_a @ N6 ) )
& ( ord_le6700572704167691815nnreal @ ( sigma_emeasure_a @ M8 ) @ ( sigma_emeasure_a @ N6 ) ) ) ) ) ) ).
% less_eq_measure.simps
thf(fact_1214_sets_Osmallest__ccdi__sets__Un,axiom,
! [A: set_a,M: sigma_measure_a,B: set_a] :
( ( member_set_a @ A @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) )
=> ( ( member_set_a @ B @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) )
=> ( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
=> ( member_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sigma_5648178489087971417sets_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) ) ) ) ) ).
% sets.smallest_ccdi_sets_Un
thf(fact_1215_sets_Osmallest__ccdi__sets__Un,axiom,
! [A: set_nat,M: sigma_measure_nat,B: set_nat] :
( ( member_set_nat @ A @ ( sigma_5553761350045521333ts_nat @ ( sigma_space_nat @ M ) @ ( sigma_sets_nat @ M ) ) )
=> ( ( member_set_nat @ B @ ( sigma_5553761350045521333ts_nat @ ( sigma_space_nat @ M ) @ ( sigma_sets_nat @ M ) ) )
=> ( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
=> ( member_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sigma_5553761350045521333ts_nat @ ( sigma_space_nat @ M ) @ ( sigma_sets_nat @ M ) ) ) ) ) ) ).
% sets.smallest_ccdi_sets_Un
thf(fact_1216_atLeastLessThan__singleton,axiom,
! [M3: nat] :
( ( set_or4665077453230672383an_nat @ M3 @ ( suc @ M3 ) )
= ( insert_nat @ M3 @ bot_bot_set_nat ) ) ).
% atLeastLessThan_singleton
thf(fact_1217_image__Suc__atLeastAtMost,axiom,
! [I2: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I2 @ J ) )
= ( set_or1269000886237332187st_nat @ ( suc @ I2 ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_1218_image__Suc__atLeastLessThan,axiom,
! [I2: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I2 @ J ) )
= ( set_or4665077453230672383an_nat @ ( suc @ I2 ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_1219_Suc__le__mono,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M3 ) )
= ( ord_less_eq_nat @ N2 @ M3 ) ) ).
% Suc_le_mono
thf(fact_1220_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_1221_Suc__mono,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_1222_Suc__less__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_less_eq
thf(fact_1223_Suc__leI,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 ) ) ).
% Suc_leI
thf(fact_1224_Suc__le__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
= ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_le_eq
thf(fact_1225_dec__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ I2 )
=> ( ! [N5: nat] :
( ( ord_less_eq_nat @ I2 @ N5 )
=> ( ( ord_less_nat @ N5 @ J )
=> ( ( P @ N5 )
=> ( P @ ( suc @ N5 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1226_inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ J )
=> ( ! [N5: nat] :
( ( ord_less_eq_nat @ I2 @ N5 )
=> ( ( ord_less_nat @ N5 @ J )
=> ( ( P @ ( suc @ N5 ) )
=> ( P @ N5 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_1227_Suc__le__lessD,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
=> ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_le_lessD
thf(fact_1228_le__less__Suc__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M3 ) )
= ( N2 = M3 ) ) ) ).
% le_less_Suc_eq
thf(fact_1229_less__Suc__eq__le,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_1230_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1231_le__imp__less__Suc,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_nat @ M3 @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_1232_Suc__diff__le,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_eq_nat @ N2 @ M3 )
=> ( ( minus_minus_nat @ ( suc @ M3 ) @ N2 )
= ( suc @ ( minus_minus_nat @ M3 @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_1233_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N2: nat,N4: nat] :
( ! [N5: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N5 ) ) @ ( F @ N5 ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1234_lift__Suc__mono__le,axiom,
! [F: nat > nat,N2: nat,N4: nat] :
( ! [N5: nat] : ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1235_Suc__leD,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% Suc_leD
thf(fact_1236_le__SucE,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M3 @ N2 )
=> ( M3
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_1237_le__SucI,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ M3 @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_1238_Suc__le__D,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M4 )
=> ? [M6: nat] :
( M4
= ( suc @ M6 ) ) ) ).
% Suc_le_D
thf(fact_1239_le__Suc__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M3 @ N2 )
| ( M3
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_1240_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_1241_not__less__eq__eq,axiom,
! [M3: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M3 @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M3 ) ) ).
% not_less_eq_eq
thf(fact_1242_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N5: nat] :
( ! [M7: nat] :
( ( ord_less_eq_nat @ ( suc @ M7 ) @ N5 )
=> ( P @ M7 ) )
=> ( P @ N5 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_1243_nat__induct__at__least,axiom,
! [M3: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( P @ M3 )
=> ( ! [N5: nat] :
( ( ord_less_eq_nat @ M3 @ N5 )
=> ( ( P @ N5 )
=> ( P @ ( suc @ N5 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_1244_transitive__stepwise__le,axiom,
! [M3: nat,N2: nat,R4: nat > nat > $o] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ! [X3: nat] : ( R4 @ X3 @ X3 )
=> ( ! [X3: nat,Y4: nat,Z4: nat] :
( ( R4 @ X3 @ Y4 )
=> ( ( R4 @ Y4 @ Z4 )
=> ( R4 @ X3 @ Z4 ) ) )
=> ( ! [N5: nat] : ( R4 @ N5 @ ( suc @ N5 ) )
=> ( R4 @ M3 @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1245_atLeastAtMost__insertL,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( insert_nat @ M3 @ ( set_or1269000886237332187st_nat @ ( suc @ M3 ) @ N2 ) )
= ( set_or1269000886237332187st_nat @ M3 @ N2 ) ) ) ).
% atLeastAtMost_insertL
thf(fact_1246_atLeastAtMostSuc__conv,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N2 ) )
=> ( ( set_or1269000886237332187st_nat @ M3 @ ( suc @ N2 ) )
= ( insert_nat @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ M3 @ N2 ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_1247_Icc__eq__insert__lb__nat,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( set_or1269000886237332187st_nat @ M3 @ N2 )
= ( insert_nat @ M3 @ ( set_or1269000886237332187st_nat @ ( suc @ M3 ) @ N2 ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_1248_atLeastLessThanSuc,axiom,
! [M3: nat,N2: nat] :
( ( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( set_or4665077453230672383an_nat @ M3 @ ( suc @ N2 ) )
= ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ M3 @ N2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( set_or4665077453230672383an_nat @ M3 @ ( suc @ N2 ) )
= bot_bot_set_nat ) ) ) ).
% atLeastLessThanSuc
thf(fact_1249_monoseq__Suc,axiom,
( topolo4902158794631467389eq_nat
= ( ^ [X2: nat > nat] :
( ! [N: nat] : ( ord_less_eq_nat @ ( X2 @ N ) @ ( X2 @ ( suc @ N ) ) )
| ! [N: nat] : ( ord_less_eq_nat @ ( X2 @ ( suc @ N ) ) @ ( X2 @ N ) ) ) ) ) ).
% monoseq_Suc
thf(fact_1250_mono__SucI2,axiom,
! [X5: nat > nat] :
( ! [N5: nat] : ( ord_less_eq_nat @ ( X5 @ ( suc @ N5 ) ) @ ( X5 @ N5 ) )
=> ( topolo4902158794631467389eq_nat @ X5 ) ) ).
% mono_SucI2
thf(fact_1251_mono__SucI1,axiom,
! [X5: nat > nat] :
( ! [N5: nat] : ( ord_less_eq_nat @ ( X5 @ N5 ) @ ( X5 @ ( suc @ N5 ) ) )
=> ( topolo4902158794631467389eq_nat @ X5 ) ) ).
% mono_SucI1
thf(fact_1252_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost_nat @ ( suc @ K ) )
= ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% atMost_Suc
thf(fact_1253_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L: nat,U3: nat] :
( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U3 )
= ( set_or5834768355832116004an_nat @ L @ U3 ) ) ).
% atLeastSucLessThan_greaterThanLessThan
thf(fact_1254_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L: nat,U3: nat] :
( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U3 )
= ( set_or6659071591806873216st_nat @ L @ U3 ) ) ).
% atLeastSucAtMost_greaterThanAtMost
thf(fact_1255_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U3: nat] :
( ( set_or4665077453230672383an_nat @ L @ ( suc @ U3 ) )
= ( set_or1269000886237332187st_nat @ L @ U3 ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_1256_lift__Suc__mono__less,axiom,
! [F: nat > nat,N2: nat,N4: nat] :
( ! [N5: nat] : ( ord_less_nat @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
=> ( ( ord_less_nat @ N2 @ N4 )
=> ( ord_less_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1257_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N2: nat,M3: nat] :
( ! [N5: nat] : ( ord_less_nat @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
=> ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M3 ) )
= ( ord_less_nat @ N2 @ M3 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1258_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1259_Suc__lessD,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ N2 )
=> ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_lessD
thf(fact_1260_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1261_Suc__lessI,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ( ( suc @ M3 )
!= N2 )
=> ( ord_less_nat @ ( suc @ M3 ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_1262_less__SucE,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M3 @ N2 )
=> ( M3 = N2 ) ) ) ).
% less_SucE
thf(fact_1263_less__SucI,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_nat @ M3 @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_1264_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I6: nat] :
( ( ord_less_nat @ I6 @ ( suc @ N2 ) )
& ( P @ I6 ) ) )
= ( ( P @ N2 )
| ? [I6: nat] :
( ( ord_less_nat @ I6 @ N2 )
& ( P @ I6 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1265_less__Suc__eq,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) ) ) ).
% less_Suc_eq
thf(fact_1266_not__less__eq,axiom,
! [M3: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M3 @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M3 ) ) ) ).
% not_less_eq
thf(fact_1267_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I6: nat] :
( ( ord_less_nat @ I6 @ ( suc @ N2 ) )
=> ( P @ I6 ) ) )
= ( ( P @ N2 )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ N2 )
=> ( P @ I6 ) ) ) ) ).
% All_less_Suc
thf(fact_1268_Suc__less__eq2,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M3 )
= ( ? [M9: nat] :
( ( M3
= ( suc @ M9 ) )
& ( ord_less_nat @ N2 @ M9 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1269_less__antisym,axiom,
! [N2: nat,M3: nat] :
( ~ ( ord_less_nat @ N2 @ M3 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M3 ) )
=> ( M3 = N2 ) ) ) ).
% less_antisym
thf(fact_1270_Suc__less__SucD,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M3 @ N2 ) ) ).
% Suc_less_SucD
% Conjectures (1)
thf(conj_0,conjecture,
member_set_a @ ( uminus_uminus_set_a @ i ) @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ).
%------------------------------------------------------------------------------